--- /srv/rebuilderd/tmp/rebuilderduFOIeD/inputs/macaulay2-common_1.25.11+ds-2_all.deb +++ /srv/rebuilderd/tmp/rebuilderduFOIeD/out/macaulay2-common_1.25.11+ds-2_all.deb ├── file list │ @@ -1,3 +1,3 @@ │ -rw-r--r-- 0 0 0 4 2025-12-14 14:09:53.000000 debian-binary │ --rw-r--r-- 0 0 0 540512 2025-12-14 14:09:53.000000 control.tar.xz │ --rw-r--r-- 0 0 0 31297916 2025-12-14 14:09:53.000000 data.tar.xz │ +-rw-r--r-- 0 0 0 540524 2025-12-14 14:09:53.000000 control.tar.xz │ +-rw-r--r-- 0 0 0 31296964 2025-12-14 14:09:53.000000 data.tar.xz ├── control.tar.xz │ ├── control.tar │ │ ├── ./control │ │ │ @@ -1,13 +1,13 @@ │ │ │ Package: macaulay2-common │ │ │ Source: macaulay2 │ │ │ Version: 1.25.11+ds-2 │ │ │ Architecture: all │ │ │ Maintainer: Debian Math Team │ │ │ -Installed-Size: 305306 │ │ │ +Installed-Size: 305291 │ │ │ Depends: fonts-katex (>= 0.16.10+~cs6.1.0), libjs-bootsidemenu (>= 1.0.0), 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+~cs9.6.11), node-fortawesome-fontawesome-free (>= 6.7.2+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 │ │ │ @@ -3353,25 +3353,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 47016 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/A1BrouwerDegrees/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15458 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/A1BrouwerDegrees/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 41444 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1026 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/___A__Infinity.out │ │ │ --rw-r--r-- 0 root (0) root (0) 918 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/___Check.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 916 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/___Check.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4374 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_a__Infinity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 56403 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_burke__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3427 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_display__Blocks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3016 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_extract__Blocks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1740 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_golod__Betti.out │ │ │ -rw-r--r-- 0 root (0) root (0) 832 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_is__Golod__A__Inf.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2183 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_picture.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 40 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 7249 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/html/___Check.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7247 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/html/___Check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14672 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_a__Infinity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 67508 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_burke__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9657 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_display__Blocks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10392 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_extract__Blocks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9306 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_golod__Betti.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5933 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_has__Minimal__Mult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6465 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_is__Golod__A__Inf.html │ │ │ @@ -3465,25 +3465,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 11993 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjointIdeal/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7615 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjointIdeal/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4178 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjointIdeal/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 37097 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1946 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1945 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1748 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out │ │ │ -rw-r--r-- 0 root (0) root (0) 558 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_expected__Dimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 558 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_linear__System__On__Rational__Surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1889 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_parametrization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1272 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_rational__Surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1596 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_slow__Adjunction__Calculation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2944 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_special__Families__Of__Sommese__Vande__Ven.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 23 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 9249 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjoint__Matrix.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9248 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjoint__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11329 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6620 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_expected__Dimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7258 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_linear__System__On__Rational__Surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9580 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9741 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_rational__Surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9319 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_slow__Adjunction__Calculation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12483 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_special__Families__Of__Sommese__Vande__Ven.html │ │ │ @@ -3723,18 +3723,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 76682 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4226 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 2909 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2927 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 423 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 433 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 29 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 5574 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5584 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5233 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4242 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 2912 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BernsteinSato/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 289851 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BernsteinSato/example-output/ │ │ │ @@ -4315,22 +4315,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5291 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Browse/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4374 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Browse/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3057 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Browse/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 19549 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 4572 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4571 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1717 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2579 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_elementary.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1678 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_evans__Griffith.out │ │ │ -rw-r--r-- 0 root (0) root (0) 570 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_is__Syzygy.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 49 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 14041 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/_bruns.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14040 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/_bruns.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8831 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/_bruns__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10497 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/_elementary.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8129 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/_evans__Griffith.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6572 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/_is__Syzygy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9591 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6809 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4075 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/toc.html │ │ │ @@ -4433,15 +4433,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1448 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Quasi__Isomorphism.out │ │ │ -rw-r--r-- 0 root (0) root (0) 771 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_koszul__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1962 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_minimize_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 694 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_nonzero__Max.out │ │ │ -rw-r--r-- 0 root (0) root (0) 684 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_prepend__Zero__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 899 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_remove__Zero__Trailing__Terms.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3453 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3451 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 541 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2570 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_scarf__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 537 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_substitute_lp__Chain__Complex_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 672 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_taylor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1333 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_taylor__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1351 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_trivial__Homological__Truncation.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/ │ │ │ @@ -4464,15 +4464,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5473 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_koszul__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7220 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_koszul__Complex_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10096 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_minimize_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6612 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_nonzero__Max.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6519 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_nonzero__Min.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5900 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_prepend__Zero__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6665 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_remove__Zero__Trailing__Terms.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12265 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12263 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7373 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9135 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10433 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_scarf__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5755 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_substitute_lp__Chain__Complex_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5714 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6750 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7255 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor__Resolution_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ @@ -4501,49 +4501,49 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12258 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8418 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4775 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 123136 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 4377 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4376 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1503 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3419 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Toric__Variety__Valid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3442 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chern.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2404 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Class__In__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 775 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Class__In__Toric__Chow__Ring.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2017 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4380 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2014 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4377 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out │ │ │ -rw-r--r-- 0 root (0) root (0) 342 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler__Affine.out │ │ │ --rw-r--r-- 0 root (0) root (0) 759 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out │ │ │ --rw-r--r-- 0 root (0) root (0) 667 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out │ │ │ --rw-r--r-- 0 root (0) root (0) 624 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 760 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 666 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 623 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1049 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Multi__Proj__Coord__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6662 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Output.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3327 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Segre.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1666 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Toric__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 570 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/_is__Multi__Homogeneous.out │ │ │ -rw-r--r-- 0 root (0) root (0) 802 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/_probabilistic_spalgorithm.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 665 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 55 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 23694 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23693 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6377 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11239 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Toric__Variety__Valid.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18179 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Chern.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9577 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6409 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Class__In__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7339 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Class__In__Toric__Chow__Ring.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10385 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html │ │ │ --rw-r--r-- 0 root (0) root (0) 18563 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10382 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 18560 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5955 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler__Affine.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5902 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6013 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6466 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5903 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6012 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6465 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8377 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Multi__Proj__Coord__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16398 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16710 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Segre.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8690 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Toric__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4910 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_bertini__Check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5128 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_configuring_sp__Bertini.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7468 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_is__Multi__Homogeneous.html │ │ │ @@ -4558,15 +4558,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2624 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/___Chordal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/___Ring__Element_sp_pc_sp__Chordal__Net.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1273 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/___Ring__Map_sp__Chordal__Net.out │ │ │ -rw-r--r-- 0 root (0) root (0) 178 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_adjacent__Minors__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2151 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Elim.out │ │ │ -rw-r--r-- 0 root (0) root (0) 969 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1787 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1202 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1196 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1633 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Tria.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2559 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal_spnetworks_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 429 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chromatic__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 237 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_codim__Count.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1454 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_components_lp__Chordal__Net_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 414 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_constraint__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 726 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_digraph_lp__Chordal__Net_rp.out │ │ │ @@ -4599,15 +4599,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3907 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/___Get__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7598 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/___Ring__Element_sp_pc_sp__Chordal__Net.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7248 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/___Ring__Map_sp__Chordal__Net.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5071 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_adjacent__Minors__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10905 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Elim.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7288 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9374 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8344 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8338 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8861 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Tria.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10852 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal_spnetworks_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5434 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chromatic__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7090 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_codim__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8141 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_components_lp__Chordal__Net_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5721 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_constraint__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7695 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_digraph_lp__Chordal__Net_rp.html │ │ │ @@ -4800,23 +4800,23 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 36467 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26490 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15240 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 13418 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 13261 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13258 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1045 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/_cohom__Calg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 982 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/_cohom__Calg_lp__Normal__Toric__Variety_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 97 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3840 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/___Silent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10103 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/_cohom__Calg.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8434 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/_cohom__Calg_lp__Normal__Toric__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25594 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25591 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5794 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3603 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 107816 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1103 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/example-output/___Coincident__Root__Locus_sp_st_sp__Coincident__Root__Locus.out │ │ │ @@ -4899,15 +4899,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 231378 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 650 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___B__G__G__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1959 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___B__Ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3164 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Complete__Intersection__Resolutions.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4598 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4595 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4116 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash__Total.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2743 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Ext__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1023 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Ext__Module__Data.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2102 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___S2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1754 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Shamash.out │ │ │ -rw-r--r-- 0 root (0) root (0) 761 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Tate__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1322 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_complexity.out │ │ │ @@ -4931,24 +4931,24 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1851 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_make__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 857 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_make__T.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1336 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_matrix__Factorization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10478 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_new__Ext.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1284 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_odd__Ext__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_regularity__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1314 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_splittings.out │ │ │ --rw-r--r-- 0 root (0) root (0) 379 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out │ │ │ --rw-r--r-- 0 root (0) root (0) 451 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 378 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 450 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5986 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___A__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4935 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Augmentation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6307 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___B__G__G__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9094 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___B__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5966 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Check.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15259 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15256 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14302 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash__Total.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11218 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Ext__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10327 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Ext__Module__Data.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5284 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5531 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Hom__With__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5130 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Layered.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4742 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Lift.html │ │ │ @@ -4993,18 +4993,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10902 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_module__As__Ext.html │ │ │ -rw-r--r-- 0 root (0) root (0) 23042 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_new__Ext.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8736 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_odd__Ext__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5888 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_psi__Maps.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7031 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_regularity__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7383 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_splittings.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5224 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_stable__Hom.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6048 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6047 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5573 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_tensor__With__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4869 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_to__Array.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6366 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6365 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 56031 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35373 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15036 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Complexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Complexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 706402 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Complexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Complexes/example-output/ │ │ │ @@ -5340,29 +5340,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 21320 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16146 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9518 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50933 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 36623 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 36624 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Gauss_sq_sphypergeometric_spfunction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3314 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.out │ │ │ -rw-r--r-- 0 root (0) root (0) 240 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_base__Fraction__Field.out │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_connection__Matrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 240 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_connection__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 494 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_gauge__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 369 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_gauge__Transform.out │ │ │ -rw-r--r-- 0 root (0) root (0) 299 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_is__Epsilon__Factorized.out │ │ │ -rw-r--r-- 0 root (0) root (0) 263 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_is__Integrable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 202 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_normal__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 244 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_standard__Monomials.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 56 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 46479 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 46480 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7850 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Gauss_sq_sphypergeometric_spfunction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9213 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5611 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_base__Fraction__Field.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6834 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_connection__Matrices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6189 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_connection__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7537 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_gauge__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7266 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_gauge__Transform.html │ │ │ @@ -5584,136 +5584,136 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 38506 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CpMackeyFunctors/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35258 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CpMackeyFunctors/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16572 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CpMackeyFunctors/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Cremona/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Cremona/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 239171 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Cremona/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) 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./usr/share/doc/Macaulay2/Elimination/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 98399 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/example-output/ │ │ │ @@ -6459,35 +6459,35 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6990 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EngineTests/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 10834 2025-12-14 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./usr/share/doc/Macaulay2/EnumerationCurves/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5203 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5211 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_multiple__Cover.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11546 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11543 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7109 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5529 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3452 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52732 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/___Equivariant__G__B.out │ │ │ -rw-r--r-- 0 root (0) root (0) 298 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/___Priority__Queue.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1050 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_build__E__Monomial__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 576 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_build__E__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 346 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_build__E__Ring_lp__Ring_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 222 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_delete__Min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 153 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1286 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb__Toric.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1283 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb__Toric.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb_lp..._cm__Algorithm_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 348 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_exponent__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 449 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_inc__Orbit.out │ │ │ -rw-r--r-- 0 root (0) root (0) 289 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_insert_lp__Priority__Queue_cm__Thing_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 248 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_length_lp__Priority__Queue_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 323 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_merge__P__Q.out │ │ │ -rw-r--r-- 0 root (0) root (0) 161 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_min_lp__Priority__Queue_rp.out 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0 root (0) root (0) 9002 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb__Toric.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8999 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb__Toric.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6615 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb_lp..._cm__Algorithm_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6126 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_exponent__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7150 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_inc__Orbit.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6020 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_insert_lp__Priority__Queue_cm__Thing_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5369 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_length_lp__Priority__Queue_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6478 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_merge__P__Q.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5290 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_min_lp__Priority__Queue_rp.html │ │ │ @@ -6751,52 +6751,52 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10138 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FGLM/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4568 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FGLM/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 2981 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FGLM/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 142955 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 29586 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 29588 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1035 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Point__Options.out │ │ │ --rw-r--r-- 0 root (0) root (0) 14278 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 14276 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1054 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Strategy__Default.out │ │ │ -rw-r--r-- 0 root (0) root (0) 337 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Good__Minors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 246 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Random__Submatrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 307 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Submatrix__Largest__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Submatrix__Smallest__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 533 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_get__Submatrix__Of__Rank.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1787 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Codim__At__Least.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1788 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Codim__At__Least.out │ │ │ -rw-r--r-- 0 root (0) root (0) 275 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Rank__At__Least.out │ │ │ -rw-r--r-- 0 root (0) root (0) 436 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_proj__Dim.out │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_recursive__Minors.out │ │ │ --rw-r--r-- 0 root (0) root (0) 25047 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_regular__In__Codimension.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 25041 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_regular__In__Codimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 273 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_reorder__Polynomial__Ring.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 586 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6134 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Det__Strategy.html │ │ │ --rw-r--r-- 0 root (0) root (0) 50410 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 50412 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5373 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Max__Minors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4415 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Min__Dimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4447 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Modulus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6808 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Point__Options.html │ │ │ --rw-r--r-- 0 root (0) root (0) 27489 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Regular__In__Codimension__Tutorial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 27487 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Regular__In__Codimension__Tutorial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14641 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Strategy__Default.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9190 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_choose__Good__Minors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5703 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_choose__Random__Submatrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6319 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_choose__Submatrix__Largest__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6235 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_choose__Submatrix__Smallest__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10115 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_get__Submatrix__Of__Rank.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11275 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Codim__At__Least.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11276 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Codim__At__Least.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6128 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Dim__At__Most.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9506 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Rank__At__Least.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5364 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Rank__At__Least_lp..._cm__Threads_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10205 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_proj__Dim.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7912 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_recursive__Minors.html │ │ │ --rw-r--r-- 0 root (0) root (0) 43514 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 43508 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6458 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_reorder__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24783 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26704 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7533 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 25936 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/rawdocumentation.dump │ │ │ @@ -7036,16 +7036,16 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 105269 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Bounds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 318 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Frobenius__Thresholds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 793 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Guess__Strategy.out │ │ │ -rw-r--r-- 0 root (0) root (0) 866 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_compare__F__P__T.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4034 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_fpt.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2459 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4032 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_fpt.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2455 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out │ │ │ -rw-r--r-- 0 root (0) root (0) 760 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__F__Jumping__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 552 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__F__P__T.out │ │ │ -rw-r--r-- 0 root (0) root (0) 828 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__Simple__Normal__Crossing.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 617 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 12 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5582 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Bounds.html │ │ │ @@ -7056,16 +7056,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4963 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Global__Frobenius__Root.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10580 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Guess__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4665 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Return__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4769 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Search.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4735 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Standard__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5945 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Use__Special__Algorithms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14756 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_compare__F__P__T.html │ │ │ --rw-r--r-- 0 root (0) root (0) 26000 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html │ │ │ --rw-r--r-- 0 root (0) root (0) 24542 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25998 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 24538 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13295 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__F__Jumping__Exponent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12519 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__F__P__T.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9887 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__Simple__Normal__Crossing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19600 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20028 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7092 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FunctionFieldDesingularization/ │ │ │ @@ -7117,15 +7117,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_make__K__Class_lp__G__K__M__Variety_cm__Flag__Matroid_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1001 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_make__K__Class_lp__G__K__M__Variety_cm__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 339 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_map_lp__G__K__M__Variety_cm__G__K__M__Variety_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1060 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 732 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph_lp__G__K__M__Variety_cm__Moment__Graph_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 170 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph_lp__G__K__M__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_normal__Toric__Variety_lp__G__K__M__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8064 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8063 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1060 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_projective__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 612 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_pullback_lp__Equivariant__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 615 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_pushforward.out │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_set__Indicator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 437 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_trivial__K__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 401 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_underlying__Graph_lp__Moment__Graph_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/ │ │ │ @@ -7168,15 +7168,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7893 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_make__K__Class_lp__G__K__M__Variety_cm__Flag__Matroid_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8393 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_make__K__Class_lp__G__K__M__Variety_cm__Toric__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9167 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_map_lp__G__K__M__Variety_cm__G__K__M__Variety_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8448 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_moment__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7456 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_moment__Graph_lp__G__K__M__Variety_cm__Moment__Graph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5700 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_moment__Graph_lp__G__K__M__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7617 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_normal__Toric__Variety_lp__G__K__M__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 19815 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 19814 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7021 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_projective__Space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7191 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_pullback_lp__Equivariant__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7746 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_pushforward.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6964 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_set__Indicator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5716 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_trivial__K__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5537 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_underlying__Graph_lp__Moment__Graph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26820 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/index.html │ │ │ @@ -8244,32 +8244,32 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 33164 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/___Groebner__Strata.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1505 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_find__Weight__Constraints.out │ │ │ -rw-r--r-- 0 root (0) root (0) 598 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_find__Weight__Vector.out │ │ │ -rw-r--r-- 0 root (0) root (0) 30368 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_groebner__Family.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2709 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_groebner__Stratum.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_linear__Part.out │ │ │ --rw-r--r-- 0 root (0) root (0) 13267 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_nonminimal__Maps.out │ │ │ --rw-r--r-- 0 root (0) root (0) 14736 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6625 2025-12-14 14:09:53.000000 ./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) 13044 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_nonminimal__Maps.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 15158 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6646 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_smaller__Monomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 960 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_standard__Monomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1344 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_tail__Monomials.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 42 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6030 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/___All__Standard.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4977 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/___Minimalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9811 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_find__Weight__Constraints.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8464 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_find__Weight__Vector.html │ │ │ -rw-r--r-- 0 root (0) root (0) 42133 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_groebner__Family.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10343 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_groebner__Stratum.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6000 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_linear__Part.html │ │ │ --rw-r--r-- 0 root (0) 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-rw-r--r-- 0 root (0) root (0) 6255 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4619 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Base.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4534 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Info.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4884 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4662 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Max__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7171 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Projection.html │ │ │ @@ -8983,21 +8983,21 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9828 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Jets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7299 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Jets/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 101880 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1949 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_all__Gradings.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2783 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2786 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5835 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_canonical__Homotopies.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1036 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2262 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3425 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1005 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out │ │ │ --rw-r--r-- 0 root (0) root (0) 269 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2261 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3426 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1004 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 267 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6823 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_correspondence__Scroll.out │ │ │ -rw-r--r-- 0 root (0) root (0) 804 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_cox__Matrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1634 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6875 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2293 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_homotopy__Ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 720 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_irrelevant__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_product__Of__Projective__Spaces.out │ │ │ @@ -9010,22 +9010,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3037 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_schreyer__Name.out │ │ │ -rw-r--r-- 0 root (0) root (0) 813 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_small__Diagonal.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 30 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4913 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/___Fine__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4489 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/___Scrolls.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7927 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_all__Gradings.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9534 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9537 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6565 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_canonical__Carpet.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13299 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_canonical__Homotopies.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11993 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9792 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10621 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6854 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6737 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9791 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10622 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6853 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6735 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18189 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_correspondence__Scroll.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7392 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_cox__Matrices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9428 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14901 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5106 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_gorenstein__Double.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8478 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_hankel__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8111 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_homotopy__Ranks.html │ │ │ @@ -9198,15 +9198,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 70564 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/___Working_spwith_splattice_sppolytopes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_adjoint__Polytope.out │ │ │ -rw-r--r-- 0 root (0) root (0) 329 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_ambient__Halfspaces.out │ │ │ --rw-r--r-- 0 root (0) root (0) 596 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 594 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 684 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_cayley.out │ │ │ -rw-r--r-- 0 root (0) root (0) 85 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_codegree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 281 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_degree__Of__Jet__Separation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 312 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_epsilon__Bounds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gauss__Fiber.out │ │ │ -rw-r--r-- 0 root (0) root (0) 373 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gauss__Image.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gaussk__Fiber.out │ │ │ @@ -9223,15 +9223,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_toric__Div.out │ │ │ -rw-r--r-- 0 root (0) root (0) 167 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_torus__Embedding.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 17 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5005 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/___Working_spwith_splattice_sppolytopes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5659 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_adjoint__Polytope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5749 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_ambient__Halfspaces.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7712 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7710 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9462 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_cayley.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4957 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_codegree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6907 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_degree__Of__Jet__Separation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6587 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_epsilon__Bounds.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6730 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gauss__Fiber.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6764 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gauss__Image.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7032 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gaussk__Fiber.html │ │ │ @@ -9711,15 +9711,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1999 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Expression.out │ │ │ -rw-r--r-- 0 root (0) root (0) 788 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Flat__Monoid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 100 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 340 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function_sp_at_at_sp__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 836 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function_sp_us_sp__Thing.out │ │ │ --rw-r--r-- 0 root (0) root (0) 414 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 415 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 611 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__F.out │ │ │ -rw-r--r-- 0 root (0) root (0) 194 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Rev__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Gamma.out │ │ │ -rw-r--r-- 0 root (0) root (0) 230 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Assign__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 371 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Release__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6869 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Grassmannian.out │ │ │ @@ -9820,15 +9820,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 238 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring__Family_sp_us_st.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1401 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 428 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring__Map_sp__Ring__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring_sp^_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring_sp^_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring_sp_sl_sp__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1854 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D.out │ │ │ --rw-r--r-- 0 root (0) root (0) 349 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 348 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5127 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Schreyer_sporders.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1024 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Schubert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Self__Initializing__Type.out │ │ │ -rw-r--r-- 0 root (0) root (0) 272 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 950 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 205 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set_sp-_sp__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 157 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set_sp_sh_qu_sp__Thing.out │ │ │ @@ -9965,38 +9965,38 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_apply_lp__Z__Z_cm__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1347 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_apropos.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ascii.out │ │ │ -rw-r--r-- 0 root (0) root (0) 112 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_asin.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_asinh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 307 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_assert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 436 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_assigning_spvalues.out │ │ │ --rw-r--r-- 0 root (0) root (0) 235 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 234 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 154 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atan.out │ │ │ -rw-r--r-- 0 root (0) root (0) 245 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atan2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 207 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atanh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 624 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_augmented_spassignment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1258 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_autoload.out │ │ │ -rw-r--r-- 0 root (0) root (0) 137 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Name.out │ │ │ -rw-r--r-- 0 root (0) root (0) 418 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2978 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_basic_sparithmetic_spof_spmatrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1001 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_basic_spconstruction_cm_spsource_spand_sptarget_spof_spa_spring_spmap.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1118 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_basic_springs_spof_spnumbers.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4584 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_basis.out │ │ │ --rw-r--r-- 0 root (0) root (0) 133 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_benchmark.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 134 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_benchmark.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3673 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_betti.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2269 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_betti_lp__Betti__Tally_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 202 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_between.out │ │ │ -rw-r--r-- 0 root (0) root (0) 332 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_binomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 609 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_block__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 223 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_borel_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 725 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_break.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cache.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1307 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_caching_spcomputation_spresults.out │ │ │ --rw-r--r-- 0 root (0) root (0) 591 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 592 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1803 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_capture.out │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ceiling_lp__Number_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_center__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 946 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Base.out │ │ │ -rw-r--r-- 0 root (0) root (0) 259 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Directory.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_char.out │ │ │ -rw-r--r-- 0 root (0) root (0) 196 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_characters.out │ │ │ @@ -10020,15 +10020,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 288 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_column__Swap.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_columnate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_combine.out │ │ │ -rw-r--r-- 0 root (0) root (0) 198 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_command__Interpreter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 149 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comments.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_common__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 465 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_commonest.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1535 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1547 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 225 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 372 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compact__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 251 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compare__Exchange.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1085 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compose.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2477 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compositions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 272 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compress.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4287 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out │ │ │ @@ -10115,15 +10115,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1720 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eagon__Northcott.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvalues.out │ │ │ -rw-r--r-- 0 root (0) root (0) 717 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvectors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eint.out │ │ │ -rw-r--r-- 0 root (0) root (0) 103 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 162 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Timing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 366 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elements.out │ │ │ --rw-r--r-- 0 root (0) root (0) 21211 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 21207 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 782 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3683 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end__Package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 487 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries.out │ │ │ -rw-r--r-- 0 root (0) root (0) 185 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_equality_spand_spcontainment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erf.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erfc.out │ │ │ @@ -10264,15 +10264,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1260 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_induced__Map_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1035 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inheritance.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1187 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inputting_spa_spmatrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 592 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_insert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 686 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_installing_spassignment_spmethods.out │ │ │ -rw-r--r-- 0 root (0) root (0) 936 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_installing_spaugmented_spassignment_spmethods.out │ │ │ --rw-r--r-- 0 root (0) root (0) 932 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 933 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out │ │ │ -rw-r--r-- 0 root (0) root (0) 316 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integers_spmodulo_spa_spprime.out │ │ │ -rw-r--r-- 0 root (0) root (0) 315 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integrate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1162 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect.out │ │ │ -rw-r--r-- 0 root (0) root (0) 997 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Ideal_cm__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 189 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Set_cm__Set_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 190 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersection_spof_spideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inverse__Erf.out │ │ │ @@ -10404,18 +10404,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 769 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Mutable__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 125 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 228 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 560 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max.out │ │ │ --rw-r--r-- 0 root (0) root (0) 82 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out │ │ │ +-rw-r--r-- 0 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2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_method__Options_lp__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6748 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2721 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods_spfor_spnormal_spforms_spand_spremainder.out │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min__Exponent.out │ │ │ @@ -10501,16 +10501,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_override.out │ │ │ -rw-r--r-- 0 root (0) root (0) 681 2025-12-14 14:09:53.000000 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2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1651 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8671 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parsing_spprecedence_cm_spin_spdetail.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3030 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1297 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_partition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1097 2025-12-14 14:09:53.000000 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./usr/share/doc/Macaulay2/ModuleDeformations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 709 2025-12-14 14:09:53.000000 ./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) 2987 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/example-output/_deform__M__C__M__Module_lp__Module_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2986 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 607 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 53 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6415 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7877 2025-12-14 14:09:53.000000 ./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) 9182 2025-12-14 14:09:53.000000 ./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) 11921 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp__Module_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11920 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3567 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/_xi.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9547 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5234 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3891 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 133490 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 447 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/___Monodromy__Solver.out │ │ │ -rw-r--r-- 0 root (0) root (0) 985 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/___Monodromy__Solver__Options.out │ │ │ -rw-r--r-- 0 root (0) root (0) 333 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_complete__Graph__Augment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 243 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_complete__Graph__Init.out │ │ │ -rw-r--r-- 0 root (0) root (0) 418 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_compute__Mixed__Volume.out │ │ │ -rw-r--r-- 0 root (0) root (0) 407 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_create__Seed__Pair.out │ │ │ --rw-r--r-- 0 root (0) root (0) 942 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 943 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_flower__Graph__Augment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 242 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_flower__Graph__Init.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9570 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Group.out │ │ │ -rw-r--r-- 0 root (0) root (0) 427 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Solve_lp__System_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 956 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_potential__E.out │ │ │ -rw-r--r-- 0 root (0) root (0) 442 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_potential__Lower__Bound.out │ │ │ @@ -13609,15 +13609,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5869 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/___Point__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4500 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_append__Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4552 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_append__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4875 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_complete__Graph__Augment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4697 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_complete__Graph__Init.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5190 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_compute__Mixed__Volume.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7814 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_create__Seed__Pair.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7624 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7625 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5009 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_flower__Graph__Augment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4670 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_flower__Graph__Init.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4889 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_get__Track__Time.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6199 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_homotopy__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4606 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_indices_lp__Point__Array_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4847 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_is__Member_lp__Abstract__Point_cm__Point__Array_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4736 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_length_lp__Point__Array_rp.html │ │ │ @@ -13855,28 +13855,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 19109 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12849 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7429 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 41382 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2242 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2240 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 719 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_compute__Component.out │ │ │ -rw-r--r-- 0 root (0) root (0) 723 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_interpolate__Component.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1000 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_max__Grading.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_trim__Basis__In__Degree.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5318 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Coefficient__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4235 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4491 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Previous__Gens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4750 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Return__Target__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5012 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Use__Interpolation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4647 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Use__Matroid.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11856 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11854 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6602 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9224 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_compute__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10098 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_interpolate__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8008 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_max__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9470 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_trim__Basis__In__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14051 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13133 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/master.html │ │ │ @@ -13884,27 +13884,27 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 35822 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 369 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/___N__P.out │ │ │ -rw-r--r-- 0 root (0) root (0) 656 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_gr__Gr.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1249 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_hilbert__Sequence.out │ │ │ --rw-r--r-- 0 root (0) root (0) 413 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 414 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 261 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out │ │ │ -rw-r--r-- 0 root (0) root (0) 573 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Reduction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 596 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_monj__Mult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 805 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_multiplicity__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1239 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_print__Hilbert__Sequence.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 594 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 47 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6190 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/___N__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6706 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_gr__Gr.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8885 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_hilbert__Sequence.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6040 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6041 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5705 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6831 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Reduction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6478 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_monj__Mult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9585 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_multiplicity__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6921 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_print__Hilbert__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15265 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9576 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/master.html │ │ │ @@ -13958,31 +13958,31 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5192 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplierIdealsDim2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 379736 2025-12-14 14:09:53.000000 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./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_vb_sp__Multirational__Map.out │ │ │ @@ -14001,49 +14001,49 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 179 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_coefficient__Ring_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3115 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_cone__Of__Lines.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_conormal__Variety_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 575 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_cycle__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 671 2025-12-14 14:09:53.000000 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14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Member_lp__Multirational__Map_cm__R__A__T_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 753 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Morphism_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 754 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Morphism_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Subset_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 678 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Well__Defined_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 790 2025-12-14 14:09:53.000000 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14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_parametrize_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 490 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_permute_lp__Multiprojective__Variety_cm__List_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1238 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_point_lp__Multiprojective__Variety_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1237 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_point_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 811 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_projection__Maps.out │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_projection__Maps_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2190 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_projections.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2823 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_projective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 370 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_projective__Variety_lp__List_cm__List_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 343 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_projective__Variety_lp__List_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1055 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_projective__Variety_lp__Multidimensional__Matrix_rp.out │ │ │ @@ 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./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre__Embedding.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6206 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multiprojective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1051 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1050 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 288 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_shape_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 813 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_shortcuts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3242 2025-12-14 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./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_ring_lp__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6808 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_schubert__Cycle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5384 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_sectional__Genus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6085 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_segre__Embedding.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11827 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_segre_lp__Multiprojective__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7309 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_segre_lp__Multirational__Map_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7308 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_segre_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5885 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_shape_lp__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9083 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_shortcuts.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8452 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_show_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5740 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_singular__Locus_lp__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5549 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_source_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6708 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_super_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6660 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_support_lp__Multiprojective__Variety_rp.html │ │ │ @@ -14560,26 +14560,26 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 28886 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NCAlgebra/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 139526 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 266 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 496 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1406 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1405 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1437 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_add__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 240 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2137 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_build__Graph__Filter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 165 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_count__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 363 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_filter__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2806 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Bipartite__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3579 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 326 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 807 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Regular__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 146 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_graph6__To__Sparse6.out │ │ │ --rw-r--r-- 0 root (0) root (0) 498 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_graph__Complement.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 499 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_graph__Complement.out │ │ │ -rw-r--r-- 0 root (0) root (0) 289 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_graph__To__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 152 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_is__Planar.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1500 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_neighborhood__Complements.out │ │ │ -rw-r--r-- 0 root (0) root (0) 290 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_new__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 215 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_only__Planar.out │ │ │ -rw-r--r-- 0 root (0) root (0) 505 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_relabel__Bipartite.out │ │ │ -rw-r--r-- 0 root (0) root (0) 280 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_relabel__Graph.out │ │ │ @@ -14589,26 +14589,26 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 179 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_string__To__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 245 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_string__To__Graph.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 575 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 18 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5988 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5714 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8522 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8521 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10316 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_add__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6991 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14195 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_build__Graph__Filter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7805 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_count__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8103 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_filter__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13715 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Bipartite__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14088 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9658 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8443 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Regular__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6502 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_graph6__To__Sparse6.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8741 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_graph__Complement.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8742 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_graph__Complement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8943 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_graph__To__String.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5998 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_is__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8315 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_neighborhood__Complements.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7139 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_new__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6373 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_only__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7363 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_relabel__Bipartite.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8940 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_relabel__Graph.html │ │ │ @@ -14622,48 +14622,48 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8104 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 131317 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 375 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 493 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1404 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1406 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 826 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_add__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 388 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1498 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_build__Graph__Filter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 326 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 124 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Regular__Graphs.out │ │ │ --rw-r--r-- 0 root (0) root (0) 497 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 498 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out │ │ │ -rw-r--r-- 0 root (0) root (0) 323 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__To__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_is__Planar.out │ │ │ -rw-r--r-- 0 root (0) root (0) 849 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_neighborhood__Complements.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_new__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 247 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_only__Planar.out │ │ │ -rw-r--r-- 0 root (0) root (0) 322 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_relabel__Bipartite.out │ │ │ -rw-r--r-- 0 root (0) root (0) 381 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_relabel__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_remove__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_remove__Isomorphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 203 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_string__To__Graph.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6195 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5706 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8546 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8548 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9550 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_add__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7633 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13153 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_build__Graph__Filter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7800 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_count__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8080 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_filter__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9132 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Bipartite__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9062 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8413 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6700 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Regular__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6470 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph6__To__Sparse6.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7731 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7732 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8914 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__To__String.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5929 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_is__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7363 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_neighborhood__Complements.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6390 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_new__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6301 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_only__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6955 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_relabel__Bipartite.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9029 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_relabel__Graph.html │ │ │ @@ -14691,15 +14691,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 179468 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 431 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Dependent__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 791 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Diff__Op.out │ │ │ -rw-r--r-- 0 root (0) root (0) 553 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Diff__Op_sp__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_amult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 931 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_coordinate__Change__Ops.out │ │ │ -rw-r--r-- 0 root (0) root (0) 354 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_diff__Op__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 403 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_diff__Op_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 667 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_differential__Primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 349 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_eliminating__Dual.out │ │ │ -rw-r--r-- 0 root (0) root (0) 270 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_evaluate_lp__Diff__Op_cm__Abstract__Point_rp.out │ │ │ @@ -14732,15 +14732,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 75 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 8524 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Dependent__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9032 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Diff__Op.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7316 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Diff__Op_sp__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5026 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Sampler.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7406 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Hybrid_dq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7805 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Macaulay__Matrix_dq.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7814 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7815 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4027 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Tolerance_sp_lp__Noetherian__Operators_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6738 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_amult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5314 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_colon.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8444 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_coordinate__Change__Ops.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4120 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7027 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6518 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op_lp__Matrix_rp.html │ │ │ @@ -14829,15 +14829,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8147 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5466 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3915 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 608995 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 3509 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3512 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3270 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___H__H^__Z__Z_lp__Normal__Toric__Variety_cm__Coherent__Sheaf_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1905 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp^_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 908 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp^_st_st_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 840 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_st_st_sp__Normal__Toric__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1922 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_us_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 708 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1977 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___O__O_sp__Toric__Divisor.out │ │ │ @@ -14901,15 +14901,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1271 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_lattice__Points_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1229 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Simplicial_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1568 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Smooth_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 601 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1011 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 912 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_matrix_lp__Toric__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 543 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_max_lp__Normal__Toric__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1560 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1559 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2228 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_nef__Generators_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1002 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3171 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1964 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 704 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1599 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_orbits_lp__Normal__Toric__Variety_cm__Z__Z_rp.out │ │ │ @@ -14939,15 +14939,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 881 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_vector_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1765 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_vertices_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1650 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weighted__Projective__Space_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 452 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weil__Divisor__Group_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 985 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weil__Divisor__Group_lp__Toric__Map_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 68 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 12847 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12850 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11949 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___H__H^__Z__Z_lp__Normal__Toric__Variety_cm__Coherent__Sheaf_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27601 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14079 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp^_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9111 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp^_st_st_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9410 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp_st_st_sp__Normal__Toric__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13026 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp_us_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8752 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp_us_sp__Z__Z.html │ │ │ @@ -15016,15 +15016,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10561 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_make__Simplicial_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13649 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_make__Smooth_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9577 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_making_spnormal_sptoric_spvarieties.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11442 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12498 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11093 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_matrix_lp__Toric__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9104 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_max_lp__Normal__Toric__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10405 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10404 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10956 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_nef__Generators_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10436 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19155 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__List_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13815 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13553 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9792 2025-12-14 14:09:53.000000 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root (0) 6350 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__Zero_lp__Free__O__I__Module__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6045 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__Zero_lp__Vector__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6249 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_lead__Coefficient_lp__Vector__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6493 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_lead__Monomial_lp__Vector__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6235 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_lead__Term_lp__Vector__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8244 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_make__Free__O__I__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6466 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_make__Polynomial__O__I__Algebra.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8932 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_minimize__O__I__G__B.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8914 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_minimize__O__I__G__B.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5976 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Free__O__I__Module__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5391 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Free__O__I__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5627 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Module__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5787 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__O__I__Resolution_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5332 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Polynomial__O__I__Algebra_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5767 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Vector__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8337 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__G__B.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6345 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Orbit.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8521 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Res.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8523 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Res.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9903 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Syz.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5814 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8298 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_reduce__O__I__G__B.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6438 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_restricted__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6328 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_terms_lp__Vector__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5333 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_to__String_lp__Free__O__I__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5423 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_to__String_lp__Polynomial__O__I__Algebra_rp.html │ │ │ @@ -15708,15 +15708,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 713 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Ring_sp^_sp__Betti__Tally.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1814 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_betti_lp..._cm__Minimize_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_betti_lp__Graded__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 518 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_chain__Complex_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 358 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_chain__Complex_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 832 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_chain__Complex_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_complete_lp__Chain__Complex_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1284 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_computing_spresolutions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1393 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_computing_spresolutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1250 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_cone_lp__Chain__Complex__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 621 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_dd.out │ │ │ -rw-r--r-- 0 root (0) root (0) 536 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_dual_lp__Chain__Complex__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1335 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_extracting_spinformation_spfrom_spchain_spcomplexes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_free_spresolutions_spof_spmodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_graded__Module__Map.out │ │ │ @@ -15790,15 +15790,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4997 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__Graded__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6585 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5865 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6228 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5591 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_complete_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4290 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_complete_lp__Graded__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4214 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_components_lp__Chain__Complex_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7475 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_computing_spresolutions.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7584 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_computing_spresolutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7151 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_cone_lp__Chain__Complex__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5466 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_dd.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5785 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_dual_lp__Chain__Complex__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3841 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_dual_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8237 2025-12-14 14:09:53.000000 ./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) 7530 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_extracting_spinformation_spfrom_spchain_spcomplexes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5900 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_free_spresolutions_spof_spmodules.html │ │ │ @@ -16254,22 +16254,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6360 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OpenMath/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4999 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OpenMath/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3143 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OpenMath/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 84707 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 17045 2025-12-14 14:09:53.000000 ./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) 17043 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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) 2540 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Oscillators.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3811 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3814 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1682 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_all__Unique__Principal__Minors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10260 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_find__Real__Solutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 151 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Angles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 429 2025-12-14 14:09:53.000000 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-rw-r--r-- 0 root (0) root (0) 241 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_standard__Sols.out │ │ │ -rw-r--r-- 0 root (0) root (0) 236 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_vertex__Spanning__Polynomial.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 33 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 25404 2025-12-14 14:09:53.000000 ./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) 25402 2025-12-14 14:09:53.000000 ./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) 6234 2025-12-14 14:09:53.000000 ./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) 7736 2025-12-14 14:09:53.000000 ./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) 7733 2025-12-14 14:09:53.000000 ./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) 7644 2025-12-14 14:09:53.000000 ./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) 10645 2025-12-14 14:09:53.000000 ./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) 6691 2025-12-14 14:09:53.000000 ./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) 3563 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Harrington-__Schenck-__Stillman.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8190 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8193 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8807 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_all__Unique__Principal__Minors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19710 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_find__Real__Solutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6864 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_get__Angles.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6738 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_get__Linearly__Stable__Solutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10573 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_identify__Stability.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7493 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_is__Stable__Solution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20003 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_osc__Jacobian.html │ │ │ @@ -16524,15 +16524,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12104 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Parsing/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9154 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Parsing/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7244 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Parsing/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 136324 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2842 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___A_spfamily_spof_sppaths_spon_spa_spcone.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2840 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___A_spfamily_spof_sppaths_spon_spa_spcone.out │ │ │ -rw-r--r-- 0 root (0) root (0) 456 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Array_sp_us_sp__N__C__Polynomial__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 863 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___C__Axis__Tensor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1364 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___C__Mon__Tensor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2634 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Path.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1530 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Path__Signatures.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1392 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Polynomial_sppaths_spof_spdegree_spm.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11000 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___The_spuniversal_spvariety_spand_sptoric_spcoordinates.out │ │ │ @@ -16558,15 +16558,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 204 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_tensor__Log.out │ │ │ -rw-r--r-- 0 root (0) root (0) 886 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_tensor__Parametrization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2023 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_word__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 387 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_word__Format.out │ │ │ -rw-r--r-- 0 root (0) root (0) 215 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_word__String.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 38 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 11402 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11400 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7073 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___Array_sp_us_sp__N__C__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7959 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___C__Axis__Tensor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8285 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___C__Mon__Tensor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3941 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___Computing_sp__Path_sp__Varieties.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7451 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___N__C__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21351 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___N__C__Ring__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15444 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___Path.html │ │ │ @@ -17404,15 +17404,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2469 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Example_co_sp__Hibi_spideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 358 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Example_co_sp__Intersection_splattices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 433 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Example_co_sp__L__C__M-lattices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset.out │ │ │ -rw-r--r-- 0 root (0) root (0) 131 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset_sp_us_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 123 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 150 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset_sp_us_st.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2466 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2467 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_adjoin__Max.out │ │ │ -rw-r--r-- 0 root (0) root (0) 306 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_adjoin__Min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 668 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_all__Relations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 315 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_antichains.out │ │ │ -rw-r--r-- 0 root (0) root (0) 320 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 157 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_atoms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 253 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_boolean__Lattice.out │ │ │ @@ -17531,15 +17531,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9486 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/html/___Example_co_sp__Hibi_spideals.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7037 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/html/___Example_co_sp__Intersection_splattices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6041 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/html/___Example_co_sp__L__C__M-lattices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 38795 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5707 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset_sp_us_sp__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5830 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset_sp_us_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5493 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset_sp_us_st.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8666 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8667 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6042 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/html/_adjoin__Max.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6073 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/html/_adjoin__Min.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6593 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/html/_all__Relations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6851 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/html/_antichains.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7024 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5632 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/html/_atoms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5773 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Posets/html/_augment__Poset.html │ │ │ @@ -18142,15 +18142,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1962 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2633 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300c].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2039 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Half_spcanonical_spdegree_sp20.out │ │ │ --rw-r--r-- 0 root (0) root (0) 28147 2025-12-14 14:09:53.000000 ./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) 21833 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Noether-__Lefschetz_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7051 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Pfaffians_spon_spquadrics.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Singularities_spof_splifting_spof_sptype_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 4091 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[331]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.out │ │ │ @@ -18184,15 +18184,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9061 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 9848 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300c].html │ │ │ -rw-r--r-- 0 root (0) root (0) 6828 2025-12-14 14:09:53.000000 ./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) 16172 2025-12-14 14:09:53.000000 ./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) 19856 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sp__Betti_spstratum_spof_spa_spgiven_spquartic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9268 2025-12-14 14:09:53.000000 ./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) 8693 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Half_spcanonical_spdegree_sp20.html │ │ │ --rw-r--r-- 0 root (0) root (0) 53795 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 47481 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10984 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Noether-__Lefschetz_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4123 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12440 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Pfaffians_spon_spquadrics.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8586 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Singularities_spof_splifting_spof_sptype_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 14692 2025-12-14 14:09:53.000000 ./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) 12230 2025-12-14 14:09:53.000000 ./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) 9284 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[310]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.html │ │ │ @@ -18311,18 +18311,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 40654 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RInterface/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 31191 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RInterface/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7199 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RInterface/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 4410 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 416 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/example-output/_canonical__Curve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 417 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/example-output/_canonical__Curve.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 61 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 6065 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/_canonical__Curve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6066 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/_canonical__Curve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5886 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4523 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3193 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 27181 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/ │ │ │ @@ -18378,42 +18378,42 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 13233 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 276 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_canonical__Curve__Genus14.out │ │ │ -rw-r--r-- 0 root (0) root (0) 279 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_curve__Genus14__Degree18in__P6.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2925 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Canonical__Curve__Genus8with8__Points.out │ │ │ --rw-r--r-- 0 root (0) root (0) 485 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Curve__Genus14__Degree18in__P6.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 484 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Curve__Genus14__Degree18in__P6.out │ │ │ -rw-r--r-- 0 root (0) root (0) 393 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Curve__Genus8__Degree14in__P6.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5382 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_canonical__Curve__Genus14.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5485 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_curve__Genus14__Degree18in__P6.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9332 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_random__Canonical__Curve__Genus8with8__Points.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7434 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_random__Curve__Genus14__Degree18in__P6.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7433 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_random__Curve__Genus14__Degree18in__P6.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6239 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_random__Curve__Genus8__Degree14in__P6.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8399 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6319 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4166 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 85866 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Finding_sp__Extreme_sp__Examples.out │ │ │ --rw-r--r-- 0 root (0) root (0) 447 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 448 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 481 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__Chain__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_is__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 250 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Addition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 894 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 564 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 479 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Elements__From__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 461 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 287 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 281 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 421 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 359 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Pure__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 733 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal__Chain.out │ │ │ -rw-r--r-- 0 root (0) root (0) 752 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 426 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Sparse__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out │ │ │ @@ -18430,28 +18430,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5789 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_is__Shelling.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7774 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Addition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7286 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Binomial__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7700 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Binomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6565 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7793 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Elements__From__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7658 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Ideal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6274 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6268 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7113 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7700 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Pure__Binomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7367 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Shellable__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7426 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Shellable__Ideal__Chain.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9696 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Shelling.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8172 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Sparse__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7458 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Monomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18095 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Step.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8317 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Toric__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5737 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_reg__Seq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6021 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_square__Free.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5742 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_square__Free_lp__Z__Z_cm__Ring_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25638 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25639 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17515 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8052 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 151393 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/example-output/___C__M__Stats.out │ │ │ @@ -18580,15 +18580,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 96387 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 356 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/___Max__Coordinates__To__Replace.out │ │ │ -rw-r--r-- 0 root (0) root (0) 268 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/___Point__Check__Attempts.out │ │ │ -rw-r--r-- 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2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Regularity/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RelativeCanonicalResolution/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RelativeCanonicalResolution/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 36804 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RelativeCanonicalResolution/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RelativeCanonicalResolution/example-output/ │ │ │ @@ -19088,80 +19088,80 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 114689 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1231 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/___Grass.out │ │ │ -rw-r--r-- 0 root (0) root (0) 454 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_affine__Discriminant.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2692 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_affine__Resultant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2422 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_cayley__Trick.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6583 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Equations.out │ │ │ --rw-r--r-- 0 root (0) root (0) 17439 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Form.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2423 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_cayley__Trick.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6585 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Equations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 17437 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1436 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_conormal__Variety.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4384 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_discriminant_lp__Ring__Element_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1758 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_dual__Variety.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4381 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_discriminant_lp__Ring__Element_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1757 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_dual__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7159 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_dualize.out │ │ │ --rw-r--r-- 0 root (0) root (0) 7652 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_from__Plucker__To__Stiefel.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 7653 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_from__Plucker__To__Stiefel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2051 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_generic__Polynomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5202 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_hurwitz__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3136 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_is__Coisotropic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1332 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_is__In__Coisotropic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6760 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_macaulay__Formula.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1903 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_plucker.out │ │ │ --rw-r--r-- 0 root (0) root (0) 7137 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp..._cm__Algorithm_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 122873 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp__Matrix_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1902 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_plucker.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 7135 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp..._cm__Algorithm_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 122874 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5789 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_tangential__Chow__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1268 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_veronese.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 41 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5949 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Affine__Chart__Grass.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5248 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Affine__Chart__Proj.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5166 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Assume__Ordinary.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4966 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Duality.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9160 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Grass.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6511 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Grass_lp..._cm__Variable_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4737 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Singular__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6413 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_affine__Discriminant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8671 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_affine__Resultant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10651 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14347 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Equations.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25098 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10652 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14349 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Equations.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25096 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7973 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_conormal__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11621 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_conormal__Variety_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12075 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_discriminant_lp__Ring__Element_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9235 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_dual__Variety.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12072 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_discriminant_lp__Ring__Element_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9234 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_dual__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13158 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_dualize.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14856 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_from__Plucker__To__Stiefel.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14857 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_from__Plucker__To__Stiefel.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8152 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_generic__Polynomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12389 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_hurwitz__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9185 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_is__Coisotropic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8219 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_is__In__Coisotropic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12572 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_macaulay__Formula.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10010 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_plucker.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14373 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp..._cm__Algorithm_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 130499 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp__Matrix_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10009 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_plucker.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14371 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp..._cm__Algorithm_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 130500 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16276 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_tangential__Chow__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7768 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/_veronese.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19593 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20713 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7336 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Resultants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 37681 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 415 2025-12-14 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./usr/share/doc/Macaulay2/RunExternalM2/html/_is__External__M2__Child.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5095 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_is__External__M2__Parent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7601 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23865 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23868 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5662 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2__Return__Answer.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6323 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2_lp..._cm__Keep__Files_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7559 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_suggestions_spfor_spusing_sp__Run__External__M2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9683 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7490 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4878 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SCMAlgebras/ │ │ │ @@ -19203,15 +19203,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8415 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SCSCP/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6794 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SCSCP/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3956 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SCSCP/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 101515 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2516 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/___S__L__Pexpressions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2515 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/___S__L__Pexpressions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 525 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_arithmetic_spwith_spcircuits.out │ │ │ -rw-r--r-- 0 root (0) root (0) 830 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_compressing_spcircuits.out │ │ │ -rw-r--r-- 0 root (0) root (0) 623 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_creating_spgates.out │ │ │ -rw-r--r-- 0 root (0) root (0) 142 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_declare__Variable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 635 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_differentiating_spcircuits.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1597 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_evaluate_lp__S__L__Program_cm__Mutable__Matrix_cm__Mutable__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 524 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_evaluating_spgates.out │ │ │ @@ -19246,15 +19246,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4088 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_make__Compiled__S__L__Program.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7422 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_make__Interpreted__S__L__Program.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4854 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_make__S__L__Program.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8288 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_measuring_spthe_spsize_spof_spcircuits.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6033 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_undeclare__Variable.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6529 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_value__Hash__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8773 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_working_spwith_spgate_spmatrices.html │ │ │ --rw-r--r-- 0 root (0) root (0) 36826 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 36825 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20897 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6550 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 42778 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 3218 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/example-output/_sl2__Equivariant__Constant__Rank__Matrix.out │ │ │ @@ -19494,29 +19494,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 44888 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2826 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11853 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complexes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3437 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Homology.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5542 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_are__Pseudo__Inverses.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2088 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_common__Entries.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6983 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_euclidean__Distance.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6984 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_euclidean__Distance.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7308 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_laplacians.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1301 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_numeric__Rank.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9529 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_project__To__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9254 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_pseudo__Inverse.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3737 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___Laplacian.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3751 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___Projection.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11976 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13012 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15236 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_are__Pseudo__Inverses.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3541 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_check__S__V__D__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10080 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14701 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14702 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14335 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_laplacians.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3882 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_new__Chain__Complex__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9100 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_numeric__Rank.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18251 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_project__To__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18702 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_pseudo__Inverse.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3961 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_pseudo__Inverse1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 28647 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/index.html │ │ │ @@ -19539,24 +19539,24 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52965 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 859 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_annihilator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 846 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_ideal_spquotients_spand_spsaturation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 259 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_is__Supported__In__Zero__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 807 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_module_spquotients_cm_spsaturation_cm_spand_spannihilator.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1834 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1836 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2582 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 837 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_saturate.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 69 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 8747 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_annihilator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6560 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_ideal_spquotients_spand_spsaturation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6876 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_is__Supported__In__Zero__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7263 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_module_spquotients_cm_spsaturation_cm_spand_spannihilator.html │ │ │ --rw-r--r-- 0 root (0) root (0) 17354 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 17356 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15598 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp__Module_cm__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11944 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_saturate.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12655 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_saturate_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13853 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11866 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4273 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/ │ │ │ @@ -19590,15 +19590,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1688 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1584 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ -rw-r--r-- 0 root (0) root (0) 256 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___O__O_sp_us_sp__Abstract__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1332 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___O__O_sp_us_sp__Ring__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 913 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Quotient__Bundles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 869 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spon_spa_spcurve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1741 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spon_spa_spsurface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7382 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spwithout_spdenominators.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Ring_sp_us_sp__Chern__Class__Variable.out │ │ │ @@ -19710,15 +19710,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5605 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Examples_spfrom_sp__Schubert_cm_sptranslated.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8983 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Flag__Bundle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6370 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Hom_lp__Abstract__Sheaf_cm__Abstract__Sheaf_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5551 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Ideas_spfor_spfuture_spdevelopment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6351 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Incidence__Correspondence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4163 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Intersection__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4202 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Isotropic.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8062 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8061 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4976 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___O__O_sp_us_sp__Abstract__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6353 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___O__O_sp_us_sp__Ring__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4277 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Pull__Back.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6529 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Quotient__Bundles.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5154 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spon_spa_spcurve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7875 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spon_spa_spsurface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11386 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spwithout_spdenominators.html │ │ │ @@ -19997,29 +19997,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 435 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_is__Multi__Hom.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_make__Chow__Ring.out │ │ │ -rw-r--r-- 0 root 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21933 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20968 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11034 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 178042 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/dump/rawdocumentation.dump │ │ │ @@ -20644,15 +20644,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 210 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_cycle__Decomposition_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 252 2025-12-14 14:09:53.000000 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root (0) root (0) 914 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/___G__Mtables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1325 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_ambient__Fivefold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 954 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1005 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1191 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_beauville__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 351 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_clean_lp__Hodge__Special__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1173 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1550 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1172 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1555 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 298 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Cubic__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 286 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 672 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_fano__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 599 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_from__Ordinary__To__Gushel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 153 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_is__Admissible.out │ │ │ -rw-r--r-- 0 root (0) root (0) 164 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_is__Admissible__G__M.out │ │ │ -rw-r--r-- 0 root (0) root (0) 716 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_map_lp__Congruence__Of__Curves_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1356 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_mirror__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 357 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_normal__Sheaf.out │ │ │ --rw-r--r-- 0 root (0) root (0) 814 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 813 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out │ │ │ -rw-r--r-- 0 root (0) root (0) 786 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Cubic__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 973 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 653 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 652 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1972 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize_lp__Hodge__Special__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6386 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6385 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 324 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 393 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold_lp__String_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1499 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1520 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 762 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold_lp__Array_cm__Array_cm__String_cm__Thing_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 469 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 530 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold_lp__String_cm__Ring_rp.out │ │ │ @@ -20812,34 +20812,34 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4945 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10359 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10574 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6841 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_beauville__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5897 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_check_lp__Z__Z_cm__Congruence__Of__Curves_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5533 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_clean_lp__Hodge__Special__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6192 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8687 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9569 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8686 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9574 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6654 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Cubic__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6745 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6642 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_fano__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5909 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_from__Ordinary__To__Gushel.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5585 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_is__Admissible.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5729 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_is__Admissible__G__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5440 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_is__Member_lp__Embedded__Projective__Variety_cm__Congruence__Of__Curves_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7955 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_map_lp__Congruence__Of__Curves_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5595 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_map_lp__Special__Cubic__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5686 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_map_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9405 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_mirror__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6394 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_normal__Sheaf.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8483 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8482 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7955 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Cubic__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8182 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8186 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8185 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8747 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize_lp__Hodge__Special__Fourfold_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15460 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15459 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6948 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7039 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold_lp__String_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10491 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6591 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold_lp__String_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12074 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11528 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold_lp__Array_cm__Array_cm__String_cm__Thing_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7954 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold_lp__Embedded__Projective__Variety_rp.html │ │ │ @@ -21787,25 +21787,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 806 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_compatible__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 378 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_decompose__Fraction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 606 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_descend__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_floor__Log.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1518 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Preimage.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1329 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1328 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1117 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Trace__On__Canonical__Module.out │ │ │ --rw-r--r-- 0 root (0) root (0) 487 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1824 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 488 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1820 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ -rw-r--r-- 0 root (0) root (0) 637 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Pure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 517 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Rational.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1492 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1491 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_multiplicative__Order.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_parameter__Test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 216 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Element.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1514 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1513 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2756 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Module.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 590 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 40 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4326 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Ascent__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4385 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__C__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5130 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__Domain.html │ │ │ @@ -21838,25 +21838,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9151 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_compatible__Ideals.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7168 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_decompose__Fraction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9334 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_descend__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5225 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_floor__Log.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10611 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12408 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5237 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Preimage.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16746 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16745 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8554 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Trace__On__Canonical__Module.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8215 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15811 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8216 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15807 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10346 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Pure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9661 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Rational.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16392 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16391 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5824 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_multiplicative__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7890 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_parameter__Test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6302 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Element.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14994 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14993 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17708 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 37577 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 34814 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12484 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Text/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Text/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 136721 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Text/dump/rawdocumentation.dump │ │ │ @@ -22056,30 +22056,30 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 37607 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27702 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11513 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 24388 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1472 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1467 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6776 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Threaded__G__B.out │ │ │ --rw-r--r-- 0 root (0) root (0) 615 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 938 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1256 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4068 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 572 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1147 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1143 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2349 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb_lp..._cm__Verbose_eq_gt..._rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6425 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Lineage__Table.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7059 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6670 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6880 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7259 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14771 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7054 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6627 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7089 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7146 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13052 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7430 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21626 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6806 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4627 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Topcom/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Topcom/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 66598 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Topcom/dump/rawdocumentation.dump │ │ │ @@ -22186,25 +22186,25 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 20320 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 944 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_cm__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 567 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_cm__Volumes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 717 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_dual__Deg__Codim.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1536 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1537 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1032 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_polar__Degrees.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 86 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4388 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Force__Amat.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4214 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4206 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Text__Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8847 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_cm__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7510 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_cm__Volumes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7471 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_dual__Deg__Codim.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8817 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8818 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8849 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_polar__Degrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9974 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7189 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4667 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricTopology/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricTopology/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 38537 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricTopology/dump/rawdocumentation.dump │ │ │ @@ -22373,17 +22373,17 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 17380 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TriangularSets/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11888 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TriangularSets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7217 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TriangularSets/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 31886 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2276 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2275 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 18048 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_all__Triangulations_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 75224 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_generate__Triangulations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 75242 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_generate__Triangulations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 714 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_is__Regular__Triangulation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1654 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_triangulation.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 42 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4460 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/___Chirotope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3951 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/___Cone__Index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7308 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/___Triangulation.html │ │ │ @@ -22391,29 +22391,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 30386 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_all__Triangulations_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3970 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_bistellar__Flip.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3884 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_chirotope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3827 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_delaunay__Subdivision.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3823 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_delaunay__Weights.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3867 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_fine__Star__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3781 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_flips.html │ │ │ --rw-r--r-- 0 root (0) root (0) 86004 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_generate__Triangulations.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 86022 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_generate__Triangulations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3911 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_gkz__Vector.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3873 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_is__Fine.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9223 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_is__Regular__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3883 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_is__Star.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3814 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_naive__Chirotope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4176 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_naive__Is__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3807 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_neighbors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3950 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_regular__Fine__Star__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3970 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_regular__Fine__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4144 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_regular__Triangulation__Weights.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9719 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3727 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_vectors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3837 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_volume__Vector.html │ │ │ --rw-r--r-- 0 root (0) root (0) 24700 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 24699 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10341 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6323 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triplets/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triplets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 49146 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triplets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triplets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triplets/example-output/___Betti1_lp__Triplet_rp.out │ │ │ @@ -23373,15 +23373,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 620 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ceiling_lp__R__Weil__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 264 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_clean__Support.out │ │ │ -rw-r--r-- 0 root (0) root (0) 470 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_clear__Cache.out │ │ │ -rw-r--r-- 0 root (0) root (0) 254 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_coefficient_lp__Ideal_cm__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 728 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_coefficients_lp__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2714 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_divisor.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1714 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_dualize.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1715 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_dualize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1174 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_embed__As__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 500 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_find__Element__Of__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_gbs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 831 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Linear__Diophantine__Solution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 501 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Prime__Count.out │ │ │ -rw-r--r-- 0 root (0) root (0) 357 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Prime__Divisors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 336 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ideal__Power.out │ │ │ @@ -23405,15 +23405,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 441 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Zero__Divisor.out │ │ │ 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./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexify.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4352 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexify.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexive__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ring_lp__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 375 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_to__Q__Weil__Divisor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 458 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_to__R__Weil__Divisor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 576 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_to__Weil__Divisor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 357 2025-12-14 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./usr/share/doc/Macaulay2/WeilDivisors/html/_coefficient_lp__Ideal_cm__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8179 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_coefficients_lp__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19737 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_divisor.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12301 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_dualize.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12302 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_dualize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13200 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_embed__As__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7595 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_find__Element__Of__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7315 2025-12-14 14:09:53.000000 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./usr/share/doc/Macaulay2/WeilDivisors/html/_non__Cartier__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6689 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_positive__Part.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7155 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_primes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9116 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10363 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_ramification__Divisor.html │ │ │ --rw-r--r-- 0 root (0) root (0) 20374 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexify.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 20372 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexify.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9556 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexive__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5137 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_ring_lp__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6136 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_to__Q__Weil__Divisor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6886 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_to__R__Weil__Divisor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7207 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_to__Weil__Divisor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6826 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_torsion__Submodule.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6176 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_trim_lp__Basic__Divisor_rp.html │ │ │ @@ -23502,15 +23502,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 365 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeylAlgebras/example-output/___Dsingular__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 380 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeylAlgebras/example-output/___Dtransposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2187 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeylAlgebras/example-output/___Fourier.out │ │ │ -rw-r--r-- 0 root (0) root (0) 404 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeylAlgebras/example-output/_characteristic__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 300 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeylAlgebras/example-output/_create__Dpairs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 264 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeylAlgebras/example-output/_extract__Diffs__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 261 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeylAlgebras/example-output/_extract__Vars__Algebra.out │ │ │ --rw-r--r-- 0 root (0) 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./usr/share/info/TriangularSets.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 13681 2025-12-14 14:09:53.000000 ./usr/share/info/Triangulations.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 13680 2025-12-14 14:09:53.000000 ./usr/share/info/Triangulations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7188 2025-12-14 14:09:53.000000 ./usr/share/info/Triplets.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 11010 2025-12-14 14:09:53.000000 ./usr/share/info/Tropical.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10698 2025-12-14 14:09:53.000000 ./usr/share/info/TropicalToric.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 5772 2025-12-14 14:09:53.000000 ./usr/share/info/Truncations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8693 2025-12-14 14:09:53.000000 ./usr/share/info/Units.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4687 2025-12-14 14:09:53.000000 ./usr/share/info/VNumber.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8877 2025-12-14 14:09:53.000000 ./usr/share/info/Valuations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 44204 2025-12-14 14:09:53.000000 ./usr/share/info/Varieties.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 19378 2025-12-14 14:09:53.000000 ./usr/share/info/VectorFields.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 51845 2025-12-14 14:09:53.000000 ./usr/share/info/VectorGraphics.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 41369 2025-12-14 14:09:53.000000 ./usr/share/info/VersalDeformations.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 41370 2025-12-14 14:09:53.000000 ./usr/share/info/VersalDeformations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 12693 2025-12-14 14:09:53.000000 ./usr/share/info/VirtualResolutions.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10443 2025-12-14 14:09:53.000000 ./usr/share/info/Visualize.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 38082 2025-12-14 14:09:53.000000 ./usr/share/info/WeilDivisors.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 10700 2025-12-14 14:09:53.000000 ./usr/share/info/WeylAlgebras.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 38098 2025-12-14 14:09:53.000000 ./usr/share/info/WeilDivisors.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 10702 2025-12-14 14:09:53.000000 ./usr/share/info/WeylAlgebras.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 33226 2025-12-14 14:09:53.000000 ./usr/share/info/WeylGroups.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 14777 2025-12-14 14:09:53.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 14775 2025-12-14 14:09:53.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8865 2025-12-14 14:09:53.000000 ./usr/share/info/XML.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 49484 2025-12-14 14:09:53.000000 ./usr/share/info/gfanInterface.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 49481 2025-12-14 14:09:53.000000 ./usr/share/info/gfanInterface.info.gz │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/lintian/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/lintian/overrides/ │ │ │ -rw-r--r-- 0 root (0) root (0) 11489 2025-12-14 14:09:53.000000 ./usr/share/lintian/overrides/macaulay2-common │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/Macaulay2/Style/katex/contrib/auto-render.min.js -> ../../../../javascript/katex/contrib/auto-render.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/Macaulay2/Style/katex/contrib/copy-tex.min.js -> ../../../../javascript/katex/contrib/copy-tex.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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) │ │ │ - -- 1.71138s elapsed │ │ │ + -- 1.5172s 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) │ │ │ - -- 2.15695s elapsed │ │ │ + -- 1.7848s 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 │ │ │ @@ -90,28 +90,28 @@ │ │ │ 1 │ │ │ o2 : R-module, quotient of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i3 : elapsedTime burkeResolution(M, 7, Check => false)
│ │ │ - -- 1.71138s elapsed
│ │ │ + -- 1.5172s 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)
│ │ │ - -- 2.15695s elapsed
│ │ │ + -- 1.7848s 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) │ │ │ │ - -- 1.71138s elapsed │ │ │ │ + -- 1.5172s 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) │ │ │ │ - -- 2.15695s elapsed │ │ │ │ + -- 1.7848s 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/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out │ │ │ @@ -49,15 +49,15 @@ │ │ │ o8 : BettiTally │ │ │ │ │ │ i9 : c=codim I │ │ │ │ │ │ o9 = 4 │ │ │ │ │ │ i10 : elapsedTime fI=res I │ │ │ - -- .0228791s elapsed │ │ │ + -- .030915s elapsed │ │ │ │ │ │ 1 14 33 28 8 │ │ │ o10 = Pn <-- Pn <-- Pn <-- Pn <-- Pn <-- 0 │ │ │ │ │ │ 0 1 2 3 4 5 │ │ │ │ │ │ o10 : ChainComplex │ │ ├── ./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) │ │ │ - -- .670263s elapsed │ │ │ + -- .583559s 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; │ │ │ - -- 5.77797s elapsed │ │ │ + -- 5.57033s 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) │ │ │ - -- .0136638s elapsed │ │ │ + -- .0147241s 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) │ │ │ - -- .0563731s elapsed │ │ │ + -- .0686235s elapsed │ │ │ │ │ │ 0 1 │ │ │ o16 = total: 1 12 │ │ │ 0: 1 . │ │ │ 1: . 12 │ │ │ │ │ │ o16 : BettiTally │ │ │ │ │ │ i17 : I'== I │ │ │ │ │ │ o17 = true │ │ │ │ │ │ i18 : elapsedTime basePts=primaryDecomposition ideal H; │ │ │ - -- 1.83465s elapsed │ │ │ + -- 1.65508s 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 │ │ │ @@ -149,15 +149,15 @@ │ │ │ │ │ │ o9 = 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i10 : elapsedTime fI=res I
│ │ │ - -- .0228791s elapsed
│ │ │ + -- .030915s elapsed
│ │ │  
│ │ │          1       14       33       28       8
│ │ │  o10 = Pn  <-- Pn   <-- Pn   <-- Pn   <-- Pn  <-- 0
│ │ │                                                    
│ │ │        0       1        2        3        4       5
│ │ │  
│ │ │  o10 : ChainComplex
│ │ │ ├── html2text {} │ │ │ │ @@ -54,15 +54,15 @@ │ │ │ │ 2: . 12 │ │ │ │ │ │ │ │ o8 : BettiTally │ │ │ │ i9 : c=codim I │ │ │ │ │ │ │ │ o9 = 4 │ │ │ │ i10 : elapsedTime fI=res I │ │ │ │ - -- .0228791s elapsed │ │ │ │ + -- .030915s elapsed │ │ │ │ │ │ │ │ 1 14 33 28 8 │ │ │ │ o10 = Pn <-- Pn <-- Pn <-- Pn <-- Pn <-- 0 │ │ │ │ │ │ │ │ 0 1 2 3 4 5 │ │ │ │ │ │ │ │ o10 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ @@ -217,15 +217,15 @@ │ │ │ │ │ │ o14 : RingMap P2 <-- Pn │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i15 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .670263s elapsed
│ │ │ + -- .583559s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o15 = total: 1 11
│ │ │            0: 1  .
│ │ │            1: .  3
│ │ │            2: .  8
│ │ │  
│ │ │ @@ -238,15 +238,15 @@
│ │ │  
│ │ │  o16 = true
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 5.77797s elapsed
│ │ │ + -- 5.57033s 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)
│ │ │ │ - -- .670263s elapsed
│ │ │ │ + -- .583559s 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;
│ │ │ │ - -- 5.77797s elapsed
│ │ │ │ + -- 5.57033s 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
│ │ │ @@ -193,15 +193,15 @@
│ │ │  
│ │ │  o13 : BettiTally
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i14 : elapsedTime sub(I,H)
│ │ │ - -- .0136638s elapsed
│ │ │ + -- .0147241s elapsed
│ │ │  
│ │ │  o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o14 : Ideal of P2
│ │ │ │ │ │ │ │ │ │ │ │ @@ -210,15 +210,15 @@ │ │ │ │ │ │ o15 : RingMap P2 <-- Pn │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i16 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .0563731s elapsed
│ │ │ + -- .0686235s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o16 = total: 1 12
│ │ │            0: 1  .
│ │ │            1: . 12
│ │ │  
│ │ │  o16 : BettiTally
│ │ │ @@ -230,15 +230,15 @@ │ │ │ │ │ │ o17 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 1.83465s elapsed
│ │ │ + -- 1.65508s 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)
│ │ │ │ - -- .0136638s elapsed
│ │ │ │ + -- .0147241s 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)
│ │ │ │ - -- .0563731s elapsed
│ │ │ │ + -- .0686235s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o16 = total: 1 12
│ │ │ │            0: 1  .
│ │ │ │            1: . 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ │ │  i17 : I'== I
│ │ │ │  
│ │ │ │  o17 = true
│ │ │ │  i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 1.83465s elapsed
│ │ │ │ + -- 1.65508s 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 0.440925s (cpu); 0.367004s (thread); 0s (gc)
│ │ │ + -- used 0.560535s (cpu); 0.454712s (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 0.477334s (cpu); 0.400537s (thread); 0s (gc)
│ │ │ + -- used 0.587028s (cpu); 0.484857s (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
│ │ │ @@ -253,15 +253,15 @@
│ │ │                4      4
│ │ │  o13 : Matrix A  <-- A
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ - -- used 0.440925s (cpu); 0.367004s (thread); 0s (gc)
│ │ │ + -- used 0.560535s (cpu); 0.454712s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o14 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │  
│ │ │ @@ -272,15 +272,15 @@
│ │ │          

│ │ │            
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 {}
│ │ │ │ @@ -192,26 +192,26 @@
│ │ │ │  o18 : ActionOnComplex
│ │ │ │  i19 : A2 = action(RI2,G,Sub=>false)
│ │ │ │  
│ │ │ │  o19 = Complex with 6 actors
│ │ │ │  
│ │ │ │  o19 : ActionOnComplex
│ │ │ │  i20 : elapsedTime a1 = character A1
│ │ │ │ - -- .852762s elapsed
│ │ │ │ + -- .839315s elapsed
│ │ │ │  
│ │ │ │  o20 = Character over R
│ │ │ │  
│ │ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │ │        (1, {8}) => | 3 -1 0 1 a4+a2+a -a4-a2-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
│ │ │ │ - -- 34.344s elapsed
│ │ │ │ + -- 27.1288s elapsed
│ │ │ │  
│ │ │ │  o21 = Character over R
│ │ │ │  
│ │ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │ │        (1, {16}) => | 6 2 0 0 -1 -1 |
│ │ │ │        (2, {19}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ │        (2, {21}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ │ @@ -308,15 +308,15 @@
│ │ │ │  i30 : M = Is2 / I2;
│ │ │ │  i31 : B = action(M,G,Sub=>false)
│ │ │ │  
│ │ │ │  o31 = Module with 6 actors
│ │ │ │  
│ │ │ │  o31 : ActionOnGradedModule
│ │ │ │  i32 : elapsedTime b = character(B,21)
│ │ │ │ - -- 14.2617s elapsed
│ │ │ │ + -- 12.529s elapsed
│ │ │ │  
│ │ │ │  o32 = Character over R
│ │ │ │  
│ │ │ │        (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 0.354218s (cpu); 0.289007s (thread); 0s (gc)
│ │ │ + -- used 0.33925s (cpu); 0.252181s (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
│ │ │ @@ -380,15 +380,15 @@
│ │ │  
│ │ │  o22 : BettiTally
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │              
│ │ │            
│ │ │          

│ │ │                
i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ - -- used 0.477334s (cpu); 0.400537s (thread); 0s (gc)
│ │ │ + -- used 0.587028s (cpu); 0.484857s (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 0.440925s (cpu); 0.367004s (thread); 0s (gc)
│ │ │ │ + -- used 0.560535s (cpu); 0.454712s (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 0.477334s (cpu); 0.400537s (thread); 0s (gc)
│ │ │ │ + -- used 0.587028s (cpu); 0.484857s (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 Sun Dec 14 15:31:42 UTC 2025
│ │ │ --- Linux sbuild 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.249  
│ │ │ +-- beginning computation Wed Jan  7 12:26:50 UTC 2026
│ │ │ +-- Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998  
│ │ │  -- Macaulay2 1.25.11, compiled with gcc 15.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .153215 seconds
│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .175224 seconds
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html
│ │ │ @@ -75,19 +75,19 @@
│ │ │          

│ │ │            
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 Sun Dec 14 15:31:42 UTC 2025
│ │ │ --- Linux sbuild 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.249  
│ │ │ +-- beginning computation Wed Jan  7 12:26:50 UTC 2026
│ │ │ +-- Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998  
│ │ │  -- Macaulay2 1.25.11, compiled with gcc 15.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .153215 seconds

│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .175224 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 Sun Dec 14 15:31:42 UTC 2025
│ │ │ │ --- Linux sbuild 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1
│ │ │ │ -(2025-11-05) x86_64 GNU/Linux
│ │ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.249
│ │ │ │ +-- beginning computation Wed Jan  7 12:26:50 UTC 2026
│ │ │ │ +-- Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998
│ │ │ │  -- Macaulay2 1.25.11, compiled with gcc 15.2.0
│ │ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .153215 seconds
│ │ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .175224 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-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Benchmark.m2:297:0.
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/dump/rawdocumentation.dump
│ │ │ @@ -515,15 +515,15 @@
│ │ │  Pi4uLikiLCJCZXJ0aW5pIn0sIlJhbmRvbUNvbXBsZXgifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwi
│ │ │  LCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwie30ifSwiLCAiLFNQQU57fX0sU1BBTntUTzJ7bmV3
│ │ │  IERvY3VtZW50VGFnIGZyb20ge1tiZXJ0aW5pVXNlckhvbW90b3B5LFJhbmRvbVJlYWxdLCJiZXJ0
│ │ │  aW5pVXNlckhvbW90b3B5KC4uLixSYW5kb21SZWFsPT4uLi4pIiwiQmVydGluaSJ9LCJSYW5kb21S
│ │ │  ZWFsIn0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZhdWx0IHZhbHVlICIsInt9
│ │ │  In0sIiwgIixTUEFOe319LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHsiVG9wRGlyZWN0
│ │ │  b3J5IiwiVG9wRGlyZWN0b3J5IiwiQmVydGluaSJ9LCJUb3BEaXJlY3RvcnkifSxUVHsiID0+ICJ9
│ │ │ -LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiXCIvdG1wL00yLTI4NzA2LTAv
│ │ │ +LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiXCIvdG1wL00yLTQwOTEzLTAv
│ │ │  MFwiIn0sIiwgIixTUEFOeyJPcHRpb24gdG8gY2hhbmdlIGRpcmVjdG9yeSBmb3IgZmlsZSBzdG9y
│ │ │  YWdlLiJ9fSxTUEFOe1RPMntuZXcgRG9jdW1lbnRUYWcgZnJvbSB7W2JlcnRpbmlVc2VySG9tb3Rv
│ │ │  cHksVmVyYm9zZV0sImJlcnRpbmlVc2VySG9tb3RvcHkoLi4uLFZlcmJvc2U9Pi4uLikiLCJCZXJ0
│ │ │  aW5pIn0sIlZlcmJvc2UifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQg
│ │ │  dmFsdWUgIiwiZmFsc2UifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBzaWxlbmNlIGFkZGl0aW9uYWwg
│ │ │  b3V0cHV0In19fSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsi
│ │ │  YmVydGluaVVzZXJIb21vdG9weSIsImJlcnRpbmlVc2VySG9tb3RvcHkiLCJCZXJ0aW5pIn0sIEtl
│ │ │ @@ -1100,15 +1100,15 @@
│ │ │  ZXJ0aW5pUGFyYW1ldGVySG9tb3RvcHkoLi4uLFJhbmRvbVJlYWw9Pi4uLikiLCJCZXJ0aW5pIn0s
│ │ │  IlJhbmRvbVJlYWwifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFs
│ │ │  dWUgIiwie30ifSwiLCAiLFNQQU57ImFuIG9wdGlvbiB3aGljaCBkZXNpZ25hdGVzIHN5bWJvbHMv
│ │ │  c3RyaW5ncy92YXJpYWJsZXMgdGhhdCB3aWxsIGJlIHNldCB0byBiZSBhIHJhbmRvbSByZWFsIG51
│ │ │  bWJlciBvciByYW5kb20gY29tcGxleCBudW1iZXIifX0sU1BBTntUTzJ7bmV3IERvY3VtZW50VGFn
│ │ │  IGZyb20geyJUb3BEaXJlY3RvcnkiLCJUb3BEaXJlY3RvcnkiLCJCZXJ0aW5pIn0sIlRvcERpcmVj
│ │ │  dG9yeSJ9LFRUeyIgPT4gIn0sVFR7Ii4uLiJ9LCIsICIsU1BBTnsiZGVmYXVsdCB2YWx1ZSAiLCJc
│ │ │ -Ii90bXAvTTItMjg3MDYtMC8wXCIifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBjaGFuZ2UgZGlyZWN0
│ │ │ +Ii90bXAvTTItNDA5MTMtMC8wXCIifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBjaGFuZ2UgZGlyZWN0
│ │ │  b3J5IGZvciBmaWxlIHN0b3JhZ2UuIn19LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHtb
│ │ │  YmVydGluaVBhcmFtZXRlckhvbW90b3B5LFZlcmJvc2VdLCJiZXJ0aW5pUGFyYW1ldGVySG9tb3Rv
│ │ │  cHkoLi4uLFZlcmJvc2U9Pi4uLikiLCJCZXJ0aW5pIn0sIlZlcmJvc2UifSxUVHsiID0+ICJ9LFRU
│ │ │  eyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiZmFsc2UifSwiLCAiLFNQQU57Ik9w
│ │ │  dGlvbiB0byBzaWxlbmNlIGFkZGl0aW9uYWwgb3V0cHV0In19fSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsiYmVydGluaVBhcmFtZXRlckhvbW90b3B5IiwiYmVy
│ │ │  dGluaVBhcmFtZXRlckhvbW90b3B5IiwiQmVydGluaSJ9LCBLZXkgPT4gYmVydGluaVBhcmFtZXRl
│ │ │ @@ -2449,15 +2449,15 @@
│ │ │  YWw9Pi4uLikiLCJCZXJ0aW5pIn0sIlJhbmRvbVJlYWwifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwi
│ │ │  LCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwie30ifSwiLCAiLFNQQU57ImFuIG9wdGlvbiB3aGlj
│ │ │  aCBkZXNpZ25hdGVzIHN5bWJvbHMvc3RyaW5ncy92YXJpYWJsZXMgdGhhdCB3aWxsIGJlIHNldCB0
│ │ │  byBiZSBhIHJhbmRvbSByZWFsIG51bWJlciBvciByYW5kb20gY29tcGxleCBudW1iZXIifX0sU1BB
│ │ │  TntUTzJ7bmV3IERvY3VtZW50VGFnIGZyb20ge1tiZXJ0aW5pWmVyb0RpbVNvbHZlLFRvcERpcmVj
│ │ │  dG9yeV0sImJlcnRpbmlaZXJvRGltU29sdmUoLi4uLFRvcERpcmVjdG9yeT0+Li4uKSIsIkJlcnRp
│ │ │  bmkifSwiVG9wRGlyZWN0b3J5In0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZh
│ │ │ -dWx0IHZhbHVlICIsIlwiL3RtcC9NMi0yODcwNi0wLzBcIiJ9LCIsICIsU1BBTnsiT3B0aW9uIHRv
│ │ │ +dWx0IHZhbHVlICIsIlwiL3RtcC9NMi00MDkxMy0wLzBcIiJ9LCIsICIsU1BBTnsiT3B0aW9uIHRv
│ │ │  IGNoYW5nZSBkaXJlY3RvcnkgZm9yIGZpbGUgc3RvcmFnZS4ifX0sU1BBTntUTzJ7bmV3IERvY3Vt
│ │ │  ZW50VGFnIGZyb20ge1tiZXJ0aW5pWmVyb0RpbVNvbHZlLFVzZVJlZ2VuZXJhdGlvbl0sImJlcnRp
│ │ │  bmlaZXJvRGltU29sdmUoLi4uLFVzZVJlZ2VuZXJhdGlvbj0+Li4uKSIsIkJlcnRpbmkifSwiVXNl
│ │ │  UmVnZW5lcmF0aW9uIn0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZhdWx0IHZh
│ │ │  bHVlICIsIi0xIn0sIiwgIixTUEFOe319LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHtb
│ │ │  YmVydGluaVplcm9EaW1Tb2x2ZSxWZXJib3NlXSwiYmVydGluaVplcm9EaW1Tb2x2ZSguLi4sVmVy
│ │ │  Ym9zZT0+Li4uKSIsIkJlcnRpbmkifSwiVmVyYm9zZSJ9LFRUeyIgPT4gIn0sVFR7Ii4uLiJ9LCIs
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Parameter__Homotopy.html
│ │ │ @@ -72,15 +72,15 @@
│ │ │              
  • 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/M2-28706-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-40913-0/0", 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/M2-28706-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-40913-0/0", 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
│ │ │ @@ -77,15 +77,15 @@
│ │ │   => ..., default value 53, 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex (missing documentation)
│ │ │   => ..., default value {}, 
    • 
│ │ │              
  • RandomReal (missing documentation)
│ │ │   => ..., default value {}, 
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-28706-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-40913-0/0", 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/M2-28706-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-40913-0/0", 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
│ │ │ @@ -79,15 +79,15 @@
│ │ │   => ..., 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/M2-28706-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-40913-0/0", 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/M2-28706-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-40913-0/0", 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
│ │ │ - -- .559135s elapsed
│ │ │ + -- .437094s elapsed
│ │ │  
│ │ │  o9 = Character over R
│ │ │        
│ │ │       (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
│ │ │ - -- .474916s elapsed
│ │ │ + -- .498653s elapsed
│ │ │  
│ │ │  o7 = Character over R
│ │ │        
│ │ │       (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
│ │ │ @@ -187,27 +187,27 @@
│ │ │  i19 : A2 = action(RI2,G,Sub=>false)
│ │ │  
│ │ │  o19 = Complex with 6 actors
│ │ │  
│ │ │  o19 : ActionOnComplex
│ │ │  
│ │ │  i20 : elapsedTime a1 = character A1
│ │ │ - -- .852762s elapsed
│ │ │ + -- .839315s elapsed
│ │ │  
│ │ │  o20 = Character over R
│ │ │         
│ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │        (1, {8}) => | 3 -1 0 1 a4+a2+a -a4-a2-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
│ │ │ - -- 34.344s elapsed
│ │ │ + -- 27.1288s elapsed
│ │ │  
│ │ │  o21 = Character over R
│ │ │         
│ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │        (1, {16}) => | 6 2 0 0 -1 -1 |
│ │ │        (2, {19}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │        (2, {21}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ @@ -297,15 +297,15 @@
│ │ │  i31 : B = action(M,G,Sub=>false)
│ │ │  
│ │ │  o31 = Module with 6 actors
│ │ │  
│ │ │  o31 : ActionOnGradedModule
│ │ │  
│ │ │  i32 : elapsedTime b = character(B,21)
│ │ │ - -- 14.2617s elapsed
│ │ │ + -- 12.529s elapsed
│ │ │  
│ │ │  o32 = Character over R
│ │ │         
│ │ │        (0, {21}) => | 1 1 1 1 1 1 |
│ │ │  
│ │ │  o32 : Character
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp1.html
│ │ │ @@ -162,15 +162,15 @@
│ │ │  
│ │ │  o8 : ActionOnComplex
    • 
│ │ │              
    
│ │ │                
    i9 : elapsedTime c = character A
│ │ │ - -- .559135s elapsed
│ │ │ + -- .437094s elapsed
│ │ │  
│ │ │  o9 = Character over R
│ │ │        
│ │ │       (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
│ │ │ │ - -- .559135s elapsed
│ │ │ │ + -- .437094s elapsed
│ │ │ │  
│ │ │ │  o9 = Character over R
│ │ │ │  
│ │ │ │       (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
│ │ │ @@ -180,15 +180,15 @@
│ │ │  
│ │ │  o6 : ActionOnComplex
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime c=character A
│ │ │ - -- .474916s elapsed
│ │ │ + -- .498653s elapsed
│ │ │  
│ │ │  o7 = Character over R
│ │ │        
│ │ │       (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
│ │ │ │ - -- .474916s elapsed
│ │ │ │ + -- .498653s elapsed
│ │ │ │  
│ │ │ │  o7 = Character over R
│ │ │ │  
│ │ │ │       (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
│ │ │ @@ -310,30 +310,30 @@
│ │ │  
│ │ │  o19 : ActionOnComplex
    
│ │ │              
    
│ │ │                
    i20 : elapsedTime a1 = character A1
│ │ │ - -- .852762s elapsed
│ │ │ + -- .839315s elapsed
│ │ │  
│ │ │  o20 = Character over R
│ │ │         
│ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │        (1, {8}) => | 3 -1 0 1 a4+a2+a -a4-a2-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
│ │ │ - -- 34.344s elapsed
│ │ │ + -- 27.1288s elapsed
│ │ │  
│ │ │  o21 = Character over R
│ │ │         
│ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │        (1, {16}) => | 6 2 0 0 -1 -1 |
│ │ │        (2, {19}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │        (2, {21}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ @@ -467,15 +467,15 @@
│ │ │  
│ │ │  o31 : ActionOnGradedModule
    
│ │ │              
    
│ │ │                
    i32 : elapsedTime b = character(B,21)
│ │ │ - -- 14.2617s elapsed
│ │ │ + -- 12.529s elapsed
│ │ │  
│ │ │  o32 = Character over R
│ │ │         
│ │ │        (0, {21}) => | 1 1 1 1 1 1 |
│ │ │  
│ │ │  o32 : Character
    
│ │ │              
    
│ │ │                
    i23 : time j=bruns F.dd_3;
│ │ │ - -- used 0.354218s (cpu); 0.289007s (thread); 0s (gc)
│ │ │ + -- used 0.33925s (cpu); 0.252181s (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 0.354218s (cpu); 0.289007s (thread); 0s (gc)
│ │ │ │ + -- used 0.33925s (cpu); 0.252181s (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/_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 3   5 2   2 4   5 3   5 4   5    5 2   5 3   5 4   4 2   4 4
│ │ │ -               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   2 4   5 3   5 4   5    5 2   5 3   5 4   4 2   4 4   5    3 3
│ │ │ +               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/_hull__Complex.out
│ │ │ @@ -19,16 +19,16 @@
│ │ │                             
│ │ │       -1     0      1      2
│ │ │  
│ │ │  o4 : Complex
│ │ │  
│ │ │  i5 : cells(1,H)/cellLabel
│ │ │  
│ │ │ -       5 3    5 4   3 5    4 5   2        2    4 4
│ │ │ -o5 = {x y z, x y , x y z, x y , x y*z, x*y z, x y z}
│ │ │ +       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/_cell__Complex_lp__Ring_cm__Simplicial__Complex_rp.html
│ │ │ @@ -129,16 +129,16 @@
│ │ │            
    
│ │ │                
    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 3   5 2   2 4   5 3   5 4   5    5 2   5 3   5 4   4 2   4 4
│ │ │ -               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   2 4   5 3   5 4   5    5 2   5 3   5 4   4 2   4 4   5    3 3
│ │ │ +               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 {}
│ │ │ │ @@ -41,18 +41,18 @@
│ │ │ │  
│ │ │ │  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 3   5 2   2 4   5 3   5 4   5    5 2   5 3   5 4
│ │ │ │ -4 2   4 4
│ │ │ │ -               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   2 4   5 3   5 4   5    5 2   5 3   5 4   4 2   4 4
│ │ │ │ +5    3 3
│ │ │ │ +               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:: **********
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_hull__Complex.html
│ │ │ @@ -109,16 +109,16 @@
│ │ │  o4 : Complex
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : cells(1,H)/cellLabel
│ │ │  
│ │ │ -       5 3    5 4   3 5    4 5   2        2    4 4
│ │ │ -o5 = {x y z, x y , x y z, x y , x y*z, x*y z, x y z}
│ │ │ +       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
│ │ │ │  
│ │ │ │ -       5 3    5 4   3 5    4 5   2        2    4 4
│ │ │ │ -o5 = {x y z, x y , x y z, x y , x y*z, x*y z, x y z}
│ │ │ │ +       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/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 0.297722s (cpu); 0.239584s (thread); 0s (gc)
│ │ │ + -- used 0.380932s (cpu); 0.292675s (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 0.0968336s (cpu); 0.0968327s (thread); 0s (gc)
│ │ │ + -- used 0.110715s (cpu); 0.110716s (thread); 0s (gc)
│ │ │  
│ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.226002s (cpu); 0.156796s (thread); 0s (gc)
│ │ │ + -- used 0.281453s (cpu); 0.192379s (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
│ │ │ @@ -181,15 +181,15 @@
│ │ │          
    
│ │ │            
    Now we minimize the result. The free summand we added to the end maps to zero, and thus is part of the minimization.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -301,15 +301,15 @@
│ │ │  
│ │ │  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 0.984576s (cpu); 0.463326s (thread); 0s (gc)
│ │ │ │ + -- used 1.42925s (cpu); 0.510427s (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 0.111403s (cpu); 0.0580925s (thread); 0s (gc)
│ │ │ │ + -- used 0.105777s (cpu); 0.0841649s (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
│ │ │ @@ -72,15 +72,15 @@
│ │ │  
│ │ │  o2 : NormalToricVariety
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i13 : time m = minimize (E[1]);
│ │ │ - -- used 0.297722s (cpu); 0.239584s (thread); 0s (gc)
    
│ │ │ + -- used 0.380932s (cpu); 0.292675s (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 0.297722s (cpu); 0.239584s (thread); 0s (gc)
│ │ │ │ + -- used 0.380932s (cpu); 0.292675s (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
│ │ │ @@ -129,21 +129,21 @@
│ │ │                  		
│ │ │                
    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 0.0968336s (cpu); 0.0968327s (thread); 0s (gc)
    
│ │ │ + -- used 0.110715s (cpu); 0.110716s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.226002s (cpu); 0.156796s (thread); 0s (gc)
    
│ │ │ + -- used 0.281453s (cpu); 0.192379s (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 0.0968336s (cpu); 0.0968327s (thread); 0s (gc)
│ │ │ │ + -- used 0.110715s (cpu); 0.110716s (thread); 0s (gc)
│ │ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ │ - -- used 0.226002s (cpu); 0.156796s (thread); 0s (gc)
│ │ │ │ + -- used 0.281453s (cpu); 0.192379s (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 0.984576s (cpu); 0.463326s (thread); 0s (gc)
│ │ │ + -- used 1.42925s (cpu); 0.510427s (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 0.111403s (cpu); 0.0580925s (thread); 0s (gc)
│ │ │ + -- used 0.105777s (cpu); 0.0841649s (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 0.246174s (cpu); 0.164624s (thread); 0s (gc)
│ │ │ + -- used 0.288745s (cpu); 0.184081s (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 0.374312s (cpu); 0.295125s (thread); 0s (gc)
│ │ │ + -- used 0.475736s (cpu); 0.375863s (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 0.629302s (cpu); 0.301613s (thread); 0s (gc)
│ │ │ + -- used 0.93162s (cpu); 0.390155s (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 2.13127s (cpu); 1.82455s (thread); 0s (gc)
│ │ │ + -- used 2.46817s (cpu); 2.19007s (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 0.279106s (cpu); 0.195272s (thread); 0s (gc)
│ │ │ + -- used 0.321391s (cpu); 0.233657s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │  o10 : ------
│ │ │           4
│ │ │          h
│ │ │           1
│ │ │  
│ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0825143s (cpu); 0.0825209s (thread); 0s (gc)
│ │ │ + -- used 0.102983s (cpu); 0.102862s (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 0.0576959s (cpu); 0.0354928s (thread); 0s (gc)
│ │ │ + -- used 0.104201s (cpu); 0.0514337s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : time Euler I
│ │ │ - -- used 0.254114s (cpu); 0.146833s (thread); 0s (gc)
│ │ │ + -- used 0.359599s (cpu); 0.201281s (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 0.182967s (cpu); 0.0690199s (thread); 0s (gc)
│ │ │ + -- used 0.197439s (cpu); 0.103033s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
│ │ │  
│ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.190575s (cpu); 0.0845693s (thread); 0s (gc)
│ │ │ + -- used 0.283546s (cpu); 0.118035s (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 0.0874131s (cpu); 0.0492159s (thread); 0s (gc)
│ │ │ + -- used 0.0851631s (cpu); 0.0688623s (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 2.58397s (cpu); 1.12003s (thread); 0s (gc)
│ │ │ + -- used 6.54387s (cpu); 1.57433s (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 2.73505s (cpu); 1.2427s (thread); 0s (gc)
│ │ │ + -- used 7.09559s (cpu); 1.57436s (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 0.852243s (cpu); 0.438332s (thread); 0s (gc)
│ │ │ + -- used 1.09918s (cpu); 0.534845s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0911228s (cpu); 0.0316495s (thread); 0s (gc)
│ │ │ + -- used 0.0864587s (cpu); 0.0478865s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o4 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i5 : time Chern I
│ │ │ - -- used 0.0751141s (cpu); 0.0309694s (thread); 0s (gc)
│ │ │ + -- used 0.0621425s (cpu); 0.0418672s (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 1.71002s (cpu); 0.984712s (thread); 0s (gc)
│ │ │ + -- used 4.08327s (cpu); 1.37892s (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 0.438602s (cpu); 0.209232s (thread); 0s (gc)
│ │ │ + -- used 0.863456s (cpu); 0.327363s (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
│ │ │ @@ -234,15 +234,15 @@
│ │ │  
│ │ │  o14 : Ideal of R
    
│ │ │              
    
│ │ │                
    i15 : time csmK=CSM(A,K)
│ │ │ - -- used 0.984576s (cpu); 0.463326s (thread); 0s (gc)
│ │ │ + -- used 1.42925s (cpu); 0.510427s (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 0.111403s (cpu); 0.0580925s (thread); 0s (gc)
│ │ │ + -- used 0.105777s (cpu); 0.0841649s (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 0.246174s (cpu); 0.164624s (thread); 0s (gc)
│ │ │ + -- used 0.288745s (cpu); 0.184081s (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
│ │ │ @@ -88,15 +88,15 @@
│ │ │       (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 0.374312s (cpu); 0.295125s (thread); 0s (gc)
│ │ │ + -- used 0.475736s (cpu); 0.375863s (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 0.246174s (cpu); 0.164624s (thread); 0s (gc)
│ │ │ │ + -- used 0.288745s (cpu); 0.184081s (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 0.374312s (cpu); 0.295125s (thread); 0s (gc)
│ │ │ │ + -- used 0.475736s (cpu); 0.375863s (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
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.629302s (cpu); 0.301613s (thread); 0s (gc)
│ │ │ + -- used 0.93162s (cpu); 0.390155s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │         1      1      1      1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -109,15 +109,15 @@
│ │ │         h
│ │ │          1
    
│ │ │              
    
│ │ │                
    i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ - -- used 2.13127s (cpu); 1.82455s (thread); 0s (gc)
│ │ │ + -- used 2.46817s (cpu); 2.19007s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          6
│ │ │ @@ -142,15 +142,15 @@
│ │ │  
│ │ │  o9 : Ideal of S
    
│ │ │              
    
│ │ │                
    i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.279106s (cpu); 0.195272s (thread); 0s (gc)
│ │ │ + -- used 0.321391s (cpu); 0.233657s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │ @@ -159,15 +159,15 @@
│ │ │          h
│ │ │           1
    
│ │ │              
    
│ │ │                
    i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0825143s (cpu); 0.0825209s (thread); 0s (gc)
│ │ │ + -- used 0.102983s (cpu); 0.102862s (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 0.629302s (cpu); 0.301613s (thread); 0s (gc)
│ │ │ │ + -- used 0.93162s (cpu); 0.390155s (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 2.13127s (cpu); 1.82455s (thread); 0s (gc)
│ │ │ │ + -- used 2.46817s (cpu); 2.19007s (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 0.279106s (cpu); 0.195272s (thread); 0s (gc)
│ │ │ │ + -- used 0.321391s (cpu); 0.233657s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o10 = 3h  + 5h
│ │ │ │          1     1
│ │ │ │  
│ │ │ │        ZZ[h ]
│ │ │ │            1
│ │ │ │  o10 : ------
│ │ │ │           4
│ │ │ │          h
│ │ │ │           1
│ │ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ │ - -- used 0.0825143s (cpu); 0.0825209s (thread); 0s (gc)
│ │ │ │ + -- used 0.102983s (cpu); 0.102862s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o11 = 3H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o11 : -----
│ │ │ │           4
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html
│ │ │ @@ -125,23 +125,23 @@
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time Euler(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0576959s (cpu); 0.0354928s (thread); 0s (gc)
│ │ │ + -- used 0.104201s (cpu); 0.0514337s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
    
│ │ │              
    
│ │ │                
    i5 : time Euler I
│ │ │ - -- used 0.254114s (cpu); 0.146833s (thread); 0s (gc)
│ │ │ + -- used 0.359599s (cpu); 0.201281s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 4
    
│ │ │              
    
│ │ │                
    i6 : EulerIHash=Euler(I,Output=>HashForm);
    
│ │ │ @@ -189,23 +189,23 @@
│ │ │          
    
│ │ │            
    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 0.182967s (cpu); 0.0690199s (thread); 0s (gc)
│ │ │ + -- used 0.197439s (cpu); 0.103033s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
    
│ │ │              
    
│ │ │                
    i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.190575s (cpu); 0.0845693s (thread); 0s (gc)
│ │ │ + -- used 0.283546s (cpu); 0.118035s (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 0.0576959s (cpu); 0.0354928s (thread); 0s (gc)
│ │ │ │ + -- used 0.104201s (cpu); 0.0514337s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 4
│ │ │ │  i5 : time Euler I
│ │ │ │ - -- used 0.254114s (cpu); 0.146833s (thread); 0s (gc)
│ │ │ │ + -- used 0.359599s (cpu); 0.201281s (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 0.182967s (cpu); 0.0690199s (thread); 0s (gc)
│ │ │ │ + -- used 0.197439s (cpu); 0.103033s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 2
│ │ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ │ - -- used 0.190575s (cpu); 0.0845693s (thread); 0s (gc)
│ │ │ │ + -- used 0.283546s (cpu); 0.118035s (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
│ │ │ @@ -95,15 +95,15 @@
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │              
    
│ │ │                
    i4 : time EulerAffine I
│ │ │ - -- used 0.0874131s (cpu); 0.0492159s (thread); 0s (gc)
│ │ │ + -- used 0.0851631s (cpu); 0.0688623s (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 0.0874131s (cpu); 0.0492159s (thread); 0s (gc)
│ │ │ │ + -- used 0.0851631s (cpu); 0.0688623s (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
│ │ │ @@ -70,15 +70,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 2.58397s (cpu); 1.12003s (thread); 0s (gc)
│ │ │ + -- used 6.54387s (cpu); 1.57433s (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
│ │ │ @@ -87,15 +87,15 @@
│ │ │        (h , h )
│ │ │          1   2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ - -- used 2.73505s (cpu); 1.2427s (thread); 0s (gc)
│ │ │ + -- used 7.09559s (cpu); 1.57436s (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 2.58397s (cpu); 1.12003s (thread); 0s (gc)
│ │ │ │ + -- used 6.54387s (cpu); 1.57433s (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 2.73505s (cpu); 1.2427s (thread); 0s (gc)
│ │ │ │ + -- used 7.09559s (cpu); 1.57436s (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
│ │ │ @@ -66,15 +66,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time CSM I
│ │ │ - -- used 0.852243s (cpu); 0.438332s (thread); 0s (gc)
│ │ │ + -- used 1.09918s (cpu); 0.534845s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -83,15 +83,15 @@
│ │ │         h
│ │ │          1
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0911228s (cpu); 0.0316495s (thread); 0s (gc)
│ │ │ + -- used 0.0864587s (cpu); 0.0478865s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -105,15 +105,15 @@
│ │ │          
    
│ │ │            
    Note that one could, equivalently, use the command Chern instead in this case.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : time Chern I
│ │ │ - -- used 0.0751141s (cpu); 0.0309694s (thread); 0s (gc)
│ │ │ + -- used 0.0621425s (cpu); 0.0418672s (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 0.852243s (cpu); 0.438332s (thread); 0s (gc)
│ │ │ │ + -- used 1.09918s (cpu); 0.534845s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o3 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o3 : ------
│ │ │ │          5
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ │ - -- used 0.0911228s (cpu); 0.0316495s (thread); 0s (gc)
│ │ │ │ + -- used 0.0864587s (cpu); 0.0478865s (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 0.0751141s (cpu); 0.0309694s (thread); 0s (gc)
│ │ │ │ + -- used 0.0621425s (cpu); 0.0418672s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o5 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : time CSM I
│ │ │ - -- used 1.71002s (cpu); 0.984712s (thread); 0s (gc)
│ │ │ + -- used 4.08327s (cpu); 1.37892s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o3 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -87,15 +87,15 @@
│ │ │         h
│ │ │          1
    
│ │ │              
    
│ │ │                
    i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ - -- used 0.438602s (cpu); 0.209232s (thread); 0s (gc)
│ │ │ + -- used 0.863456s (cpu); 0.327363s (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 1.71002s (cpu); 0.984712s (thread); 0s (gc)
│ │ │ │ + -- used 4.08327s (cpu); 1.37892s (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 0.438602s (cpu); 0.209232s (thread); 0s (gc)
│ │ │ │ + -- used 0.863456s (cpu); 0.327363s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          5      4     3
│ │ │ │  o4 = 12h  + 10h  + 6h
│ │ │ │          1      1     1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.out
│ │ │ @@ -16,32 +16,32 @@
│ │ │  
│ │ │  o2 : Digraph
│ │ │  
│ │ │  i3 : G = chordalGraph digraph hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};
│ │ │  
│ │ │  i4 : tree = elimTree G
│ │ │  
│ │ │ -o4 = ElimTree{a => b   }
│ │ │ +o4 = ElimTree{a => c}
│ │ │                b => c
│ │ │                c => d
│ │ │ -              d => null
│ │ │ +              d => b
│ │ │  
│ │ │  o4 : ElimTree
│ │ │  
│ │ │  i5 : rnk = hashTable{"a0"=>a, "a1"=>a, "b0"=>b, "b1"=>b, "b2"=>b,
│ │ │                       "c0"=>c, "d0"=>d, "c1"=>c, "d1"=>d};
│ │ │  
│ │ │  i6 : eqs = hashTable{"a0" => ({a},{}), "a1" => ({},{}),
│ │ │                       "b0" => ({b},{}), "b1" => ({},{}), "b2" => ({b},{}),
│ │ │                       "c0" => ({c},{}), "c1" => ({},{}),
│ │ │                       "d0" => ({},{}), "d1" => ({d},{}) };
│ │ │  
│ │ │  i7 : chordalNet(eqs,rnk,tree,DG)
│ │ │  
│ │ │ -o7 = ChordalNet{ d => { , d}    }
│ │ │ -                 b => {b,  , b}
│ │ │ -                 a => {a,  }
│ │ │ +o7 = ChordalNet{ a => {a,  }    }
│ │ │                   c => { , c}
│ │ │ +                 d => { , d}
│ │ │ +                 b => {b,  , b}
│ │ │  
│ │ │  o7 : ChordalNet
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.html
│ │ │ @@ -102,18 +102,18 @@
│ │ │                
    i3 : G = chordalGraph digraph hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};
    
│ │ │              
    
│ │ │                
    i4 : tree = elimTree G
│ │ │  
│ │ │ -o4 = ElimTree{a => b   }
│ │ │ +o4 = ElimTree{a => c}
│ │ │                b => c
│ │ │                c => d
│ │ │ -              d => null
│ │ │ +              d => b
│ │ │  
│ │ │  o4 : ElimTree
    
│ │ │              
    
│ │ │                
    i5 : rnk = hashTable{"a0"=>a, "a1"=>a, "b0"=>b, "b1"=>b, "b2"=>b,
│ │ │ @@ -128,18 +128,18 @@
│ │ │                       "d0" => ({},{}), "d1" => ({d},{}) };
    
│ │ │              
    
│ │ │                
    i7 : chordalNet(eqs,rnk,tree,DG)
│ │ │  
│ │ │ -o7 = ChordalNet{ d => { , d}    }
│ │ │ -                 b => {b,  , b}
│ │ │ -                 a => {a,  }
│ │ │ +o7 = ChordalNet{ a => {a,  }    }
│ │ │                   c => { , c}
│ │ │ +                 d => { , d}
│ │ │ +                 b => {b,  , b}
│ │ │  
│ │ │  o7 : ChordalNet
    
│ │ │              
    
│ │ │          
    
│ │ │        
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,32 +32,32 @@
│ │ │ │               d0 => {}
│ │ │ │               d1 => {}
│ │ │ │  
│ │ │ │  o2 : Digraph
│ │ │ │  i3 : G = chordalGraph digraph hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};
│ │ │ │  i4 : tree = elimTree G
│ │ │ │  
│ │ │ │ -o4 = ElimTree{a => b   }
│ │ │ │ +o4 = ElimTree{a => c}
│ │ │ │                b => c
│ │ │ │                c => d
│ │ │ │ -              d => null
│ │ │ │ +              d => b
│ │ │ │  
│ │ │ │  o4 : ElimTree
│ │ │ │  i5 : rnk = hashTable{"a0"=>a, "a1"=>a, "b0"=>b, "b1"=>b, "b2"=>b,
│ │ │ │                       "c0"=>c, "d0"=>d, "c1"=>c, "d1"=>d};
│ │ │ │  i6 : eqs = hashTable{"a0" => ({a},{}), "a1" => ({},{}),
│ │ │ │                       "b0" => ({b},{}), "b1" => ({},{}), "b2" => ({b},{}),
│ │ │ │                       "c0" => ({c},{}), "c1" => ({},{}),
│ │ │ │                       "d0" => ({},{}), "d1" => ({d},{}) };
│ │ │ │  i7 : chordalNet(eqs,rnk,tree,DG)
│ │ │ │  
│ │ │ │ -o7 = ChordalNet{ d => { , d}    }
│ │ │ │ -                 b => {b,  , b}
│ │ │ │ -                 a => {a,  }
│ │ │ │ +o7 = ChordalNet{ a => {a,  }    }
│ │ │ │                   c => { , c}
│ │ │ │ +                 d => { , d}
│ │ │ │ +                 b => {b,  , b}
│ │ │ │  
│ │ │ │  o7 : ChordalNet
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_i_s_p_l_a_y_N_e_t -- displays a chordal network using Graphivz
│ │ │ │      * _d_i_g_r_a_p_h_(_C_h_o_r_d_a_l_N_e_t_) -- digraph associated to a chordal network
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _c_h_o_r_d_a_l_N_e_t_(_H_a_s_h_T_a_b_l_e_,_H_a_s_h_T_a_b_l_e_,_E_l_i_m_T_r_e_e_,_D_i_g_r_a_p_h_) -- construct chordal
│ │ ├── ./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)
│ │ │ - -- 2.67408s elapsed
│ │ │ + -- 3.33257s 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)
│ │ │ - -- .302915s elapsed
│ │ │ + -- .531464s 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))
│ │ │ - -- 10.9792s elapsed
│ │ │ + -- 10.2716s 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)
│ │ │ - -- .342126s elapsed
│ │ │ + -- .5629s 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))
│ │ │ - -- .517346s elapsed
│ │ │ - -- .517373s elapsed
│ │ │ + -- .46318s elapsed
│ │ │ + -- .463211s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
│ │ │  
│ │ │  i29 : assert(cohomvec1 == cohomvec2)
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/html/index.html
│ │ │ @@ -309,15 +309,15 @@
│ │ │  
│ │ │  o19 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ - -- 2.67408s elapsed
│ │ │ + -- 3.33257s 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,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -399,25 +399,25 @@
│ │ │  
│ │ │  o22 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ - -- .302915s elapsed
│ │ │ + -- .531464s 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))
│ │ │ - -- 10.9792s elapsed
│ │ │ + -- 10.2716s elapsed
│ │ │  
│ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o24 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -433,26 +433,26 @@
│ │ │  
│ │ │  o26 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ - -- .342126s elapsed
│ │ │ + -- .5629s 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))
│ │ │ - -- .517346s elapsed
│ │ │ - -- .517373s elapsed
│ │ │ + -- .46318s elapsed
│ │ │ + -- .463211s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : 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)
│ │ │ │ - -- 2.67408s elapsed
│ │ │ │ + -- 3.33257s 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)
│ │ │ │ - -- .302915s elapsed
│ │ │ │ + -- .531464s 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))
│ │ │ │ - -- 10.9792s elapsed
│ │ │ │ + -- 10.2716s 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)
│ │ │ │ - -- .342126s elapsed
│ │ │ │ + -- .5629s 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))
│ │ │ │ - -- .517346s elapsed
│ │ │ │ - -- .517373s elapsed
│ │ │ │ + -- .46318s elapsed
│ │ │ │ + -- .463211s 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 6.66292s (cpu); 4.89797s (thread); 0s (gc)
│ │ │ + -- used 8.49326s (cpu); 6.42483s (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 0.169356s (cpu); 0.0960445s (thread); 0s (gc)
│ │ │ + -- used 0.284269s (cpu); 0.185904s (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 0.937459s (cpu); 0.720912s (thread); 0s (gc)
│ │ │ + -- used 1.30926s (cpu); 1.01458s (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 0.93026s (cpu); 0.702132s (thread); 0s (gc)
│ │ │ + -- used 1.28157s (cpu); 0.966654s (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 0.370239s (cpu); 0.319739s (thread); 0s (gc)
│ │ │ + -- used 0.576534s (cpu); 0.451409s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │ - -- used 0.210104s (cpu); 0.138159s (thread); 0s (gc)
│ │ │ + -- used 0.30923s (cpu); 0.169573s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │ - -- used 3.697e-06s (cpu); 3.326e-06s (thread); 0s (gc)
│ │ │ + -- used 3.505e-06s (cpu); 2.743e-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 0.802517s (cpu); 0.585358s (thread); 0s (gc)
│ │ │ + -- used 1.28068s (cpu); 0.802488s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.401303s (cpu); 0.335055s (thread); 0s (gc)
│ │ │ + -- used 0.653202s (cpu); 0.438906s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.202221s (cpu); 0.138045s (thread); 0s (gc)
│ │ │ + -- used 0.301374s (cpu); 0.160576s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o6 = 10
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ - -- used 6.66292s (cpu); 4.89797s (thread); 0s (gc)
│ │ │ + -- used 8.49326s (cpu); 6.42483s (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 /
│ │ │                                                                                                                                                                                
│ │ │ @@ -295,28 +295,28 @@
│ │ │  
│ │ │  o19 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : FF = time Shamash(R1,F,4)
│ │ │ - -- used 0.169356s (cpu); 0.0960445s (thread); 0s (gc)
│ │ │ + -- used 0.284269s (cpu); 0.185904s (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 0.937459s (cpu); 0.720912s (thread); 0s (gc)
│ │ │ + -- used 1.30926s (cpu); 1.01458s (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
│ │ │ @@ -328,15 +328,15 @@
│ │ │          
    
│ │ │            
    The function also deals correctly with complexes F where min F is not 0:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ - -- used 0.93026s (cpu); 0.702132s (thread); 0s (gc)
│ │ │ + -- used 1.28157s (cpu); 0.966654s (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 6.66292s (cpu); 4.89797s (thread); 0s (gc)
│ │ │ │ + -- used 8.49326s (cpu); 6.42483s (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 0.169356s (cpu); 0.0960445s (thread); 0s (gc)
│ │ │ │ + -- used 0.284269s (cpu); 0.185904s (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 0.937459s (cpu); 0.720912s (thread); 0s (gc)
│ │ │ │ + -- used 1.30926s (cpu); 1.01458s (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 0.93026s (cpu); 0.702132s (thread); 0s (gc)
│ │ │ │ + -- used 1.28157s (cpu); 0.966654s (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
│ │ │ @@ -79,23 +79,23 @@
│ │ │  
│ │ │  o1 = 0
│ │ │              
    
│ │ │                
    i2 : sumTwoMonomials(2,3)
│ │ │ - -- used 0.370239s (cpu); 0.319739s (thread); 0s (gc)
│ │ │ + -- used 0.576534s (cpu); 0.451409s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │ - -- used 0.210104s (cpu); 0.138159s (thread); 0s (gc)
│ │ │ + -- used 0.30923s (cpu); 0.169573s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │ - -- used 3.697e-06s (cpu); 3.326e-06s (thread); 0s (gc)
│ │ │ + -- used 3.505e-06s (cpu); 2.743e-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 0.370239s (cpu); 0.319739s (thread); 0s (gc)
│ │ │ │ + -- used 0.576534s (cpu); 0.451409s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │ │  
│ │ │ │ - -- used 0.210104s (cpu); 0.138159s (thread); 0s (gc)
│ │ │ │ + -- used 0.30923s (cpu); 0.169573s (thread); 0s (gc)
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │ │  
│ │ │ │ - -- used 3.697e-06s (cpu); 3.326e-06s (thread); 0s (gc)
│ │ │ │ + -- used 3.505e-06s (cpu); 2.743e-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
│ │ │ @@ -83,25 +83,25 @@
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : twoMonomials(2,3)
│ │ │ - -- used 0.802517s (cpu); 0.585358s (thread); 0s (gc)
│ │ │ + -- used 1.28068s (cpu); 0.802488s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.401303s (cpu); 0.335055s (thread); 0s (gc)
│ │ │ + -- used 0.653202s (cpu); 0.438906s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.202221s (cpu); 0.138045s (thread); 0s (gc)
│ │ │ + -- used 0.301374s (cpu); 0.160576s (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 0.802517s (cpu); 0.585358s (thread); 0s (gc)
│ │ │ │ + -- used 1.28068s (cpu); 0.802488s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{1, 1}} => 2        }
│ │ │ │        {{2, 2}, {1, 2}} => 4
│ │ │ │  
│ │ │ │ - -- used 0.401303s (cpu); 0.335055s (thread); 0s (gc)
│ │ │ │ + -- used 0.653202s (cpu); 0.438906s (thread); 0s (gc)
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │ │        {{3, 3}, {2, 3}} => 1
│ │ │ │  
│ │ │ │ - -- used 0.202221s (cpu); 0.138045s (thread); 0s (gc)
│ │ │ │ + -- used 0.301374s (cpu); 0.160576s (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/ConformalBlocks/dump/rawdocumentation.dump
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sun Dec 14 14:09:54 2025
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sun Dec 14 14:09:53 2025
│ │ │  #: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
│ │ │  Y2Fub25pY2FsRGl2aXNvck0wbmJhcg==
│ │ ├── ./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;
│ │ │ - -- 2.81078s elapsed
│ │ │ + -- 2.60629s elapsed
│ │ │  
│ │ │  i10 : elapsedTime assert isIntegrable A
│ │ │ - -- 5.94013s elapsed
│ │ │ + -- 4.47441s 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);
│ │ │ - -- .73523s elapsed
│ │ │ + -- .538947s elapsed
│ │ │  
│ │ │                4      4
│ │ │  o14 : Matrix F  <-- F
│ │ │  
│ │ │  i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ - -- 1.57042s elapsed
│ │ │ + -- 1.21076s elapsed
│ │ │  
│ │ │  i16 : elapsedTime assert isIntegrable A1
│ │ │ - -- .808615s elapsed
│ │ │ + -- .955087s 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);
│ │ │ - -- .496173s elapsed
│ │ │ + -- .388146s elapsed
│ │ │  
│ │ │  i20 : elapsedTime assert isIntegrable A2
│ │ │ - -- .830479s elapsed
│ │ │ + -- .712828s 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;
│ │ │ - -- .267226s elapsed
│ │ │ + -- .219553s elapsed
│ │ │  
│ │ │  i8 : elapsedTime assert isIntegrable A
│ │ │ - -- .203883s elapsed
│ │ │ + -- .181528s 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
│ │ │ @@ -118,21 +118,21 @@
│ │ │          
    
│ │ │            
    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 0.157531s (cpu); 0.157527s (thread); 0s (gc)
│ │ │ │ + -- used 0.184367s (cpu); 0.184366s (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
│ │ │ @@ -134,59 +134,59 @@
│ │ │                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;
│ │ │ - -- 2.81078s elapsed
    
│ │ │ + -- 2.60629s elapsed
│ │ │              
    
│ │ │                
    i10 : elapsedTime assert isIntegrable A
│ │ │ - -- 5.94013s elapsed
    
│ │ │ + -- 4.47441s elapsed
│ │ │              
    
│ │ │                
    i11 : netList A
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -167,30 +167,30 @@
│ │ │              
    		
│ │ │                
    i13 : B = {1_D,dx,dy,dx*dy};
    
│ │ │              
    
│ │ │                
    i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ - -- .73523s elapsed
│ │ │ + -- .538947s elapsed
│ │ │  
│ │ │                4      4
│ │ │  o14 : Matrix F  <-- F
    
│ │ │              
    
│ │ │                
    i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ - -- 1.57042s elapsed
    
│ │ │ + -- 1.21076s elapsed
│ │ │              
    
│ │ │                
    i16 : elapsedTime assert isIntegrable A1
│ │ │ - -- .808615s elapsed
    
│ │ │ + -- .955087s elapsed
│ │ │              
    
│ │ │                
    i17 : netList A1
│ │ │  
│ │ │        +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -227,21 +227,21 @@
│ │ │                4      4
│ │ │  o18 : Matrix F  <-- F
    
│ │ │              
    
│ │ │                
    i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ - -- .496173s elapsed
    
│ │ │ + -- .388146s elapsed
│ │ │              
    
│ │ │                
    i20 : elapsedTime assert isIntegrable A2
│ │ │ - -- .830479s elapsed
    
│ │ │ + -- .712828s 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;
│ │ │ │ - -- 2.81078s elapsed
│ │ │ │ + -- 2.60629s elapsed
│ │ │ │  i10 : elapsedTime assert isIntegrable A
│ │ │ │ - -- 5.94013s elapsed
│ │ │ │ + -- 4.47441s 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);
│ │ │ │ - -- .73523s elapsed
│ │ │ │ + -- .538947s elapsed
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o14 : Matrix F  <-- F
│ │ │ │  i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ │ - -- 1.57042s elapsed
│ │ │ │ + -- 1.21076s elapsed
│ │ │ │  i16 : elapsedTime assert isIntegrable A1
│ │ │ │ - -- .808615s elapsed
│ │ │ │ + -- .955087s 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);
│ │ │ │ - -- .496173s elapsed
│ │ │ │ + -- .388146s elapsed
│ │ │ │  i20 : elapsedTime assert isIntegrable A2
│ │ │ │ - -- .830479s elapsed
│ │ │ │ + -- .712828s 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
│ │ │ @@ -100,21 +100,21 @@
│ │ │          
    
│ │ │            
    Finally, we can compute the connection matrices.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : elapsedTime A = connectionMatrices I;
│ │ │ - -- .267226s elapsed
    
│ │ │ + -- .219553s elapsed
│ │ │              
    
│ │ │                
    i8 : elapsedTime assert isIntegrable A
│ │ │ - -- .203883s elapsed
    
│ │ │ + -- .181528s 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;
│ │ │ │ - -- .267226s elapsed
│ │ │ │ + -- .219553s elapsed
│ │ │ │  i8 : elapsedTime assert isIntegrable A
│ │ │ │ - -- .203883s elapsed
│ │ │ │ + -- .181528s elapsed
│ │ │ │  i9 : netList A
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------+
│ │ │ │  o9 = || 0                                                       1
│ │ ├── ./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 2.25862s (cpu); 1.18322s (thread); 0s (gc)
│ │ │ + -- used 2.54343s (cpu); 1.34304s (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 1.61784s (cpu); 1.14031s (thread); 0s (gc)
│ │ │ + -- used 1.53812s (cpu); 1.05304s (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 0.439952s (cpu); 0.258932s (thread); 0s (gc)
│ │ │ + -- used 0.39165s (cpu); 0.214069s (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 0.132347s (cpu); 0.0449789s (thread); 0s (gc)
│ │ │ + -- used 0.154793s (cpu); 0.0449216s (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 0.00430215s (cpu); 0.00429842s (thread); 0s (gc)
│ │ │ + -- used 0.00651371s (cpu); 0.00651227s (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 0.137231s (cpu); 0.0699679s (thread); 0s (gc)
│ │ │ + -- used 0.187395s (cpu); 0.0938856s (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 0.02944s (cpu); 0.0294445s (thread); 0s (gc)
│ │ │ + -- used 0.0586502s (cpu); 0.0586499s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projectiveDegrees phi
│ │ │ - -- used 0.68756s (cpu); 0.487586s (thread); 0s (gc)
│ │ │ + -- used 0.744311s (cpu); 0.55853s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.0623207s (cpu); 0.0622653s (thread); 0s (gc)
│ │ │ + -- used 0.103903s (cpu); 0.103704s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.0021486s (cpu); 0.00214939s (thread); 0s (gc)
│ │ │ + -- used 0.00294027s (cpu); 0.00294735s (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 0.474252s (cpu); 0.394534s (thread); 0s (gc)
│ │ │ + -- used 0.447739s (cpu); 0.447744s (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 0.00931603s (cpu); 0.00931855s (thread); 0s (gc)
│ │ │ + -- used 0.0113264s (cpu); 0.0113318s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time degreeMap psi
│ │ │ - -- used 0.458493s (cpu); 0.294259s (thread); 0s (gc)
│ │ │ + -- used 0.563595s (cpu); 0.294308s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
│ │ │  
│ │ │  i11 : time projectiveDegrees psi
│ │ │ - -- used 5.29004s (cpu); 4.63968s (thread); 0s (gc)
│ │ │ + -- used 6.43014s (cpu); 5.97295s (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 0.00220238s (cpu); 0.00220313s (thread); 0s (gc)
│ │ │ + -- used 0.00250238s (cpu); 0.00250843s (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 0.15731s (cpu); 0.0849869s (thread); 0s (gc)
│ │ │ + -- used 0.198282s (cpu); 0.100168s (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 0.512059s (cpu); 0.426868s (thread); 0s (gc)
│ │ │ + -- used 0.493489s (cpu); 0.493164s (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 0.347749s (cpu); 0.274141s (thread); 0s (gc)
│ │ │ + -- used 0.472144s (cpu); 0.366608s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : time degrees phi
│ │ │ - -- used 0.0180113s (cpu); 0.0176993s (thread); 0s (gc)
│ │ │ + -- used 0.0875272s (cpu); 0.0251346s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
│ │ │  
│ │ │  i17 : time describe phi
│ │ │ - -- used 0.00320718s (cpu); 0.00320744s (thread); 0s (gc)
│ │ │ + -- used 0.00421075s (cpu); 0.00422119s (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 0.00997997s (cpu); 0.0099807s (thread); 0s (gc)
│ │ │ + -- used 0.011936s (cpu); 0.0119473s (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 0.00950571s (cpu); 0.00950659s (thread); 0s (gc)
│ │ │ + -- used 0.0120656s (cpu); 0.0120784s (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 1.33327s (cpu); 0.95512s (thread); 0s (gc)
│ │ │ + -- used 1.19811s (cpu); 1.02906s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
│ │ │  
│ │ │  i22 : time degree f
│ │ │ - -- used 1.625e-05s (cpu); 1.593e-05s (thread); 0s (gc)
│ │ │ + -- used 1.7684e-05s (cpu); 1.7206e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
│ │ │  
│ │ │  i23 : time describe f
│ │ │ - -- used 0.00161465s (cpu); 0.00161555s (thread); 0s (gc)
│ │ │ + -- used 0.00172246s (cpu); 0.00172799s (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 0.265746s (cpu); 0.15298s (thread); 0s (gc)
│ │ │ + -- used 0.358491s (cpu); 0.192059s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
│ │ │  
│ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0115158s (cpu); 0.0109715s (thread); 0s (gc)
│ │ │ + -- used 0.0303717s (cpu); 0.0172716s (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 0.0532429s (cpu); 0.0528992s (thread); 0s (gc)
│ │ │ + -- used 0.118638s (cpu); 0.0709907s (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 0.0360627s (cpu); 0.0357049s (thread); 0s (gc)
│ │ │ + -- used 0.0580553s (cpu); 0.044975s (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 0.157531s (cpu); 0.157527s (thread); 0s (gc)
│ │ │ + -- used 0.184367s (cpu); 0.184366s (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 0.865825s (cpu); 0.519842s (thread); 0s (gc)
│ │ │ + -- used 0.827915s (cpu); 0.549293s (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 0.565093s (cpu); 0.36901s (thread); 0s (gc)
│ │ │ + -- used 0.67955s (cpu); 0.379962s (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 0.0212909s (cpu); 0.0208755s (thread); 0s (gc)
│ │ │ + -- used 0.0508406s (cpu); 0.026628s (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 0.097715s (cpu); 0.0973659s (thread); 0s (gc)
│ │ │ + -- used 0.152985s (cpu); 0.124011s (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
│ │ │ +       ZZ
│ │ │ +o9 = ------[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 0.216303s (cpu); 0.10276s (thread); 0s (gc)
│ │ │ + -- used 0.280103s (cpu); 0.120852s (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 0.338604s (cpu); 0.228164s (thread); 0s (gc)
│ │ │ + -- used 0.227822s (cpu); 0.227831s (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 0.39646s (cpu); 0.292129s (thread); 0s (gc)
│ │ │ + -- used 0.450925s (cpu); 0.310725s (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 1.41797s (cpu); 0.900612s (thread); 0s (gc)
│ │ │ + -- used 1.99007s (cpu); 1.0831s (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 0.000410019s (cpu); 0.000406292s (thread); 0s (gc)
│ │ │ + -- used 0.000493397s (cpu); 0.000485585s (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 0.298853s (cpu); 0.18527s (thread); 0s (gc)
│ │ │ + -- used 0.414107s (cpu); 0.242631s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time rationalMap psi
│ │ │ - -- used 0.504026s (cpu); 0.366499s (thread); 0s (gc)
│ │ │ + -- used 0.604915s (cpu); 0.504893s (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 0.149185s (cpu); 0.0769299s (thread); 0s (gc)
│ │ │ + -- used 0.18261s (cpu); 0.0872618s (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 4.07764s (cpu); 2.12022s (thread); 0s (gc)
│ │ │ + -- used 5.3433s (cpu); 2.46689s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
│ │ │  
│ │ │  i16 : time T2 = T * T
│ │ │ - -- used 2.8564e-05s (cpu); 2.8273e-05s (thread); 0s (gc)
│ │ │ + -- used 2.8546e-05s (cpu); 2.671e-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 6.65901s (cpu); 3.45699s (thread); 0s (gc)
│ │ │ + -- used 7.99029s (cpu); 3.92144s (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 5.38367s (cpu); 2.92119s (thread); 0s (gc)
│ │ │ + -- used 6.20953s (cpu); 3.31023s (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 0.272017s (cpu); 0.206969s (thread); 0s (gc)
│ │ │ + -- used 0.319869s (cpu); 0.243932s (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 0.227935s (cpu); 0.16233s (thread); 0s (gc)
│ │ │ + -- used 0.326886s (cpu); 0.246914s (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 2.31467s (cpu); 1.78736s (thread); 0s (gc)
│ │ │ + -- used 2.228s (cpu); 1.88917s (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 3.88333s (cpu); 3.11658s (thread); 0s (gc)
│ │ │ + -- used 3.23525s (cpu); 2.80126s (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 0.0453696s (cpu); 0.0453707s (thread); 0s (gc)
│ │ │ + -- used 0.0537572s (cpu); 0.0537559s (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 1.24889s (cpu); 0.706401s (thread); 0s (gc)
│ │ │ + -- used 1.56205s (cpu); 0.862278s (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 0.000607219s (cpu); 0.000601939s (thread); 0s (gc)
│ │ │ + -- used 0.00081254s (cpu); 0.000804349s (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 0.0188178s (cpu); 0.0184354s (thread); 0s (gc)
│ │ │ + -- used 0.0922538s (cpu); 0.0295831s (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 0.0317165s (cpu); 0.0312624s (thread); 0s (gc)
│ │ │ + -- used 0.0977423s (cpu); 0.0447804s (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 0.0035599s (cpu); 0.00355522s (thread); 0s (gc)
│ │ │ + -- used 0.00502322s (cpu); 0.00502123s (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 0.0930691s (cpu); 0.0930488s (thread); 0s (gc)
│ │ │ + -- used 0.114269s (cpu); 0.114268s (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 0.184232s (cpu); 0.12115s (thread); 0s (gc)
│ │ │ + -- used 0.20957s (cpu); 0.116482s (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 0.371428s (cpu); 0.224639s (thread); 0s (gc)
│ │ │ + -- used 0.471849s (cpu); 0.249085s (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 0.056969s (cpu); 0.0569687s (thread); 0s (gc)
│ │ │ + -- used 0.0645625s (cpu); 0.0645621s (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 0.0193201s (cpu); 0.0193206s (thread); 0s (gc)
│ │ │ + -- used 0.0253448s (cpu); 0.0253471s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0136925s (cpu); 0.0132915s (thread); 0s (gc)
│ │ │ + -- used 0.0281203s (cpu); 0.0158762s (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 2.58345s (cpu); 2.01134s (thread); 0s (gc)
│ │ │ + -- used 2.65473s (cpu); 2.31996s (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 3.88844s (cpu); 2.54882s (thread); 0s (gc)
│ │ │ + -- used 3.94783s (cpu); 2.94981s (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 0.0174446s (cpu); 0.0174411s (thread); 0s (gc)
│ │ │ + -- used 0.0215404s (cpu); 0.0215418s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
│ │ │  
│ │ │  i3 : time kernel(phi,2)
│ │ │ - -- used 0.911685s (cpu); 0.449335s (thread); 0s (gc)
│ │ │ + -- used 1.14536s (cpu); 0.538298s (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 0.00500618s (cpu); 0.00500168s (thread); 0s (gc)
│ │ │ + -- used 0.00636444s (cpu); 0.00638338s (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 0.5466s (cpu); 0.394392s (thread); 0s (gc)
│ │ │ + -- used 0.632307s (cpu); 0.466254s (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 0.463068s (cpu); 0.208194s (thread); 0s (gc)
│ │ │ + -- used 0.497934s (cpu); 0.238097s (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 0.212468s (cpu); 0.135812s (thread); 0s (gc)
│ │ │ + -- used 0.230663s (cpu); 0.142s (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 0.0151376s (cpu); 0.0148069s (thread); 0s (gc)
│ │ │ + -- used 0.0412574s (cpu); 0.0211614s (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 0.0116489s (cpu); 0.0113647s (thread); 0s (gc)
│ │ │ + -- used 0.0279848s (cpu); 0.0140302s (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 5.859e-05s (cpu); 5.361e-05s (thread); 0s (gc)
│ │ │ + -- used 7.3527e-05s (cpu); 6.2036e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 2.6109e-05s (cpu); 2.5939e-05s (thread); 0s (gc)
│ │ │ + -- used 3.9904e-05s (cpu); 3.7457e-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 0.095321s (cpu); 0.0953215s (thread); 0s (gc)
│ │ │ + -- used 0.133048s (cpu); 0.133048s (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 0.0301481s (cpu); 0.0301432s (thread); 0s (gc)
│ │ │ + -- used 0.0396531s (cpu); 0.0396503s (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 1.24116s (cpu); 0.701199s (thread); 0s (gc)
│ │ │ + -- used 1.62842s (cpu); 0.70643s (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 1.59376s (cpu); 1.16591s (thread); 0s (gc)
│ │ │ + -- used 1.64986s (cpu); 1.26429s (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 0.0954111s (cpu); 0.0954104s (thread); 0s (gc)
│ │ │ + -- used 0.10227s (cpu); 0.102271s (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 0.0182984s (cpu); 0.0182857s (thread); 0s (gc)
│ │ │ + -- used 0.0195273s (cpu); 0.0195271s (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 0.0733927s (cpu); 0.0733921s (thread); 0s (gc)
│ │ │ + -- used 0.0782691s (cpu); 0.0782673s (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 0.110232s (cpu); 0.0307456s (thread); 0s (gc)
│ │ │ + -- used 0.117603s (cpu); 0.0334491s (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 0.0234197s (cpu); 0.0234189s (thread); 0s (gc)
│ │ │ + -- used 0.0280019s (cpu); 0.0280041s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │  
│ │ │  i5 : time describe phi'
│ │ │ - -- used 0.00540039s (cpu); 0.00540078s (thread); 0s (gc)
│ │ │ + -- used 0.00670417s (cpu); 0.00671332s (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 0.0044104s (cpu); 0.00441117s (thread); 0s (gc)
│ │ │ + -- used 0.00554705s (cpu); 0.00555551s (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
│ │ │ @@ -97,30 +97,30 @@
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
    
│ │ │              
    
│ │ │                
    i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 2.25862s (cpu); 1.18322s (thread); 0s (gc)
│ │ │ + -- used 2.54343s (cpu); 1.34304s (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 1.61784s (cpu); 1.14031s (thread); 0s (gc)
│ │ │ + -- used 1.53812s (cpu); 1.05304s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          5
│ │ │ @@ -167,30 +167,30 @@
│ │ │  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 0.439952s (cpu); 0.258932s (thread); 0s (gc)
│ │ │ + -- used 0.39165s (cpu); 0.214069s (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 0.132347s (cpu); 0.0449789s (thread); 0s (gc)
│ │ │ + -- used 0.154793s (cpu); 0.0449216s (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 2.25862s (cpu); 1.18322s (thread); 0s (gc)
│ │ │ │ + -- used 2.54343s (cpu); 1.34304s (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 1.61784s (cpu); 1.14031s (thread); 0s (gc)
│ │ │ │ + -- used 1.53812s (cpu); 1.05304s (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 0.439952s (cpu); 0.258932s (thread); 0s (gc)
│ │ │ │ + -- used 0.39165s (cpu); 0.214069s (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 0.132347s (cpu); 0.0449789s (thread); 0s (gc)
│ │ │ │ + -- used 0.154793s (cpu); 0.0449216s (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
│ │ │ @@ -85,24 +85,24 @@
│ │ │  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 0.265746s (cpu); 0.15298s (thread); 0s (gc)
│ │ │ + -- used 0.358491s (cpu); 0.192059s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
    
│ │ │              
    
│ │ │                
    i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0115158s (cpu); 0.0109715s (thread); 0s (gc)
│ │ │ + -- used 0.0303717s (cpu); 0.0172716s (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 0.265746s (cpu); 0.15298s (thread); 0s (gc)
│ │ │ │ + -- used 0.358491s (cpu); 0.192059s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 10
│ │ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0115158s (cpu); 0.0109715s (thread); 0s (gc)
│ │ │ │ + -- used 0.0303717s (cpu); 0.0172716s (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
│ │ │ @@ -86,15 +86,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    
│ │ │              
    
│ │ │                
    i4 : time phi! ;
│ │ │ - -- used 0.0532429s (cpu); 0.0528992s (thread); 0s (gc)
│ │ │ + -- used 0.118638s (cpu); 0.0709907s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
    
│ │ │              
    
│ │ │                
    i5 : describe phi
│ │ │ @@ -127,15 +127,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    
│ │ │              
    
│ │ │                
    i9 : time phi! ;
│ │ │ - -- used 0.0360627s (cpu); 0.0357049s (thread); 0s (gc)
│ │ │ + -- used 0.0580553s (cpu); 0.044975s (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 0.0532429s (cpu); 0.0528992s (thread); 0s (gc)
│ │ │ │ + -- used 0.118638s (cpu); 0.0709907s (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 0.0360627s (cpu); 0.0357049s (thread); 0s (gc)
│ │ │ │ + -- used 0.0580553s (cpu); 0.044975s (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
│ │ │ @@ -153,15 +153,15 @@
│ │ │  
│ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
    
│ │ │              
    
│ │ │                
    i6 : time phi^** q
│ │ │ - -- used 0.157531s (cpu); 0.157527s (thread); 0s (gc)
│ │ │ + -- used 0.184367s (cpu); 0.184366s (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 0.865825s (cpu); 0.519842s (thread); 0s (gc)
│ │ │ + -- used 0.827915s (cpu); 0.549293s (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 0.565093s (cpu); 0.36901s (thread); 0s (gc)
│ │ │ + -- used 0.67955s (cpu); 0.379962s (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 0.0212909s (cpu); 0.0208755s (thread); 0s (gc)
│ │ │ + -- used 0.0508406s (cpu); 0.026628s (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 0.097715s (cpu); 0.0973659s (thread); 0s (gc)
│ │ │ + -- used 0.152985s (cpu); 0.124011s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │          8
│ │ │ @@ -203,25 +203,25 @@
│ │ │          
    
│ │ │          
    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
│ │ │ +       ZZ
│ │ │ +o9 = ------[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 0.216303s (cpu); 0.10276s (thread); 0s (gc)
│ │ │ + -- used 0.280103s (cpu); 0.120852s (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
│ │ │ @@ -233,15 +233,15 @@
│ │ │                (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 0.338604s (cpu); 0.228164s (thread); 0s (gc)
│ │ │ + -- used 0.227822s (cpu); 0.227831s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │          10
│ │ │ @@ -267,30 +267,30 @@
│ │ │                   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 0.39646s (cpu); 0.292129s (thread); 0s (gc)
│ │ │ + -- used 0.450925s (cpu); 0.310725s (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 1.41797s (cpu); 0.900612s (thread); 0s (gc)
│ │ │ + -- used 1.99007s (cpu); 1.0831s (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 0.865825s (cpu); 0.519842s (thread); 0s (gc)
│ │ │ │ + -- used 0.827915s (cpu); 0.549293s (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 0.565093s (cpu); 0.36901s (thread); 0s (gc)
│ │ │ │ + -- used 0.67955s (cpu); 0.379962s (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 0.0212909s (cpu); 0.0208755s (thread); 0s (gc)
│ │ │ │ + -- used 0.0508406s (cpu); 0.026628s (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 0.097715s (cpu); 0.0973659s (thread); 0s (gc)
│ │ │ │ + -- used 0.152985s (cpu); 0.124011s (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
│ │ │ │ +       ZZ
│ │ │ │ +o9 = ------[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 0.216303s (cpu); 0.10276s (thread); 0s (gc)
│ │ │ │ + -- used 0.280103s (cpu); 0.120852s (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 0.338604s (cpu); 0.228164s (thread); 0s (gc)
│ │ │ │ + -- used 0.227822s (cpu); 0.227831s (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 0.39646s (cpu); 0.292129s (thread); 0s (gc)
│ │ │ │ + -- used 0.450925s (cpu); 0.310725s (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 1.41797s (cpu); 0.900612s (thread); 0s (gc)
│ │ │ │ + -- used 1.99007s (cpu); 1.0831s (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
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ - -- used 0.000410019s (cpu); 0.000406292s (thread); 0s (gc)
│ │ │ + -- used 0.000493397s (cpu); 0.000485585s (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
│ │ │ @@ -119,23 +119,23 @@
│ │ │            
    
│ │ │          
    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 0.298853s (cpu); 0.18527s (thread); 0s (gc)
│ │ │ + -- used 0.414107s (cpu); 0.242631s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    
│ │ │              
    
│ │ │                
    i6 : time rationalMap psi
│ │ │ - -- used 0.504026s (cpu); 0.366499s (thread); 0s (gc)
│ │ │ + -- used 0.604915s (cpu); 0.504893s (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: {
│ │ │ @@ -233,15 +233,15 @@
│ │ │  o13 : Ideal of -----[x ..x ]
│ │ │                 65521  0   3
    
│ │ │              
    
│ │ │                
    i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ - -- used 0.149185s (cpu); 0.0769299s (thread); 0s (gc)
│ │ │ + -- used 0.18261s (cpu); 0.0872618s (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 ])
│ │ │ @@ -253,26 +253,26 @@
│ │ │            
    
│ │ │          
    The degree of the forms defining the abstract map T can be obtained by the following command:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i15 : time projectiveDegrees(T,2)
│ │ │ - -- used 4.07764s (cpu); 2.12022s (thread); 0s (gc)
│ │ │ + -- used 5.3433s (cpu); 2.46689s (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 2.8564e-05s (cpu); 2.8273e-05s (thread); 0s (gc)
│ │ │ + -- used 2.8546e-05s (cpu); 2.671e-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 ])
│ │ │ @@ -281,15 +281,15 @@
│ │ │  
│ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
    
│ │ │              
    
│ │ │                
    i17 : time projectiveDegrees(T2,2)
│ │ │ - -- used 6.65901s (cpu); 3.45699s (thread); 0s (gc)
│ │ │ + -- used 7.99029s (cpu); 3.92144s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 1
    
│ │ │              
    
│ │ │          
    We verify that the composition of T with itself leaves a random point fixed:
    
│ │ │          
│ │ │ @@ -322,15 +322,15 @@
│ │ │            
│ │ │          
    
│ │ │          
    We now compute the concrete rational map corresponding to T:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i21 : time f = rationalMap T
│ │ │ - -- used 5.38367s (cpu); 2.92119s (thread); 0s (gc)
│ │ │ + -- used 6.20953s (cpu); 3.31023s (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 0.000410019s (cpu); 0.000406292s (thread); 0s (gc)
│ │ │ │ + -- used 0.000493397s (cpu); 0.000485585s (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 0.298853s (cpu); 0.18527s (thread); 0s (gc)
│ │ │ │ + -- used 0.414107s (cpu); 0.242631s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : time rationalMap psi
│ │ │ │ - -- used 0.504026s (cpu); 0.366499s (thread); 0s (gc)
│ │ │ │ + -- used 0.604915s (cpu); 0.504893s (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 0.149185s (cpu); 0.0769299s (thread); 0s (gc)
│ │ │ │ + -- used 0.18261s (cpu); 0.0872618s (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 4.07764s (cpu); 2.12022s (thread); 0s (gc)
│ │ │ │ + -- used 5.3433s (cpu); 2.46689s (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 2.8564e-05s (cpu); 2.8273e-05s (thread); 0s (gc)
│ │ │ │ + -- used 2.8546e-05s (cpu); 2.671e-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 6.65901s (cpu); 3.45699s (thread); 0s (gc)
│ │ │ │ + -- used 7.99029s (cpu); 3.92144s (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 5.38367s (cpu); 2.92119s (thread); 0s (gc)
│ │ │ │ + -- used 6.20953s (cpu); 3.31023s (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
│ │ │ @@ -139,15 +139,15 @@
│ │ │  -- 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 0.272017s (cpu); 0.206969s (thread); 0s (gc)
│ │ │ + -- used 0.319869s (cpu); 0.243932s (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
│ │ │ @@ -200,15 +200,15 @@
│ │ │            
    
│ │ │                
    i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 0.227935s (cpu); 0.16233s (thread); 0s (gc)
│ │ │ + -- used 0.326886s (cpu); 0.246914s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
    
│ │ │              
    
│ │ │                
    i6 : assert(psi == psi')
    
│ │ │ @@ -295,15 +295,15 @@
│ │ │            
    
│ │ │                
    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 2.31467s (cpu); 1.78736s (thread); 0s (gc)
│ │ │ + -- used 2.228s (cpu); 1.88917s (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
│ │ │ @@ -367,15 +367,15 @@
│ │ │              
    		
│ │ │                
    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 3.88333s (cpu); 3.11658s (thread); 0s (gc)
│ │ │ + -- used 3.23525s (cpu); 2.80126s (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 0.272017s (cpu); 0.206969s (thread); 0s (gc)
│ │ │ │ + -- used 0.319869s (cpu); 0.243932s (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 0.227935s (cpu); 0.16233s (thread); 0s (gc)
│ │ │ │ + -- used 0.326886s (cpu); 0.246914s (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 2.31467s (cpu); 1.78736s (thread); 0s (gc)
│ │ │ │ + -- used 2.228s (cpu); 1.88917s (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 3.88333s (cpu); 3.11658s (thread); 0s (gc)
│ │ │ │ + -- used 3.23525s (cpu); 2.80126s (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
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o4 : RingMap ringP8 <-- ringP14
    
│ │ │              
    
│ │ │                
    i5 : time degreeMap phi
│ │ │ - -- used 0.0453696s (cpu); 0.0453707s (thread); 0s (gc)
│ │ │ + -- used 0.0537572s (cpu); 0.0537559s (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 
│ │ │ @@ -113,15 +113,15 @@
│ │ │  
│ │ │  o6 : RingMap ringP8 <-- ringP8
    
│ │ │              
    
│ │ │                
    i7 : time degreeMap phi'
│ │ │ - -- used 1.24889s (cpu); 0.706401s (thread); 0s (gc)
│ │ │ + -- used 1.56205s (cpu); 0.862278s (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 0.0453696s (cpu); 0.0453707s (thread); 0s (gc)
│ │ │ │ + -- used 0.0537572s (cpu); 0.0537559s (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 1.24889s (cpu); 0.706401s (thread); 0s (gc)
│ │ │ │ + -- used 1.56205s (cpu); 0.862278s (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
│ │ │ @@ -83,15 +83,15 @@
│ │ │  
│ │ │  o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ - -- used 0.000607219s (cpu); 0.000601939s (thread); 0s (gc)
    
│ │ │ + -- used 0.00081254s (cpu); 0.000804349s (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 0.000607219s (cpu); 0.000601939s (thread); 0s (gc)
│ │ │ │ + -- used 0.00081254s (cpu); 0.000804349s (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
│ │ │ @@ -113,15 +113,15 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time (p1,p2) = graph phi;
│ │ │ - -- used 0.0188178s (cpu); 0.0184354s (thread); 0s (gc)
    
│ │ │ + -- used 0.0922538s (cpu); 0.0295831s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : p1
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │ @@ -272,15 +272,15 @@
│ │ │            
│ │ │          
│ │ │          
    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 0.184232s (cpu); 0.12115s (thread); 0s (gc)
│ │ │ │ + -- used 0.20957s (cpu); 0.116482s (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 0.371428s (cpu); 0.224639s (thread); 0s (gc)
│ │ │ │ + -- used 0.471849s (cpu); 0.249085s (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
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i9 : time g = graph p2;
│ │ │ - -- used 0.0317165s (cpu); 0.0312624s (thread); 0s (gc)
    
│ │ │ + -- used 0.0977423s (cpu); 0.0447804s (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 0.0188178s (cpu); 0.0184354s (thread); 0s (gc)
│ │ │ │ + -- used 0.0922538s (cpu); 0.0295831s (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 0.0317165s (cpu); 0.0312624s (thread); 0s (gc)
│ │ │ │ + -- used 0.0977423s (cpu); 0.0447804s (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
│ │ │ @@ -111,15 +111,15 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │              
    
│ │ │                
    i3 : time ideal phi
│ │ │ - -- used 0.0035599s (cpu); 0.00355522s (thread); 0s (gc)
│ │ │ + -- used 0.00502322s (cpu); 0.00502123s (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 )
│ │ │ @@ -195,15 +195,15 @@
│ │ │  
│ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4)
    
│ │ │              
    
│ │ │                
    i6 : time ideal phi'
│ │ │ - -- used 0.0930691s (cpu); 0.0930488s (thread); 0s (gc)
│ │ │ + -- used 0.114269s (cpu); 0.114268s (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 0.0035599s (cpu); 0.00355522s (thread); 0s (gc)
│ │ │ │ + -- used 0.00502322s (cpu); 0.00502123s (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 0.0930691s (cpu); 0.0930488s (thread); 0s (gc)
│ │ │ │ + -- used 0.114269s (cpu); 0.114268s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = ideal 1
│ │ │ │  
│ │ │ │  
│ │ │ │  QQ[x ..x , y ..y ]
│ │ │ │  
│ │ │ │  0   5   0   4
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_inverse__Map.html
│ │ │ @@ -153,15 +153,15 @@
│ │ │  
│ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
    
│ │ │              
    
│ │ │                
    i2 : time psi = inverseMap phi
│ │ │ - -- used 0.184232s (cpu); 0.12115s (thread); 0s (gc)
│ │ │ + -- used 0.20957s (cpu); 0.116482s (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: {
│ │ │ @@ -251,15 +251,15 @@
│ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
    
│ │ │              
    
│ │ │                
    i5 : time psi = inverseMap phi
│ │ │ - -- used 0.371428s (cpu); 0.224639s (thread); 0s (gc)
│ │ │ + -- used 0.471849s (cpu); 0.249085s (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 0.056969s (cpu); 0.0569687s (thread); 0s (gc)
│ │ │ + -- used 0.0645625s (cpu); 0.0645621s (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 0.056969s (cpu); 0.0569687s (thread); 0s (gc)
│ │ │ │ + -- used 0.0645625s (cpu); 0.0645621s (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
│ │ │ @@ -123,24 +123,24 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
    
│ │ │              
    
│ │ │                
    i3 : time isBirational phi
│ │ │ - -- used 0.0193201s (cpu); 0.0193206s (thread); 0s (gc)
│ │ │ + -- used 0.0253448s (cpu); 0.0253471s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
    
│ │ │                
    i4 : time isBirational(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0136925s (cpu); 0.0132915s (thread); 0s (gc)
│ │ │ + -- used 0.0281203s (cpu); 0.0158762s (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 0.0193201s (cpu); 0.0193206s (thread); 0s (gc)
│ │ │ │ + -- used 0.0253448s (cpu); 0.0253471s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0136925s (cpu); 0.0132915s (thread); 0s (gc)
│ │ │ │ + -- used 0.0281203s (cpu); 0.0158762s (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
│ │ │ @@ -86,15 +86,15 @@
│ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 2.58345s (cpu); 2.01134s (thread); 0s (gc)
│ │ │ + -- used 2.65473s (cpu); 2.31996s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : P7 = ZZ/101[x_0..x_7];
    
│ │ │ @@ -115,15 +115,15 @@
│ │ │  o6 : RationalMap (cubic rational map from PP^7 to PP^7)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 3.88844s (cpu); 2.54882s (thread); 0s (gc)
│ │ │ + -- used 3.94783s (cpu); 2.94981s (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 2.58345s (cpu); 2.01134s (thread); 0s (gc)
│ │ │ │ + -- used 2.65473s (cpu); 2.31996s (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 3.88844s (cpu); 2.54882s (thread); 0s (gc)
│ │ │ │ + -- used 3.94783s (cpu); 2.94981s (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
│ │ │ @@ -90,26 +90,26 @@
│ │ │  o1 : RingMap QQ[x ..x ] <-- QQ[y ..y  ]
│ │ │                   0   8          0   11
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time kernel(phi,1)
│ │ │ - -- used 0.0174446s (cpu); 0.0174411s (thread); 0s (gc)
│ │ │ + -- used 0.0215404s (cpu); 0.0215418s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time kernel(phi,2)
│ │ │ - -- used 0.911685s (cpu); 0.449335s (thread); 0s (gc)
│ │ │ + -- used 1.14536s (cpu); 0.538298s (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 0.0174446s (cpu); 0.0174411s (thread); 0s (gc)
│ │ │ │ + -- used 0.0215404s (cpu); 0.0215418s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = ideal ()
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │ │                    0   11
│ │ │ │  i3 : time kernel(phi,2)
│ │ │ │ - -- used 0.911685s (cpu); 0.449335s (thread); 0s (gc)
│ │ │ │ + -- used 1.14536s (cpu); 0.538298s (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
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o2 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time parametrize L
│ │ │ - -- used 0.00500618s (cpu); 0.00500168s (thread); 0s (gc)
│ │ │ + -- used 0.00636444s (cpu); 0.00638338s (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 ])
│ │ │ @@ -201,15 +201,15 @@
│ │ │  o4 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time parametrize Q
│ │ │ - -- used 0.5466s (cpu); 0.394392s (thread); 0s (gc)
│ │ │ + -- used 0.632307s (cpu); 0.466254s (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 0.00500618s (cpu); 0.00500168s (thread); 0s (gc)
│ │ │ │ + -- used 0.00636444s (cpu); 0.00638338s (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 0.5466s (cpu); 0.394392s (thread); 0s (gc)
│ │ │ │ + -- used 0.632307s (cpu); 0.466254s (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
│ │ │ @@ -78,15 +78,15 @@
│ │ │  
│ │ │  o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to PP^8)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time p = point source f
│ │ │ - -- used 0.463068s (cpu); 0.208194s (thread); 0s (gc)
│ │ │ + -- used 0.497934s (cpu); 0.238097s (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  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -100,15 +100,15 @@
│ │ │                (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 0.212468s (cpu); 0.135812s (thread); 0s (gc)
│ │ │ + -- used 0.230663s (cpu); 0.142s (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 0.463068s (cpu); 0.208194s (thread); 0s (gc)
│ │ │ │ + -- used 0.497934s (cpu); 0.238097s (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 0.212468s (cpu); 0.135812s (thread); 0s (gc)
│ │ │ │ + -- used 0.230663s (cpu); 0.142s (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
│ │ │ @@ -89,15 +89,15 @@
│ │ │                          0   4                 0   5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0151376s (cpu); 0.0148069s (thread); 0s (gc)
│ │ │ + -- used 0.0412574s (cpu); 0.0211614s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -117,15 +117,15 @@
│ │ │                2 3    1 4    0 5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0116489s (cpu); 0.0113647s (thread); 0s (gc)
│ │ │ + -- used 0.0279848s (cpu); 0.0140302s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ @@ -143,25 +143,25 @@
│ │ │  o6 : RingMap ------[x ..x ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   6
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time projectiveDegrees phi
│ │ │ - -- used 5.859e-05s (cpu); 5.361e-05s (thread); 0s (gc)
│ │ │ + -- used 7.3527e-05s (cpu); 6.2036e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 2.6109e-05s (cpu); 2.5939e-05s (thread); 0s (gc)
│ │ │ + -- used 3.9904e-05s (cpu); 3.7457e-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 0.0151376s (cpu); 0.0148069s (thread); 0s (gc)
│ │ │ │ + -- used 0.0412574s (cpu); 0.0211614s (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 0.0116489s (cpu); 0.0113647s (thread); 0s (gc)
│ │ │ │ + -- used 0.0279848s (cpu); 0.0140302s (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 5.859e-05s (cpu); 5.361e-05s (thread); 0s (gc)
│ │ │ │ + -- used 7.3527e-05s (cpu); 6.2036e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ │ - -- used 2.6109e-05s (cpu); 2.5939e-05s (thread); 0s (gc)
│ │ │ │ + -- used 3.9904e-05s (cpu); 3.7457e-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
│ │ │ @@ -88,15 +88,15 @@
│ │ │  o2 : Ideal of -----[x ..x ]
│ │ │                33331  0   6
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time phi = rationalMap(V,3,2)
│ │ │ - -- used 0.095321s (cpu); 0.0953215s (thread); 0s (gc)
│ │ │ + -- used 0.133048s (cpu); 0.133048s (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 0.095321s (cpu); 0.0953215s (thread); 0s (gc)
│ │ │ │ + -- used 0.133048s (cpu); 0.133048s (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
│ │ │ @@ -111,15 +111,15 @@
│ │ │              
│ │ │                
    i5 : D = new Tally from {H => 2,C => 1};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time phi = rationalMap D
│ │ │ - -- used 0.0301481s (cpu); 0.0301432s (thread); 0s (gc)
│ │ │ + -- used 0.0396531s (cpu); 0.0396503s (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
│ │ │ @@ -219,15 +219,15 @@
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time ? image(phi,"F4")
│ │ │ - -- used 1.24116s (cpu); 0.701199s (thread); 0s (gc)
│ │ │ + -- used 1.62842s (cpu); 0.70643s (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 0.0301481s (cpu); 0.0301432s (thread); 0s (gc)
│ │ │ │ + -- used 0.0396531s (cpu); 0.0396503s (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 1.24116s (cpu); 0.701199s (thread); 0s (gc)
│ │ │ │ + -- used 1.62842s (cpu); 0.70643s (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
│ │ │ @@ -70,15 +70,15 @@
│ │ │        
    
│ │ │          
    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 1.59376s (cpu); 1.16591s (thread); 0s (gc)
│ │ │ + -- used 1.64986s (cpu); 1.26429s (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 1.59376s (cpu); 1.16591s (thread); 0s (gc)
│ │ │ │ + -- used 1.64986s (cpu); 1.26429s (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
│ │ │ @@ -70,15 +70,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The field K is required to be large enough.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time specialCubicTransformation 9
│ │ │ - -- used 0.0954111s (cpu); 0.0954104s (thread); 0s (gc)
│ │ │ + -- used 0.10227s (cpu); 0.102271s (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
│ │ │               {
│ │ │ @@ -138,15 +138,15 @@
│ │ │  
│ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i2 : time describe oo
│ │ │ - -- used 0.0182984s (cpu); 0.0182857s (thread); 0s (gc)
│ │ │ + -- used 0.0195273s (cpu); 0.0195271s (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 0.0954111s (cpu); 0.0954104s (thread); 0s (gc)
│ │ │ │ + -- used 0.10227s (cpu); 0.102271s (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 0.0182984s (cpu); 0.0182857s (thread); 0s (gc)
│ │ │ │ + -- used 0.0195273s (cpu); 0.0195271s (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
│ │ │ @@ -70,15 +70,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The field K is required to be large enough.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time specialQuadraticTransformation 4
│ │ │ - -- used 0.0733927s (cpu); 0.0733921s (thread); 0s (gc)
│ │ │ + -- used 0.0782691s (cpu); 0.0782673s (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
│ │ │               {
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
    
│ │ │              
    
│ │ │                
    i2 : time describe oo
│ │ │ - -- used 0.110232s (cpu); 0.0307456s (thread); 0s (gc)
│ │ │ + -- used 0.117603s (cpu); 0.0334491s (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 0.0733927s (cpu); 0.0733921s (thread); 0s (gc)
│ │ │ │ + -- used 0.0782691s (cpu); 0.0782673s (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 0.110232s (cpu); 0.0307456s (thread); 0s (gc)
│ │ │ │ + -- used 0.117603s (cpu); 0.0334491s (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
│ │ │ @@ -88,23 +88,23 @@
│ │ │  
│ │ │  o3 = 6927
    
│ │ │              
    
│ │ │                
    i4 : time phi' = value str;
│ │ │ - -- used 0.0234197s (cpu); 0.0234189s (thread); 0s (gc)
│ │ │ + -- used 0.0280019s (cpu); 0.0280041s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
    
│ │ │              
    
│ │ │                
    i5 : time describe phi'
│ │ │ - -- used 0.00540039s (cpu); 0.00540078s (thread); 0s (gc)
│ │ │ + -- used 0.00670417s (cpu); 0.00671332s (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}
│ │ │ @@ -113,15 +113,15 @@
│ │ │       degree base locus: 5
│ │ │       coefficient ring: ZZ/33331
    
│ │ │              
    
│ │ │                
    i6 : time describe inverse phi'
│ │ │ - -- used 0.0044104s (cpu); 0.00441117s (thread); 0s (gc)
│ │ │ + -- used 0.00554705s (cpu); 0.00555551s (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 0.0234197s (cpu); 0.0234189s (thread); 0s (gc)
│ │ │ │ + -- used 0.0280019s (cpu); 0.0280041s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │ │  i5 : time describe phi'
│ │ │ │ - -- used 0.00540039s (cpu); 0.00540078s (thread); 0s (gc)
│ │ │ │ + -- used 0.00670417s (cpu); 0.00671332s (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 0.0044104s (cpu); 0.00441117s (thread); 0s (gc)
│ │ │ │ + -- used 0.00554705s (cpu); 0.00555551s (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
│ │ │ @@ -58,29 +58,29 @@
│ │ │              
    		
│ │ │                
    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 0.00430215s (cpu); 0.00429842s (thread); 0s (gc)
│ │ │ + -- used 0.00651371s (cpu); 0.00651227s (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 0.137231s (cpu); 0.0699679s (thread); 0s (gc)
│ │ │ + -- used 0.187395s (cpu); 0.0938856s (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
│ │ │  
│ │ │ @@ -88,43 +88,43 @@
│ │ │  o3 : Ideal of ------[x ..x ]
│ │ │                300007  0   9
    
│ │ │              
    
│ │ │                
    i4 : time degreeMap phi
│ │ │ - -- used 0.02944s (cpu); 0.0294445s (thread); 0s (gc)
│ │ │ + -- used 0.0586502s (cpu); 0.0586499s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
    
│ │ │                
    i5 : time projectiveDegrees phi
│ │ │ - -- used 0.68756s (cpu); 0.487586s (thread); 0s (gc)
│ │ │ + -- used 0.744311s (cpu); 0.55853s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.0623207s (cpu); 0.0622653s (thread); 0s (gc)
│ │ │ + -- used 0.103903s (cpu); 0.103704s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.0021486s (cpu); 0.00214939s (thread); 0s (gc)
│ │ │ + -- used 0.00294027s (cpu); 0.00294735s (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
│ │ │ @@ -136,15 +136,15 @@
│ │ │               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 0.474252s (cpu); 0.394534s (thread); 0s (gc)
│ │ │ + -- used 0.447739s (cpu); 0.447744s (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
│ │ │ @@ -156,44 +156,44 @@
│ │ │               (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 0.00931603s (cpu); 0.00931855s (thread); 0s (gc)
│ │ │ + -- used 0.0113264s (cpu); 0.0113318s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time degreeMap psi
│ │ │ - -- used 0.458493s (cpu); 0.294259s (thread); 0s (gc)
│ │ │ + -- used 0.563595s (cpu); 0.294308s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
    
│ │ │              
    
│ │ │                
    i11 : time projectiveDegrees psi
│ │ │ - -- used 5.29004s (cpu); 4.63968s (thread); 0s (gc)
│ │ │ + -- used 6.43014s (cpu); 5.97295s (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
│ │ │ │          2
│ │ │ │  
│ │ │ │  o6 : R[T ..T ]
│ │ │ │          1   3
│ │ │ │  i7 : H = HH(KR)
│ │ │ │ -Finding easy relations           :  -- used 0.0137422s (cpu); 0.0130223s
│ │ │ │ +Finding easy relations           :  -- used 0.0402553s (cpu); 0.0193477s
│ │ │ │  (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
│ │ │ @@ -129,15 +129,15 @@
│ │ │  
│ │ │  o5 : Complex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00220238s (cpu); 0.00220313s (thread); 0s (gc)
│ │ │ + -- used 0.00250238s (cpu); 0.00250843s (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 ])
│ │ │ @@ -242,15 +242,15 @@
│ │ │  
│ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
    
│ │ │              
    
│ │ │                
    i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ - -- used 0.15731s (cpu); 0.0849869s (thread); 0s (gc)
│ │ │ + -- used 0.198282s (cpu); 0.100168s (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
│ │ │ @@ -315,15 +315,15 @@
│ │ │  
│ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i14 : time phi^(-1)
│ │ │ - -- used 0.512059s (cpu); 0.426868s (thread); 0s (gc)
│ │ │ + -- used 0.493489s (cpu); 0.493164s (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 ,
│ │ │ @@ -376,49 +376,49 @@
│ │ │  
│ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9 to PP^6)
    
│ │ │              
    
│ │ │                
    i15 : time degrees phi^(-1)
│ │ │ - -- used 0.347749s (cpu); 0.274141s (thread); 0s (gc)
│ │ │ + -- used 0.472144s (cpu); 0.366608s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
    
│ │ │              
    
│ │ │                
    i16 : time degrees phi
│ │ │ - -- used 0.0180113s (cpu); 0.0176993s (thread); 0s (gc)
│ │ │ + -- used 0.0875272s (cpu); 0.0251346s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
    
│ │ │              
    
│ │ │                
    i17 : time describe phi
│ │ │ - -- used 0.00320718s (cpu); 0.00320744s (thread); 0s (gc)
│ │ │ + -- used 0.00421075s (cpu); 0.00422119s (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 0.00997997s (cpu); 0.0099807s (thread); 0s (gc)
│ │ │ + -- used 0.011936s (cpu); 0.0119473s (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}
│ │ │ @@ -427,41 +427,41 @@
│ │ │        degree base locus: 24
│ │ │        coefficient ring: ZZ/300007
    
│ │ │              
    
│ │ │                
    i19 : time (f,g) = graph phi^-1; f;
│ │ │ - -- used 0.00950571s (cpu); 0.00950659s (thread); 0s (gc)
│ │ │ + -- used 0.0120656s (cpu); 0.0120784s (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 1.33327s (cpu); 0.95512s (thread); 0s (gc)
│ │ │ + -- used 1.19811s (cpu); 1.02906s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
    
│ │ │              
    
│ │ │                
    i22 : time degree f
│ │ │ - -- used 1.625e-05s (cpu); 1.593e-05s (thread); 0s (gc)
│ │ │ + -- used 1.7684e-05s (cpu); 1.7206e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
    
│ │ │              
    
│ │ │                
    i23 : time describe f
│ │ │ - -- used 0.00161465s (cpu); 0.00161555s (thread); 0s (gc)
│ │ │ + -- used 0.00172246s (cpu); 0.00172799s (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 0.00430215s (cpu); 0.00429842s (thread); 0s (gc)
│ │ │ │ + -- used 0.00651371s (cpu); 0.00651227s (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 0.137231s (cpu); 0.0699679s (thread); 0s (gc)
│ │ │ │ + -- used 0.187395s (cpu); 0.0938856s (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 0.02944s (cpu); 0.0294445s (thread); 0s (gc)
│ │ │ │ + -- used 0.0586502s (cpu); 0.0586499s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projectiveDegrees phi
│ │ │ │ - -- used 0.68756s (cpu); 0.487586s (thread); 0s (gc)
│ │ │ │ + -- used 0.744311s (cpu); 0.55853s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ │ - -- used 0.0623207s (cpu); 0.0622653s (thread); 0s (gc)
│ │ │ │ + -- used 0.103903s (cpu); 0.103704s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = {5}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ │ - -- used 0.0021486s (cpu); 0.00214939s (thread); 0s (gc)
│ │ │ │ + -- used 0.00294027s (cpu); 0.00294735s (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 0.474252s (cpu); 0.394534s (thread); 0s (gc)
│ │ │ │ + -- used 0.447739s (cpu); 0.447744s (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 0.00931603s (cpu); 0.00931855s (thread); 0s (gc)
│ │ │ │ + -- used 0.0113264s (cpu); 0.0113318s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time degreeMap psi
│ │ │ │ - -- used 0.458493s (cpu); 0.294259s (thread); 0s (gc)
│ │ │ │ + -- used 0.563595s (cpu); 0.294308s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 1
│ │ │ │  i11 : time projectiveDegrees psi
│ │ │ │ - -- used 5.29004s (cpu); 4.63968s (thread); 0s (gc)
│ │ │ │ + -- used 6.43014s (cpu); 5.97295s (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 0.00220238s (cpu); 0.00220313s (thread); 0s (gc)
│ │ │ │ + -- used 0.00250238s (cpu); 0.00250843s (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 0.15731s (cpu); 0.0849869s (thread); 0s (gc)
│ │ │ │ + -- used 0.198282s (cpu); 0.100168s (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 0.512059s (cpu); 0.426868s (thread); 0s (gc)
│ │ │ │ + -- used 0.493489s (cpu); 0.493164s (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 0.347749s (cpu); 0.274141s (thread); 0s (gc)
│ │ │ │ + -- used 0.472144s (cpu); 0.366608s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : time degrees phi
│ │ │ │ - -- used 0.0180113s (cpu); 0.0176993s (thread); 0s (gc)
│ │ │ │ + -- used 0.0875272s (cpu); 0.0251346s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o16 : List
│ │ │ │  i17 : time describe phi
│ │ │ │ - -- used 0.00320718s (cpu); 0.00320744s (thread); 0s (gc)
│ │ │ │ + -- used 0.00421075s (cpu); 0.00422119s (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 0.00997997s (cpu); 0.0099807s (thread); 0s (gc)
│ │ │ │ + -- used 0.011936s (cpu); 0.0119473s (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 0.00950571s (cpu); 0.00950659s (thread); 0s (gc)
│ │ │ │ + -- used 0.0120656s (cpu); 0.0120784s (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 1.33327s (cpu); 0.95512s (thread); 0s (gc)
│ │ │ │ + -- used 1.19811s (cpu); 1.02906s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │ │  
│ │ │ │  o21 : List
│ │ │ │  i22 : time degree f
│ │ │ │ - -- used 1.625e-05s (cpu); 1.593e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.7684e-05s (cpu); 1.7206e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = 1
│ │ │ │  i23 : time describe f
│ │ │ │ - -- used 0.00161465s (cpu); 0.00161555s (thread); 0s (gc)
│ │ │ │ + -- used 0.00172246s (cpu); 0.00172799s (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     2     3         1       4        6        5       3 4        3 5       2 4      1 7     3 7     2 7
│ │ │  
│ │ │  o16 : DGAlgebra
│ │ │  
│ │ │  i17 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0138129s (cpu); 0.0130958s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0344827s (cpu); 0.0183202s (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   4
│ │ │        Differential => {a, b, c, d}
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : HB = HH B
│ │ │ -Finding easy relations           :  -- used 0.016973s (cpu); 0.0160828s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0345135s (cpu); 0.0216576s (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 T }
│ │ │                                        1
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │  
│ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ -Finding easy relations           :  -- used 0.0188901s (cpu); 0.0173443s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.321028s (cpu); 0.0626935s (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__Map.out
│ │ │ @@ -55,15 +55,15 @@
│ │ │                 {2} | 0 |
│ │ │                 {2} | 0 |
│ │ │                 {2} | 1 |
│ │ │  
│ │ │  o6 : ComplexMap
│ │ │  
│ │ │  i7 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0145697s (cpu); 0.0138207s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0307627s (cpu); 0.017651s (thread); 0s (gc)
│ │ │  
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ ├── ./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 0.126962s (cpu); 0.0524544s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.151101s (cpu); 0.0558439s (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 0.54717s (cpu); 0.472623s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.771115s (cpu); 0.68909s (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 0.019887s (cpu); 0.0171466s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0499479s (cpu); 0.0242867s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i5 : numgens HA
│ │ ├── ./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 0.0169473s (cpu); 0.0162102s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0342352s (cpu); 0.0218532s (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 0.0853296s (cpu); 0.0826839s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.127307s (cpu); 0.103034s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
│ │ │  
│ │ │  i9 : numgens HA
│ │ │  
│ │ │ @@ -114,15 +114,15 @@
│ │ │  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 0.0511448s (cpu); 0.0499304s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.117266s (cpu); 0.0950784s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
│ │ │  
│ │ │  i17 : R = ZZ/101[a,b,c,d]
│ │ │  
│ │ │ @@ -151,14 +151,14 @@
│ │ │         Underlying algebra => S[T ..T ]
│ │ │                                  1   4
│ │ │         Differential => {a, b, c, d}
│ │ │  
│ │ │  o20 : DGAlgebra
│ │ │  
│ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ -Finding easy relations           :  -- used 0.0169588s (cpu); 0.016221s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.035201s (cpu); 0.0223172s (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
│ │ │          2
│ │ │  
│ │ │  o6 : R[T ..T ]
│ │ │          1   3
│ │ │  
│ │ │  i7 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.0137422s (cpu); 0.0130223s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0402553s (cpu); 0.0193477s (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 0.27818s (cpu); 0.202362s (thread); 0s (gc)
│ │ │ + -- used 0.290677s (cpu); 0.180575s (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 2 3    1 2 2 2 3 4
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ -Finding easy relations           :  -- used 0.517042s (cpu); 0.445082s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.851888s (cpu); 0.65181s (thread); 0s (gc)
│ │ │  
│ │ │               2
│ │ │  o10 = x x y z T T T T
│ │ │         1 2 2   2 3 4 5
│ │ │  
│ │ │  o10 : R[T ..T ]
│ │ │           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   4
│ │ │        Differential => {t , t , t , t }
│ │ │                          1   2   3   4
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.138884s (cpu); 0.136232s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.18848s (cpu); 0.174573s (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)
│ │ │ - -- used 0.601766s (cpu); 0.506804s (thread); 0s (gc)
│ │ │ + -- used 0.642579s (cpu); 0.530767s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
│ │ │  
│ │ │  i5 : numgens HB
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.html
│ │ │ @@ -289,15 +289,15 @@
│ │ │          
    
│ │ │            
    One can also obtain the map on homology induced by a DGAlgebra map.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i17 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0138129s (cpu); 0.0130958s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0344827s (cpu); 0.0183202s (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 {}
│ │ │ │ @@ -210,15 +210,15 @@
│ │ │ │  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     2     3         1       4        6
│ │ │ │  5       3 4        3 5       2 4      1 7     3 7     2 7
│ │ │ │  
│ │ │ │  o16 : DGAlgebra
│ │ │ │  One can also obtain the map on homology induced by a DGAlgebra map.
│ │ │ │  i17 : HHg = HH g
│ │ │ │ -Finding easy relations           :  -- used 0.0138129s (cpu); 0.0130958s
│ │ │ │ +Finding easy relations           :  -- used 0.0344827s (cpu); 0.0183202s
│ │ │ │  (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
│ │ │ @@ -113,15 +113,15 @@
│ │ │          
    
│ │ │            
    One can compute the homology algebra of a DGAlgebra using the homology (or HH) command.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -174,15 +174,15 @@
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,15 +42,15 @@
│ │ │ │                                 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 0.016973s (cpu); 0.0160828s
│ │ │ │ +Finding easy relations           :  -- used 0.0345135s (cpu); 0.0216576s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HB
│ │ │ │  
│ │ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i6 : describe HB
│ │ │ │  
│ │ │ │ @@ -87,15 +87,15 @@
│ │ │ │                                 1   5
│ │ │ │                                      2
│ │ │ │        Differential => {a, b, c, d, a T }
│ │ │ │                                        1
│ │ │ │  
│ │ │ │  o9 : DGAlgebra
│ │ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ │ -Finding easy relations           :  -- used 0.0188901s (cpu); 0.0173443s
│ │ │ │ +Finding easy relations           :  -- used 0.321028s (cpu); 0.0626935s
│ │ │ │  (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__Map.html
│ │ │ @@ -144,15 +144,15 @@
│ │ │  
│ │ │  o6 : ComplexMap
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -242,15 +242,15 @@
│ │ │  
│ │ │  o15 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : HB = HH B
│ │ │ -Finding easy relations           :  -- used 0.016973s (cpu); 0.0160828s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0345135s (cpu); 0.0216576s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ -Finding easy relations           :  -- used 0.0188901s (cpu); 0.0173443s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.321028s (cpu); 0.0626935s (thread); 0s (gc)
│ │ │  
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i7 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0145697s (cpu); 0.0138207s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0307627s (cpu); 0.017651s (thread); 0s (gc)
│ │ │  
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -62,15 +62,15 @@
│ │ │ │       2 : R  <------------- R  : 2
│ │ │ │                 {2} | 0 |
│ │ │ │                 {2} | 0 |
│ │ │ │                 {2} | 1 |
│ │ │ │  
│ │ │ │  o6 : ComplexMap
│ │ │ │  i7 : HHg = HH g
│ │ │ │ -Finding easy relations           :  -- used 0.0145697s (cpu); 0.0138207s
│ │ │ │ +Finding easy relations           :  -- used 0.0307627s (cpu); 0.017651s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │                           ZZ
│ │ │ │                          ---[a..c]
│ │ │ │             ZZ           101
│ │ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │            101  1   2           3   1     1
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.html
│ │ │ @@ -138,15 +138,15 @@
│ │ │          
    
│ │ │            
    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.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  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 0.126962s (cpu); 0.0524544s
│ │ │ │ +Finding easy relations           :  -- used 0.151101s (cpu); 0.0558439s
│ │ │ │  (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 0.54717s (cpu); 0.472623s (thread);
│ │ │ │ +Finding easy relations           :  -- used 0.771115s (cpu); 0.68909s (thread);
│ │ │ │  0s (gc)
│ │ │ │  
│ │ │ │  o10 = HKR'
│ │ │ │  
│ │ │ │  o10 : QuotientRing
│ │ │ │  i11 : numgens HKR'
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_cycles.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │  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 0.019887s (cpu); 0.0171466s
│ │ │ │ +Finding easy relations           :  -- used 0.0499479s (cpu); 0.0242867s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = HA
│ │ │ │  
│ │ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i5 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Algebra.html
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : HKR = HH KR
│ │ │ -Finding easy relations           :  -- used 0.126962s (cpu); 0.0524544s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.151101s (cpu); 0.0558439s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = HKR
│ │ │  
│ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : HKR' = HH koszulComplexDGA R'
│ │ │ -Finding easy relations           :  -- used 0.54717s (cpu); 0.472623s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.771115s (cpu); 0.68909s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = HKR'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.019887s (cpu); 0.0171466s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0499479s (cpu); 0.0242867s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0169473s (cpu); 0.0162102s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0342352s (cpu); 0.0218532s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │ @@ -148,15 +148,15 @@
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │                
    i8 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0853296s (cpu); 0.0826839s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.127307s (cpu); 0.103034s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
    
│ │ │              
    
│ │ │                
    i16 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0511448s (cpu); 0.0499304s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.117266s (cpu); 0.0950784s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │              
    
│ │ │ @@ -302,15 +302,15 @@
│ │ │  
│ │ │  o20 : DGAlgebra
    
│ │ │              
    
│ │ │                
    i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ -Finding easy relations           :  -- used 0.0169588s (cpu); 0.016221s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.035201s (cpu); 0.0223172s (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 0.0169473s (cpu); 0.0162102s
│ │ │ │ +Finding easy relations           :  -- used 0.0342352s (cpu); 0.0218532s
│ │ │ │  (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 0.0853296s (cpu); 0.0826839s
│ │ │ │ +Finding easy relations           :  -- used 0.127307s (cpu); 0.103034s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = HA
│ │ │ │  
│ │ │ │  o8 : QuotientRing
│ │ │ │  i9 : numgens HA
│ │ │ │  
│ │ │ │ @@ -122,15 +122,15 @@
│ │ │ │  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 0.0511448s (cpu); 0.0499304s
│ │ │ │ +Finding easy relations           :  -- used 0.117266s (cpu); 0.0950784s
│ │ │ │  (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
│ │ │ │ @@ -158,15 +158,15 @@
│ │ │ │  o20 = {Ring => S                      }
│ │ │ │         Underlying algebra => S[T ..T ]
│ │ │ │                                  1   4
│ │ │ │         Differential => {a, b, c, d}
│ │ │ │  
│ │ │ │  o20 : DGAlgebra
│ │ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ │ -Finding easy relations           :  -- used 0.0169588s (cpu); 0.016221s
│ │ │ │ +Finding easy relations           :  -- used 0.035201s (cpu); 0.0223172s
│ │ │ │  (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
│ │ │ @@ -135,15 +135,15 @@
│ │ │  o6 : R[T ..T ]
│ │ │          1   3
    
│ │ │              
    
│ │ │                
    i7 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.0137422s (cpu); 0.0130223s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0402553s (cpu); 0.0193477s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = H
│ │ │  
│ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i6 : HKR = HH(KR)
│ │ │ - -- used 0.27818s (cpu); 0.202362s (thread); 0s (gc)
│ │ │ + -- used 0.290677s (cpu); 0.180575s (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 0.27818s (cpu); 0.202362s (thread); 0s (gc)
│ │ │ │ + -- used 0.290677s (cpu); 0.180575s (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
│ │ │ @@ -192,15 +192,15 @@
│ │ │          
    
│ │ │            
    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 {}
│ │ │ │ @@ -49,16 +49,16 @@
│ │ │ │        Underlying algebra => R[T ..T ]
│ │ │ │                                 1   4
│ │ │ │        Differential => {t , t , t , t }
│ │ │ │                          1   2   3   4
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  i5 : H = HH(KR)
│ │ │ │ -Finding easy relations           :  -- used 0.138884s (cpu); 0.136232s
│ │ │ │ -(thread); 0s (gc)
│ │ │ │ +Finding easy relations           :  -- used 0.18848s (cpu); 0.174573s (thread);
│ │ │ │ +0s (gc)
│ │ │ │  
│ │ │ │  o5 = H
│ │ │ │  
│ │ │ │  o5 : QuotientRing
│ │ │ │  i6 : masseys = masseyTripleProduct(KR,1,1,1);
│ │ │ │  
│ │ │ │                5       343
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_tor__Algebra_lp__Ring_cm__Ring_rp.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  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)
│ │ │ │ - -- used 0.601766s (cpu); 0.506804s (thread); 0s (gc)
│ │ │ │ + -- used 0.642579s (cpu); 0.530767s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o4 = HB
│ │ │ │  
│ │ │ │  o4 : QuotientRing
│ │ │ │  i5 : numgens HB
│ │ │ │  
│ │ │ │  o5 = 35
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_complement__Graph.out
│ │ │ @@ -2,15 +2,15 @@
│ │ │  
│ │ │  i1 : R = QQ[a,b,c,d,e];
│ │ │  
│ │ │  i2 : c5 = graph {a*b,b*c,c*d,d*e,e*a}; -- graph of the 5-cycle
│ │ │  
│ │ │  i3 : complementGraph c5 -- the graph complement of the 5-cycle
│ │ │  
│ │ │ -o3 = Graph{"edges" => {{a, c}, {b, e}, {b, d}, {c, e}, {a, d}}}
│ │ │ +o3 = Graph{"edges" => {{a, d}, {a, c}, {b, e}, {b, d}, {c, e}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o3 : Graph
│ │ │  
│ │ │  i4 : c5hypergraph = hyperGraph c5 -- the 5-cycle, but viewed as a hypergraph
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_random__Hyper__Graph.out
│ │ │ @@ -3,25 +3,25 @@
│ │ │  i1 : R = QQ[x_1..x_5];
│ │ │  
│ │ │  i2 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  i3 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              1   3   4     1   5     2   3   4   5
│ │ │ +                              2   3   5     1   3     1   2   4   5
│ │ │                  "ring" => R
│ │ │                  "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 }}}
│ │ │ -                              2   3   4     1   2     1   3   4   5
│ │ │ +                              1   3   4     3   5     1   2   4   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     3   4     4   5     1   6     5   6
│ │ │ +                         1   2     2   3     3   4     4   5     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   3     2   3     4   6     5   6
│ │ │ +                         1   2     1   3     4   5     4   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/_complement__Graph.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │                
    i2 : c5 = graph {a*b,b*c,c*d,d*e,e*a}; -- graph of the 5-cycle
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  When applied to a graph, complementGraph returns the graph whose edge set is
│ │ │ │  the set of edges not in G. When applied to a hypergraph, the edge set is found
│ │ │ │  by taking the complement of each edge of H in the vertex set.
│ │ │ │  i1 : R = QQ[a,b,c,d,e];
│ │ │ │  i2 : c5 = graph {a*b,b*c,c*d,d*e,e*a}; -- graph of the 5-cycle
│ │ │ │  i3 : complementGraph c5 -- the graph complement of the 5-cycle
│ │ │ │  
│ │ │ │ -o3 = Graph{"edges" => {{a, c}, {b, e}, {b, d}, {c, e}, {a, d}}}
│ │ │ │ +o3 = Graph{"edges" => {{a, d}, {a, c}, {b, e}, {b, d}, {c, e}}}
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ │ │  
│ │ │ │  o3 : Graph
│ │ │ │  i4 : c5hypergraph = hyperGraph c5 -- the 5-cycle, but viewed as a hypergraph
│ │ │ │  
│ │ │ │  o4 = HyperGraph{"edges" => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}}
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph.html
│ │ │ @@ -88,28 +88,28 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,24 +25,24 @@
│ │ │ │  _T_i_m_e_L_i_m_i_t). The method will return null if it cannot find a hypergraph within
│ │ │ │  the branch and time limits.
│ │ │ │  i1 : R = QQ[x_1..x_5];
│ │ │ │  i2 : randomHyperGraph(R,{3,2,4})
│ │ │ │  i3 : randomHyperGraph(R,{3,2,4})
│ │ │ │  
│ │ │ │  o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ │ -                              1   3   4     1   5     2   3   4   5
│ │ │ │ +                              2   3   5     1   3     1   2   4   5
│ │ │ │                  "ring" => R
│ │ │ │                  "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 }}}
│ │ │ │ -                              2   3   4     1   2     1   3   4   5
│ │ │ │ +                              1   3   4     3   5     1   2   4   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
│ │ │ @@ -82,15 +82,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -112,15 +112,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── 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     3   4     4   5     1   6     5   6
│ │ │ │ +                         1   2     2   3     3   4     4   5     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   3     2   3     4   6     5   6
│ │ │ │ +                         1   2     1   3     4   5     4   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;
│ │ │ - -- .34144s elapsed
│ │ │ + -- .278901s elapsed
│ │ │  
│ │ │  i4 : #sols -- 156 solutions
│ │ │  
│ │ │  o4 = 156
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/html/index.html
│ │ │ @@ -80,15 +80,15 @@
│ │ │  
│ │ │  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 0.00276448s (cpu); 0.00276172s (thread); 0s (gc)
│ │ │ │ + -- used 0.00372932s (cpu); 0.00372495s (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 0.00161961s (cpu); 0.00162013s (thread); 0s (gc)
│ │ │ │ + -- used 0.00197103s (cpu); 0.00197342s (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
│ │ │ @@ -97,26 +97,26 @@
│ │ │  
│ │ │  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 0.00357906s (cpu); 0.00357709s (thread); 0s (gc)
│ │ │ │ + -- used 0.0036454s (cpu); 0.00364396s (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 0.00211786s (cpu); 0.0021216s (thread); 0s (gc)
│ │ │ │ + -- used 0.00199918s (cpu); 0.0020023s (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
│ │ │ @@ -105,15 +105,15 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ -Finding easy relations           :  -- used 0.517042s (cpu); 0.445082s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.851888s (cpu); 0.65181s (thread); 0s (gc)
│ │ │  
│ │ │               2
│ │ │  o10 = x x y z T T T T
│ │ │         1 2 2   2 3 4 5
│ │ │  
│ │ │  o10 : R[T ..T ]
│ │ │           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 0.517042s (cpu); 0.445082s
│ │ │ │ -(thread); 0s (gc)
│ │ │ │ +Finding easy relations           :  -- used 0.851888s (cpu); 0.65181s (thread);
│ │ │ │ +0s (gc)
│ │ │ │  
│ │ │ │               2
│ │ │ │  o10 = x x y z T T T T
│ │ │ │         1 2 2   2 3 4 5
│ │ │ │  
│ │ │ │  o10 : R[T ..T ]
│ │ │ │           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
│ │ │ @@ -119,15 +119,15 @@
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │              
    
│ │ │                
    i5 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.138884s (cpu); 0.136232s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.18848s (cpu); 0.174573s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ - -- used 0.601766s (cpu); 0.506804s (thread); 0s (gc)
│ │ │ + -- used 0.642579s (cpu); 0.530767s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
    
│ │ │                
    i3 : complementGraph c5 -- the graph complement of the 5-cycle
│ │ │  
│ │ │ -o3 = Graph{"edges" => {{a, c}, {b, e}, {b, d}, {c, e}, {a, d}}}
│ │ │ +o3 = Graph{"edges" => {{a, d}, {a, c}, {b, e}, {b, d}, {c, e}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o3 : Graph
    
│ │ │              
    
│ │ │                
    i3 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              1   3   4     1   5     2   3   4   5
│ │ │ +                              2   3   5     1   3     1   2   4   5
│ │ │                  "ring" => R
│ │ │                  "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 }}}
│ │ │ -                              2   3   4     1   2     1   3   4   5
│ │ │ +                              1   3   4     3   5     1   2   4   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     3   4     4   5     1   6     5   6
│ │ │ +                         1   2     2   3     3   4     4   5     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   3     2   3     4   6     5   6
│ │ │ +                         1   2     1   3     4   5     4   6
│ │ │             "ring" => R
│ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │                             1   2   3   4   5   6
│ │ │  
│ │ │  o7 : Graph
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ - -- .34144s elapsed
    
│ │ │ + -- .278901s 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;
│ │ │ │ - -- .34144s elapsed
│ │ │ │ + -- .278901s 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 0.00276448s (cpu); 0.00276172s (thread); 0s (gc)
│ │ │ + -- used 0.00372932s (cpu); 0.00372495s (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 0.00161961s (cpu); 0.00162013s (thread); 0s (gc)
│ │ │ + -- used 0.00197103s (cpu); 0.00197342s (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 0.00357906s (cpu); 0.00357709s (thread); 0s (gc)
│ │ │ + -- used 0.0036454s (cpu); 0.00364396s (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 0.00211786s (cpu); 0.0021216s (thread); 0s (gc)
│ │ │ + -- used 0.00199918s (cpu); 0.0020023s (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 1.89068s (cpu); 1.67085s (thread); 0s (gc)
│ │ │ + -- used 1.71728s (cpu); 1.5103s (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 0.0240253s (cpu); 0.0240283s (thread); 0s (gc)
│ │ │ + -- used 0.0173944s (cpu); 0.0173975s (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 1.67634s (cpu); 1.45099s (thread); 0s (gc)
│ │ │ + -- used 1.6159s (cpu); 1.46189s (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 0.01611s (cpu); 0.016112s (thread); 0s (gc)
│ │ │ + -- used 0.0169919s (cpu); 0.0169951s (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
│ │ │ @@ -103,26 +103,26 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
    
│ │ │                
    i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00276448s (cpu); 0.00276172s (thread); 0s (gc)
│ │ │ + -- used 0.00372932s (cpu); 0.00372495s (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 0.00161961s (cpu); 0.00162013s (thread); 0s (gc)
│ │ │ + -- used 0.00197103s (cpu); 0.00197342s (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 0.00357906s (cpu); 0.00357709s (thread); 0s (gc)
│ │ │ + -- used 0.0036454s (cpu); 0.00364396s (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 0.00211786s (cpu); 0.0021216s (thread); 0s (gc)
│ │ │ + -- used 0.00199918s (cpu); 0.0020023s (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 1.89068s (cpu); 1.67085s (thread); 0s (gc)
│ │ │ + -- used 1.71728s (cpu); 1.5103s (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  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -164,15 +164,15 @@
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0240253s (cpu); 0.0240283s (thread); 0s (gc)
│ │ │ + -- used 0.0173944s (cpu); 0.0173975s (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 1.89068s (cpu); 1.67085s (thread); 0s (gc)
│ │ │ │ + -- used 1.71728s (cpu); 1.5103s (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 0.0240253s (cpu); 0.0240283s (thread); 0s (gc)
│ │ │ │ + -- used 0.0173944s (cpu); 0.0173975s (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
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o4 : R
    
│ │ │              
    
│ │ │                
    i5 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.67634s (cpu); 1.45099s (thread); 0s (gc)
│ │ │ + -- used 1.6159s (cpu); 1.46189s (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  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -163,15 +163,15 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.01611s (cpu); 0.016112s (thread); 0s (gc)
│ │ │ + -- used 0.0169919s (cpu); 0.0169951s (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 1.67634s (cpu); 1.45099s (thread); 0s (gc)
│ │ │ │ + -- used 1.6159s (cpu); 1.46189s (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 0.01611s (cpu); 0.016112s (thread); 0s (gc)
│ │ │ │ + -- used 0.0169919s (cpu); 0.0169951s (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 0.0280835s (cpu); 0.0280868s (thread); 0s (gc)
│ │ │ + -- used 0.0363058s (cpu); 0.0363077s (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 0.328797s (cpu); 0.276936s (thread); 0s (gc)
│ │ │ + -- used 0.402089s (cpu); 0.34762s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time rationalCurve(3)
│ │ │ - -- used 0.228142s (cpu); 0.168847s (thread); 0s (gc)
│ │ │ + -- used 0.141634s (cpu); 0.141644s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
│ │ │  
│ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 5.32989s (cpu); 4.61697s (thread); 0s (gc)
│ │ │ + -- used 5.28842s (cpu); 4.64979s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.217214s (cpu); 0.167544s (thread); 0s (gc)
│ │ │ + -- used 0.145437s (cpu); 0.145447s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
│ │ │  
│ │ │  i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 5.31342s (cpu); 4.64208s (thread); 0s (gc)
│ │ │ + -- used 5.55897s (cpu); 4.90851s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : time rationalCurve(4)
│ │ │ - -- used 1.64976s (cpu); 1.4421s (thread); 0s (gc)
│ │ │ + -- used 1.59288s (cpu); 1.4533s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
│ │ │  
│ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 7.46787s (cpu); 5.79404s (thread); 0s (gc)
│ │ │ + -- used 7.24919s (cpu); 6.00017s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
│ │ │  
│ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 1.65931s (cpu); 1.42972s (thread); 0s (gc)
│ │ │ + -- used 1.71893s (cpu); 1.501s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
│ │ │  
│ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 7.68075s (cpu); 6.13623s (thread); 0s (gc)
│ │ │ + -- used 7.6756s (cpu); 6.42546s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
│ │ │  
│ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 7.9262s (cpu); 6.01692s (thread); 0s (gc)
│ │ │ + -- used 7.18198s (cpu); 5.93205s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 1139448384
│ │ │  
│ │ │  o16 : QQ
│ │ │  
│ │ │  i17 :
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html
│ │ │ @@ -71,15 +71,15 @@
│ │ │            
    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 0.0280835s (cpu); 0.0280868s (thread); 0s (gc)
│ │ │ │ + -- used 0.0363058s (cpu); 0.0363077s (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
│ │ │ @@ -152,15 +152,15 @@
│ │ │            
    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 0.0280835s (cpu); 0.0280868s (thread); 0s (gc)
│ │ │ + -- used 0.0363058s (cpu); 0.0363077s (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 0.328797s (cpu); 0.276936s (thread); 0s (gc)
│ │ │ + -- used 0.402089s (cpu); 0.34762s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ @@ -168,27 +168,27 @@
│ │ │            
    For rational curves of degree 3:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -198,15 +198,15 @@
│ │ │            
    The number of rational curves of degree 3 on a general quintic threefold can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
    
│ │ │                
    i8 : time rationalCurve(3)
│ │ │ - -- used 0.228142s (cpu); 0.168847s (thread); 0s (gc)
│ │ │ + -- used 0.141634s (cpu); 0.141644s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
    
│ │ │              
    
│ │ │                
    i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 5.32989s (cpu); 4.61697s (thread); 0s (gc)
│ │ │ + -- used 5.28842s (cpu); 4.64979s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.217214s (cpu); 0.167544s (thread); 0s (gc)
│ │ │ + -- used 0.145437s (cpu); 0.145447s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
    
│ │ │              
    
│ │ │ @@ -214,15 +214,15 @@
│ │ │            
    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 5.31342s (cpu); 4.64208s (thread); 0s (gc)
│ │ │ + -- used 5.55897s (cpu); 4.90851s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
    
│ │ │ @@ -230,27 +230,27 @@
│ │ │            
    For rational curves of degree 4:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : time rationalCurve(4)
│ │ │ - -- used 1.64976s (cpu); 1.4421s (thread); 0s (gc)
│ │ │ + -- used 1.59288s (cpu); 1.4533s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
    
│ │ │              
    
│ │ │                
    i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 7.46787s (cpu); 5.79404s (thread); 0s (gc)
│ │ │ + -- used 7.24919s (cpu); 6.00017s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
    
│ │ │              
    
│ │ │ @@ -258,15 +258,15 @@
│ │ │            
    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 1.65931s (cpu); 1.42972s (thread); 0s (gc)
│ │ │ + -- used 1.71893s (cpu); 1.501s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
    
│ │ │              
    
│ │ │ @@ -274,25 +274,25 @@
│ │ │            
    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 7.68075s (cpu); 6.13623s (thread); 0s (gc)
│ │ │ + -- used 7.6756s (cpu); 6.42546s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
    
│ │ │              
    
│ │ │                
    i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 7.9262s (cpu); 6.01692s (thread); 0s (gc)
│ │ │ + -- used 7.18198s (cpu); 5.93205s (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 0.328797s (cpu); 0.276936s (thread); 0s (gc)
│ │ │ │ + -- used 0.402089s (cpu); 0.34762s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  For rational curves of degree 3:
│ │ │ │  i8 : time rationalCurve(3)
│ │ │ │ - -- used 0.228142s (cpu); 0.168847s (thread); 0s (gc)
│ │ │ │ + -- used 0.141634s (cpu); 0.141644s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       8564575000
│ │ │ │  o8 = ----------
│ │ │ │           27
│ │ │ │  
│ │ │ │  o8 : QQ
│ │ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ │ - -- used 5.32989s (cpu); 4.61697s (thread); 0s (gc)
│ │ │ │ + -- used 5.28842s (cpu); 4.64979s (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 0.217214s (cpu); 0.167544s (thread); 0s (gc)
│ │ │ │ + -- used 0.145437s (cpu); 0.145447s (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 5.31342s (cpu); 4.64208s (thread); 0s (gc)
│ │ │ │ + -- used 5.55897s (cpu); 4.90851s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  For rational curves of degree 4:
│ │ │ │  i12 : time rationalCurve(4)
│ │ │ │ - -- used 1.64976s (cpu); 1.4421s (thread); 0s (gc)
│ │ │ │ + -- used 1.59288s (cpu); 1.4533s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        15517926796875
│ │ │ │  o12 = --------------
│ │ │ │              64
│ │ │ │  
│ │ │ │  o12 : QQ
│ │ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ │ - -- used 7.46787s (cpu); 5.79404s (thread); 0s (gc)
│ │ │ │ + -- used 7.24919s (cpu); 6.00017s (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 1.65931s (cpu); 1.42972s (thread); 0s (gc)
│ │ │ │ + -- used 1.71893s (cpu); 1.501s (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 7.68075s (cpu); 6.13623s (thread); 0s (gc)
│ │ │ │ + -- used 7.6756s (cpu); 6.42546s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3883902528
│ │ │ │  
│ │ │ │  o15 : QQ
│ │ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ │ - -- used 7.9262s (cpu); 6.01692s (thread); 0s (gc)
│ │ │ │ + -- used 7.18198s (cpu); 5.93205s (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 .00198437 seconds
│ │ │ -     -- used .000540734 seconds
│ │ │ +     -- used .0027049 seconds
│ │ │ +     -- used .00112103 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00328788 seconds
│ │ │ -     -- used .00432848 seconds
│ │ │ +     -- used .00487625 seconds
│ │ │ +     -- used .00506632 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00788476 seconds
│ │ │ -     -- used .0260362 seconds
│ │ │ +     -- used .0134169 seconds
│ │ │ +     -- used .0288429 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0175752 seconds
│ │ │ -     -- used .212106 seconds
│ │ │ +     -- used .0226244 seconds
│ │ │ +     -- used .236642 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .0370974 seconds
│ │ │ -     -- used .791376 seconds
│ │ │ +     -- used .0454796 seconds
│ │ │ +     -- used .874367 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
│ │ │ @@ -101,34 +101,34 @@
│ │ │  o3 : RingMap R <-- S
    
│ │ │              
    
│ │ │                
    i4 : G = egbToric(m, OutFile=>stdio)
│ │ │  3
│ │ │ -     -- used .00198437 seconds
│ │ │ -     -- used .000540734 seconds
│ │ │ +     -- used .0027049 seconds
│ │ │ +     -- used .00112103 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00328788 seconds
│ │ │ -     -- used .00432848 seconds
│ │ │ +     -- used .00487625 seconds
│ │ │ +     -- used .00506632 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00788476 seconds
│ │ │ -     -- used .0260362 seconds
│ │ │ +     -- used .0134169 seconds
│ │ │ +     -- used .0288429 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0175752 seconds
│ │ │ -     -- used .212106 seconds
│ │ │ +     -- used .0226244 seconds
│ │ │ +     -- used .236642 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .0370974 seconds
│ │ │ -     -- used .791376 seconds
│ │ │ +     -- used .0454796 seconds
│ │ │ +     -- used .874367 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 .00198437 seconds
│ │ │ │ -     -- used .000540734 seconds
│ │ │ │ +     -- used .0027049 seconds
│ │ │ │ +     -- used .00112103 seconds
│ │ │ │  (9, 9)
│ │ │ │  new stuff found
│ │ │ │  4
│ │ │ │ -     -- used .00328788 seconds
│ │ │ │ -     -- used .00432848 seconds
│ │ │ │ +     -- used .00487625 seconds
│ │ │ │ +     -- used .00506632 seconds
│ │ │ │  (16, 26)
│ │ │ │  new stuff found
│ │ │ │  5
│ │ │ │ -     -- used .00788476 seconds
│ │ │ │ -     -- used .0260362 seconds
│ │ │ │ +     -- used .0134169 seconds
│ │ │ │ +     -- used .0288429 seconds
│ │ │ │  (25, 60)
│ │ │ │  6
│ │ │ │ -     -- used .0175752 seconds
│ │ │ │ -     -- used .212106 seconds
│ │ │ │ +     -- used .0226244 seconds
│ │ │ │ +     -- used .236642 seconds
│ │ │ │  (36, 120)
│ │ │ │  7
│ │ │ │ -     -- used .0370974 seconds
│ │ │ │ -     -- used .791376 seconds
│ │ │ │ +     -- used .0454796 seconds
│ │ │ │ +     -- used .874367 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
│ │ │ @@ -462,50 +462,50 @@
│ │ │                 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 0.177969s (cpu); 0.122004s (thread); 0s (gc)
│ │ │ + -- used 0.235066s (cpu); 0.172562s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
│ │ │  
│ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.324762s (cpu); 0.212652s (thread); 0s (gc)
│ │ │ + -- used 0.377864s (cpu); 0.243329s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
│ │ │  
│ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.505075s (cpu); 0.32834s (thread); 0s (gc)
│ │ │ + -- used 0.581966s (cpu); 0.368048s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
│ │ │  
│ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.226825s (cpu); 0.185587s (thread); 0s (gc)
│ │ │ + -- used 0.300822s (cpu); 0.233416s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
│ │ │  
│ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.386378s (cpu); 0.21271s (thread); 0s (gc)
│ │ │ + -- used 0.464162s (cpu); 0.248254s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
│ │ │  
│ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.356033s (cpu); 0.24901s (thread); 0s (gc)
│ │ │ + -- used 0.405066s (cpu); 0.272011s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
│ │ │  
│ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.299885s (cpu); 0.188636s (thread); 0s (gc)
│ │ │ + -- used 0.362747s (cpu); 0.240964s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
│ │ │  
│ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 15.4766s (cpu); 10.5375s (thread); 0s (gc)
│ │ │ + -- used 19.5605s (cpu); 12.84s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 1
│ │ │  
│ │ │  i36 : peek StrategyDefault
│ │ │  
│ │ │  o36 = OptionTable{GRevLexLargest => 0      }
│ │ │                    GRevLexSmallest => 16
│ │ │ @@ -514,15 +514,15 @@
│ │ │                    LexSmallest => 16
│ │ │                    LexSmallestTerm => 16
│ │ │                    Points => 0
│ │ │                    Random => 16
│ │ │                    RandomNonzero => 16
│ │ │  
│ │ │  i37 : time chooseGoodMinors(20, 6, M, J, Strategy=>StrategyDefault, Verbose=>true);
│ │ │ - -- used 0.383136s (cpu); 0.324174s (thread); 0s (gc)
│ │ │ + -- used 0.45547s (cpu); 0.380341s (thread); 0s (gc)
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ @@ -582,15 +582,15 @@
│ │ │  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 0.487214s (cpu); 0.426333s (thread); 0s (gc)
│ │ │ + -- used 0.760153s (cpu); 0.611235s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
│ │ │  
│ │ │  i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │  
│ │ │  o44 = OptionTable{DecompositionStrategy => Decompose}
│ │ │                    DimensionFunction => dim
│ │ │ @@ -605,47 +605,47 @@
│ │ │  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 0.506683s (cpu); 0.379698s (thread); 0s (gc)
│ │ │ + -- used 0.521084s (cpu); 0.450882s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
│ │ │  
│ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 3.28944s (cpu); 3.0128s (thread); 0s (gc)
│ │ │ + -- used 4.35788s (cpu); 3.91067s (thread); 0s (gc)
│ │ │  
│ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 0.905703s (cpu); 0.789473s (thread); 0s (gc)
│ │ │ + -- used 0.979538s (cpu); 0.824092s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
│ │ │  
│ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 3.58264s (cpu); 3.36454s (thread); 0s (gc)
│ │ │ + -- used 3.58625s (cpu); 3.36342s (thread); 0s (gc)
│ │ │  
│ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 2.38279s (cpu); 1.9883s (thread); 0s (gc)
│ │ │ + -- used 2.94573s (cpu); 2.43591s (thread); 0s (gc)
│ │ │  
│ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 0.831174s (cpu); 0.71578s (thread); 0s (gc)
│ │ │ + -- used 1.01229s (cpu); 0.93891s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
│ │ │  
│ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 2.6018s (cpu); 2.15961s (thread); 0s (gc)
│ │ │ + -- used 3.31185s (cpu); 2.72932s (thread); 0s (gc)
│ │ │  
│ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 3.03003s (cpu); 2.62369s (thread); 0s (gc)
│ │ │ + -- used 4.04259s (cpu); 3.43991s (thread); 0s (gc)
│ │ │  
│ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 9.15531s (cpu); 7.58445s (thread); 0s (gc)
│ │ │ + -- used 11.6132s (cpu); 9.62585s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
│ │ │  
│ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 7.17456s (cpu); 5.85532s (thread); 0s (gc)
│ │ │ + -- used 8.96829s (cpu); 7.36393s (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 0.954389s (cpu); 0.635206s (thread); 0s (gc)
│ │ │ + -- used 1.18993s (cpu); 0.756142s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 11.1794s (cpu); 8.10432s (thread); 0s (gc)
│ │ │ + -- used 13.3074s (cpu); 9.34486s (thread); 0s (gc)
│ │ │  
│ │ │  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 LexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ @@ -87,21 +87,21 @@
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 49, and computed = 39
│ │ │  regularInCodimension:  singularLocus dimension verified by isCodimAtLeast
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 2
│ │ │ -regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.43458s (cpu); 1.01606s (thread); 0s (gc)
│ │ │ +regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.69305s (cpu); 1.1984s (thread); 0s (gc)
│ │ │  d = 39.  singular locus dimension appears to be = 2
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : time regularInCodimension(1, S/J, MaxMinors=>10, Verbose=>true)
│ │ │ - -- used 0.175899s (cpu); 0.12452s (thread); 0s (gc)
│ │ │ + -- used 0.221683s (cpu); 0.15609s (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 GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -115,15 +115,15 @@
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  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 0.15707s (cpu); 0.106723s (thread); 0s (gc)
│ │ │ + -- used 0.193756s (cpu); 0.133139s (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
│ │ │ @@ -137,15 +137,15 @@
│ │ │  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 0.627779s (cpu); 0.442058s (thread); 0s (gc)
│ │ │ + -- used 0.95771s (cpu); 0.637674s (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 Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 3, and computed = 3
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -165,15 +165,15 @@
│ │ │  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
│ │ │  
│ │ │  i10 : time regularInCodimension(1, S/J, MaxMinors=>25, CodimCheckFunction => t->t/5, MinMinorsFunction => t->2, Verbose=>true)
│ │ │ - -- used 0.739844s (cpu); 0.504085s (thread); 0s (gc)
│ │ │ + -- used 0.852016s (cpu); 0.601959s (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 GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  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
│ │ │ @@ -214,15 +214,15 @@
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 25, and computed = 23
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 25, and computed = 23.  singular locus dimension appears to be = 3
│ │ │  
│ │ │  i11 : time regularInCodimension(1, S/J, MaxMinors=>25, UseOnlyFastCodim => true, Verbose=>true)
│ │ │ - -- used 0.482077s (cpu); 0.314107s (thread); 0s (gc)
│ │ │ + -- used 0.546003s (cpu); 0.373622s (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 RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ ├── ./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)
│ │ │ - -- 1.73127s elapsed
│ │ │ + -- 1.61205s 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)
│ │ │ - -- 1.19064s elapsed
│ │ │ + -- .971402s 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 0.00394593s (cpu); 0.00280101s (thread); 0s (gc)
│ │ │ + -- used 0.00400021s (cpu); 0.0034092s (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 0.00231368s (cpu); 0.00253938s (thread); 0s (gc)
│ │ │ + -- used 0.000452752s (cpu); 0.00325482s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 6.0604e-05s (cpu); 0.00243851s (thread); 0s (gc)
│ │ │ + -- used 0.000330497s (cpu); 0.00307954s (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 0.267277s (cpu); 0.158683s (thread); 0s (gc)
│ │ │ + -- used 0.320407s (cpu); 0.180761s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.0107026s (cpu); 0.0126789s (thread); 0s (gc)
│ │ │ + -- used 0.0148508s (cpu); 0.0160678s (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 0.514952s (cpu); 0.461544s (thread); 0s (gc)
│ │ │ + -- used 0.568394s (cpu); 0.505724s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.47881s (cpu); 1.27687s (thread); 0s (gc)
│ │ │ + -- used 1.41864s (cpu); 1.28351s (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 0.654392s (cpu); 0.499543s (thread); 0s (gc)
│ │ │ + -- used 0.813424s (cpu); 0.62154s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : time regularInCodimension(2, S)
│ │ │ - -- used 7.00197s (cpu); 5.27802s (thread); 0s (gc)
│ │ │ + -- used 8.1655s (cpu); 6.11118s (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 0.0199988s (cpu); 0.0198096s (thread); 0s (gc)
│ │ │ + -- used 0.0240005s (cpu); 0.0218344s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
│ │ │  
│ │ │  i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.182885s (cpu); 0.135147s (thread); 0s (gc)
│ │ │ + -- used 0.233636s (cpu); 0.161639s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
│ │ │  
│ │ │  i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.91949s (cpu); 0.572332s (thread); 0s (gc)
│ │ │ + -- used 1.2013s (cpu); 0.735407s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : time regularInCodimension(2, R)
│ │ │ - -- used 1.24103s (cpu); 0.866303s (thread); 0s (gc)
│ │ │ + -- used 1.59361s (cpu); 1.0549s (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 GRevLexSmallestTerm
│ │ │ @@ -386,15 +386,15 @@
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ -internalChooseMinor: Ch -- used 6.56865s (cpu); 4.94928s (thread); 0s (gc)
│ │ │ +internalChooseMinor: Ch -- used 8.60426s (cpu); 6.49853s (thread); 0s (gc)
│ │ │  oosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ @@ -430,15 +430,15 @@
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 328, and computed = 180
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 328, and computed = 180.  singular locus dimension appears to be = 1
│ │ │  
│ │ │  i17 : time regularInCodimension(2, S, Verbose=>true, MaxMinors=>30)
│ │ │ - -- used 1.27203s (cpu); 0.95852s (thread); 0s (gc)
│ │ │ + -- used 1.62811s (cpu); 1.24193s (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 LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ @@ -490,59 +490,59 @@
│ │ │  i18 : StrategyCurrent#Random = 0;
│ │ │  
│ │ │  i19 : StrategyCurrent#LexSmallest = 100;
│ │ │  
│ │ │  i20 : StrategyCurrent#LexSmallestTerm = 0;
│ │ │  
│ │ │  i21 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.30306s (cpu); 0.217176s (thread); 0s (gc)
│ │ │ + -- used 0.384636s (cpu); 0.263254s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
│ │ │  
│ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.113658s (cpu); 0.0792881s (thread); 0s (gc)
│ │ │ + -- used 0.145733s (cpu); 0.0894967s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
│ │ │  
│ │ │  i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.366044s (cpu); 0.272723s (thread); 0s (gc)
│ │ │ + -- used 0.481996s (cpu); 0.333493s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.73381s (cpu); 1.2512s (thread); 0s (gc)
│ │ │ + -- used 2.27333s (cpu); 1.64218s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 : StrategyCurrent#LexSmallest = 0;
│ │ │  
│ │ │  i26 : StrategyCurrent#LexSmallestTerm = 100;
│ │ │  
│ │ │  i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.32962s (cpu); 1.64458s (thread); 0s (gc)
│ │ │ + -- used 3.10047s (cpu); 2.1406s (thread); 0s (gc)
│ │ │  
│ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.38741s (cpu); 1.61396s (thread); 0s (gc)
│ │ │ + -- used 2.96386s (cpu); 2.0487s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
│ │ │  
│ │ │  i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.462956s (cpu); 0.370163s (thread); 0s (gc)
│ │ │ + -- used 0.514021s (cpu); 0.377889s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = true
│ │ │  
│ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.770202s (cpu); 0.604272s (thread); 0s (gc)
│ │ │ + -- used 0.898605s (cpu); 0.702043s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.07872s (cpu); 0.871235s (thread); 0s (gc)
│ │ │ + -- used 1.24569s (cpu); 1.01534s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
│ │ │  
│ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.85936s (cpu); 1.48073s (thread); 0s (gc)
│ │ │ + -- used 2.08977s (cpu); 1.64234s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = true
│ │ │  
│ │ │  i33 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html
│ │ │ @@ -620,71 +620,71 @@
│ │ │          
    
│ │ │            
    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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│ │ │              
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│ │ │              
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│ │ │              
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│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.177969s (cpu); 0.122004s (thread); 0s (gc)
│ │ │ + -- used 0.235066s (cpu); 0.172562s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
    
│ │ │              
    
│ │ │                
    i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.324762s (cpu); 0.212652s (thread); 0s (gc)
│ │ │ + -- used 0.377864s (cpu); 0.243329s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
    
│ │ │              
    
│ │ │                
    i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.505075s (cpu); 0.32834s (thread); 0s (gc)
│ │ │ + -- used 0.581966s (cpu); 0.368048s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
    
│ │ │              
    
│ │ │                
    i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.226825s (cpu); 0.185587s (thread); 0s (gc)
│ │ │ + -- used 0.300822s (cpu); 0.233416s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
    
│ │ │              
    
│ │ │                
    i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.386378s (cpu); 0.21271s (thread); 0s (gc)
│ │ │ + -- used 0.464162s (cpu); 0.248254s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
    
│ │ │              
    
│ │ │                
    i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.356033s (cpu); 0.24901s (thread); 0s (gc)
│ │ │ + -- used 0.405066s (cpu); 0.272011s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
    
│ │ │              
    
│ │ │                
    i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.299885s (cpu); 0.188636s (thread); 0s (gc)
│ │ │ + -- used 0.362747s (cpu); 0.240964s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
    
│ │ │              
    
│ │ │                
    i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 15.4766s (cpu); 10.5375s (thread); 0s (gc)
│ │ │ + -- used 19.5605s (cpu); 12.84s (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,
    
│ │ │ @@ -709,15 +709,15 @@
│ │ │          
    
│ │ │            
    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 0.383136s (cpu); 0.324174s (thread); 0s (gc)
│ │ │ + -- used 0.45547s (cpu); 0.380341s (thread); 0s (gc)
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ @@ -820,15 +820,15 @@
│ │ │  
│ │ │  o42 = true
    
│ │ │              
    
│ │ │                
    i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratGeometric))
│ │ │ - -- used 0.487214s (cpu); 0.426333s (thread); 0s (gc)
│ │ │ + -- used 0.760153s (cpu); 0.611235s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
    
│ │ │              
    
│ │ │                
    i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │ @@ -852,15 +852,15 @@
│ │ │  
│ │ │  o45 = false
    
│ │ │              
    
│ │ │                
    i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratRational))
│ │ │ - -- used 0.506683s (cpu); 0.379698s (thread); 0s (gc)
│ │ │ + -- used 0.521084s (cpu); 0.450882s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Other options may also be passed to the RandomPoints package via the PointOptions option.
    
│ │ │ @@ -868,69 +868,69 @@
│ │ │          
    
│ │ │            
    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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│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
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│ │ │            
│ │ │            
│ │ │              
│ │ │            
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│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 3.28944s (cpu); 3.0128s (thread); 0s (gc)
    
│ │ │ + -- used 4.35788s (cpu); 3.91067s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 0.905703s (cpu); 0.789473s (thread); 0s (gc)
│ │ │ + -- used 0.979538s (cpu); 0.824092s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
    
│ │ │              
    
│ │ │                
    i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 3.58264s (cpu); 3.36454s (thread); 0s (gc)
    
│ │ │ + -- used 3.58625s (cpu); 3.36342s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 2.38279s (cpu); 1.9883s (thread); 0s (gc)
    
│ │ │ + -- used 2.94573s (cpu); 2.43591s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 0.831174s (cpu); 0.71578s (thread); 0s (gc)
│ │ │ + -- used 1.01229s (cpu); 0.93891s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
    
│ │ │              
    
│ │ │                
    i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 2.6018s (cpu); 2.15961s (thread); 0s (gc)
    
│ │ │ + -- used 3.31185s (cpu); 2.72932s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 3.03003s (cpu); 2.62369s (thread); 0s (gc)
    
│ │ │ + -- used 4.04259s (cpu); 3.43991s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 9.15531s (cpu); 7.58445s (thread); 0s (gc)
│ │ │ + -- used 11.6132s (cpu); 9.62585s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
    
│ │ │              
    
│ │ │                
    i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 7.17456s (cpu); 5.85532s (thread); 0s (gc)
│ │ │ + -- used 8.96829s (cpu); 7.36393s (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 {}
│ │ │ │ @@ -486,44 +486,44 @@
│ │ │ │  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 0.177969s (cpu); 0.122004s (thread); 0s (gc)
│ │ │ │ + -- used 0.235066s (cpu); 0.172562s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = 2
│ │ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ │ - -- used 0.324762s (cpu); 0.212652s (thread); 0s (gc)
│ │ │ │ + -- used 0.377864s (cpu); 0.243329s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = 3
│ │ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ │ - -- used 0.505075s (cpu); 0.32834s (thread); 0s (gc)
│ │ │ │ + -- used 0.581966s (cpu); 0.368048s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 1
│ │ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ │ - -- used 0.226825s (cpu); 0.185587s (thread); 0s (gc)
│ │ │ │ + -- used 0.300822s (cpu); 0.233416s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = 2
│ │ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ │ - -- used 0.386378s (cpu); 0.21271s (thread); 0s (gc)
│ │ │ │ + -- used 0.464162s (cpu); 0.248254s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o32 = 3
│ │ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J,
│ │ │ │  Strategy=>GRevLexSmallestTerm))
│ │ │ │ - -- used 0.356033s (cpu); 0.24901s (thread); 0s (gc)
│ │ │ │ + -- used 0.405066s (cpu); 0.272011s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o33 = 3
│ │ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ │ - -- used 0.299885s (cpu); 0.188636s (thread); 0s (gc)
│ │ │ │ + -- used 0.362747s (cpu); 0.240964s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o34 = 3
│ │ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ │ - -- used 15.4766s (cpu); 10.5375s (thread); 0s (gc)
│ │ │ │ + -- used 19.5605s (cpu); 12.84s (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
│ │ │ │ @@ -544,15 +544,15 @@
│ │ │ │  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 0.383136s (cpu); 0.324174s (thread); 0s (gc)
│ │ │ │ + -- used 0.45547s (cpu); 0.380341s (thread); 0s (gc)
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │ @@ -633,15 +633,15 @@
│ │ │ │  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 0.487214s (cpu); 0.426333s (thread); 0s (gc)
│ │ │ │ + -- used 0.760153s (cpu); 0.611235s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o43 = 2
│ │ │ │  i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that
│ │ │ │  value
│ │ │ │  
│ │ │ │  o44 = OptionTable{DecompositionStrategy => Decompose}
│ │ │ │                    DimensionFunction => dim
│ │ │ │ @@ -655,58 +655,58 @@
│ │ │ │  
│ │ │ │  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 0.506683s (cpu); 0.379698s (thread); 0s (gc)
│ │ │ │ + -- used 0.521084s (cpu); 0.450882s (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 3.28944s (cpu); 3.0128s (thread); 0s (gc)
│ │ │ │ + -- used 4.35788s (cpu); 3.91067s (thread); 0s (gc)
│ │ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultNonRandom)
│ │ │ │ - -- used 0.905703s (cpu); 0.789473s (thread); 0s (gc)
│ │ │ │ + -- used 0.979538s (cpu); 0.824092s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o48 = true
│ │ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ │ - -- used 3.58264s (cpu); 3.36454s (thread); 0s (gc)
│ │ │ │ + -- used 3.58625s (cpu); 3.36342s (thread); 0s (gc)
│ │ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallest)
│ │ │ │ - -- used 2.38279s (cpu); 1.9883s (thread); 0s (gc)
│ │ │ │ + -- used 2.94573s (cpu); 2.43591s (thread); 0s (gc)
│ │ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallestTerm)
│ │ │ │ - -- used 0.831174s (cpu); 0.71578s (thread); 0s (gc)
│ │ │ │ + -- used 1.01229s (cpu); 0.93891s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o51 = true
│ │ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallest)
│ │ │ │ - -- used 2.6018s (cpu); 2.15961s (thread); 0s (gc)
│ │ │ │ + -- used 3.31185s (cpu); 2.72932s (thread); 0s (gc)
│ │ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallestTerm)
│ │ │ │ - -- used 3.03003s (cpu); 2.62369s (thread); 0s (gc)
│ │ │ │ + -- used 4.04259s (cpu); 3.43991s (thread); 0s (gc)
│ │ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ │ - -- used 9.15531s (cpu); 7.58445s (thread); 0s (gc)
│ │ │ │ + -- used 11.6132s (cpu); 9.62585s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o54 = true
│ │ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultWithPoints)
│ │ │ │ - -- used 7.17456s (cpu); 5.85532s (thread); 0s (gc)
│ │ │ │ + -- used 8.96829s (cpu); 7.36393s (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
│ │ │ @@ -81,23 +81,23 @@
│ │ │          
    
│ │ │            
    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 0.954389s (cpu); 0.635206s (thread); 0s (gc)
│ │ │ + -- used 1.18993s (cpu); 0.756142s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │                
    i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 11.1794s (cpu); 8.10432s (thread); 0s (gc)
    
│ │ │ + -- used 13.3074s (cpu); 9.34486s (thread); 0s (gc)
│ │ │              
    
│ │ │          
    
│ │ │            
    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.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -172,29 +172,29 @@
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 49, and computed = 39
│ │ │  regularInCodimension:  singularLocus dimension verified by isCodimAtLeast
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 2
│ │ │ -regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.43458s (cpu); 1.01606s (thread); 0s (gc)
│ │ │ +regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.69305s (cpu); 1.1984s (thread); 0s (gc)
│ │ │  d = 39.  singular locus dimension appears to be = 2
│ │ │  
│ │ │  o6 = true
│ │ │              
│ │ │            
│ │ │          
    
│ │ │          
    
│ │ │            
    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 0.175899s (cpu); 0.12452s (thread); 0s (gc)
│ │ │ + -- used 0.221683s (cpu); 0.15609s (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 GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -219,15 +219,15 @@
│ │ │          
    
│ │ │            
    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 0.15707s (cpu); 0.106723s (thread); 0s (gc)
│ │ │ + -- used 0.193756s (cpu); 0.133139s (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
│ │ │ @@ -252,15 +252,15 @@
│ │ │          
    
│ │ │            
    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 0.627779s (cpu); 0.442058s (thread); 0s (gc)
│ │ │ + -- used 0.95771s (cpu); 0.637674s (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 Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 3, and computed = 3
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -291,15 +291,15 @@
│ │ │          
    
│ │ │            
    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 0.739844s (cpu); 0.504085s (thread); 0s (gc)
│ │ │ + -- used 0.852016s (cpu); 0.601959s (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 GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  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
│ │ │ @@ -348,15 +348,15 @@
│ │ │          
    
│ │ │            
    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 0.482077s (cpu); 0.314107s (thread); 0s (gc)
│ │ │ + -- used 0.546003s (cpu); 0.373622s (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 RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ ├── 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 0.954389s (cpu); 0.635206s (thread); 0s (gc)
│ │ │ │ + -- used 1.18993s (cpu); 0.756142s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ │ - -- used 11.1794s (cpu); 8.10432s (thread); 0s (gc)
│ │ │ │ + -- used 13.3074s (cpu); 9.34486s (thread); 0s (gc)
│ │ │ │  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
│ │ │ │ @@ -121,22 +121,22 @@
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 49, and computed = 39
│ │ │ │  regularInCodimension:  singularLocus dimension verified by isCodimAtLeast
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 2
│ │ │ │  regularInCodimension:  Loop completed, submatrices considered = 49, and compute
│ │ │ │ --- used 1.43458s (cpu); 1.01606s (thread); 0s (gc)
│ │ │ │ +-- used 1.69305s (cpu); 1.1984s (thread); 0s (gc)
│ │ │ │  d = 39.  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 0.175899s (cpu); 0.12452s (thread); 0s (gc)
│ │ │ │ + -- used 0.221683s (cpu); 0.15609s (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 GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │ @@ -159,15 +159,15 @@
│ │ │ │  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 0.15707s (cpu); 0.106723s (thread); 0s (gc)
│ │ │ │ + -- used 0.193756s (cpu); 0.133139s (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
│ │ │ │ @@ -197,15 +197,15 @@
│ │ │ │  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 0.627779s (cpu); 0.442058s (thread); 0s (gc)
│ │ │ │ + -- used 0.95771s (cpu); 0.637674s (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 Random
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │ @@ -243,15 +243,15 @@
│ │ │ │  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 0.739844s (cpu); 0.504085s (thread); 0s (gc)
│ │ │ │ + -- used 0.852016s (cpu); 0.601959s (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 GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 2, and computed = 2
│ │ │ │ @@ -308,15 +308,15 @@
│ │ │ │  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 0.482077s (cpu); 0.314107s (thread); 0s (gc)
│ │ │ │ + -- used 0.546003s (cpu); 0.373622s (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 RandomNonZero
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Strategy__Default.html
│ │ │ @@ -68,15 +68,15 @@
│ │ │              
    		
│ │ │                
    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)
│ │ │ - -- 1.73127s elapsed
│ │ │ + -- 1.61205s 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
      • 
│ │ │ @@ -122,15 +122,15 @@
│ │ │            
    • 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)
│ │ │ - -- 1.19064s elapsed
│ │ │ + -- .971402s 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)
│ │ │ │ - -- 1.73127s elapsed
│ │ │ │ + -- 1.61205s 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)
│ │ │ │ - -- 1.19064s elapsed
│ │ │ │ + -- .971402s 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-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/FastMinors.m2:1993:0.
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_is__Codim__At__Least.html
│ │ │ @@ -114,15 +114,15 @@
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i7 : time isCodimAtLeast(3, J)
│ │ │ - -- used 0.00394593s (cpu); 0.00280101s (thread); 0s (gc)
│ │ │ + -- used 0.00400021s (cpu); 0.0034092s (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.
    
│ │ │ @@ -136,24 +136,24 @@
│ │ │  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 0.00231368s (cpu); 0.00253938s (thread); 0s (gc)
│ │ │ + -- used 0.000452752s (cpu); 0.00325482s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 6.0604e-05s (cpu); 0.00243851s (thread); 0s (gc)
│ │ │ + -- used 0.000330497s (cpu); 0.00307954s (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 0.00394593s (cpu); 0.00280101s (thread); 0s (gc)
│ │ │ │ + -- used 0.00400021s (cpu); 0.0034092s (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 0.00231368s (cpu); 0.00253938s (thread); 0s (gc)
│ │ │ │ + -- used 0.000452752s (cpu); 0.00325482s (thread); 0s (gc)
│ │ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ │ - -- used 6.0604e-05s (cpu); 0.00243851s (thread); 0s (gc)
│ │ │ │ + -- used 0.000330497s (cpu); 0.00307954s (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
│ │ │ @@ -99,23 +99,23 @@
│ │ │  
│ │ │  o3 = 2
    
│ │ │              
    
│ │ │                
    i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ - -- used 0.267277s (cpu); 0.158683s (thread); 0s (gc)
│ │ │ + -- used 0.320407s (cpu); 0.180761s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
    
│ │ │                
    i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.0107026s (cpu); 0.0126789s (thread); 0s (gc)
│ │ │ + -- used 0.0148508s (cpu); 0.0160678s (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 0.267277s (cpu); 0.158683s (thread); 0s (gc)
│ │ │ │ + -- used 0.320407s (cpu); 0.180761s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ │ - -- used 0.0107026s (cpu); 0.0126789s (thread); 0s (gc)
│ │ │ │ + -- used 0.0148508s (cpu); 0.0160678s (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
│ │ │ @@ -92,23 +92,23 @@
│ │ │               6      7
│ │ │  o2 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ - -- used 0.514952s (cpu); 0.461544s (thread); 0s (gc)
│ │ │ + -- used 0.568394s (cpu); 0.505724s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.47881s (cpu); 1.27687s (thread); 0s (gc)
│ │ │ + -- used 1.41864s (cpu); 1.28351s (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 0.514952s (cpu); 0.461544s (thread); 0s (gc)
│ │ │ │ + -- used 0.568394s (cpu); 0.505724s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ │ - -- used 1.47881s (cpu); 1.27687s (thread); 0s (gc)
│ │ │ │ + -- used 1.41864s (cpu); 1.28351s (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
│ │ │ @@ -131,23 +131,23 @@
│ │ │  
│ │ │  o7 = 3
    
│ │ │              
    
│ │ │                
    i8 : time regularInCodimension(1, S)
│ │ │ - -- used 0.654392s (cpu); 0.499543s (thread); 0s (gc)
│ │ │ + -- used 0.813424s (cpu); 0.62154s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : time regularInCodimension(2, S)
│ │ │ - -- used 7.00197s (cpu); 5.27802s (thread); 0s (gc)
    
│ │ │ + -- used 8.1655s (cpu); 6.11118s (thread); 0s (gc)
│ │ │              
    
│ │ │          
    
│ │ │            
    There are numerous examples where regularInCodimension is several orders of magnitude faster that calls of dim singularLocus.
    
│ │ │          
    
│ │ │          
    
│ │ │ @@ -165,39 +165,39 @@
│ │ │  
│ │ │  o11 = 2
    
│ │ │              
    
│ │ │                
    i12 : time (dim singularLocus (R))
│ │ │ - -- used 0.0199988s (cpu); 0.0198096s (thread); 0s (gc)
│ │ │ + -- used 0.0240005s (cpu); 0.0218344s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
    
│ │ │              
    
│ │ │                
    i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.182885s (cpu); 0.135147s (thread); 0s (gc)
│ │ │ + -- used 0.233636s (cpu); 0.161639s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
    
│ │ │              
    
│ │ │                
    i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.91949s (cpu); 0.572332s (thread); 0s (gc)
│ │ │ + -- used 1.2013s (cpu); 0.735407s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
    
│ │ │              
    
│ │ │                
    i15 : time regularInCodimension(2, R)
│ │ │ - -- used 1.24103s (cpu); 0.866303s (thread); 0s (gc)
│ │ │ + -- used 1.59361s (cpu); 1.0549s (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.
    
│ │ │ @@ -537,15 +537,15 @@
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ -internalChooseMinor: Ch -- used 6.56865s (cpu); 4.94928s (thread); 0s (gc)
│ │ │ +internalChooseMinor: Ch -- used 8.60426s (cpu); 6.49853s (thread); 0s (gc)
│ │ │  oosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ @@ -589,15 +589,15 @@
│ │ │          
    
│ │ │            
    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 1.27203s (cpu); 0.95852s (thread); 0s (gc)
│ │ │ + -- used 1.62811s (cpu); 1.24193s (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 LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ @@ -666,39 +666,39 @@
│ │ │              
    		
│ │ │                
    i20 : StrategyCurrent#LexSmallestTerm = 0;
    
│ │ │              
    
│ │ │                
    i21 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.30306s (cpu); 0.217176s (thread); 0s (gc)
│ │ │ + -- used 0.384636s (cpu); 0.263254s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
    
│ │ │              
    
│ │ │                
    i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.113658s (cpu); 0.0792881s (thread); 0s (gc)
│ │ │ + -- used 0.145733s (cpu); 0.0894967s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
    
│ │ │              
    
│ │ │                
    i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.366044s (cpu); 0.272723s (thread); 0s (gc)
│ │ │ + -- used 0.481996s (cpu); 0.333493s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.73381s (cpu); 1.2512s (thread); 0s (gc)
│ │ │ + -- used 2.27333s (cpu); 1.64218s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
    
│ │ │              
    
│ │ │                
    i25 : StrategyCurrent#LexSmallest = 0;
    
│ │ │ @@ -708,53 +708,53 @@
│ │ │                  		
│ │ │                
    i26 : StrategyCurrent#LexSmallestTerm = 100;
    
│ │ │              
    
│ │ │                
    i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.32962s (cpu); 1.64458s (thread); 0s (gc)
    
│ │ │ + -- used 3.10047s (cpu); 2.1406s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.38741s (cpu); 1.61396s (thread); 0s (gc)
│ │ │ + -- used 2.96386s (cpu); 2.0487s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
    
│ │ │              
    
│ │ │                
    i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.462956s (cpu); 0.370163s (thread); 0s (gc)
│ │ │ + -- used 0.514021s (cpu); 0.377889s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = true
    
│ │ │              
    
│ │ │                
    i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.770202s (cpu); 0.604272s (thread); 0s (gc)
│ │ │ + -- used 0.898605s (cpu); 0.702043s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
    
│ │ │              
    
│ │ │                
    i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.07872s (cpu); 0.871235s (thread); 0s (gc)
│ │ │ + -- used 1.24569s (cpu); 1.01534s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
    
│ │ │              
    
│ │ │                
    i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.85936s (cpu); 1.48073s (thread); 0s (gc)
│ │ │ + -- used 2.08977s (cpu); 1.64234s (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 0.654392s (cpu); 0.499543s (thread); 0s (gc)
│ │ │ │ + -- used 0.813424s (cpu); 0.62154s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : time regularInCodimension(2, S)
│ │ │ │ - -- used 7.00197s (cpu); 5.27802s (thread); 0s (gc)
│ │ │ │ + -- used 8.1655s (cpu); 6.11118s (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 0.0199988s (cpu); 0.0198096s (thread); 0s (gc)
│ │ │ │ + -- used 0.0240005s (cpu); 0.0218344s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 = -1
│ │ │ │  i13 : time regularInCodimension(2, R)
│ │ │ │ - -- used 0.182885s (cpu); 0.135147s (thread); 0s (gc)
│ │ │ │ + -- used 0.233636s (cpu); 0.161639s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = true
│ │ │ │  i14 : time regularInCodimension(2, R)
│ │ │ │ - -- used 0.91949s (cpu); 0.572332s (thread); 0s (gc)
│ │ │ │ + -- used 1.2013s (cpu); 0.735407s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : time regularInCodimension(2, R)
│ │ │ │ - -- used 1.24103s (cpu); 0.866303s (thread); 0s (gc)
│ │ │ │ + -- used 1.59361s (cpu); 1.0549s (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)
│ │ │ │ @@ -461,15 +461,15 @@
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │ -internalChooseMinor: Ch -- used 6.56865s (cpu); 4.94928s (thread); 0s (gc)
│ │ │ │ +internalChooseMinor: Ch -- used 8.60426s (cpu); 6.49853s (thread); 0s (gc)
│ │ │ │  oosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │ @@ -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 1.27203s (cpu); 0.95852s (thread); 0s (gc)
│ │ │ │ + -- used 1.62811s (cpu); 1.24193s (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 LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │ @@ -590,51 +590,51 @@
│ │ │ │  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 0.30306s (cpu); 0.217176s (thread); 0s (gc)
│ │ │ │ + -- used 0.384636s (cpu); 0.263254s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = true
│ │ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.113658s (cpu); 0.0792881s (thread); 0s (gc)
│ │ │ │ + -- used 0.145733s (cpu); 0.0894967s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = true
│ │ │ │  i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.366044s (cpu); 0.272723s (thread); 0s (gc)
│ │ │ │ + -- used 0.481996s (cpu); 0.333493s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 1.73381s (cpu); 1.2512s (thread); 0s (gc)
│ │ │ │ + -- used 2.27333s (cpu); 1.64218s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  i25 : StrategyCurrent#LexSmallest = 0;
│ │ │ │  i26 : StrategyCurrent#LexSmallestTerm = 100;
│ │ │ │  i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 2.32962s (cpu); 1.64458s (thread); 0s (gc)
│ │ │ │ + -- used 3.10047s (cpu); 2.1406s (thread); 0s (gc)
│ │ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 2.38741s (cpu); 1.61396s (thread); 0s (gc)
│ │ │ │ + -- used 2.96386s (cpu); 2.0487s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = true
│ │ │ │  i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.462956s (cpu); 0.370163s (thread); 0s (gc)
│ │ │ │ + -- used 0.514021s (cpu); 0.377889s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = true
│ │ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.770202s (cpu); 0.604272s (thread); 0s (gc)
│ │ │ │ + -- used 0.898605s (cpu); 0.702043s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = true
│ │ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ │ - -- used 1.07872s (cpu); 0.871235s (thread); 0s (gc)
│ │ │ │ + -- used 1.24569s (cpu); 1.01534s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = true
│ │ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ │ - -- used 1.85936s (cpu); 1.48073s (thread); 0s (gc)
│ │ │ │ + -- used 2.08977s (cpu); 1.64234s (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 0.00282615s (cpu); 0.00282256s (thread); 0s (gc)
│ │ │ + -- used 0.00317807s (cpu); 0.00317454s (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 0.00631438s (cpu); 0.00631331s (thread); 0s (gc)
│ │ │ + -- used 0.00692976s (cpu); 0.00693301s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : I==J
│ │ │  
│ │ │  o17 = true
│ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html
│ │ │ @@ -202,26 +202,26 @@
│ │ │                8      8
│ │ │  o14 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i15 : time I=co1Fitting(K3)
│ │ │ - -- used 0.00282615s (cpu); 0.00282256s (thread); 0s (gc)
│ │ │ + -- used 0.00317807s (cpu); 0.00317454s (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 0.00631438s (cpu); 0.00631331s (thread); 0s (gc)
│ │ │ + -- used 0.00692976s (cpu); 0.00693301s (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 0.00282615s (cpu); 0.00282256s (thread); 0s (gc)
│ │ │ │ + -- used 0.00317807s (cpu); 0.00317454s (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 0.00631438s (cpu); 0.00631331s (thread); 0s (gc)
│ │ │ │ + -- used 0.00692976s (cpu); 0.00693301s (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 => 0x7f7f285265f0}
│ │ │ +o2 = int32{Address => 0x7fd66b07f530}
│ │ │  
│ │ │  i3 : address x
│ │ │  
│ │ │ -o3 = 0x7f7f285265f0
│ │ │ +o3 = 0x7fd66b07f530
│ │ │  
│ │ │  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 = {0x7f7f2854a030, 0x7f7f2854a020, 0x7f7f2854a010}
│ │ │ +o3 = {0x7fd66b12bbe0, 0x7fd66b12bbd0, 0x7fd66b12bbb0}
│ │ │  
│ │ │  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 = {0x7f7f28567e20, 0x7f7f28567e10, 0x7f7f28567e00}
│ │ │ +o2 = {0x7fd66b12b9b0, 0x7fd66b12b9a0, 0x7fd66b12b990}
│ │ │  
│ │ │  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 = 0x7f7f1f69e700
│ │ │ +o1 = 0x7fd66dbe97b0
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7f7f1f69e700
│ │ │ +o2 = 0x7fd66dbe97b0
│ │ │  
│ │ │  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 = 0x7f7f28526ca0
│ │ │ +o2 = 0x7fd66b0e7d50
│ │ │  
│ │ │  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 = 0x7f7f2854aa70
│ │ │ +o1 = 0x7fd66b0e7ee0
│ │ │  
│ │ │  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" => 6.92598e-310}
│ │ │ +o2 = HashTable{"bar" => 6.94444e-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 => 0x7f7f285264f0}
│ │ │ +o2 = int32{Address => 0x7fd66b0e7f90}
│ │ │  
│ │ │  i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7f7f285264f0
│ │ │ +o3 = 0x7fd66b0e7f90
│ │ │  
│ │ │  o3 : Pointer
│ │ │  
│ │ │  i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7f7f285264f5
│ │ │ +o4 = 0x7fd66b0e7f95
│ │ │  
│ │ │  o4 : Pointer
│ │ │  
│ │ │  i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7f7f285264ed
│ │ │ +o5 = 0x7fd66b0e7f8d
│ │ │  
│ │ │  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{0x7f7f2f6e4550, mpfr}
│ │ │ +o2 = SharedLibrary{0x7fd67f288550, 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 = 0x7f7f285268c0
│ │ │ +o2 = 0x7fd66b0e79a0
│ │ │  
│ │ │  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 = 0x563c19575b40
│ │ │ +o1 = 0x55a9e9e1fb40
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : address int 5
│ │ │  
│ │ │ -o2 = 0x7f7f28526a20
│ │ │ +o2 = 0x7fd66b0e7fd0
│ │ │  
│ │ │  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 = 0x7f443c06a4f0
│ │ │ +o17 = 0x7f7cc006a4f0
│ │ │  
│ │ │  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 = 0x7f7f2b093240
│ │ │ +o1 = 0x7fd67aa7e490
│ │ │  
│ │ │  o1 : ForeignObject of type void*
│ │ │  
│ │ │  i2 : ptr = getMemory(8, Atomic => true)
│ │ │  
│ │ │ -o2 = 0x7f7f285260a0
│ │ │ +o2 = 0x7fd66b0e7dc0
│ │ │  
│ │ │  o2 : ForeignObject of type void*
│ │ │  
│ │ │  i3 : ptr = getMemory int
│ │ │  
│ │ │ -o3 = 0x7f7f2854afc0
│ │ │ +o3 = 0x7fd66b0e7ca0
│ │ │  
│ │ │  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 0x7f7f1407f910
│ │ │ -freeing memory at 0x7f7f1407f930
│ │ │ -freeing memory at 0x7f7f1407f950
│ │ │ -freeing memory at 0x7f7f1407f990
│ │ │ -freeing memory at 0x7f7f1407f250
│ │ │ -freeing memory at 0x7f7f1407f230
│ │ │ -freeing memory at 0x7f7f1407f9b0
│ │ │ -freeing memory at 0x7f7f1407f970
│ │ │ -freeing memory at 0x7f7f1407f8f0
│ │ │ +freeing memory at 0x7fd65407f8f0
│ │ │ +freeing memory at 0x7fd65407f910
│ │ │ +freeing memory at 0x7fd65407f930
│ │ │ +freeing memory at 0x7fd65407f950
│ │ │ +freeing memory at 0x7fd65407f230
│ │ │ +freeing memory at 0x7fd65407f990
│ │ │ +freeing memory at 0x7fd65407f9b0
│ │ │ +freeing memory at 0x7fd65407f250
│ │ │ +freeing memory at 0x7fd65407f970
│ │ │  
│ │ │  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 = 0x7f7f2854ad20
│ │ │ +o5 = 0x7fd66b0e7220
│ │ │  
│ │ │  o5 : ForeignObject of type void*
│ │ │  
│ │ │  i6 : value x
│ │ │  
│ │ │ -o6 = 0x7f7f2854ad20
│ │ │ +o6 = 0x7fd66b0e7220
│ │ │  
│ │ │  o6 : Pointer
│ │ │  
│ │ │  i7 : x = charstar "Hello, world!"
│ │ │  
│ │ │  o7 = Hello, world!
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Object.html
│ │ │ @@ -64,27 +64,27 @@
│ │ │  o1 : ForeignObject of type int32
    
│ │ │              
    
│ │ │                
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f7f285265f0}
    
│ │ │ +o2 = int32{Address => 0x7fd66b07f530}
│ │ │              
    
│ │ │          
    
│ │ │            
    To get this, use address.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : address x
│ │ │  
│ │ │ -o3 = 0x7f7f285265f0
│ │ │ +o3 = 0x7fd66b07f530
│ │ │  
│ │ │  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 => 0x7f7f285265f0}
│ │ │ │ +o2 = int32{Address => 0x7fd66b07f530}
│ │ │ │  To get this, use _a_d_d_r_e_s_s.
│ │ │ │  i3 : address x
│ │ │ │  
│ │ │ │ -o3 = 0x7f7f285265f0
│ │ │ │ +o3 = 0x7fd66b07f530
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -74,15 +74,15 @@
│ │ │  o2 : ForeignObject of type char**
    
│ │ │              
    
│ │ │                
    i3 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o3 = {0x7f7f2854a030, 0x7f7f2854a020, 0x7f7f2854a010}
│ │ │ +o3 = {0x7fd66b12bbe0, 0x7fd66b12bbd0, 0x7fd66b12bbb0}
│ │ │  
│ │ │  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 = {0x7f7f2854a030, 0x7f7f2854a020, 0x7f7f2854a010}
│ │ │ │ +o3 = {0x7fd66b12bbe0, 0x7fd66b12bbd0, 0x7fd66b12bbb0}
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -82,15 +82,15 @@
│ │ │  o1 : ForeignObject of type char**
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o2 = {0x7f7f28567e20, 0x7f7f28567e10, 0x7f7f28567e00}
│ │ │ +o2 = {0x7fd66b12b9b0, 0x7fd66b12b9a0, 0x7fd66b12b990}
│ │ │  
│ │ │  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 = {0x7f7f28567e20, 0x7f7f28567e10, 0x7f7f28567e00}
│ │ │ │ +o2 = {0x7fd66b12b9b0, 0x7fd66b12b9a0, 0x7fd66b12b990}
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -73,24 +73,24 @@
│ │ │            
    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 = 0x7f7f1f69e700
│ │ │ +o1 = 0x7fd66dbe97b0
│ │ │  
│ │ │  o1 : Pointer
    
│ │ │              
    
│ │ │                
    i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7f7f1f69e700
│ │ │ +o2 = 0x7fd66dbe97b0
│ │ │  
│ │ │  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 = 0x7f7f1f69e700
│ │ │ │ +o1 = 0x7fd66dbe97b0
│ │ │ │  
│ │ │ │  o1 : Pointer
│ │ │ │  i2 : voidstar ptr
│ │ │ │  
│ │ │ │ -o2 = 0x7f7f1f69e700
│ │ │ │ +o2 = 0x7fd66dbe97b0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -82,15 +82,15 @@
│ │ │  o1 : ForeignObject of type int32
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7f7f28526ca0
│ │ │ +o2 = 0x7fd66b0e7d50
│ │ │  
│ │ │  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 = 0x7f7f28526ca0
│ │ │ │ +o2 = 0x7fd66b0e7d50
│ │ │ │  
│ │ │ │  o2 : Pointer
│ │ │ │  i3 : int ptr
│ │ │ │  
│ │ │ │  o3 = 5
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp_st_spvoidstar.html
│ │ │ @@ -73,15 +73,15 @@
│ │ │            
    This is syntactic sugar for T value ptr (see ForeignType Pointer) for dereferencing pointers.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : ptr = voidstar address int 5
│ │ │  
│ │ │ -o1 = 0x7f7f2854aa70
│ │ │ +o1 = 0x7fd66b0e7ee0
│ │ │  
│ │ │  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 = 0x7f7f2854aa70
│ │ │ │ +o1 = 0x7fd66b0e7ee0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -82,15 +82,15 @@
│ │ │  o1 : ForeignUnionType
    
│ │ │              
    
│ │ │                
    i2 : myunion 27
│ │ │  
│ │ │ -o2 = HashTable{"bar" => 6.92598e-310}
│ │ │ +o2 = HashTable{"bar" => 6.94444e-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" => 6.92598e-310}
│ │ │ │ +o2 = HashTable{"bar" => 6.94444e-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
│ │ │ @@ -64,50 +64,50 @@
│ │ │  o1 : ForeignObject of type int32
│ │ │              
    
│ │ │                
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f7f285264f0}
    
│ │ │ +o2 = int32{Address => 0x7fd66b0e7f90}
│ │ │              
    
│ │ │          
    
│ │ │            
    These pointers can be accessed using address.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7f7f285264f0
│ │ │ +o3 = 0x7fd66b0e7f90
│ │ │  
│ │ │  o3 : Pointer
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Simple arithmetic can be performed on pointers.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7f7f285264f5
│ │ │ +o4 = 0x7fd66b0e7f95
│ │ │  
│ │ │  o4 : Pointer
    
│ │ │              
    
│ │ │                
    i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7f7f285264ed
│ │ │ +o5 = 0x7fd66b0e7f8d
│ │ │  
│ │ │  o5 : Pointer
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,30 +10,30 @@
│ │ │ │  i1 : x = int 20
│ │ │ │  
│ │ │ │  o1 = 20
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : peek x
│ │ │ │  
│ │ │ │ -o2 = int32{Address => 0x7f7f285264f0}
│ │ │ │ +o2 = int32{Address => 0x7fd66b0e7f90}
│ │ │ │  These pointers can be accessed using _a_d_d_r_e_s_s.
│ │ │ │  i3 : ptr = address x
│ │ │ │  
│ │ │ │ -o3 = 0x7f7f285264f0
│ │ │ │ +o3 = 0x7fd66b0e7f90
│ │ │ │  
│ │ │ │  o3 : Pointer
│ │ │ │  Simple arithmetic can be performed on pointers.
│ │ │ │  i4 : ptr + 5
│ │ │ │  
│ │ │ │ -o4 = 0x7f7f285264f5
│ │ │ │ +o4 = 0x7fd66b0e7f95
│ │ │ │  
│ │ │ │  o4 : Pointer
│ │ │ │  i5 : ptr - 3
│ │ │ │  
│ │ │ │ -o5 = 0x7f7f285264ed
│ │ │ │ +o5 = 0x7fd66b0e7f8d
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -64,15 +64,15 @@
│ │ │  o1 : SharedLibrary
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : peek mpfr
│ │ │  
│ │ │ -o2 = SharedLibrary{0x7f7f2f6e4550, mpfr}
    
│ │ │ +o2 = SharedLibrary{0x7fd67f288550, mpfr}
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    Menu
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : address int
│ │ │  
│ │ │ -o1 = 0x563c19575b40
│ │ │ +o1 = 0x55a9e9e1fb40
│ │ │  
│ │ │  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 = 0x7f7f28526a20
│ │ │ +o2 = 0x7fd66b0e7fd0
│ │ │  
│ │ │  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 = 0x563c19575b40
│ │ │ │ +o1 = 0x55a9e9e1fb40
│ │ │ │  
│ │ │ │  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 = 0x7f7f28526a20
│ │ │ │ +o2 = 0x7fd66b0e7fd0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -232,15 +232,15 @@
│ │ │  o16 : ForeignFunction
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : x = malloc 8
│ │ │  
│ │ │ -o17 = 0x7f443c06a4f0
│ │ │ +o17 = 0x7f7cc006a4f0
│ │ │  
│ │ │  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 = 0x7f443c06a4f0
│ │ │ │ +o17 = 0x7f7cc006a4f0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -77,43 +77,43 @@
│ │ │            
    Allocate n bytes of memory using the GC garbage collector.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : ptr = getMemory 8
│ │ │  
│ │ │ -o1 = 0x7f7f2b093240
│ │ │ +o1 = 0x7fd67aa7e490
│ │ │  
│ │ │  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 = 0x7f7f285260a0
│ │ │ +o2 = 0x7fd66b0e7dc0
│ │ │  
│ │ │  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 = 0x7f7f2854afc0
│ │ │ +o3 = 0x7fd66b0e7ca0
│ │ │  
│ │ │  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 = 0x7f7f2b093240
│ │ │ │ +o1 = 0x7fd67aa7e490
│ │ │ │  
│ │ │ │  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 = 0x7f7f285260a0
│ │ │ │ +o2 = 0x7fd66b0e7dc0
│ │ │ │  
│ │ │ │  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 = 0x7f7f2854afc0
│ │ │ │ +o3 = 0x7fd66b0e7ca0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -100,23 +100,23 @@
│ │ │              
│ │ │                
    i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer))
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : collectGarbage()
│ │ │ -freeing memory at 0x7f7f1407f910
│ │ │ -freeing memory at 0x7f7f1407f930
│ │ │ -freeing memory at 0x7f7f1407f950
│ │ │ -freeing memory at 0x7f7f1407f990
│ │ │ -freeing memory at 0x7f7f1407f250
│ │ │ -freeing memory at 0x7f7f1407f230
│ │ │ -freeing memory at 0x7f7f1407f9b0
│ │ │ -freeing memory at 0x7f7f1407f970
│ │ │ -freeing memory at 0x7f7f1407f8f0
    
│ │ │ +freeing memory at 0x7fd65407f8f0
│ │ │ +freeing memory at 0x7fd65407f910
│ │ │ +freeing memory at 0x7fd65407f930
│ │ │ +freeing memory at 0x7fd65407f950
│ │ │ +freeing memory at 0x7fd65407f230
│ │ │ +freeing memory at 0x7fd65407f990
│ │ │ +freeing memory at 0x7fd65407f9b0
│ │ │ +freeing memory at 0x7fd65407f250
│ │ │ +freeing memory at 0x7fd65407f970
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : x = voidstar address int 5
│ │ │  
│ │ │ -o5 = 0x7f7f2854ad20
│ │ │ +o5 = 0x7fd66b0e7220
│ │ │  
│ │ │  o5 : ForeignObject of type void*
    
│ │ │              
    
│ │ │                
    i6 : value x
│ │ │  
│ │ │ -o6 = 0x7f7f2854ad20
│ │ │ +o6 = 0x7fd66b0e7220
│ │ │  
│ │ │  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 = 0x7f7f2854ad20
│ │ │ │ +o5 = 0x7fd66b0e7220
│ │ │ │  
│ │ │ │  o5 : ForeignObject of type void*
│ │ │ │  i6 : value x
│ │ │ │  
│ │ │ │ -o6 = 0x7f7f2854ad20
│ │ │ │ +o6 = 0x7fd66b0e7220
│ │ │ │  
│ │ │ │  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/M2-15935-0/0
│ │ │ +o2 = /tmp/M2-20955-0/0
│ │ │  
│ │ │  i3 : F = openOut(s)
│ │ │  
│ │ │ -o3 = /tmp/M2-15935-0/0
│ │ │ +o3 = /tmp/M2-20955-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : putMatrix(F,A)
│ │ │  
│ │ │  i5 : close(F)
│ │ │  
│ │ │ -o5 = /tmp/M2-15935-0/0
│ │ │ +o5 = /tmp/M2-20955-0/0
│ │ │  
│ │ │  o5 : File
│ │ │  
│ │ │  i6 : getMatrix(s)
│ │ │  
│ │ │  o6 = | 1 1 1 1 |
│ │ │       | 1 2 3 4 |
│ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/html/_put__Matrix.html
│ │ │ @@ -79,36 +79,36 @@
│ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : s = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-15935-0/0
    
│ │ │ +o2 = /tmp/M2-20955-0/0
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : F = openOut(s)
│ │ │  
│ │ │ -o3 = /tmp/M2-15935-0/0
│ │ │ +o3 = /tmp/M2-20955-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : putMatrix(F,A)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : close(F)
│ │ │  
│ │ │ -o5 = /tmp/M2-15935-0/0
│ │ │ +o5 = /tmp/M2-20955-0/0
│ │ │  
│ │ │  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/M2-15935-0/0
│ │ │ │ +o2 = /tmp/M2-20955-0/0
│ │ │ │  i3 : F = openOut(s)
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-15935-0/0
│ │ │ │ +o3 = /tmp/M2-20955-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : putMatrix(F,A)
│ │ │ │  i5 : close(F)
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-15935-0/0
│ │ │ │ +o5 = /tmp/M2-20955-0/0
│ │ │ │  
│ │ │ │  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 2.21552s (cpu); 1.26312s (thread); 0s (gc)
│ │ │ + -- used 2.88922s (cpu); 1.61258s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.34032s (cpu); 0.819238s (thread); 0s (gc)
│ │ │ + -- used 1.6027s (cpu); 0.967968s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
│ │ │  
│ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 1.09779s (cpu); 0.697093s (thread); 0s (gc)
│ │ │ + -- used 1.3763s (cpu); 0.829928s (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 0.00399957s (cpu); 0.00414499s (thread); 0s (gc)
│ │ │ + -- used 0.00401294s (cpu); 0.00700304s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
│ │ │  
│ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.512037s (cpu); 0.349891s (thread); 0s (gc)
│ │ │ + -- used 0.497064s (cpu); 0.33561s (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 0.302679s (cpu); 0.200173s (thread); 0s (gc)
│ │ │ + -- used 0.571606s (cpu); 0.323695s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
│ │ │  
│ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.56248s (cpu); 1.20583s (thread); 0s (gc)
│ │ │ + -- used 1.611s (cpu); 1.33367s (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 1.02964s (cpu); 0.633188s (thread); 0s (gc)
│ │ │ + -- used 1.31285s (cpu); 0.764832s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
│ │ │  
│ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.53299s (cpu); 0.899133s (thread); 0s (gc)
│ │ │ + -- used 1.9806s (cpu); 1.20629s (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 2.15426s (cpu); 1.79821s (thread); 0s (gc)
│ │ │ + -- used 2.08596s (cpu); 1.72915s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
│ │ │  
│ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.692075s (cpu); 0.57169s (thread); 0s (gc)
│ │ │ + -- used 0.616618s (cpu); 0.530041s (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
│ │ │ @@ -363,37 +363,37 @@
│ │ │          
    
│ │ │            
    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 2.21552s (cpu); 1.26312s (thread); 0s (gc)
│ │ │ │ + -- used 2.88922s (cpu); 1.61258s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = {.142067, .144}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ │ - -- used 1.34032s (cpu); 0.819238s (thread); 0s (gc)
│ │ │ │ + -- used 1.6027s (cpu); 0.967968s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        1
│ │ │ │  o28 = -
│ │ │ │        7
│ │ │ │  
│ │ │ │  o28 : QQ
│ │ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ │ - -- used 1.09779s (cpu); 0.697093s (thread); 0s (gc)
│ │ │ │ + -- used 1.3763s (cpu); 0.829928s (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
│ │ │ @@ -192,23 +192,23 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ - -- used 2.21552s (cpu); 1.26312s (thread); 0s (gc)
│ │ │ + -- used 2.88922s (cpu); 1.61258s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
    
│ │ │              
    
│ │ │                
    i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.34032s (cpu); 0.819238s (thread); 0s (gc)
│ │ │ + -- used 1.6027s (cpu); 0.967968s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
    
│ │ │              
    
│ │ │                
    i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 1.09779s (cpu); 0.697093s (thread); 0s (gc)
│ │ │ + -- used 1.3763s (cpu); 0.829928s (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 0.00399957s (cpu); 0.00414499s (thread); 0s (gc)
│ │ │ + -- used 0.00401294s (cpu); 0.00700304s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
    
│ │ │              
    
│ │ │                
    i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.512037s (cpu); 0.349891s (thread); 0s (gc)
│ │ │ + -- used 0.497064s (cpu); 0.33561s (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.
    
│ │ │ @@ -225,23 +225,23 @@
│ │ │              
│ │ │                
    i18 : f = x^3 + y^3 + z^3 + x*y*z;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by default
│ │ │ - -- used 0.302679s (cpu); 0.200173s (thread); 0s (gc)
│ │ │ + -- used 0.571606s (cpu); 0.323695s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.56248s (cpu); 1.20583s (thread); 0s (gc)
│ │ │ + -- used 1.611s (cpu); 1.33367s (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.
    
│ │ │ @@ -287,46 +287,46 @@
│ │ │              
│ │ │                
    i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ - -- used 1.02964s (cpu); 0.633188s (thread); 0s (gc)
│ │ │ + -- used 1.31285s (cpu); 0.764832s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.53299s (cpu); 0.899133s (thread); 0s (gc)
│ │ │ + -- used 1.9806s (cpu); 1.20629s (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 2.15426s (cpu); 1.79821s (thread); 0s (gc)
│ │ │ + -- used 2.08596s (cpu); 1.72915s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.692075s (cpu); 0.57169s (thread); 0s (gc)
│ │ │ + -- used 0.616618s (cpu); 0.530041s (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 0.00399957s (cpu); 0.00414499s (thread); 0s (gc)
│ │ │ │ + -- used 0.00401294s (cpu); 0.00700304s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3756
│ │ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ │ - -- used 0.512037s (cpu); 0.349891s (thread); 0s (gc)
│ │ │ │ + -- used 0.497064s (cpu); 0.33561s (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 0.302679s (cpu); 0.200173s (thread); 0s (gc)
│ │ │ │ + -- used 0.571606s (cpu); 0.323695s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o19 = 499
│ │ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ │ - -- used 1.56248s (cpu); 1.20583s (thread); 0s (gc)
│ │ │ │ + -- used 1.611s (cpu); 1.33367s (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 1.02964s (cpu); 0.633188s (thread); 0s (gc)
│ │ │ │ + -- used 1.31285s (cpu); 0.764832s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = 1124
│ │ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ │ - -- used 1.53299s (cpu); 0.899133s (thread); 0s (gc)
│ │ │ │ + -- used 1.9806s (cpu); 1.20629s (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 2.15426s (cpu); 1.79821s (thread); 0s (gc)
│ │ │ │ + -- used 2.08596s (cpu); 1.72915s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 97
│ │ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets
│ │ │ │  luckier
│ │ │ │ - -- used 0.692075s (cpu); 0.57169s (thread); 0s (gc)
│ │ │ │ + -- used 0.616618s (cpu); 0.530041s (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 0.595776s (cpu); 0.359545s (thread); 0s (gc)
│ │ │ + -- used 1.33207s (cpu); 0.434282s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
│ │ │  
│ │ │  i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 1.80107s (cpu); 1.03657s (thread); 0s (gc)
│ │ │ + -- used 3.22057s (cpu); 1.13974s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o28 : KClass
│ │ │  
│ │ │  i29 :
│ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html
│ │ │ @@ -386,25 +386,25 @@
│ │ │                 3       4
│ │ │  o26 : Matrix QQ  <-- QQ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time C = orbitClosure(X,Mat)
│ │ │ - -- used 0.595776s (cpu); 0.359545s (thread); 0s (gc)
│ │ │ + -- used 1.33207s (cpu); 0.434282s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 1.80107s (cpu); 1.03657s (thread); 0s (gc)
│ │ │ + -- used 3.22057s (cpu); 1.13974s (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 0.595776s (cpu); 0.359545s (thread); 0s (gc)
│ │ │ │ + -- used 1.33207s (cpu); 0.434282s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = an "equivariant K-class" on a GKM variety
│ │ │ │  
│ │ │ │  o27 : KClass
│ │ │ │  i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ │ - -- used 1.80107s (cpu); 1.03657s (thread); 0s (gc)
│ │ │ │ + -- used 3.22057s (cpu); 1.13974s (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 = {map, newDigraph, digraph}
│ │ │ +o3 = {map, digraph, newDigraph}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Graphs/html/_new__Digraph.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  o2 : SortedDigraph
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : keys H
│ │ │  
│ │ │ -o3 = {map, newDigraph, digraph}
│ │ │ +o3 = {map, digraph, newDigraph}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │                                           4 => {}
│ │ │ │                                           5 => {6}
│ │ │ │                                           6 => {}
│ │ │ │  
│ │ │ │  o2 : SortedDigraph
│ │ │ │  i3 : keys H
│ │ │ │  
│ │ │ │ -o3 = {map, newDigraph, digraph}
│ │ │ │ +o3 = {map, digraph, 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/_nonminimal__Maps.out
│ │ │ @@ -103,46 +103,46 @@
│ │ │  
│ │ │  i13 : #compsJ
│ │ │  
│ │ │  o13 = 2
│ │ │  
│ │ │  i14 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o14 = | 42 9 39 9 34 7 -12 -17 -29 -35 50 2 13 19 -44 50 2 -29 15 2 -27 21
│ │ │ +o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -36 -29 -39 -10 24 -16 19 -29 39 -38 -22 -8 -30 -24 |
│ │ │ +      40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 |
│ │ │  
│ │ │                 1       36
│ │ │  o14 : Matrix kk  <-- kk
│ │ │  
│ │ │  i15 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16
│ │ │ +o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 |
│ │ │ +      28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 |
│ │ │  
│ │ │                 1       36
│ │ │  o15 : Matrix kk  <-- kk
│ │ │  
│ │ │  i16 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                            2               
│ │ │ -o16 = ideal (a  - 44b*c - 35c  + 2a*d + 7b*d + 39c*d + 42d , a*b - 39b*c +
│ │ │ +              2              2                             2               
│ │ │ +o16 = ideal (a  + 25b*c - 43c  + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2                   2                 
│ │ │ -      15c  - 27a*d + 13b*d - 29c*d + 9d , a*c - 38b*c - 10c  - 16a*d + 2b*d +
│ │ │ +         2                              2                  2                
│ │ │ +      46c  + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c  + 39a*d - 20b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2   2              2                              2     2  
│ │ │ -      19c*d + 34d , b  - 30b*c + 19c  - 22a*d + 21b*d + 50c*d - 12d , b*c  -
│ │ │ +                  2   2              2                              2     2  
│ │ │ +      - 22c*d - 2d , b  + 24b*c + 19c  - 10a*d + 21b*d - 16c*d - 35d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2       2        2     3   3                2   
│ │ │ -      29b*c*d - 36c d + 24a*d  + 2b*d  + 50c*d  + 9d , c  - 24b*c*d + 39c d -
│ │ │ +                   2         2        2        2      3   3                2 
│ │ │ +      29b*c*d - 30c d - 36a*d  + 34b*d  + 48c*d  - 23d , c  - 24b*c*d - 16c d
│ │ │        -----------------------------------------------------------------------
│ │ │ -          2        2        2      3
│ │ │ -      8a*d  - 29b*d  - 29c*d  - 17d )
│ │ │ +             2        2        2      3
│ │ │ +      + 19a*d  - 38b*d  - 29c*d  - 26d )
│ │ │  
│ │ │  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  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c -
│ │ │ +              2             2                              2               
│ │ │ +o18 = ideal (a  - 9b*c - 18c  - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2                  2                
│ │ │ -      45c  - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c  + 33a*d + 16b*d
│ │ │ +        2                              2                   2                
│ │ │ +      6c  + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c  + 27a*d - 47b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                   2     2            
│ │ │ -      - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , b*c  - 43b*c*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      3   3                2         2
│ │ │ -      19c d + 34a*d  + 21b*d  + 46c*d  + 20d , c  + 38b*c*d + 22c d - 15a*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      3
│ │ │ -      - 47b*d  - 39c*d  + 40d )
│ │ │ +             2        2    3
│ │ │ +      + 38b*d  - 39c*d  - d )
│ │ │  
│ │ │  o18 : Ideal of S
│ │ │  
│ │ │  i19 : betti res F2
│ │ │  
│ │ │               0 1 2 3
│ │ │  o19 = total: 1 6 8 3
│ │ │ @@ -179,27 +179,30 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o19 : BettiTally
│ │ │  
│ │ │  i20 : netList decompose F1
│ │ │  
│ │ │ -      +------------------------------------------------------+
│ │ │ -o20 = |ideal (c - 13d, b + 32d, a + 36d)                     |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c - 16d, b + d, a + 16d)                       |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |                                    2            2    |
│ │ │ -      |ideal (b - 6c + 33d, a - 36c + 2d, c  + 43c*d - d )   |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |                                     2              2 |
│ │ │ -      |ideal (b + 29c + 7d, a - 19c + 24d, c  - 20c*d - 30d )|
│ │ │ -      +------------------------------------------------------+
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +o20 = |ideal (c + 39d, b + 27d, a - 18d)                                                                                                                        |
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                            2              2                     2   3                2        2        2      3     2                2         2      3 |
│ │ │ +      |ideal (a - 29b - 8c - 13d, b  + 24b*c + 19c  + 34b*d + 5c*d + 37d , c  - 24b*c*d - 16c d + 8b*d  + 22c*d  + 19d , b*c  - 29b*c*d - 30c d - 38c*d  + 14d )|
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i21 : netList decompose F2
│ │ │  
│ │ │ -      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                       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 + 2b*c - 13c  + 33a*d + 16b*d - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , a*b - 20b*c - 45c  - 35a*d + 21b*d - 34c*d + 10d , a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , c  + 38b*c*d + 22c d - 15a*d  - 47b*d  - 39c*d  + 40d , b*c  - 43b*c*d + 19c d + 34a*d  + 21b*d  + 46c*d  + 20d )|
│ │ │ -      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      +-------------------------------------------------------+
│ │ │ +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 )|
│ │ │ +      +-------------------------------------------------------+
│ │ │  
│ │ │  i22 :
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out
│ │ │ @@ -200,67 +200,72 @@
│ │ │  
│ │ │  o12 = {11, 8}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o13 = | 50 15 46 -33 2 -43 -46 8 33 19 -2 -18 -8 -22 43 -29 19 3 -16 -29 -38
│ │ │ +o13 = | 13 48 43 23 41 36 -4 -12 -30 -16 -33 -36 19 19 30 -10 -38 32 -29 -8
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -24 -10 -29 |
│ │ │ +      -29 -22 -29 -24 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
│ │ │  
│ │ │  i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o14 = ideal (a  + 33b*c - 33c  + 19a*d + 2b*d + 15c*d + 50d , a*b + 43b*c -
│ │ │ +              2              2                              2               
│ │ │ +o14 = ideal (a  - 30b*c + 23c  - 16a*d + 41b*d + 48c*d + 13d , a*b + 30b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2                   2                 
│ │ │ -      2c  - 29a*d - 18b*d - 43c*d + 46d , a*c - 38b*c + 19c  - 24a*d + 3b*d -
│ │ │ +         2                              2                   2                
│ │ │ +      33c  - 10a*d - 36b*d + 36c*d + 43d , a*c - 29b*c - 38c  - 22a*d + 32b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                2   2              2                             2
│ │ │ -      8c*d - 46d , b  - 10b*c - 16c  - 29a*d - 29b*d - 22c*d + 8d )
│ │ │ +                  2   2              2                             2
│ │ │ +      + 19c*d - 4d , b  - 29b*c - 29c  - 24a*d - 8b*d + 19c*d - 12d )
│ │ │  
│ │ │  o14 : Ideal of S
│ │ │  
│ │ │  i15 : decompose F1
│ │ │  
│ │ │ -                                    2              2                      2
│ │ │ -o15 = {ideal (a - 38b + 19c + 44d, b  - 10b*c - 16c  - 20b*d + 24c*d - 29d ),
│ │ │ +                                   2              2                     2
│ │ │ +o15 = {ideal (a - 29b - 38c - 9d, b  - 29b*c - 29c  + 3b*d + 16c*d - 26d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 24d, b - 38d, a + 15d)}
│ │ │ +      ideal (c - 22d, b - 21d, a + 8d)}
│ │ │  
│ │ │  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 = | 46 -2 16 -20 -1 -30 -43 -41 17 -4 -16 -29 -39 40 49 -39 -18 -13 -47
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -47 -39 34 0 |
│ │ │ +      34 19 21 39 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                   2
│ │ │ +o17 = ideal (a  + 17b*c - 20c  - 4a*d - b*d - 2c*d + 46d , a*b + 49b*c - 16c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2                   2                
│ │ │ -      38c  + 16a*d + 31b*d - 26c*d - 5d , a*c - 47b*c + 39c  - 39a*d + 21b*d
│ │ │ +                                   2                   2                  
│ │ │ +      - 39a*d - 29b*d - 30c*d + 16d , a*c + 19b*c - 18c  + 21a*d - 13b*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                      2
│ │ │ -      - 13c*d - 35d , b  + 34b*c - 18c  + 19b*d + 29c*d + 41d )
│ │ │ +                 2   2              2                      2
│ │ │ +      39c*d - 43d , b  + 39b*c - 47c  + 34b*d + 40c*d - 41d )
│ │ │  
│ │ │  o17 : Ideal of S
│ │ │  
│ │ │  i18 : decompose F2
│ │ │  
│ │ │ -o18 = {ideal (b + 19c - 18d, a + 23c + 43d), ideal (b + 15c + 37d, a + 37c +
│ │ │ +                               2                              2   2          
│ │ │ +o18 = {ideal (a*c + 19b*c - 18c  + 21a*d - 13b*d - 39c*d - 43d , b  + 39b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -      26d)}
│ │ │ +         2                      2                   2                        
│ │ │ +      47c  + 34b*d + 40c*d - 41d , a*b + 49b*c - 16c  - 39a*d - 29b*d - 30c*d
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +           2   2              2                          2
│ │ │ +      + 16d , a  + 17b*c - 20c  - 4a*d - b*d - 2c*d + 46d )}
│ │ │  
│ │ │  o18 : List
│ │ │  
│ │ │  i19 :
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.out
│ │ │ @@ -68,54 +68,54 @@
│ │ │  
│ │ │  o12 = {11, 8}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : netList randomPointsOnRationalVariety(compsJ_0, 10)
│ │ │  
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -o13 = || 29 -40 15 -49 3 -13 -6 -39 2 39 47 15 19 -47 -46 -39 -16 32 -43 34 -13 -18 21 -38 | |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 37 -7 -24 8 -26 38 9 -31 24 -47 -34 12 16 22 -22 45 -28 16 -47 2 -48 -34 38 -15 |   |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 6 1 -31 -7 44 8 -50 24 -48 -16 23 23 -23 39 -5 43 19 -15 48 15 -11 -17 7 47 |       |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -41 -49 6 -16 -12 31 23 6 -7 11 3 -42 40 11 -28 46 35 -28 -3 33 1 -28 -38 36 |      |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -11 -27 -4 40 -34 6 44 -2 19 -23 -29 21 29 -47 -37 15 -47 -24 -10 2 -13 -37 -7 22 | |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -50 42 20 -30 -46 -48 -5 40 -47 39 13 47 32 -9 41 -32 -18 25 -30 -22 24 -20 27 30 | |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 50 22 -30 3 -43 -29 -33 -18 6 39 -29 24 -49 -33 -15 -19 -15 -37 44 33 -20 17 0 -48 ||
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -9 31 -37 -42 -7 -8 -11 -21 12 9 13 -9 13 -26 11 22 36 34 -8 4 -11 -49 -39 -39 |    |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 47 14 -11 -16 -20 -40 42 5 -2 36 8 -45 -30 41 -26 16 -8 -34 35 -22 -6 -28 -3 43 |   |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 23 -8 -3 -17 38 0 11 -33 -7 6 -31 -4 -31 25 6 -2 -35 -11 -13 3 -49 -41 40 -9 |      |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -
│ │ │ -i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ -
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -o14 = || 38 -31 49 39 4 46 -29 -5 -39 -40 14 -11 -31 46 43 -26 4 30 -35 27 -40 37 -47 0 |   |
│ │ │ +o13 = || 13 15 3 36 2 48 44 -35 -34 39 5 -32 34 19 -42 -47 -16 -34 -39 -13 -18 -43 21 -38 | |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 -48 30 -48 0 ||
│ │ │ +      || -43 48 14 29 -47 -10 47 22 8 -47 15 -26 2 16 -49 22 -28 -18 45 -48 -34 -47 38 -15 ||
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1 40 -18 0 |    |
│ │ │ +      || -3 45 42 47 -50 16 -30 28 43 -16 24 19 15 -23 37 39 19 -8 43 -11 -17 48 7 47 |     |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13 3 13 0 |     |
│ │ │ +      || -49 7 32 -6 -30 -41 -10 2 44 11 -25 4 33 40 -19 11 35 -17 46 1 -28 -3 -38 36 |     |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 -18 30 0 |    |
│ │ │ +      || 35 -48 -2 45 -35 29 34 12 -32 -23 50 2 2 29 -3 -47 -47 -34 15 -13 -37 -10 -7 22 |  |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46 12 18 0 |     |
│ │ │ +      || 47 8 -14 6 -1 -13 -7 16 -20 39 -34 -22 -22 32 17 -9 -18 -6 -32 24 -20 -30 27 30 |  |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39 -37 0 |      |
│ │ │ +      || -2 -36 -39 41 -6 34 -10 42 5 39 20 33 33 -49 -15 -33 -15 41 -19 -20 17 44 0 -48 |  |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28 6 -28 0 |    |
│ │ │ +      || -30 37 -9 16 -36 19 -13 -14 -19 9 -33 5 4 13 44 -26 36 -12 22 -11 -49 -8 -39 -39 | |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 -33 26 0 |    |
│ │ │ +      || 27 41 32 -44 40 -20 41 33 28 36 44 31 -22 -30 9 41 -8 30 16 -6 -28 35 -3 43 |      |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5 -20 4 0 |   |
│ │ │ +      || 37 -2 17 -42 -42 -12 18 -31 33 6 19 -31 3 -31 -11 25 -35 28 -2 -49 -41 -13 40 -9 | |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │  
│ │ │ +i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ +
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +o14 = || -41 -1 -48 25 40 4 35 16 26 -41 -28 -16 27 -14 -39 4 4 30 -40 37 -31 -35 -47 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -1 19 -3 12 50 3 4 25 48 50 34 -6 -29 6 -5 36 -39 -31 -48 30 47 -37 -48 0 |          |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -27 -3 -40 22 27 3 -28 -41 -12 -34 -10 40 46 29 30 24 -49 28 1 40 10 -22 -18 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -26 -6 24 28 -27 26 34 47 13 50 3 -42 -17 5 4 -35 7 30 -13 3 8 -41 13 0 |            |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || 49 -7 48 1 48 25 25 -10 49 36 -16 35 -46 -5 25 -33 8 -29 49 -18 23 42 30 0 |         |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -35 28 -6 22 50 -49 2 -5 -11 -39 30 27 -16 34 -9 -34 -28 15 -46 12 27 -18 18 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -49 -44 -16 -10 48 18 22 33 -35 -48 -28 -8 -23 -48 -25 -3 -21 23 44 -39 19 20 -37 0 ||
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -33 -14 -18 10 2 -43 -26 45 10 19 -15 25 47 9 -15 -22 0 -47 -28 6 -33 -9 -28 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || 20 -27 -17 2 -47 -23 13 40 -19 -13 39 -23 5 -3 47 -6 28 -29 -37 -33 42 -28 26 0 |    |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || 19 10 -10 47 41 20 -43 -34 -43 2 44 29 22 35 -42 16 44 30 5 -20 -29 -13 4 0 |        |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +
│ │ │  i15 :
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html
│ │ │ @@ -232,52 +232,52 @@
│ │ │  o13 = 2
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o14 = | 42 9 39 9 34 7 -12 -17 -29 -35 50 2 13 19 -44 50 2 -29 15 2 -27 21
│ │ │ +o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -36 -29 -39 -10 24 -16 19 -29 39 -38 -22 -8 -30 -24 |
│ │ │ +      40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 |
│ │ │  
│ │ │                 1       36
│ │ │  o14 : Matrix kk  <-- kk
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16
│ │ │ +o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 |
│ │ │ +      28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 |
│ │ │  
│ │ │                 1       36
│ │ │  o15 : Matrix kk  <-- kk
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                            2               
│ │ │ -o16 = ideal (a  - 44b*c - 35c  + 2a*d + 7b*d + 39c*d + 42d , a*b - 39b*c +
│ │ │ +              2              2                             2               
│ │ │ +o16 = ideal (a  + 25b*c - 43c  + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2                   2                 
│ │ │ -      15c  - 27a*d + 13b*d - 29c*d + 9d , a*c - 38b*c - 10c  - 16a*d + 2b*d +
│ │ │ +         2                              2                  2                
│ │ │ +      46c  + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c  + 39a*d - 20b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2   2              2                              2     2  
│ │ │ -      19c*d + 34d , b  - 30b*c + 19c  - 22a*d + 21b*d + 50c*d - 12d , b*c  -
│ │ │ +                  2   2              2                              2     2  
│ │ │ +      - 22c*d - 2d , b  + 24b*c + 19c  - 10a*d + 21b*d - 16c*d - 35d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2       2        2     3   3                2   
│ │ │ -      29b*c*d - 36c d + 24a*d  + 2b*d  + 50c*d  + 9d , c  - 24b*c*d + 39c d -
│ │ │ +                   2         2        2        2      3   3                2 
│ │ │ +      29b*c*d - 30c d - 36a*d  + 34b*d  + 48c*d  - 23d , c  - 24b*c*d - 16c d
│ │ │        -----------------------------------------------------------------------
│ │ │ -          2        2        2      3
│ │ │ -      8a*d  - 29b*d  - 29c*d  - 17d )
│ │ │ +             2        2        2      3
│ │ │ +      + 19a*d  - 38b*d  - 29c*d  - 26d )
│ │ │  
│ │ │  o16 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : betti res F1
│ │ │ @@ -291,28 +291,28 @@
│ │ │  o17 : BettiTally
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i18 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o18 = ideal (a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c -
│ │ │ +              2             2                              2               
│ │ │ +o18 = ideal (a  - 9b*c - 18c  - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2                  2                
│ │ │ -      45c  - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c  + 33a*d + 16b*d
│ │ │ +        2                              2                   2                
│ │ │ +      6c  + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c  + 27a*d - 47b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                   2     2            
│ │ │ -      - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , b*c  - 43b*c*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      3   3                2         2
│ │ │ -      19c d + 34a*d  + 21b*d  + 46c*d  + 20d , c  + 38b*c*d + 22c d - 15a*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      3
│ │ │ -      - 47b*d  - 39c*d  + 40d )
│ │ │ +             2        2    3
│ │ │ +      + 38b*d  - 39c*d  - d )
│ │ │  
│ │ │  o18 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i19 : betti res F2
│ │ │ @@ -331,35 +331,38 @@
│ │ │            
    What are the ideals F1 and F2?
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i20 : netList decompose F1
│ │ │  
│ │ │ -      +------------------------------------------------------+
│ │ │ -o20 = |ideal (c - 13d, b + 32d, a + 36d)                     |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c - 16d, b + d, a + 16d)                       |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |                                    2            2    |
│ │ │ -      |ideal (b - 6c + 33d, a - 36c + 2d, c  + 43c*d - d )   |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |                                     2              2 |
│ │ │ -      |ideal (b + 29c + 7d, a - 19c + 24d, c  - 20c*d - 30d )|
│ │ │ -      +------------------------------------------------------+
    
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +o20 = |ideal (c + 39d, b + 27d, a - 18d)                                                                                                                        |
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                            2              2                     2   3                2        2        2      3     2                2         2      3 |
│ │ │ +      |ideal (a - 29b - 8c - 13d, b  + 24b*c + 19c  + 34b*d + 5c*d + 37d , c  - 24b*c*d - 16c d + 8b*d  + 22c*d  + 19d , b*c  - 29b*c*d - 30c d - 38c*d  + 14d )|
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │              
    
│ │ │                
    i21 : netList decompose F2
│ │ │  
│ │ │ -      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                       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 + 2b*c - 13c  + 33a*d + 16b*d - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , a*b - 20b*c - 45c  - 35a*d + 21b*d - 34c*d + 10d , a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , c  + 38b*c*d + 22c d - 15a*d  - 47b*d  - 39c*d  + 40d , b*c  - 43b*c*d + 19c d + 34a*d  + 21b*d  + 46c*d  + 20d )|
│ │ │ -      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
    
│ │ │ +      +-------------------------------------------------------+
│ │ │ +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 )|
│ │ │ +      +-------------------------------------------------------+
│ │ │              
    
│ │ │          
    
│ │ │            
    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 {}
│ │ │ │ @@ -170,117 +170,115 @@
│ │ │ │  32   13   21   33   19   31
│ │ │ │  i12 : compsJ = decompose J;
│ │ │ │  i13 : #compsJ
│ │ │ │  
│ │ │ │  o13 = 2
│ │ │ │  i14 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │ │  
│ │ │ │ -o14 = | 42 9 39 9 34 7 -12 -17 -29 -35 50 2 13 19 -44 50 2 -29 15 2 -27 21
│ │ │ │ +o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -36 -29 -39 -10 24 -16 19 -29 39 -38 -22 -8 -30 -24 |
│ │ │ │ +      40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o14 : Matrix kk  <-- kk
│ │ │ │  i15 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │ │  
│ │ │ │ -o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16
│ │ │ │ +o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 |
│ │ │ │ +      28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o15 : Matrix kk  <-- kk
│ │ │ │  i16 : F1 = sub(F, (vars S)|pt1)
│ │ │ │  
│ │ │ │ -              2              2                            2
│ │ │ │ -o16 = ideal (a  - 44b*c - 35c  + 2a*d + 7b*d + 39c*d + 42d , a*b - 39b*c +
│ │ │ │ +              2              2                             2
│ │ │ │ +o16 = ideal (a  + 25b*c - 43c  + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                             2                   2
│ │ │ │ -      15c  - 27a*d + 13b*d - 29c*d + 9d , a*c - 38b*c - 10c  - 16a*d + 2b*d +
│ │ │ │ +         2                              2                  2
│ │ │ │ +      46c  + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c  + 39a*d - 20b*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                 2   2              2                              2     2
│ │ │ │ -      19c*d + 34d , b  - 30b*c + 19c  - 22a*d + 21b*d + 50c*d - 12d , b*c  -
│ │ │ │ +                  2   2              2                              2     2
│ │ │ │ +      - 22c*d - 2d , b  + 24b*c + 19c  - 10a*d + 21b*d - 16c*d - 35d , b*c  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2         2       2        2     3   3                2
│ │ │ │ -      29b*c*d - 36c d + 24a*d  + 2b*d  + 50c*d  + 9d , c  - 24b*c*d + 39c d -
│ │ │ │ +                   2         2        2        2      3   3                2
│ │ │ │ +      29b*c*d - 30c d - 36a*d  + 34b*d  + 48c*d  - 23d , c  - 24b*c*d - 16c d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -          2        2        2      3
│ │ │ │ -      8a*d  - 29b*d  - 29c*d  - 17d )
│ │ │ │ +             2        2        2      3
│ │ │ │ +      + 19a*d  - 38b*d  - 29c*d  - 26d )
│ │ │ │  
│ │ │ │  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  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c -
│ │ │ │ +              2             2                              2
│ │ │ │ +o18 = ideal (a  - 9b*c - 18c  - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                              2                  2
│ │ │ │ -      45c  - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c  + 33a*d + 16b*d
│ │ │ │ +        2                              2                   2
│ │ │ │ +      6c  + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c  + 27a*d - 47b*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2   2              2                   2     2
│ │ │ │ -      - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , b*c  - 43b*c*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      3   3                2         2
│ │ │ │ -      19c d + 34a*d  + 21b*d  + 46c*d  + 20d , c  + 38b*c*d + 22c d - 15a*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      3
│ │ │ │ -      - 47b*d  - 39c*d  + 40d )
│ │ │ │ +             2        2    3
│ │ │ │ +      + 38b*d  - 39c*d  - d )
│ │ │ │  
│ │ │ │  o18 : Ideal of S
│ │ │ │  i19 : betti res F2
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o19 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │            1: . 4 4 1
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o19 : BettiTally
│ │ │ │  What are the ideals F1 and F2?
│ │ │ │  i20 : netList decompose F1
│ │ │ │  
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ -o20 = |ideal (c - 13d, b + 32d, a + 36d)                     |
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ -      |ideal (c - 16d, b + d, a + 16d)                       |
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ -      |                                    2            2    |
│ │ │ │ -      |ideal (b - 6c + 33d, a - 36c + 2d, c  + 43c*d - d )   |
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ -      |                                     2              2 |
│ │ │ │ -      |ideal (b + 29c + 7d, a - 19c + 24d, c  - 20c*d - 30d )|
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ -i21 : netList decompose F2
│ │ │ │ -
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ +--+
│ │ │ │ +o20 = |ideal (c + 39d, b + 27d, a - 18d)
│ │ │ │ +|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ --+
│ │ │ │ -      |                       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 + 2b*c - 13c  + 33a*d + 16b*d - 18c*d - 39d , b  - 47b*c -
│ │ │ │ -28c  - 5b*d - c*d - 30d , a*b - 20b*c - 45c  - 35a*d + 21b*d - 34c*d + 10d , a
│ │ │ │ -- 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , c  + 38b*c*d + 22c d - 15a*d  -
│ │ │ │ -47b*d  - 39c*d  + 40d , b*c  - 43b*c*d + 19c d + 34a*d  + 21b*d  + 46c*d  + 20d
│ │ │ │ +--+
│ │ │ │ +      |                            2              2                     2   3
│ │ │ │ +2        2        2      3     2                2         2      3 |
│ │ │ │ +      |ideal (a - 29b - 8c - 13d, b  + 24b*c + 19c  + 34b*d + 5c*d + 37d , c  -
│ │ │ │ +24b*c*d - 16c d + 8b*d  + 22c*d  + 19d , b*c  - 29b*c*d - 30c d - 38c*d  + 14d
│ │ │ │  )|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ --+
│ │ │ │ +--+
│ │ │ │ +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 )|
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │  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
│ │ │ @@ -318,90 +318,95 @@
│ │ │            
    There are 2 components.  We attempt to find a point on the first component
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i13 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o13 = | 50 15 46 -33 2 -43 -46 8 33 19 -2 -18 -8 -22 43 -29 19 3 -16 -29 -38
│ │ │ +o13 = | 13 48 43 23 41 36 -4 -12 -30 -16 -33 -36 19 19 30 -10 -38 32 -29 -8
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -24 -10 -29 |
│ │ │ +      -29 -22 -29 -24 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
    
│ │ │              
    
│ │ │                
    i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o14 = ideal (a  + 33b*c - 33c  + 19a*d + 2b*d + 15c*d + 50d , a*b + 43b*c -
│ │ │ +              2              2                              2               
│ │ │ +o14 = ideal (a  - 30b*c + 23c  - 16a*d + 41b*d + 48c*d + 13d , a*b + 30b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2                   2                 
│ │ │ -      2c  - 29a*d - 18b*d - 43c*d + 46d , a*c - 38b*c + 19c  - 24a*d + 3b*d -
│ │ │ +         2                              2                   2                
│ │ │ +      33c  - 10a*d - 36b*d + 36c*d + 43d , a*c - 29b*c - 38c  - 22a*d + 32b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                2   2              2                             2
│ │ │ -      8c*d - 46d , b  - 10b*c - 16c  - 29a*d - 29b*d - 22c*d + 8d )
│ │ │ +                  2   2              2                             2
│ │ │ +      + 19c*d - 4d , b  - 29b*c - 29c  - 24a*d - 8b*d + 19c*d - 12d )
│ │ │  
│ │ │  o14 : Ideal of S
    
│ │ │              
    
│ │ │                
    i15 : decompose F1
│ │ │  
│ │ │ -                                    2              2                      2
│ │ │ -o15 = {ideal (a - 38b + 19c + 44d, b  - 10b*c - 16c  - 20b*d + 24c*d - 29d ),
│ │ │ +                                   2              2                     2
│ │ │ +o15 = {ideal (a - 29b - 38c - 9d, b  - 29b*c - 29c  + 3b*d + 16c*d - 26d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 24d, b - 38d, a + 15d)}
│ │ │ +      ideal (c - 22d, b - 21d, a + 8d)}
│ │ │  
│ │ │  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 = | 46 -2 16 -20 -1 -30 -43 -41 17 -4 -16 -29 -39 40 49 -39 -18 -13 -47
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -47 -39 34 0 |
│ │ │ +      34 19 21 39 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                   2
│ │ │ +o17 = ideal (a  + 17b*c - 20c  - 4a*d - b*d - 2c*d + 46d , a*b + 49b*c - 16c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2                   2                
│ │ │ -      38c  + 16a*d + 31b*d - 26c*d - 5d , a*c - 47b*c + 39c  - 39a*d + 21b*d
│ │ │ +                                   2                   2                  
│ │ │ +      - 39a*d - 29b*d - 30c*d + 16d , a*c + 19b*c - 18c  + 21a*d - 13b*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                      2
│ │ │ -      - 13c*d - 35d , b  + 34b*c - 18c  + 19b*d + 29c*d + 41d )
│ │ │ +                 2   2              2                      2
│ │ │ +      39c*d - 43d , b  + 39b*c - 47c  + 34b*d + 40c*d - 41d )
│ │ │  
│ │ │  o17 : Ideal of S
    
│ │ │              
    
│ │ │                
    i18 : decompose F2
│ │ │  
│ │ │ -o18 = {ideal (b + 19c - 18d, a + 23c + 43d), ideal (b + 15c + 37d, a + 37c +
│ │ │ +                               2                              2   2          
│ │ │ +o18 = {ideal (a*c + 19b*c - 18c  + 21a*d - 13b*d - 39c*d - 43d , b  + 39b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -      26d)}
│ │ │ +         2                      2                   2                        
│ │ │ +      47c  + 34b*d + 40c*d - 41d , a*b + 49b*c - 16c  - 39a*d - 29b*d - 30c*d
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +           2   2              2                          2
│ │ │ +      + 16d , a  + 17b*c - 20c  - 4a*d - b*d - 2c*d + 46d )}
│ │ │  
│ │ │  o18 : List
    
│ │ │              
    
│ │ │          
    
│ │ │            
    It turns out that this is the ideal of 2 skew lines, just not defined over this field.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -212,67 +212,72 @@
│ │ │ │  
│ │ │ │  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 = | 50 15 46 -33 2 -43 -46 8 33 19 -2 -18 -8 -22 43 -29 19 3 -16 -29 -38
│ │ │ │ +o13 = | 13 48 43 23 41 36 -4 -12 -30 -16 -33 -36 19 19 30 -10 -38 32 -29 -8
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -24 -10 -29 |
│ │ │ │ +      -29 -22 -29 -24 |
│ │ │ │  
│ │ │ │                 1       24
│ │ │ │  o13 : Matrix kk  <-- kk
│ │ │ │  i14 : F1 = sub(F, (vars S)|pt1)
│ │ │ │  
│ │ │ │ -              2              2                             2
│ │ │ │ -o14 = ideal (a  + 33b*c - 33c  + 19a*d + 2b*d + 15c*d + 50d , a*b + 43b*c -
│ │ │ │ +              2              2                              2
│ │ │ │ +o14 = ideal (a  - 30b*c + 23c  - 16a*d + 41b*d + 48c*d + 13d , a*b + 30b*c -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -        2                              2                   2
│ │ │ │ -      2c  - 29a*d - 18b*d - 43c*d + 46d , a*c - 38b*c + 19c  - 24a*d + 3b*d -
│ │ │ │ +         2                              2                   2
│ │ │ │ +      33c  - 10a*d - 36b*d + 36c*d + 43d , a*c - 29b*c - 38c  - 22a*d + 32b*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                2   2              2                             2
│ │ │ │ -      8c*d - 46d , b  - 10b*c - 16c  - 29a*d - 29b*d - 22c*d + 8d )
│ │ │ │ +                  2   2              2                             2
│ │ │ │ +      + 19c*d - 4d , b  - 29b*c - 29c  - 24a*d - 8b*d + 19c*d - 12d )
│ │ │ │  
│ │ │ │  o14 : Ideal of S
│ │ │ │  i15 : decompose F1
│ │ │ │  
│ │ │ │ -                                    2              2                      2
│ │ │ │ -o15 = {ideal (a - 38b + 19c + 44d, b  - 10b*c - 16c  - 20b*d + 24c*d - 29d ),
│ │ │ │ +                                   2              2                     2
│ │ │ │ +o15 = {ideal (a - 29b - 38c - 9d, b  - 29b*c - 29c  + 3b*d + 16c*d - 26d ),
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ideal (c - 24d, b - 38d, a + 15d)}
│ │ │ │ +      ideal (c - 22d, b - 21d, a + 8d)}
│ │ │ │  
│ │ │ │  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 = | 46 -2 16 -20 -1 -30 -43 -41 17 -4 -16 -29 -39 40 49 -39 -18 -13 -47
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -47 -39 34 0 |
│ │ │ │ +      34 19 21 39 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                   2
│ │ │ │ +o17 = ideal (a  + 17b*c - 20c  - 4a*d - b*d - 2c*d + 46d , a*b + 49b*c - 16c
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                             2                   2
│ │ │ │ -      38c  + 16a*d + 31b*d - 26c*d - 5d , a*c - 47b*c + 39c  - 39a*d + 21b*d
│ │ │ │ +                                   2                   2
│ │ │ │ +      - 39a*d - 29b*d - 30c*d + 16d , a*c + 19b*c - 18c  + 21a*d - 13b*d -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2   2              2                      2
│ │ │ │ -      - 13c*d - 35d , b  + 34b*c - 18c  + 19b*d + 29c*d + 41d )
│ │ │ │ +                 2   2              2                      2
│ │ │ │ +      39c*d - 43d , b  + 39b*c - 47c  + 34b*d + 40c*d - 41d )
│ │ │ │  
│ │ │ │  o17 : Ideal of S
│ │ │ │  i18 : decompose F2
│ │ │ │  
│ │ │ │ -o18 = {ideal (b + 19c - 18d, a + 23c + 43d), ideal (b + 15c + 37d, a + 37c +
│ │ │ │ +                               2                              2   2
│ │ │ │ +o18 = {ideal (a*c + 19b*c - 18c  + 21a*d - 13b*d - 39c*d - 43d , b  + 39b*c -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      26d)}
│ │ │ │ +         2                      2                   2
│ │ │ │ +      47c  + 34b*d + 40c*d - 41d , a*b + 49b*c - 16c  - 39a*d - 29b*d - 30c*d
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +           2   2              2                          2
│ │ │ │ +      + 16d , a  + 17b*c - 20c  - 4a*d - b*d - 2c*d + 46d )}
│ │ │ │  
│ │ │ │  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/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.html
│ │ │ @@ -186,64 +186,64 @@
│ │ │            
    There are 2 components.  We attempt to find points on each of these two components. We are successful.  This indicates that the corresponding varieties are both rational. Also, if we can find one point, we can find as many as we want.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │ -          
│ │ │ -          
│ │ │ -            
│ │ │            
│ │ │ +          
│ │ │ +            
│ │ │ +          
│ │ │          
    
│ │ │                
    i13 : netList randomPointsOnRationalVariety(compsJ_0, 10)
│ │ │  
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -o13 = || 29 -40 15 -49 3 -13 -6 -39 2 39 47 15 19 -47 -46 -39 -16 32 -43 34 -13 -18 21 -38 | |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 37 -7 -24 8 -26 38 9 -31 24 -47 -34 12 16 22 -22 45 -28 16 -47 2 -48 -34 38 -15 |   |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 6 1 -31 -7 44 8 -50 24 -48 -16 23 23 -23 39 -5 43 19 -15 48 15 -11 -17 7 47 |       |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -41 -49 6 -16 -12 31 23 6 -7 11 3 -42 40 11 -28 46 35 -28 -3 33 1 -28 -38 36 |      |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -11 -27 -4 40 -34 6 44 -2 19 -23 -29 21 29 -47 -37 15 -47 -24 -10 2 -13 -37 -7 22 | |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -50 42 20 -30 -46 -48 -5 40 -47 39 13 47 32 -9 41 -32 -18 25 -30 -22 24 -20 27 30 | |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 50 22 -30 3 -43 -29 -33 -18 6 39 -29 24 -49 -33 -15 -19 -15 -37 44 33 -20 17 0 -48 ||
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -9 31 -37 -42 -7 -8 -11 -21 12 9 13 -9 13 -26 11 22 36 34 -8 4 -11 -49 -39 -39 |    |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 47 14 -11 -16 -20 -40 42 5 -2 36 8 -45 -30 41 -26 16 -8 -34 35 -22 -6 -28 -3 43 |   |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 23 -8 -3 -17 38 0 11 -33 -7 6 -31 -4 -31 25 6 -2 -35 -11 -13 3 -49 -41 40 -9 |      |
│ │ │ -      +--------------------------------------------------------------------------------------+
    
│ │ │ -            
    
│ │ │ -              
    i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ -
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -o14 = || 38 -31 49 39 4 46 -29 -5 -39 -40 14 -11 -31 46 43 -26 4 30 -35 27 -40 37 -47 0 |   |
│ │ │ +o13 = || 13 15 3 36 2 48 44 -35 -34 39 5 -32 34 19 -42 -47 -16 -34 -39 -13 -18 -43 21 -38 | |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 -48 30 -48 0 ||
│ │ │ +      || -43 48 14 29 -47 -10 47 22 8 -47 15 -26 2 16 -49 22 -28 -18 45 -48 -34 -47 38 -15 ||
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1 40 -18 0 |    |
│ │ │ +      || -3 45 42 47 -50 16 -30 28 43 -16 24 19 15 -23 37 39 19 -8 43 -11 -17 48 7 47 |     |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13 3 13 0 |     |
│ │ │ +      || -49 7 32 -6 -30 -41 -10 2 44 11 -25 4 33 40 -19 11 35 -17 46 1 -28 -3 -38 36 |     |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 -18 30 0 |    |
│ │ │ +      || 35 -48 -2 45 -35 29 34 12 -32 -23 50 2 2 29 -3 -47 -47 -34 15 -13 -37 -10 -7 22 |  |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46 12 18 0 |     |
│ │ │ +      || 47 8 -14 6 -1 -13 -7 16 -20 39 -34 -22 -22 32 17 -9 -18 -6 -32 24 -20 -30 27 30 |  |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39 -37 0 |      |
│ │ │ +      || -2 -36 -39 41 -6 34 -10 42 5 39 20 33 33 -49 -15 -33 -15 41 -19 -20 17 44 0 -48 |  |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28 6 -28 0 |    |
│ │ │ +      || -30 37 -9 16 -36 19 -13 -14 -19 9 -33 5 4 13 44 -26 36 -12 22 -11 -49 -8 -39 -39 | |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 -33 26 0 |    |
│ │ │ +      || 27 41 32 -44 40 -20 41 33 28 36 44 31 -22 -30 9 41 -8 30 16 -6 -28 35 -3 43 |      |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5 -20 4 0 |   |
│ │ │ +      || 37 -2 17 -42 -42 -12 18 -31 33 6 19 -31 3 -31 -11 25 -35 28 -2 -49 -41 -13 40 -9 | |
│ │ │        +-------------------------------------------------------------------------------------+
    
│ │ │              
    
│ │ │ +              
    i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ +
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +o14 = || -41 -1 -48 25 40 4 35 16 26 -41 -28 -16 27 -14 -39 4 4 30 -40 37 -31 -35 -47 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -1 19 -3 12 50 3 4 25 48 50 34 -6 -29 6 -5 36 -39 -31 -48 30 47 -37 -48 0 |          |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -27 -3 -40 22 27 3 -28 -41 -12 -34 -10 40 46 29 30 24 -49 28 1 40 10 -22 -18 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -26 -6 24 28 -27 26 34 47 13 50 3 -42 -17 5 4 -35 7 30 -13 3 8 -41 13 0 |            |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || 49 -7 48 1 48 25 25 -10 49 36 -16 35 -46 -5 25 -33 8 -29 49 -18 23 42 30 0 |         |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -35 28 -6 22 50 -49 2 -5 -11 -39 30 27 -16 34 -9 -34 -28 15 -46 12 27 -18 18 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -49 -44 -16 -10 48 18 22 33 -35 -48 -28 -8 -23 -48 -25 -3 -21 23 44 -39 19 20 -37 0 ||
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -33 -14 -18 10 2 -43 -26 45 10 19 -15 25 47 9 -15 -22 0 -47 -28 6 -33 -9 -28 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || 20 -27 -17 2 -47 -23 13 40 -19 -13 39 -23 5 -3 47 -6 28 -29 -37 -33 42 -28 26 0 |    |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || 19 10 -10 47 41 20 -43 -34 -43 2 44 29 22 35 -42 16 44 30 5 -20 -29 -13 4 0 |        |
│ │ │ +      +---------------------------------------------------------------------------------------+
    
│ │ │ +            
    
│ │ │        
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │          
    
│ │ │            
    This routine expects the input to represent an irreducible variety
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -84,99 +84,99 @@
│ │ │ │  There are 2 components. We attempt to find points on each of these two
│ │ │ │  components. We are successful. This indicates that the corresponding varieties
│ │ │ │  are both rational. Also, if we can find one point, we can find as many as we
│ │ │ │  want.
│ │ │ │  i13 : netList randomPointsOnRationalVariety(compsJ_0, 10)
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -o13 = || 29 -40 15 -49 3 -13 -6 -39 2 39 47 15 19 -47 -46 -39 -16 32 -43 34 -13
│ │ │ │ --18 21 -38 | |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || 37 -7 -24 8 -26 38 9 -31 24 -47 -34 12 16 22 -22 45 -28 16 -47 2 -48 -
│ │ │ │ -34 38 -15 |   |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || 6 1 -31 -7 44 8 -50 24 -48 -16 23 23 -23 39 -5 43 19 -15 48 15 -11 -17
│ │ │ │ -7 47 |       |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || -41 -49 6 -16 -12 31 23 6 -7 11 3 -42 40 11 -28 46 35 -28 -3 33 1 -28
│ │ │ │ --38 36 |      |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || -11 -27 -4 40 -34 6 44 -2 19 -23 -29 21 29 -47 -37 15 -47 -24 -10 2 -
│ │ │ │ -13 -37 -7 22 | |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || -50 42 20 -30 -46 -48 -5 40 -47 39 13 47 32 -9 41 -32 -18 25 -30 -22
│ │ │ │ -24 -20 27 30 | |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || 50 22 -30 3 -43 -29 -33 -18 6 39 -29 24 -49 -33 -15 -19 -15 -37 44 33
│ │ │ │ --20 17 0 -48 ||
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || -9 31 -37 -42 -7 -8 -11 -21 12 9 13 -9 13 -26 11 22 36 34 -8 4 -11 -49
│ │ │ │ --39 -39 |    |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || 47 14 -11 -16 -20 -40 42 5 -2 36 8 -45 -30 41 -26 16 -8 -34 35 -22 -
│ │ │ │ -6 -28 -3 43 |   |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || 23 -8 -3 -17 38 0 11 -33 -7 6 -31 -4 -31 25 6 -2 -35 -11 -13 3 -49 -41
│ │ │ │ -40 -9 |      |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ │ -
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -o14 = || 38 -31 49 39 4 46 -29 -5 -39 -40 14 -11 -31 46 43 -26 4 30 -35 27 -40
│ │ │ │ -37 -47 0 |   |
│ │ │ │ +o13 = || 13 15 3 36 2 48 44 -35 -34 39 5 -32 34 19 -42 -47 -16 -34 -39 -13 -18
│ │ │ │ +-43 21 -38 | |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 -
│ │ │ │ -48 30 -48 0 ||
│ │ │ │ +      || -43 48 14 29 -47 -10 47 22 8 -47 15 -26 2 16 -49 22 -28 -18 45 -48 -34
│ │ │ │ +-47 38 -15 ||
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1
│ │ │ │ -40 -18 0 |    |
│ │ │ │ +      || -3 45 42 47 -50 16 -30 28 43 -16 24 19 15 -23 37 39 19 -8 43 -11 -17
│ │ │ │ +48 7 47 |     |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13
│ │ │ │ -3 13 0 |     |
│ │ │ │ +      || -49 7 32 -6 -30 -41 -10 2 44 11 -25 4 33 40 -19 11 35 -17 46 1 -28 -
│ │ │ │ +3 -38 36 |     |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 -
│ │ │ │ -18 30 0 |    |
│ │ │ │ +      || 35 -48 -2 45 -35 29 34 12 -32 -23 50 2 2 29 -3 -47 -47 -34 15 -13 -37
│ │ │ │ +-10 -7 22 |  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46
│ │ │ │ -12 18 0 |     |
│ │ │ │ +      || 47 8 -14 6 -1 -13 -7 16 -20 39 -34 -22 -22 32 17 -9 -18 -6 -32 24 -20
│ │ │ │ +-30 27 30 |  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39
│ │ │ │ --37 0 |      |
│ │ │ │ +      || -2 -36 -39 41 -6 34 -10 42 5 39 20 33 33 -49 -15 -33 -15 41 -19 -20 17
│ │ │ │ +44 0 -48 |  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28
│ │ │ │ -6 -28 0 |    |
│ │ │ │ +      || -30 37 -9 16 -36 19 -13 -14 -19 9 -33 5 4 13 44 -26 36 -12 22 -11 -49
│ │ │ │ +-8 -39 -39 | |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 -
│ │ │ │ -33 26 0 |    |
│ │ │ │ +      || 27 41 32 -44 40 -20 41 33 28 36 44 31 -22 -30 9 41 -8 30 16 -6 -28 35
│ │ │ │ +-3 43 |      |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5
│ │ │ │ --20 4 0 |   |
│ │ │ │ +      || 37 -2 17 -42 -42 -12 18 -31 33 6 19 -31 3 -31 -11 25 -35 28 -2 -49 -41
│ │ │ │ +-13 40 -9 | |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ +i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ │ +
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +o14 = || -41 -1 -48 25 40 4 35 16 26 -41 -28 -16 27 -14 -39 4 4 30 -40 37 -31 -
│ │ │ │ +35 -47 0 |     |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || -1 19 -3 12 50 3 4 25 48 50 34 -6 -29 6 -5 36 -39 -31 -48 30 47 -37 -
│ │ │ │ +48 0 |          |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || -27 -3 -40 22 27 3 -28 -41 -12 -34 -10 40 46 29 30 24 -49 28 1 40 10 -
│ │ │ │ +22 -18 0 |     |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || -26 -6 24 28 -27 26 34 47 13 50 3 -42 -17 5 4 -35 7 30 -13 3 8 -41 13
│ │ │ │ +0 |            |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || 49 -7 48 1 48 25 25 -10 49 36 -16 35 -46 -5 25 -33 8 -29 49 -18 23 42
│ │ │ │ +30 0 |         |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || -35 28 -6 22 50 -49 2 -5 -11 -39 30 27 -16 34 -9 -34 -28 15 -46 12 27
│ │ │ │ +-18 18 0 |     |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || -49 -44 -16 -10 48 18 22 33 -35 -48 -28 -8 -23 -48 -25 -3 -21 23 44 -
│ │ │ │ +39 19 20 -37 0 ||
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || -33 -14 -18 10 2 -43 -26 45 10 19 -15 25 47 9 -15 -22 0 -47 -28 6 -33
│ │ │ │ +-9 -28 0 |     |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || 20 -27 -17 2 -47 -23 13 40 -19 -13 39 -23 5 -3 47 -6 28 -29 -37 -33 42
│ │ │ │ +-28 26 0 |    |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || 19 10 -10 47 41 20 -43 -34 -43 2 44 29 22 35 -42 16 44 30 5 -20 -29 -
│ │ │ │ +13 4 0 |        |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  This routine expects the input to represent an irreducible variety
│ │ │ │  ********** 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_(_I_d_e_a_l_) -- find a random point on a variety
│ │ │ │        that can be detected to be rational
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _r_a_n_d_o_m_P_o_i_n_t_s_O_n_R_a_t_i_o_n_a_l_V_a_r_i_e_t_y_(_I_d_e_a_l_,_Z_Z_) -- find random points on a
│ │ ├── ./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
│ │ │ - -- 3.00191s elapsed
│ │ │ + -- 2.40056s elapsed
│ │ │  
│ │ │  o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16]
│ │ │  
│ │ │  o5 : GroebnerBasis
│ │ │  
│ │ │  i6 : elapsedTime groebnerWalk(gb I1, R2)
│ │ │ - -- 2.08916s elapsed
│ │ │ + -- 2.00513s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]
│ │ │  
│ │ │  o6 : GroebnerBasis
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/html/index.html
│ │ │ @@ -92,30 +92,30 @@
│ │ │  
│ │ │  o4 : Ideal of R2    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime gb I2
│ │ │ - -- 3.00191s elapsed
│ │ │ + -- 2.40056s 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)
│ │ │ - -- 2.08916s elapsed
│ │ │ + -- 2.00513s 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
│ │ │ │ - -- 3.00191s elapsed
│ │ │ │ + -- 2.40056s 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)
│ │ │ │ - -- 2.08916s elapsed
│ │ │ │ + -- 2.00513s 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/Hadamard/example-output/_hadamard__Power_lp__List_cm__Z__Z_rp.out
│ │ │ @@ -6,20 +6,22 @@
│ │ │  o1 = {Point{1, 1, -}, Point{1, 0, 1}, Point{1, 2, 4}}
│ │ │                    2
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : hadamardPower(L,3)
│ │ │  
│ │ │ -                  1                                                    
│ │ │ -o2 = {Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ -                  4                                                    
│ │ │ +                  1                                                  
│ │ │ +o2 = {Point{1, 0, -}, Point{1, 0, 2}, Point{1, 2, 1}, Point{1, 0, 4},
│ │ │ +                  2                                                  
│ │ │       ------------------------------------------------------------------------
│ │ │ -                                 1                               1
│ │ │ -     Point{1, 0, 1}, Point{1, 1, -}, Point{1, 0, 2}, Point{1, 0, -}, Point{1,
│ │ │ -                                 8                               2
│ │ │ +                 1                                                    
│ │ │ +     Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ +                 4                                                    
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 1}, Point{1, 0, 4}}
│ │ │ +                                 1
│ │ │ +     Point{1, 0, 1}, Point{1, 1, -}}
│ │ │ +                                 8
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Hadamard/example-output/_hadamard__Product_lp__List_cm__List_rp.out
│ │ │ @@ -2,12 +2,12 @@
│ │ │  
│ │ │  i1 : L = {point{0,1}, point{1,2}};
│ │ │  
│ │ │  i2 : M = {point{1,0}, point{2,2}};
│ │ │  
│ │ │  i3 : hadamardProduct(L,M)
│ │ │  
│ │ │ -o3 = {Point{1, 0}, Point{0, 2}, Point{2, 4}}
│ │ │ +o3 = {Point{2, 4}, Point{1, 0}, Point{0, 2}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Power_lp__List_cm__Z__Z_rp.html
│ │ │ @@ -84,23 +84,25 @@
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : hadamardPower(L,3)
│ │ │  
│ │ │ -                  1                                                    
│ │ │ -o2 = {Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ -                  4                                                    
│ │ │ +                  1                                                  
│ │ │ +o2 = {Point{1, 0, -}, Point{1, 0, 2}, Point{1, 2, 1}, Point{1, 0, 4},
│ │ │ +                  2                                                  
│ │ │       ------------------------------------------------------------------------
│ │ │ -                                 1                               1
│ │ │ -     Point{1, 0, 1}, Point{1, 1, -}, Point{1, 0, 2}, Point{1, 0, -}, Point{1,
│ │ │ -                                 8                               2
│ │ │ +                 1                                                    
│ │ │ +     Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ +                 4                                                    
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 1}, Point{1, 0, 4}}
│ │ │ +                                 1
│ │ │ +     Point{1, 0, 1}, Point{1, 1, -}}
│ │ │ +                                 8
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,22 +22,24 @@
│ │ │ │  o1 = {Point{1, 1, -}, Point{1, 0, 1}, Point{1, 2, 4}}
│ │ │ │                    2
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : hadamardPower(L,3)
│ │ │ │  
│ │ │ │                    1
│ │ │ │ -o2 = {Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ │ -                  4
│ │ │ │ +o2 = {Point{1, 0, -}, Point{1, 0, 2}, Point{1, 2, 1}, Point{1, 0, 4},
│ │ │ │ +                  2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -                                 1                               1
│ │ │ │ -     Point{1, 0, 1}, Point{1, 1, -}, Point{1, 0, 2}, Point{1, 0, -}, Point{1,
│ │ │ │ -                                 8                               2
│ │ │ │ +                 1
│ │ │ │ +     Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ │ +                 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     2, 1}, Point{1, 0, 4}}
│ │ │ │ +                                 1
│ │ │ │ +     Point{1, 0, 1}, Point{1, 1, -}}
│ │ │ │ +                                 8
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _h_a_d_a_m_a_r_d_P_o_w_e_r_(_L_i_s_t_,_Z_Z_) -- computes the $r$-th Hadmard powers of a set
│ │ │ │        points
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Product_lp__List_cm__List_rp.html
│ │ │ @@ -83,15 +83,15 @@
│ │ │                
    i2 : M = {point{1,0}, point{2,2}};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : hadamardProduct(L,M)
│ │ │  
│ │ │ -o3 = {Point{1, 0}, Point{0, 2}, Point{2, 4}}
│ │ │ +o3 = {Point{2, 4}, Point{1, 0}, Point{0, 2}}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  Given two sets of points $L$ and $M$ returns the list of (well-defined)
│ │ │ │  entrywise multiplication of pairs of points in the cartesian product $L\times
│ │ │ │  M$.
│ │ │ │  i1 : L = {point{0,1}, point{1,2}};
│ │ │ │  i2 : M = {point{1,0}, point{2,2}};
│ │ │ │  i3 : hadamardProduct(L,M)
│ │ │ │  
│ │ │ │ -o3 = {Point{1, 0}, Point{0, 2}, Point{2, 4}}
│ │ │ │ +o3 = {Point{2, 4}, Point{1, 0}, Point{0, 2}}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _h_a_d_a_m_a_r_d_P_r_o_d_u_c_t_(_L_i_s_t_,_L_i_s_t_) -- Hadamard product of two sets of points
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Hadamard.m2:345:0.
│ │ ├── ./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.
│ │ │ - -- .000004479s elapsed
│ │ │ - -- .000004238s elapsed
│ │ │ - -- .000003977s elapsed
│ │ │ - -- .000001883s elapsed
│ │ │ - -- .000001392s elapsed
│ │ │ + -- .00000821s elapsed
│ │ │ + -- .00000697s elapsed
│ │ │ + -- .000007544s elapsed
│ │ │ + -- .000026503s elapsed
│ │ │ + -- .000008587s 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
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : R = QQ[t_1..t_5];
│ │ │  
│ │ │  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
│ │ │ - -- .000004088s elapsed
│ │ │ + -- .000008661s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │                                                                               
│ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -24,15 +24,15 @@
│ │ │         2    4    0   4    4    1   2    4    2   4    4    3       4
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
│ │ │  
│ │ │  i5 : solveFrobeniusIdeal(I, W)
│ │ │ - -- .00000535s elapsed
│ │ │ + -- .000007972s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │                                                                               
│ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html
│ │ │ @@ -134,19 +134,19 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : netList cssLeadTerm(Hbeta, w)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000004479s elapsed
│ │ │ - -- .000004238s elapsed
│ │ │ - -- .000003977s elapsed
│ │ │ - -- .000001883s elapsed
│ │ │ - -- .000001392s elapsed
│ │ │ + -- .00000821s elapsed
│ │ │ + -- .00000697s elapsed
│ │ │ + -- .000007544s elapsed
│ │ │ + -- .000026503s elapsed
│ │ │ + -- .000008587s 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.
│ │ │ │ - -- .000004479s elapsed
│ │ │ │ - -- .000004238s elapsed
│ │ │ │ - -- .000003977s elapsed
│ │ │ │ - -- .000001883s elapsed
│ │ │ │ - -- .000001392s elapsed
│ │ │ │ + -- .00000821s elapsed
│ │ │ │ + -- .00000697s elapsed
│ │ │ │ + -- .000007544s elapsed
│ │ │ │ + -- .000026503s elapsed
│ │ │ │ + -- .000008587s 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
│ │ │ @@ -84,15 +84,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : solveFrobeniusIdeal I
│ │ │ - -- .000004088s elapsed
│ │ │ + -- .000008661s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │                                                                               
│ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -113,15 +113,15 @@
│ │ │              
│ │ │                
    i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : solveFrobeniusIdeal(I, W)
│ │ │ - -- .00000535s elapsed
│ │ │ + -- .000007972s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │                                                                               
│ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  Here is [_S_S_T, Example 2.3.16]:
│ │ │ │  i1 : R = QQ[t_1..t_5];
│ │ │ │  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
│ │ │ │ - -- .000004088s elapsed
│ │ │ │ + -- .000008661s elapsed
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │  
│ │ │ │  
│ │ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │ │                  0        1        2       3        0       1       2       4
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │         1             1             1             3                 2
│ │ │ │       - -logX logX  - -logX logX  - -logX logX  - -logX logX  + logX }
│ │ │ │         2    4    0   4    4    1   2    4    2   4    4    3       4
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
│ │ │ │  i5 : solveFrobeniusIdeal(I, W)
│ │ │ │ - -- .00000535s elapsed
│ │ │ │ + -- .000007972s elapsed
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │  
│ │ │ │  
│ │ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │ │                  0        1        2       3        0       1       2       4
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/example-output/_bracket.out
│ │ │ @@ -85,19 +85,16 @@
│ │ │  
│ │ │  o13 = 600
│ │ │  
│ │ │  i14 : H' = select(keys H, k->H#k != 0);
│ │ │  
│ │ │  i15 : H'
│ │ │  
│ │ │ -o15 = {({T , T }, T T  + T T  - z*T   + y*T  ), ({T , T }, - T T  + y*T  ),
│ │ │ -          3   7    4 6    3 7      11      13      2   9      2 9      16  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      ({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ -         1   9      5 6    1 9      14      17      4   7      4 7      13  
│ │ │ +o15 = {({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ +          1   9      5 6    1 9      14      17      4   7      4 7      13  
│ │ │        -----------------------------------------------------------------------
│ │ │        ({T , T  }, T T  - T T   + x*T  ), ({T , T }, - T T  + z*T  ), ({T ,
│ │ │           1   10    4 6    1 10      20      5   9      5 9      16      3 
│ │ │        -----------------------------------------------------------------------
│ │ │        T }, T T  - z*T   + x*T  ), ({T , T }, - T T  - T T  - z*T   + x*T  ),
│ │ │         7    3 7      11      12      5   6      5 6    1 9      14      17  
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -151,22 +148,25 @@
│ │ │        -----------------------------------------------------------------------
│ │ │        z*T  ), ({T , T  }, T T  - T T   - z*T   + z*T  ), ({T , T }, T T  +
│ │ │           17      5   10    4 9    5 10      17      19      3   8    2 6  
│ │ │        -----------------------------------------------------------------------
│ │ │        T T  + T T  + y*T   - z*T  ), ({T , T }, T T  + y*T   - z*T  ), ({T ,
│ │ │         3 8    4 9      14      17      3   6    3 6      11      12      5 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  - T T   + z*T   + z*T  )}
│ │ │ -       7      5 7    4 10      12      20
│ │ │ +      T }, - T T  - T T   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ +       7      5 7    4 10      12      20      3   7    4 6    3 7      11  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      y*T  ), ({T , T }, - T T  + y*T  )}
│ │ │ +         13      2   9      2 9      16
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : H#(H'_0)
│ │ │  
│ │ │ -o16 = 1
│ │ │ +o16 = -1
│ │ │  
│ │ │  o16 : S[T ..T  ]
│ │ │           1   99
│ │ │  
│ │ │  i17 : bracketMatrix(A,1,2)
│ │ │  
│ │ │  o17 = | 0    -T_8 -T_6 -T_7 -T_10 |
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/_bracket.html
│ │ │ @@ -215,19 +215,16 @@
│ │ │                
    i14 : H' = select(keys H, k->H#k != 0);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : H'
│ │ │  
│ │ │ -o15 = {({T , T }, T T  + T T  - z*T   + y*T  ), ({T , T }, - T T  + y*T  ),
│ │ │ -          3   7    4 6    3 7      11      13      2   9      2 9      16  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      ({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ -         1   9      5 6    1 9      14      17      4   7      4 7      13  
│ │ │ +o15 = {({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ +          1   9      5 6    1 9      14      17      4   7      4 7      13  
│ │ │        -----------------------------------------------------------------------
│ │ │        ({T , T  }, T T  - T T   + x*T  ), ({T , T }, - T T  + z*T  ), ({T ,
│ │ │           1   10    4 6    1 10      20      5   9      5 9      16      3 
│ │ │        -----------------------------------------------------------------------
│ │ │        T }, T T  - z*T   + x*T  ), ({T , T }, - T T  - T T  - z*T   + x*T  ),
│ │ │         7    3 7      11      12      5   6      5 6    1 9      14      17  
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -281,25 +278,28 @@
│ │ │        -----------------------------------------------------------------------
│ │ │        z*T  ), ({T , T  }, T T  - T T   - z*T   + z*T  ), ({T , T }, T T  +
│ │ │           17      5   10    4 9    5 10      17      19      3   8    2 6  
│ │ │        -----------------------------------------------------------------------
│ │ │        T T  + T T  + y*T   - z*T  ), ({T , T }, T T  + y*T   - z*T  ), ({T ,
│ │ │         3 8    4 9      14      17      3   6    3 6      11      12      5 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  - T T   + z*T   + z*T  )}
│ │ │ -       7      5 7    4 10      12      20
│ │ │ +      T }, - T T  - T T   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ +       7      5 7    4 10      12      20      3   7    4 6    3 7      11  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      y*T  ), ({T , T }, - T T  + y*T  )}
│ │ │ +         13      2   9      2 9      16
│ │ │  
│ │ │  o15 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : H#(H'_0)
│ │ │  
│ │ │ -o16 = 1
│ │ │ +o16 = -1
│ │ │  
│ │ │  o16 : S[T ..T  ]
│ │ │           1   99
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -118,19 +118,16 @@
│ │ │ │  37   38   39   40   41   42   43   44
│ │ │ │  i13 : #keys H
│ │ │ │  
│ │ │ │  o13 = 600
│ │ │ │  i14 : H' = select(keys H, k->H#k != 0);
│ │ │ │  i15 : H'
│ │ │ │  
│ │ │ │ -o15 = {({T , T }, T T  + T T  - z*T   + y*T  ), ({T , T }, - T T  + y*T  ),
│ │ │ │ -          3   7    4 6    3 7      11      13      2   9      2 9      16
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      ({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ │ -         1   9      5 6    1 9      14      17      4   7      4 7      13
│ │ │ │ +o15 = {({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ │ +          1   9      5 6    1 9      14      17      4   7      4 7      13
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        ({T , T  }, T T  - T T   + x*T  ), ({T , T }, - T T  + z*T  ), ({T ,
│ │ │ │           1   10    4 6    1 10      20      5   9      5 9      16      3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        T }, T T  - z*T   + x*T  ), ({T , T }, - T T  - T T  - z*T   + x*T  ),
│ │ │ │         7    3 7      11      12      5   6      5 6    1 9      14      17
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -184,21 +181,24 @@
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        z*T  ), ({T , T  }, T T  - T T   - z*T   + z*T  ), ({T , T }, T T  +
│ │ │ │           17      5   10    4 9    5 10      17      19      3   8    2 6
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        T T  + T T  + y*T   - z*T  ), ({T , T }, T T  + y*T   - z*T  ), ({T ,
│ │ │ │         3 8    4 9      14      17      3   6    3 6      11      12      5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T }, - T T  - T T   + z*T   + z*T  )}
│ │ │ │ -       7      5 7    4 10      12      20
│ │ │ │ +      T }, - T T  - T T   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ │ +       7      5 7    4 10      12      20      3   7    4 6    3 7      11
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      y*T  ), ({T , T }, - T T  + y*T  )}
│ │ │ │ +         13      2   9      2 9      16
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : H#(H'_0)
│ │ │ │  
│ │ │ │ -o16 = 1
│ │ │ │ +o16 = -1
│ │ │ │  
│ │ │ │  o16 : S[T ..T  ]
│ │ │ │           1   99
│ │ │ │  From this we see that [T_5, T_6] sends T_37 to -1 in kk.
│ │ │ │  Another, often simpler view of the pairing is given by _b_r_a_c_k_e_t_M_a_t_r_i_x, where the
│ │ │ │  rows and columns correspond to the generators of Pi^d and Pi^e, and the entries
│ │ │ │  are the bracket products, interpreted as elements of Pi^{d+e}. Note the anti-
│ │ ├── ./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/_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 }
│ │ │ -       2   1   1    2
│ │ │ +o6 = {x  + x , x , x }
│ │ │ +       1    2   2   1
│ │ │  
│ │ │  o6 : Hyperplane Arrangement 
│ │ │  
│ │ │  i7 : ring A3
│ │ │  
│ │ │       ZZ
│ │ │  o7 = --[x ..x ]
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_cone_lp__Arrangement_cm__Ring__Element_rp.html
│ │ │ @@ -167,15 +167,15 @@
│ │ │  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/_type__B_lp__Z__Z_cm__Ring_rp.html
│ │ │ @@ -125,16 +125,16 @@
│ │ │  o5 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : trim A3
│ │ │  
│ │ │ -o6 = {x , x , x  + x }
│ │ │ -       2   1   1    2
│ │ │ +o6 = {x  + x , x , x }
│ │ │ +       1    2   2   1
│ │ │  
│ │ │  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 }
│ │ │ │ -       2   1   1    2
│ │ │ │ +o6 = {x  + x , x , x }
│ │ │ │ +       1    2   2   1
│ │ │ │  
│ │ │ │  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 0.69883s (cpu); 0.430243s (thread); 0s (gc)
│ │ │ + -- used 0.957969s (cpu); 0.462046s (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 0.788474s (cpu); 0.418463s (thread); 0s (gc)
│ │ │ + -- used 1.03599s (cpu); 0.477573s (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 0.874799s (cpu); 0.497563s (thread); 0s (gc)
│ │ │ + -- used 1.1629s (cpu); 0.50279s (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 0.948566s (cpu); 0.526929s (thread); 0s (gc)
│ │ │ + -- used 1.15527s (cpu); 0.526915s (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 1.74854s (cpu); 0.873262s (thread); 0s (gc)
│ │ │ + -- used 2.10417s (cpu); 0.878076s (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 0.549469s (cpu); 0.440596s (thread); 0s (gc)
│ │ │ + -- used 0.867777s (cpu); 0.472968s (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 0.044023s (cpu); 0.0440201s (thread); 0s (gc)
│ │ │ + -- used 0.0576698s (cpu); 0.0576711s (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 0.0436521s (cpu); 0.0436527s (thread); 0s (gc)
│ │ │ + -- used 0.187278s (cpu); 0.0970815s (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 0.0621814s (cpu); 0.0621828s (thread); 0s (gc)
│ │ │ + -- used 0.0812437s (cpu); 0.0812363s (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 0.0426811s (cpu); 0.0426768s (thread); 0s (gc)
│ │ │ + -- used 0.0551067s (cpu); 0.0551075s (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 0.0565581s (cpu); 0.0565591s (thread); 0s (gc)
│ │ │ + -- used 0.234603s (cpu); 0.137397s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
│ │ │  
│ │ │  i57 : icFractions R
│ │ │  
│ │ │ @@ -632,15 +632,15 @@
│ │ │  i66 : R = S/I
│ │ │  
│ │ │  o66 = R
│ │ │  
│ │ │  o66 : QuotientRing
│ │ │  
│ │ │  i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0601817s (cpu); 0.0601808s (thread); 0s (gc)
│ │ │ + -- used 0.113255s (cpu); 0.113035s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
│ │ │  
│ │ │  i68 : icFractions R
│ │ │  
│ │ │ @@ -721,15 +721,15 @@
│ │ │  i77 : R = S/I
│ │ │  
│ │ │  o77 = R
│ │ │  
│ │ │  o77 : QuotientRing
│ │ │  
│ │ │  i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.399687s (cpu); 0.347773s (thread); 0s (gc)
│ │ │ + -- used 0.667243s (cpu); 0.46373s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
│ │ │  
│ │ │  i79 : icFractions R
│ │ │  
│ │ │ @@ -749,15 +749,15 @@
│ │ │  i81 : R = S/sub(I,S)
│ │ │  
│ │ │  o81 = R
│ │ │  
│ │ │  o81 : QuotientRing
│ │ │  
│ │ │  i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.511397s (cpu); 0.373479s (thread); 0s (gc)
│ │ │ + -- used 0.688685s (cpu); 0.475841s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
│ │ │  
│ │ │  i83 : icFractions R
│ │ │  
│ │ │ @@ -777,20 +777,20 @@
│ │ │  i85 : R = S/sub(I,S)
│ │ │  
│ │ │  o85 = R
│ │ │  
│ │ │  o85 : QuotientRing
│ │ │  
│ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time .000568686 sec #minors 4]
│ │ │ + [jacobian time .000716405 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .205692 sec  #fractions 6]
│ │ │ - [step 1:   time .231823 sec  #fractions 6]
│ │ │ - -- used 0.441545s (cpu); 0.311876s (thread); 0s (gc)
│ │ │ + [step 0:   time .279362 sec  #fractions 6]
│ │ │ + [step 1:   time .156924 sec  #fractions 6]
│ │ │ + -- used 0.441098s (cpu); 0.327209s (thread); 0s (gc)
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
│ │ │  
│ │ │  i87 : icFractions R
│ │ │  
│ │ │ @@ -810,20 +810,20 @@
│ │ │  i89 : R = S/sub(I,S)
│ │ │  
│ │ │  o89 = R
│ │ │  
│ │ │  o89 : QuotientRing
│ │ │  
│ │ │  i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time .000531076 sec #minors 4]
│ │ │ + [jacobian time .000670116 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .0900187 sec  #fractions 6]
│ │ │ - [step 1:   time .361551 sec  #fractions 6]
│ │ │ - -- used 0.455508s (cpu); 0.332486s (thread); 0s (gc)
│ │ │ + [step 0:   time .266951 sec  #fractions 6]
│ │ │ + [step 1:   time .382568 sec  #fractions 6]
│ │ │ + -- used 0.65424s (cpu); 0.425013s (thread); 0s (gc)
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
│ │ │  
│ │ │  i91 : icFractions R
│ │ │  
│ │ │ @@ -843,20 +843,20 @@
│ │ │  i93 : R = S/sub(I,S)
│ │ │  
│ │ │  o93 = R
│ │ │  
│ │ │  o93 : QuotientRing
│ │ │  
│ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time .000614342 sec #minors 1]
│ │ │ + [jacobian time .000885563 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .115349 sec  #fractions 6]
│ │ │ - [step 1:   time .476597 sec  #fractions 6]
│ │ │ - -- used 0.595576s (cpu); 0.43448s (thread); 0s (gc)
│ │ │ + [step 0:   time .301729 sec  #fractions 6]
│ │ │ + [step 1:   time .68322 sec  #fractions 6]
│ │ │ + -- used 0.990036s (cpu); 0.590066s (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 .000590017 sec #minors 3]
│ │ │ + [jacobian time .00117774 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00261628 seconds
│ │ │ -      idlizer1:  .00964141 seconds
│ │ │ -      idlizer2:  .00985369 seconds
│ │ │ -      minpres:   .00867646 seconds
│ │ │ -  time .0425597 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00339736 seconds
│ │ │ +      idlizer1:  .0096856 seconds
│ │ │ +      idlizer2:  .0112375 seconds
│ │ │ +      minpres:   .0105021 seconds
│ │ │ +  time .0508319 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00239377 seconds
│ │ │ -      idlizer1:  .0111541 seconds
│ │ │ -      idlizer2:  .00992828 seconds
│ │ │ -      minpres:   .0111938 seconds
│ │ │ -  time .0454337 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00478433 seconds
│ │ │ +      idlizer1:  .0146722 seconds
│ │ │ +      idlizer2:  .0132138 seconds
│ │ │ +      minpres:   .0139497 seconds
│ │ │ +  time .0634958 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) .00239345 seconds
│ │ │ -      idlizer1:  .0115122 seconds
│ │ │ -      idlizer2:  .00971861 seconds
│ │ │ -      minpres:   .00890232 seconds
│ │ │ -  time .0434159 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00303693 seconds
│ │ │ +      idlizer1:  .0164851 seconds
│ │ │ +      idlizer2:  .0143606 seconds
│ │ │ +      minpres:   .0120521 seconds
│ │ │ +  time .0596535 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00252848 seconds
│ │ │ -      idlizer1:  .118442 seconds
│ │ │ -      idlizer2:  .0133284 seconds
│ │ │ -      minpres:   .0156925 seconds
│ │ │ -  time .162067 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00323109 seconds
│ │ │ +      idlizer1:  .160633 seconds
│ │ │ +      idlizer2:  .0178752 seconds
│ │ │ +      minpres:   .0218933 seconds
│ │ │ +  time .220674 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .00282827 seconds
│ │ │ -      idlizer1:  .00924351 seconds
│ │ │ -      idlizer2:  .0162379 seconds
│ │ │ -      minpres:   .0120262 seconds
│ │ │ -  time .0536223 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00364484 seconds
│ │ │ +      idlizer1:  .0115651 seconds
│ │ │ +      idlizer2:  .0189379 seconds
│ │ │ +      minpres:   .015161 seconds
│ │ │ +  time .0667249 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00232149 seconds
│ │ │ -      idlizer1:  .00789534 seconds
│ │ │ -  time .0169089 sec  #fractions 5]
│ │ │ - -- used 0.368261s (cpu); 0.301585s (thread); 0s (gc)
│ │ │ +      radical (use minprimes) .00347184 seconds
│ │ │ +      idlizer1:  .0127651 seconds
│ │ │ +  time .0259012 sec  #fractions 5]
│ │ │ + -- used 0.49625s (cpu); 0.407521s (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 1.00532s (cpu); 0.717095s (thread); 0s (gc)
│ │ │ + -- used 1.81097s (cpu); 1.03007s (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 0.637826s (cpu); 0.4847s (thread); 0s (gc)
│ │ │ + -- used 1.45146s (cpu); 0.722064s (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
│ │ │ @@ -99,15 +99,15 @@
│ │ │  
│ │ │  o3 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time R' = integralClosure R
│ │ │ - -- used 0.69883s (cpu); 0.430243s (thread); 0s (gc)
│ │ │ + -- used 0.957969s (cpu); 0.462046s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = R'
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -186,15 +186,15 @@
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.788474s (cpu); 0.418463s (thread); 0s (gc)
│ │ │ + -- used 1.03599s (cpu); 0.477573s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = R'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -273,15 +273,15 @@
│ │ │  
│ │ │  o15 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.874799s (cpu); 0.497563s (thread); 0s (gc)
│ │ │ + -- used 1.1629s (cpu); 0.50279s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = R'
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -348,15 +348,15 @@
│ │ │  
│ │ │  o20 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ - -- used 0.948566s (cpu); 0.526929s (thread); 0s (gc)
│ │ │ + -- used 1.15527s (cpu); 0.526915s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -423,15 +423,15 @@
│ │ │  
│ │ │  o25 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 1.74854s (cpu); 0.873262s (thread); 0s (gc)
│ │ │ + -- used 2.10417s (cpu); 0.878076s (thread); 0s (gc)
│ │ │  
│ │ │  o26 = R'
│ │ │  
│ │ │  o26 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -498,15 +498,15 @@
│ │ │  
│ │ │  o30 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.549469s (cpu); 0.440596s (thread); 0s (gc)
│ │ │ + -- used 0.867777s (cpu); 0.472968s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = R'
│ │ │  
│ │ │  o31 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -573,15 +573,15 @@
│ │ │  
│ │ │  o35 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i36 : time R' = integralClosure R
│ │ │ - -- used 0.044023s (cpu); 0.0440201s (thread); 0s (gc)
│ │ │ + -- used 0.0576698s (cpu); 0.0576711s (thread); 0s (gc)
│ │ │  
│ │ │  o36 = R'
│ │ │  
│ │ │  o36 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -643,15 +643,15 @@
│ │ │  
│ │ │  o40 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0436521s (cpu); 0.0436527s (thread); 0s (gc)
│ │ │ + -- used 0.187278s (cpu); 0.0970815s (thread); 0s (gc)
│ │ │  
│ │ │  o41 = R'
│ │ │  
│ │ │  o41 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -695,15 +695,15 @@
│ │ │  
│ │ │  o45 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.0621814s (cpu); 0.0621828s (thread); 0s (gc)
│ │ │ + -- used 0.0812437s (cpu); 0.0812363s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = R'
│ │ │  
│ │ │  o46 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -746,15 +746,15 @@
│ │ │  
│ │ │  o50 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.0426811s (cpu); 0.0426768s (thread); 0s (gc)
│ │ │ + -- used 0.0551067s (cpu); 0.0551075s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = R'
│ │ │  
│ │ │  o51 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -798,15 +798,15 @@
│ │ │  
│ │ │  o55 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.0565581s (cpu); 0.0565591s (thread); 0s (gc)
│ │ │ + -- used 0.234603s (cpu); 0.137397s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -932,15 +932,15 @@
│ │ │  
│ │ │  o66 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0601817s (cpu); 0.0601808s (thread); 0s (gc)
│ │ │ + -- used 0.113255s (cpu); 0.113035s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1056,15 +1056,15 @@
│ │ │  
│ │ │  o77 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.399687s (cpu); 0.347773s (thread); 0s (gc)
│ │ │ + -- used 0.667243s (cpu); 0.46373s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1098,15 +1098,15 @@
│ │ │  
│ │ │  o81 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.511397s (cpu); 0.373479s (thread); 0s (gc)
│ │ │ + -- used 0.688685s (cpu); 0.475841s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1140,20 +1140,20 @@
│ │ │  
│ │ │  o85 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time .000568686 sec #minors 4]
│ │ │ + [jacobian time .000716405 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .205692 sec  #fractions 6]
│ │ │ - [step 1:   time .231823 sec  #fractions 6]
│ │ │ - -- used 0.441545s (cpu); 0.311876s (thread); 0s (gc)
│ │ │ + [step 0:   time .279362 sec  #fractions 6]
│ │ │ + [step 1:   time .156924 sec  #fractions 6]
│ │ │ + -- used 0.441098s (cpu); 0.327209s (thread); 0s (gc)
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1187,20 +1187,20 @@
│ │ │  
│ │ │  o89 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time .000531076 sec #minors 4]
│ │ │ + [jacobian time .000670116 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .0900187 sec  #fractions 6]
│ │ │ - [step 1:   time .361551 sec  #fractions 6]
│ │ │ - -- used 0.455508s (cpu); 0.332486s (thread); 0s (gc)
│ │ │ + [step 0:   time .266951 sec  #fractions 6]
│ │ │ + [step 1:   time .382568 sec  #fractions 6]
│ │ │ + -- used 0.65424s (cpu); 0.425013s (thread); 0s (gc)
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1237,20 +1237,20 @@
│ │ │  
│ │ │  o93 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time .000614342 sec #minors 1]
│ │ │ + [jacobian time .000885563 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .115349 sec  #fractions 6]
│ │ │ - [step 1:   time .476597 sec  #fractions 6]
│ │ │ - -- used 0.595576s (cpu); 0.43448s (thread); 0s (gc)
│ │ │ + [step 0:   time .301729 sec  #fractions 6]
│ │ │ + [step 1:   time .68322 sec  #fractions 6]
│ │ │ + -- used 0.990036s (cpu); 0.590066s (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 0.69883s (cpu); 0.430243s (thread); 0s (gc)
│ │ │ │ + -- used 0.957969s (cpu); 0.462046s (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 0.788474s (cpu); 0.418463s (thread); 0s (gc)
│ │ │ │ + -- used 1.03599s (cpu); 0.477573s (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 0.874799s (cpu); 0.497563s (thread); 0s (gc)
│ │ │ │ + -- used 1.1629s (cpu); 0.50279s (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 0.948566s (cpu); 0.526929s (thread); 0s (gc)
│ │ │ │ + -- used 1.15527s (cpu); 0.526915s (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 1.74854s (cpu); 0.873262s (thread); 0s (gc)
│ │ │ │ + -- used 2.10417s (cpu); 0.878076s (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 0.549469s (cpu); 0.440596s (thread); 0s (gc)
│ │ │ │ + -- used 0.867777s (cpu); 0.472968s (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 0.044023s (cpu); 0.0440201s (thread); 0s (gc)
│ │ │ │ + -- used 0.0576698s (cpu); 0.0576711s (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 0.0436521s (cpu); 0.0436527s (thread); 0s (gc)
│ │ │ │ + -- used 0.187278s (cpu); 0.0970815s (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 0.0621814s (cpu); 0.0621828s (thread); 0s (gc)
│ │ │ │ + -- used 0.0812437s (cpu); 0.0812363s (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 0.0426811s (cpu); 0.0426768s (thread); 0s (gc)
│ │ │ │ + -- used 0.0551067s (cpu); 0.0551075s (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 0.0565581s (cpu); 0.0565591s (thread); 0s (gc)
│ │ │ │ + -- used 0.234603s (cpu); 0.137397s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o56 = R'
│ │ │ │  
│ │ │ │  o56 : QuotientRing
│ │ │ │  i57 : icFractions R
│ │ │ │  
│ │ │ │         b*d
│ │ │ │ @@ -754,15 +754,15 @@
│ │ │ │  o65 : BettiTally
│ │ │ │  i66 : R = S/I
│ │ │ │  
│ │ │ │  o66 = R
│ │ │ │  
│ │ │ │  o66 : QuotientRing
│ │ │ │  i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.0601817s (cpu); 0.0601808s (thread); 0s (gc)
│ │ │ │ + -- used 0.113255s (cpu); 0.113035s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o67 = R'
│ │ │ │  
│ │ │ │  o67 : QuotientRing
│ │ │ │  i68 : icFractions R
│ │ │ │  
│ │ │ │          2    2
│ │ │ │ @@ -838,15 +838,15 @@
│ │ │ │  o76 : BettiTally
│ │ │ │  i77 : R = S/I
│ │ │ │  
│ │ │ │  o77 = R
│ │ │ │  
│ │ │ │  o77 : QuotientRing
│ │ │ │  i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.399687s (cpu); 0.347773s (thread); 0s (gc)
│ │ │ │ + -- used 0.667243s (cpu); 0.46373s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o78 = R'
│ │ │ │  
│ │ │ │  o78 : QuotientRing
│ │ │ │  i79 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -862,15 +862,15 @@
│ │ │ │  o80 : PolynomialRing
│ │ │ │  i81 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o81 = R
│ │ │ │  
│ │ │ │  o81 : QuotientRing
│ │ │ │  i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ │ - -- used 0.511397s (cpu); 0.373479s (thread); 0s (gc)
│ │ │ │ + -- used 0.688685s (cpu); 0.475841s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o82 = R'
│ │ │ │  
│ │ │ │  o82 : QuotientRing
│ │ │ │  i83 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -886,20 +886,20 @@
│ │ │ │  o84 : PolynomialRing
│ │ │ │  i85 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o85 = R
│ │ │ │  
│ │ │ │  o85 : QuotientRing
│ │ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ │ - [jacobian time .000568686 sec #minors 4]
│ │ │ │ + [jacobian time .000716405 sec #minors 4]
│ │ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │ - [step 0:   time .205692 sec  #fractions 6]
│ │ │ │ - [step 1:   time .231823 sec  #fractions 6]
│ │ │ │ - -- used 0.441545s (cpu); 0.311876s (thread); 0s (gc)
│ │ │ │ + [step 0:   time .279362 sec  #fractions 6]
│ │ │ │ + [step 1:   time .156924 sec  #fractions 6]
│ │ │ │ + -- used 0.441098s (cpu); 0.327209s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o86 = R'
│ │ │ │  
│ │ │ │  o86 : QuotientRing
│ │ │ │  i87 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -915,20 +915,20 @@
│ │ │ │  o88 : PolynomialRing
│ │ │ │  i89 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o89 = R
│ │ │ │  
│ │ │ │  o89 : QuotientRing
│ │ │ │  i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ │ - [jacobian time .000531076 sec #minors 4]
│ │ │ │ + [jacobian time .000670116 sec #minors 4]
│ │ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │ - [step 0:   time .0900187 sec  #fractions 6]
│ │ │ │ - [step 1:   time .361551 sec  #fractions 6]
│ │ │ │ - -- used 0.455508s (cpu); 0.332486s (thread); 0s (gc)
│ │ │ │ + [step 0:   time .266951 sec  #fractions 6]
│ │ │ │ + [step 1:   time .382568 sec  #fractions 6]
│ │ │ │ + -- used 0.65424s (cpu); 0.425013s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o90 = R'
│ │ │ │  
│ │ │ │  o90 : QuotientRing
│ │ │ │  i91 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -947,20 +947,20 @@
│ │ │ │  i93 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o93 = R
│ │ │ │  
│ │ │ │  o93 : QuotientRing
│ │ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos,
│ │ │ │  StartWithOneMinor}, Verbosity => 1)
│ │ │ │ - [jacobian time .000614342 sec #minors 1]
│ │ │ │ + [jacobian time .000885563 sec #minors 1]
│ │ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │ - [step 0:   time .115349 sec  #fractions 6]
│ │ │ │ - [step 1:   time .476597 sec  #fractions 6]
│ │ │ │ - -- used 0.595576s (cpu); 0.43448s (thread); 0s (gc)
│ │ │ │ + [step 0:   time .301729 sec  #fractions 6]
│ │ │ │ + [step 1:   time .68322 sec  #fractions 6]
│ │ │ │ + -- used 0.990036s (cpu); 0.590066s (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
│ │ │ @@ -71,52 +71,52 @@
│ │ │              
│ │ │                
    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 .000590017 sec #minors 3]
│ │ │ + [jacobian time .00117774 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00261628 seconds
│ │ │ -      idlizer1:  .00964141 seconds
│ │ │ -      idlizer2:  .00985369 seconds
│ │ │ -      minpres:   .00867646 seconds
│ │ │ -  time .0425597 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00339736 seconds
│ │ │ +      idlizer1:  .0096856 seconds
│ │ │ +      idlizer2:  .0112375 seconds
│ │ │ +      minpres:   .0105021 seconds
│ │ │ +  time .0508319 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00239377 seconds
│ │ │ -      idlizer1:  .0111541 seconds
│ │ │ -      idlizer2:  .00992828 seconds
│ │ │ -      minpres:   .0111938 seconds
│ │ │ -  time .0454337 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00478433 seconds
│ │ │ +      idlizer1:  .0146722 seconds
│ │ │ +      idlizer2:  .0132138 seconds
│ │ │ +      minpres:   .0139497 seconds
│ │ │ +  time .0634958 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) .00239345 seconds
│ │ │ -      idlizer1:  .0115122 seconds
│ │ │ -      idlizer2:  .00971861 seconds
│ │ │ -      minpres:   .00890232 seconds
│ │ │ -  time .0434159 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00303693 seconds
│ │ │ +      idlizer1:  .0164851 seconds
│ │ │ +      idlizer2:  .0143606 seconds
│ │ │ +      minpres:   .0120521 seconds
│ │ │ +  time .0596535 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00252848 seconds
│ │ │ -      idlizer1:  .118442 seconds
│ │ │ -      idlizer2:  .0133284 seconds
│ │ │ -      minpres:   .0156925 seconds
│ │ │ -  time .162067 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00323109 seconds
│ │ │ +      idlizer1:  .160633 seconds
│ │ │ +      idlizer2:  .0178752 seconds
│ │ │ +      minpres:   .0218933 seconds
│ │ │ +  time .220674 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .00282827 seconds
│ │ │ -      idlizer1:  .00924351 seconds
│ │ │ -      idlizer2:  .0162379 seconds
│ │ │ -      minpres:   .0120262 seconds
│ │ │ -  time .0536223 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00364484 seconds
│ │ │ +      idlizer1:  .0115651 seconds
│ │ │ +      idlizer2:  .0189379 seconds
│ │ │ +      minpres:   .015161 seconds
│ │ │ +  time .0667249 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00232149 seconds
│ │ │ -      idlizer1:  .00789534 seconds
│ │ │ -  time .0169089 sec  #fractions 5]
│ │ │ - -- used 0.368261s (cpu); 0.301585s (thread); 0s (gc)
│ │ │ +      radical (use minprimes) .00347184 seconds
│ │ │ +      idlizer1:  .0127651 seconds
│ │ │ +  time .0259012 sec  #fractions 5]
│ │ │ + -- used 0.49625s (cpu); 0.407521s (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 .000590017 sec #minors 3]
│ │ │ │ + [jacobian time .00117774 sec #minors 3]
│ │ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │   [step 0:
│ │ │ │ -      radical (use minprimes) .00261628 seconds
│ │ │ │ -      idlizer1:  .00964141 seconds
│ │ │ │ -      idlizer2:  .00985369 seconds
│ │ │ │ -      minpres:   .00867646 seconds
│ │ │ │ -  time .0425597 sec  #fractions 4]
│ │ │ │ +      radical (use minprimes) .00339736 seconds
│ │ │ │ +      idlizer1:  .0096856 seconds
│ │ │ │ +      idlizer2:  .0112375 seconds
│ │ │ │ +      minpres:   .0105021 seconds
│ │ │ │ +  time .0508319 sec  #fractions 4]
│ │ │ │   [step 1:
│ │ │ │ -      radical (use minprimes) .00239377 seconds
│ │ │ │ -      idlizer1:  .0111541 seconds
│ │ │ │ -      idlizer2:  .00992828 seconds
│ │ │ │ -      minpres:   .0111938 seconds
│ │ │ │ -  time .0454337 sec  #fractions 4]
│ │ │ │ +      radical (use minprimes) .00478433 seconds
│ │ │ │ +      idlizer1:  .0146722 seconds
│ │ │ │ +      idlizer2:  .0132138 seconds
│ │ │ │ +      minpres:   .0139497 seconds
│ │ │ │ +  time .0634958 sec  #fractions 4]
│ │ │ │   [step 2:
│ │ │ │ -      radical (use minprimes) .00239345 seconds
│ │ │ │ -      idlizer1:  .0115122 seconds
│ │ │ │ -      idlizer2:  .00971861 seconds
│ │ │ │ -      minpres:   .00890232 seconds
│ │ │ │ -  time .0434159 sec  #fractions 5]
│ │ │ │ +      radical (use minprimes) .00303693 seconds
│ │ │ │ +      idlizer1:  .0164851 seconds
│ │ │ │ +      idlizer2:  .0143606 seconds
│ │ │ │ +      minpres:   .0120521 seconds
│ │ │ │ +  time .0596535 sec  #fractions 5]
│ │ │ │   [step 3:
│ │ │ │ -      radical (use minprimes) .00252848 seconds
│ │ │ │ -      idlizer1:  .118442 seconds
│ │ │ │ -      idlizer2:  .0133284 seconds
│ │ │ │ -      minpres:   .0156925 seconds
│ │ │ │ -  time .162067 sec  #fractions 5]
│ │ │ │ +      radical (use minprimes) .00323109 seconds
│ │ │ │ +      idlizer1:  .160633 seconds
│ │ │ │ +      idlizer2:  .0178752 seconds
│ │ │ │ +      minpres:   .0218933 seconds
│ │ │ │ +  time .220674 sec  #fractions 5]
│ │ │ │   [step 4:
│ │ │ │ -      radical (use minprimes) .00282827 seconds
│ │ │ │ -      idlizer1:  .00924351 seconds
│ │ │ │ -      idlizer2:  .0162379 seconds
│ │ │ │ -      minpres:   .0120262 seconds
│ │ │ │ -  time .0536223 sec  #fractions 5]
│ │ │ │ +      radical (use minprimes) .00364484 seconds
│ │ │ │ +      idlizer1:  .0115651 seconds
│ │ │ │ +      idlizer2:  .0189379 seconds
│ │ │ │ +      minpres:   .015161 seconds
│ │ │ │ +  time .0667249 sec  #fractions 5]
│ │ │ │   [step 5:
│ │ │ │ -      radical (use minprimes) .00232149 seconds
│ │ │ │ -      idlizer1:  .00789534 seconds
│ │ │ │ -  time .0169089 sec  #fractions 5]
│ │ │ │ - -- used 0.368261s (cpu); 0.301585s (thread); 0s (gc)
│ │ │ │ +      radical (use minprimes) .00347184 seconds
│ │ │ │ +      idlizer1:  .0127651 seconds
│ │ │ │ +  time .0259012 sec  #fractions 5]
│ │ │ │ + -- used 0.49625s (cpu); 0.407521s (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
│ │ │ @@ -109,29 +109,29 @@
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time integralClosure J
│ │ │ - -- used 1.00532s (cpu); 0.717095s (thread); 0s (gc)
│ │ │ + -- used 1.81097s (cpu); 1.03007s (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 0.637826s (cpu); 0.4847s (thread); 0s (gc)
│ │ │ + -- used 1.45146s (cpu); 0.722064s (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 1.00532s (cpu); 0.717095s (thread); 0s (gc)
│ │ │ │ + -- used 1.81097s (cpu); 1.03007s (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 0.637826s (cpu); 0.4847s (thread); 0s (gc)
│ │ │ │ + -- used 1.45146s (cpu); 0.722064s (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/example-output/_equivariant__Hilbert.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o3 : DiagonalAction
│ │ │  
│ │ │  i4 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o4 = false
│ │ │  
│ │ │  i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00274702s elapsed
│ │ │ + -- .00317537s elapsed
│ │ │  
│ │ │                    -1    -1       2 2              -2    -1 -1    -2  2  
│ │ │  o5 = 1 + (z z  + z   + z  )T + (z z  + z  + z  + z   + z  z   + z  )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
│ │ │       (z z  + z z  + z z  + z z   + 1 + z   + z  z  + z  z   + z  z   + z  )T 
│ │ │ @@ -51,10 +51,10 @@
│ │ │           0   1
│ │ │  
│ │ │  i6 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .000427798s elapsed
│ │ │ + -- .000606858s 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 0.797659s (cpu); 0.519949s (thread); 0s (gc)
│ │ │ + -- used 0.986781s (cpu); 0.63802s (thread); 0s (gc)
│ │ │  
│ │ │         2   2    2   3       3
│ │ │  o4 = {z , x  + y , x y - x*y }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.671422s (cpu); 0.364464s (thread); 0s (gc)
│ │ │ + -- used 0.940831s (cpu); 0.480619s (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 0.0219117s (cpu); 0.0219168s (thread); 0s (gc)
│ │ │ + -- used 0.0319392s (cpu); 0.0319437s (thread); 0s (gc)
│ │ │  
│ │ │            4    4
│ │ │  o6 = {1, x  + y }
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 2.00343s (cpu); 1.26795s (thread); 0s (gc)
│ │ │ + -- used 2.75814s (cpu); 1.69362s (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
│ │ │ - -- .779956s elapsed
│ │ │ + -- .728545s 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.
│ │ │  
│ │ │ - -- .524625s elapsed
│ │ │ + -- .571132s 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__Use__Linear__Algebra_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
│ │ │ - -- .677194s elapsed
│ │ │ + -- .593653s 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,UseLinearAlgebra=>true)
│ │ │ - -- .104964s elapsed
│ │ │ + -- .0736658s 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/_equivariant__Hilbert.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o4 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00274702s elapsed
│ │ │ + -- .00317537s elapsed
│ │ │  
│ │ │                    -1    -1       2 2              -2    -1 -1    -2  2  
│ │ │  o5 = 1 + (z z  + z   + z  )T + (z z  + z  + z  + z   + z  z   + z  )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
│ │ │       (z z  + z z  + z z  + z z   + 1 + z   + z  z  + z  z   + z  z   + z  )T 
│ │ │ @@ -124,15 +124,15 @@
│ │ │  
│ │ │  o6 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .000427798s elapsed
    
│ │ │ + -- .000606858s 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)
│ │ │ │ - -- .00274702s elapsed
│ │ │ │ + -- .00317537s elapsed
│ │ │ │  
│ │ │ │                    -1    -1       2 2              -2    -1 -1    -2  2
│ │ │ │  o5 = 1 + (z z  + z   + z  )T + (z z  + z  + z  + z   + z  z   + z  )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
│ │ │ │       (z z  + z z  + z z  + z z   + 1 + z   + z  z  + z  z   + z  z   + z  )T
│ │ │ │ @@ -54,13 +54,13 @@
│ │ │ │  
│ │ │ │  o5 : ZZ[z ..z ][T]
│ │ │ │           0   1
│ │ │ │  i6 : T.cache.?equivariantHilbert
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ │ - -- .000427798s elapsed
│ │ │ │ + -- .000606858s 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-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/InvariantRing/AbelianGroupsDoc.m2:185:0.
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_hsop_spalgorithms.html
│ │ │ @@ -92,26 +92,26 @@
│ │ │            
│ │ │          
│ │ │          
    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 0.797659s (cpu); 0.519949s (thread); 0s (gc)
│ │ │ + -- used 0.986781s (cpu); 0.63802s (thread); 0s (gc)
│ │ │  
│ │ │         2   2    2   3       3
│ │ │  o4 = {z , x  + y , x y - x*y }
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.671422s (cpu); 0.364464s (thread); 0s (gc)
│ │ │ + -- used 0.940831s (cpu); 0.480619s (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  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -168,26 +168,26 @@
│ │ │            
    
│ │ │          
    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 0.0219117s (cpu); 0.0219168s (thread); 0s (gc)
│ │ │ + -- used 0.0319392s (cpu); 0.0319437s (thread); 0s (gc)
│ │ │  
│ │ │            4    4
│ │ │  o6 = {1, x  + y }
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 2.00343s (cpu); 1.26795s (thread); 0s (gc)
│ │ │ + -- used 2.75814s (cpu); 1.69362s (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 0.797659s (cpu); 0.519949s (thread); 0s (gc)
│ │ │ │ + -- used 0.986781s (cpu); 0.63802s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2   2    2   3       3
│ │ │ │  o4 = {z , x  + y , x y - x*y }
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ │ - -- used 0.671422s (cpu); 0.364464s (thread); 0s (gc)
│ │ │ │ + -- used 0.940831s (cpu); 0.480619s (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 0.0219117s (cpu); 0.0219168s (thread); 0s (gc)
│ │ │ │ + -- used 0.0319392s (cpu); 0.0319437s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            4    4
│ │ │ │  o6 = {1, x  + y }
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ │ - -- used 2.00343s (cpu); 1.26795s (thread); 0s (gc)
│ │ │ │ + -- used 2.75814s (cpu); 1.69362s (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
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime invariants S4
│ │ │ - -- .779956s elapsed
│ │ │ + -- .728545s 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 }
│ │ │ @@ -117,15 +117,15 @@
│ │ │              
    		
│ │ │                
    i5 : elapsedTime invariants(S4,DegreeBound=>4)
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │ - -- .524625s elapsed
│ │ │ + -- .571132s 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
│ │ │ │ - -- .779956s elapsed
│ │ │ │ + -- .728545s 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.
│ │ │ │  
│ │ │ │ - -- .524625s elapsed
│ │ │ │ + -- .571132s 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__Use__Linear__Algebra_eq_gt..._rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime invariants S4
│ │ │ - -- .677194s elapsed
│ │ │ + -- .593653s 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 }
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime invariants(S4,UseLinearAlgebra=>true)
│ │ │ - -- .104964s elapsed
│ │ │ + -- .0736658s 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 {}
│ │ │ │ @@ -35,27 +35,27 @@
│ │ │ │  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
│ │ │ │ - -- .677194s elapsed
│ │ │ │ + -- .593653s 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,UseLinearAlgebra=>true)
│ │ │ │ - -- .104964s elapsed
│ │ │ │ + -- .0736658s 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/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_2 x_1 x_0^2 |
│ │ │                     {-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_3 x_2 x_1^2 |
│ │ │  
│ │ │                                  40
│ │ │  o22 : S-module, subquotient of S
│ │ │  
│ │ │  i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ - -- 1.4004s elapsed
│ │ │ + -- 1.62622s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : elapsedTime isomorphism(T1, T2)
│ │ │ - -- .00002087s elapsed
│ │ │ + -- .000064127s 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
│ │ │ @@ -328,23 +328,23 @@
│ │ │                                  40
│ │ │  o22 : S-module, subquotient of S
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ - -- 1.4004s elapsed
│ │ │ + -- 1.62622s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime isomorphism(T1, T2)
│ │ │ - -- .00002087s elapsed
│ │ │ + -- .000064127s 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 {}
│ │ │ │ @@ -684,19 +684,19 @@
│ │ │ │  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     0   0   0     0   0   0     0   0   0
│ │ │ │  0   0   0     x_3 x_2 x_1^2 |
│ │ │ │  
│ │ │ │                                  40
│ │ │ │  o22 : S-module, subquotient of S
│ │ │ │  i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ │ - -- 1.4004s elapsed
│ │ │ │ + -- 1.62622s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isomorphism(T1, T2)
│ │ │ │ - -- .00002087s elapsed
│ │ │ │ + -- .000064127s 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
│ │ │ @@ -39,19 +39,19 @@
│ │ │  
│ │ │  o8 = {1, 2, 3}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-50412-0/0.json
│ │ │ +o9 = /tmp/M2-79250-0/0.json
│ │ │  
│ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-50412-0/0.json
│ │ │ +o10 = /tmp/M2-79250-0/0.json
│ │ │  
│ │ │  o10 : File
│ │ │  
│ │ │  i11 : fromJSON openIn jsonFile
│ │ │  
│ │ │  o11 = {1, 2, 3}
│ │ ├── ./usr/share/doc/Macaulay2/JSON/html/_from__J__S__O__N.html
│ │ │ @@ -167,22 +167,22 @@
│ │ │            
    The input may also be a file containing JSON data.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-50412-0/0.json
    
│ │ │ +o9 = /tmp/M2-79250-0/0.json
│ │ │              
    
│ │ │                
    i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-50412-0/0.json
│ │ │ +o10 = /tmp/M2-79250-0/0.json
│ │ │  
│ │ │  o10 : File
    
│ │ │              
    
│ │ │                
    i11 : fromJSON openIn jsonFile
│ │ │ ├── html2text {}
│ │ │ │ @@ -53,18 +53,18 @@
│ │ │ │  
│ │ │ │  o8 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  The input may also be a file containing JSON data.
│ │ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-50412-0/0.json
│ │ │ │ +o9 = /tmp/M2-79250-0/0.json
│ │ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-50412-0/0.json
│ │ │ │ +o10 = /tmp/M2-79250-0/0.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)
│ │ │ - -- .00226289s elapsed
│ │ │ + -- .00298849s 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
│ │ │ - -- .297009s elapsed
│ │ │ + -- .277696s 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)
│ │ │ - -- .0247993s elapsed
│ │ │ + -- .0114759s 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)
│ │ │ - -- .0146144s elapsed
│ │ │ + -- .00320824s 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)
│ │ │ - -- .00634335s elapsed
│ │ │ + -- .00286022s 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)
│ │ │ - -- .0116862s elapsed
│ │ │ + -- .0154046s 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)
│ │ │ - -- .000613795s elapsed
│ │ │ + -- .000814701s 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
│ │ │ @@ -87,27 +87,27 @@
│ │ │          
    
│ │ │            
    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)
│ │ │ │ - -- .00226289s elapsed
│ │ │ │ + -- .00298849s 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
│ │ │ │ - -- .297009s elapsed
│ │ │ │ + -- .277696s 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
│ │ │ @@ -117,15 +117,15 @@
│ │ │  
│ │ │  o7 = false
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -146,26 +146,26 @@
│ │ │                                 | x0^2-y0        |
│ │ │                   jetsMaxOrder => 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -290,15 +290,15 @@
│ │ │  
│ │ │  o23 = false
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime jetsRadical(2,I)
│ │ │ - -- .00226289s elapsed
│ │ │ + -- .00298849s 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
│ │ │ - -- .297009s elapsed
│ │ │ + -- .277696s 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)
│ │ │ - -- .0247993s elapsed
│ │ │ + -- .0114759s 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)
│ │ │ - -- .0146144s elapsed
│ │ │ + -- .00320824s 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)
│ │ │ - -- .00634335s elapsed
│ │ │ + -- .00286022s 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)
│ │ │ - -- .0116862s elapsed
│ │ │ + -- .0154046s 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]
│ │ │ @@ -324,15 +324,15 @@
│ │ │                                 | t0 t0^2        |
│ │ │                   jetsMaxOrder => 3
    
│ │ │              
    
│ │ │                
    i27 : elapsedTime jets(2,f)
│ │ │ - -- .000613795s elapsed
│ │ │ + -- .000814701s 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)
│ │ │ │ - -- .0247993s elapsed
│ │ │ │ + -- .0114759s 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)
│ │ │ │ - -- .0146144s elapsed
│ │ │ │ + -- .00320824s 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)
│ │ │ │ - -- .00634335s elapsed
│ │ │ │ + -- .00286022s 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)
│ │ │ │ - -- .0116862s elapsed
│ │ │ │ + -- .0154046s 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)
│ │ │ │ - -- .000613795s elapsed
│ │ │ │ + -- .000814701s 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
│ │ │ - -- .0269396s elapsed
│ │ │ + -- .0305541s 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)
│ │ │ - -- .0024042s elapsed
│ │ │ - -- .00650777s elapsed
│ │ │ - -- .0527064s elapsed
│ │ │ - -- .0736751s elapsed
│ │ │ - -- .016221s elapsed
│ │ │ + -- .00296112s elapsed
│ │ │ + -- .00805075s elapsed
│ │ │ + -- .0275992s elapsed
│ │ │ + -- .0462432s elapsed
│ │ │ + -- .00436879s 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)
│ │ │ - -- .00223679s elapsed
│ │ │ - -- .00804055s elapsed
│ │ │ - -- .0226273s elapsed
│ │ │ - -- .00963621s elapsed
│ │ │ - -- .00349231s elapsed
│ │ │ - -- .474407s elapsed
│ │ │ + -- .00312888s elapsed
│ │ │ + -- .00768868s elapsed
│ │ │ + -- .0279775s elapsed
│ │ │ + -- .0113343s elapsed
│ │ │ + -- .00439147s elapsed
│ │ │ + -- .400449s 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
│ │ │ - -- .247702s elapsed
│ │ │ + -- .216672s 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);
│ │ │ - -- .00450803s elapsed
│ │ │ - -- .017145s elapsed
│ │ │ - -- .132278s elapsed
│ │ │ - -- 1.14139s elapsed
│ │ │ - -- .517143s elapsed
│ │ │ - -- .0710902s elapsed
│ │ │ - -- .00652267s elapsed
│ │ │ - -- 6.47678s elapsed
│ │ │ + -- .00538201s elapsed
│ │ │ + -- .0208165s elapsed
│ │ │ + -- .121611s elapsed
│ │ │ + -- 1.05198s elapsed
│ │ │ + -- .47187s elapsed
│ │ │ + -- .0569017s elapsed
│ │ │ + -- .00829579s elapsed
│ │ │ + -- 6.28684s 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)
│ │ │ - -- .0103685s elapsed
│ │ │ - -- .00662677s elapsed
│ │ │ - -- .0228435s elapsed
│ │ │ - -- .0337852s elapsed
│ │ │ - -- .00363175s elapsed
│ │ │ + -- .00305476s elapsed
│ │ │ + -- .00733376s elapsed
│ │ │ + -- .0277025s elapsed
│ │ │ + -- .0113175s elapsed
│ │ │ + -- .00420875s 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
│ │ │ - -- .248736s elapsed
│ │ │ + -- .209665s 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);
│ │ │ - -- .00536115s elapsed
│ │ │ - -- .0370005s elapsed
│ │ │ - -- .204983s elapsed
│ │ │ - -- 1.32933s elapsed
│ │ │ - -- .430969s elapsed
│ │ │ - -- .0511654s elapsed
│ │ │ - -- .00654992s elapsed
│ │ │ - -- 6.84149s elapsed
│ │ │ + -- .00543162s elapsed
│ │ │ + -- .0202663s elapsed
│ │ │ + -- .109769s elapsed
│ │ │ + -- 1.14465s elapsed
│ │ │ + -- .473105s elapsed
│ │ │ + -- .0474626s elapsed
│ │ │ + -- .00805537s elapsed
│ │ │ + -- 6.27442s 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)
│ │ │ - -- .00694639s elapsed
│ │ │ - -- .0123481s elapsed
│ │ │ + -- .0080706s elapsed
│ │ │ + -- .0140967s elapsed
│ │ │  (number Of blocks, 26)
│ │ │ - -- .000297184s elapsed
│ │ │ + -- .000283788s elapsed
│ │ │  1
│ │ │ - -- .000143437s elapsed
│ │ │ + -- .000228178s elapsed
│ │ │  1
│ │ │ - -- .000131295s elapsed
│ │ │ + -- .000173857s elapsed
│ │ │  1
│ │ │ - -- .000128579s elapsed
│ │ │ + -- .000179947s elapsed
│ │ │  1
│ │ │ - -- .000143327s elapsed
│ │ │ + -- .000246311s elapsed
│ │ │  2
│ │ │ - -- .000142685s elapsed
│ │ │ + -- .000188789s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000154078s elapsed
│ │ │ + -- .000197423s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000160099s elapsed
│ │ │ + -- .000235954s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000140462s elapsed
│ │ │ + -- .000260644s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .00013971s elapsed
│ │ │ + -- .000183743s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000138028s elapsed
│ │ │ + -- .000170697s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000132678s elapsed
│ │ │ + -- .000167883s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000120584s elapsed
│ │ │ + -- .000173951s elapsed
│ │ │  2
│ │ │ - -- .000122919s elapsed
│ │ │ + -- .000145589s elapsed
│ │ │  2
│ │ │ - -- .000134922s elapsed
│ │ │ + -- .000155031s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .00012317s elapsed
│ │ │ + -- .00016271s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000140812s elapsed
│ │ │ + -- .000180971s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000133859s elapsed
│ │ │ + -- .000168531s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000142265s elapsed
│ │ │ + -- .00017714s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000130654s elapsed
│ │ │ + -- .000195081s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000132116s elapsed
│ │ │ + -- .000172724s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000117308s elapsed
│ │ │ + -- .000153133s elapsed
│ │ │  2
│ │ │ - -- .000119734s elapsed
│ │ │ + -- .000171883s elapsed
│ │ │  1
│ │ │ - -- .000126405s elapsed
│ │ │ + -- .000167409s elapsed
│ │ │  1
│ │ │ - -- .000127438s elapsed
│ │ │ + -- .000163484s elapsed
│ │ │  1
│ │ │ - -- .000133549s elapsed
│ │ │ + -- .000151628s 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
│ │ │ - -- .193447s elapsed
│ │ │ - -- .265378s elapsed
│ │ │ - -- .211761s elapsed
│ │ │ - -- .284175s elapsed
│ │ │ - -- .322238s elapsed
│ │ │ - -- .410165s elapsed
│ │ │ + -- .177918s elapsed
│ │ │ + -- .19592s elapsed
│ │ │ + -- .182155s elapsed
│ │ │ + -- .29462s elapsed
│ │ │ + -- .227293s elapsed
│ │ │ + -- .192633s 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)
│ │ │ - -- .0634893s elapsed
│ │ │ - -- .00657433s elapsed
│ │ │ - -- .0254523s elapsed
│ │ │ - -- .00879327s elapsed
│ │ │ - -- .00336788s elapsed
│ │ │ + -- .0334179s elapsed
│ │ │ + -- .00737552s elapsed
│ │ │ + -- .0302602s elapsed
│ │ │ + -- .00995625s elapsed
│ │ │ + -- .00408811s 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)
│ │ │ - -- .309166s elapsed
│ │ │ + -- .257533s 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)
│ │ │ - -- .00250595s elapsed
│ │ │ - -- .00659358s elapsed
│ │ │ - -- .0231823s elapsed
│ │ │ - -- .0102019s elapsed
│ │ │ - -- .00377573s elapsed
│ │ │ + -- .00304241s elapsed
│ │ │ + -- .00755411s elapsed
│ │ │ + -- .0305028s elapsed
│ │ │ + -- .0162895s elapsed
│ │ │ + -- .00449834s 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)
│ │ │ - -- .0177477s elapsed
│ │ │ + -- .0210588s 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
│ │ │ - -- .000051325s elapsed
│ │ │ + -- .000055192s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . 2
│ │ │ - -- .000095639s elapsed
│ │ │ + -- .000119865s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . .
│ │ │      9: . 2
│ │ │ - -- .000076743s elapsed
│ │ │ + -- .000083476s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │      7: 2 .
│ │ │      8: 1 .
│ │ │      9: . 1
│ │ │     10: . 2
│ │ │ - -- .000079969s elapsed
│ │ │ + -- .000106085s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (-3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      7: 1 .
│ │ │      8: 1 .
│ │ │      9: 2 2
│ │ │     10: . 1
│ │ │     11: . 1
│ │ │ - -- .000083625s elapsed
│ │ │ + -- .000107413s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      8: 1 .
│ │ │      9: 2 1
│ │ │     10: 1 2
│ │ │     11: . 1
│ │ │ - -- .000093434s elapsed
│ │ │ + -- .00010274s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      9: 1 1
│ │ │ - -- .000024326s elapsed
│ │ │ + -- .000032908s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      9: 1 1
│ │ │     10: 1 1
│ │ │ - -- .000067026s elapsed
│ │ │ + -- .000083798s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .00009138s elapsed
│ │ │ + -- .000112368s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 2 1
│ │ │     11: 1 2
│ │ │     12: . 1
│ │ │ - -- .000095969s elapsed
│ │ │ + -- .000115386s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 1 .
│ │ │     11: 2 2
│ │ │     12: . 1
│ │ │     13: . 1
│ │ │ - -- .000090488s elapsed
│ │ │ + -- .000111017s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000081492s elapsed
│ │ │ + -- .000105723s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │     10: 2 .
│ │ │     11: 1 .
│ │ │     12: . 1
│ │ │     13: . 2
│ │ │ - -- .00009612s elapsed
│ │ │ + -- .000112633s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     10: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000065421s elapsed
│ │ │ + -- .000095768s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     11: 2 .
│ │ │     12: . .
│ │ │     13: . 2
│ │ │ - -- .000077685s elapsed
│ │ │ + -- .000086287s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000024636s elapsed
│ │ │ + -- .000034508s elapsed
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     12: 2 .
│ │ │     13: . 2
│ │ │ - -- .000076392s elapsed
│ │ │ + -- .00009826s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     13: 1 1
│ │ │ - -- .000027041s elapsed
│ │ │ + -- .000032758s elapsed
│ │ │  (e )
│ │ │    1
│ │ │  
│ │ │         6      32    32
│ │ │  o2 = (3 , (e )  (e )  )
│ │ │              1     2
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o3 : Complex
    
│ │ │              
    
│ │ │                
    i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .0269396s elapsed
│ │ │ + -- .0305541s elapsed
│ │ │  
│ │ │  o5 = 350
    
│ │ │              
    
│ │ │                
    i6 : betti F_a, betti F
│ │ │ @@ -141,19 +141,19 @@
│ │ │  
│ │ │  o9 = 14
    
│ │ │              
    
│ │ │                
    i10 : carpetBettiTable(a,b,3)
│ │ │ - -- .0024042s elapsed
│ │ │ - -- .00650777s elapsed
│ │ │ - -- .0527064s elapsed
│ │ │ - -- .0736751s elapsed
│ │ │ - -- .016221s elapsed
│ │ │ + -- .00296112s elapsed
│ │ │ + -- .00805075s elapsed
│ │ │ + -- .0275992s elapsed
│ │ │ + -- .0462432s elapsed
│ │ │ + -- .00436879s 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
│ │ │ │ - -- .0269396s elapsed
│ │ │ │ + -- .0305541s 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)
│ │ │ │ - -- .0024042s elapsed
│ │ │ │ - -- .00650777s elapsed
│ │ │ │ - -- .0527064s elapsed
│ │ │ │ - -- .0736751s elapsed
│ │ │ │ - -- .016221s elapsed
│ │ │ │ + -- .00296112s elapsed
│ │ │ │ + -- .00805075s elapsed
│ │ │ │ + -- .0275992s elapsed
│ │ │ │ + -- .0462432s elapsed
│ │ │ │ + -- .00436879s 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
│ │ │ @@ -83,20 +83,20 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ - -- .00223679s elapsed
│ │ │ - -- .00804055s elapsed
│ │ │ - -- .0226273s elapsed
│ │ │ - -- .00963621s elapsed
│ │ │ - -- .00349231s elapsed
│ │ │ - -- .474407s elapsed
│ │ │ + -- .00312888s elapsed
│ │ │ + -- .00768868s elapsed
│ │ │ + -- .0279775s elapsed
│ │ │ + -- .0113343s elapsed
│ │ │ + -- .00439147s elapsed
│ │ │ + -- .400449s 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
│ │ │ @@ -112,15 +112,15 @@
│ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime T'=minimalBetti J
│ │ │ - -- .247702s elapsed
│ │ │ + -- .216672s 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
│ │ │ @@ -140,22 +140,22 @@
│ │ │  
│ │ │  o5 : BettiTally
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00450803s elapsed
│ │ │ - -- .017145s elapsed
│ │ │ - -- .132278s elapsed
│ │ │ - -- 1.14139s elapsed
│ │ │ - -- .517143s elapsed
│ │ │ - -- .0710902s elapsed
│ │ │ - -- .00652267s elapsed
│ │ │ - -- 6.47678s elapsed
    
│ │ │ + -- .00538201s elapsed
│ │ │ + -- .0208165s elapsed
│ │ │ + -- .121611s elapsed
│ │ │ + -- 1.05198s elapsed
│ │ │ + -- .47187s elapsed
│ │ │ + -- .0569017s elapsed
│ │ │ + -- .00829579s elapsed
│ │ │ + -- 6.28684s 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)
│ │ │ │ - -- .00223679s elapsed
│ │ │ │ - -- .00804055s elapsed
│ │ │ │ - -- .0226273s elapsed
│ │ │ │ - -- .00963621s elapsed
│ │ │ │ - -- .00349231s elapsed
│ │ │ │ - -- .474407s elapsed
│ │ │ │ + -- .00312888s elapsed
│ │ │ │ + -- .00768868s elapsed
│ │ │ │ + -- .0279775s elapsed
│ │ │ │ + -- .0113343s elapsed
│ │ │ │ + -- .00439147s elapsed
│ │ │ │ + -- .400449s 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
│ │ │ │ - -- .247702s elapsed
│ │ │ │ + -- .216672s 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);
│ │ │ │ - -- .00450803s elapsed
│ │ │ │ - -- .017145s elapsed
│ │ │ │ - -- .132278s elapsed
│ │ │ │ - -- 1.14139s elapsed
│ │ │ │ - -- .517143s elapsed
│ │ │ │ - -- .0710902s elapsed
│ │ │ │ - -- .00652267s elapsed
│ │ │ │ - -- 6.47678s elapsed
│ │ │ │ + -- .00538201s elapsed
│ │ │ │ + -- .0208165s elapsed
│ │ │ │ + -- .121611s elapsed
│ │ │ │ + -- 1.05198s elapsed
│ │ │ │ + -- .47187s elapsed
│ │ │ │ + -- .0569017s elapsed
│ │ │ │ + -- .00829579s elapsed
│ │ │ │ + -- 6.28684s 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
│ │ │ @@ -80,19 +80,19 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : h=carpetBettiTables(a,b)
│ │ │ - -- .0103685s elapsed
│ │ │ - -- .00662677s elapsed
│ │ │ - -- .0228435s elapsed
│ │ │ - -- .0337852s elapsed
│ │ │ - -- .00363175s elapsed
│ │ │ + -- .00305476s elapsed
│ │ │ + -- .00733376s elapsed
│ │ │ + -- .0277025s elapsed
│ │ │ + -- .0113175s elapsed
│ │ │ + -- .00420875s 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
│ │ │ @@ -134,15 +134,15 @@
│ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime T'=minimalBetti J
│ │ │ - -- .248736s elapsed
│ │ │ + -- .209665s 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
│ │ │ @@ -162,22 +162,22 @@
│ │ │  
│ │ │  o6 : BettiTally
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00536115s elapsed
│ │ │ - -- .0370005s elapsed
│ │ │ - -- .204983s elapsed
│ │ │ - -- 1.32933s elapsed
│ │ │ - -- .430969s elapsed
│ │ │ - -- .0511654s elapsed
│ │ │ - -- .00654992s elapsed
│ │ │ - -- 6.84149s elapsed
    
│ │ │ + -- .00543162s elapsed
│ │ │ + -- .0202663s elapsed
│ │ │ + -- .109769s elapsed
│ │ │ + -- 1.14465s elapsed
│ │ │ + -- .473105s elapsed
│ │ │ + -- .0474626s elapsed
│ │ │ + -- .00805537s elapsed
│ │ │ + -- 6.27442s 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)
│ │ │ │ - -- .0103685s elapsed
│ │ │ │ - -- .00662677s elapsed
│ │ │ │ - -- .0228435s elapsed
│ │ │ │ - -- .0337852s elapsed
│ │ │ │ - -- .00363175s elapsed
│ │ │ │ + -- .00305476s elapsed
│ │ │ │ + -- .00733376s elapsed
│ │ │ │ + -- .0277025s elapsed
│ │ │ │ + -- .0113175s elapsed
│ │ │ │ + -- .00420875s 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
│ │ │ │ - -- .248736s elapsed
│ │ │ │ + -- .209665s 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);
│ │ │ │ - -- .00536115s elapsed
│ │ │ │ - -- .0370005s elapsed
│ │ │ │ - -- .204983s elapsed
│ │ │ │ - -- 1.32933s elapsed
│ │ │ │ - -- .430969s elapsed
│ │ │ │ - -- .0511654s elapsed
│ │ │ │ - -- .00654992s elapsed
│ │ │ │ - -- 6.84149s elapsed
│ │ │ │ + -- .00543162s elapsed
│ │ │ │ + -- .0202663s elapsed
│ │ │ │ + -- .109769s elapsed
│ │ │ │ + -- 1.14465s elapsed
│ │ │ │ + -- .473105s elapsed
│ │ │ │ + -- .0474626s elapsed
│ │ │ │ + -- .00805537s elapsed
│ │ │ │ + -- 6.27442s elapsed
│ │ │ │  i8 : keys h
│ │ │ │  
│ │ │ │  o8 = {0, 2, 3, 5}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  i9 : carpetBettiTable(h,7)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html
│ │ │ @@ -80,82 +80,82 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : d=carpetDet(a,b)
│ │ │ - -- .00694639s elapsed
│ │ │ - -- .0123481s elapsed
│ │ │ + -- .0080706s elapsed
│ │ │ + -- .0140967s elapsed
│ │ │  (number Of blocks, 26)
│ │ │ - -- .000297184s elapsed
│ │ │ + -- .000283788s elapsed
│ │ │  1
│ │ │ - -- .000143437s elapsed
│ │ │ + -- .000228178s elapsed
│ │ │  1
│ │ │ - -- .000131295s elapsed
│ │ │ + -- .000173857s elapsed
│ │ │  1
│ │ │ - -- .000128579s elapsed
│ │ │ + -- .000179947s elapsed
│ │ │  1
│ │ │ - -- .000143327s elapsed
│ │ │ + -- .000246311s elapsed
│ │ │  2
│ │ │ - -- .000142685s elapsed
│ │ │ + -- .000188789s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000154078s elapsed
│ │ │ + -- .000197423s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000160099s elapsed
│ │ │ + -- .000235954s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000140462s elapsed
│ │ │ + -- .000260644s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .00013971s elapsed
│ │ │ + -- .000183743s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000138028s elapsed
│ │ │ + -- .000170697s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000132678s elapsed
│ │ │ + -- .000167883s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000120584s elapsed
│ │ │ + -- .000173951s elapsed
│ │ │  2
│ │ │ - -- .000122919s elapsed
│ │ │ + -- .000145589s elapsed
│ │ │  2
│ │ │ - -- .000134922s elapsed
│ │ │ + -- .000155031s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .00012317s elapsed
│ │ │ + -- .00016271s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000140812s elapsed
│ │ │ + -- .000180971s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000133859s elapsed
│ │ │ + -- .000168531s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000142265s elapsed
│ │ │ + -- .00017714s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000130654s elapsed
│ │ │ + -- .000195081s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000132116s elapsed
│ │ │ + -- .000172724s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000117308s elapsed
│ │ │ + -- .000153133s elapsed
│ │ │  2
│ │ │ - -- .000119734s elapsed
│ │ │ + -- .000171883s elapsed
│ │ │  1
│ │ │ - -- .000126405s elapsed
│ │ │ + -- .000167409s elapsed
│ │ │  1
│ │ │ - -- .000127438s elapsed
│ │ │ + -- .000163484s elapsed
│ │ │  1
│ │ │ - -- .000133549s elapsed
│ │ │ + -- .000151628s 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)
│ │ │ │ - -- .00694639s elapsed
│ │ │ │ - -- .0123481s elapsed
│ │ │ │ + -- .0080706s elapsed
│ │ │ │ + -- .0140967s elapsed
│ │ │ │  (number Of blocks, 26)
│ │ │ │ - -- .000297184s elapsed
│ │ │ │ + -- .000283788s elapsed
│ │ │ │  1
│ │ │ │ - -- .000143437s elapsed
│ │ │ │ + -- .000228178s elapsed
│ │ │ │  1
│ │ │ │ - -- .000131295s elapsed
│ │ │ │ + -- .000173857s elapsed
│ │ │ │  1
│ │ │ │ - -- .000128579s elapsed
│ │ │ │ + -- .000179947s elapsed
│ │ │ │  1
│ │ │ │ - -- .000143327s elapsed
│ │ │ │ + -- .000246311s elapsed
│ │ │ │  2
│ │ │ │ - -- .000142685s elapsed
│ │ │ │ + -- .000188789s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000154078s elapsed
│ │ │ │ + -- .000197423s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000160099s elapsed
│ │ │ │ + -- .000235954s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000140462s elapsed
│ │ │ │ + -- .000260644s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .00013971s elapsed
│ │ │ │ + -- .000183743s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000138028s elapsed
│ │ │ │ + -- .000170697s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000132678s elapsed
│ │ │ │ + -- .000167883s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000120584s elapsed
│ │ │ │ + -- .000173951s elapsed
│ │ │ │  2
│ │ │ │ - -- .000122919s elapsed
│ │ │ │ + -- .000145589s elapsed
│ │ │ │  2
│ │ │ │ - -- .000134922s elapsed
│ │ │ │ + -- .000155031s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .00012317s elapsed
│ │ │ │ + -- .00016271s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000140812s elapsed
│ │ │ │ + -- .000180971s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000133859s elapsed
│ │ │ │ + -- .000168531s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000142265s elapsed
│ │ │ │ + -- .00017714s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000130654s elapsed
│ │ │ │ + -- .000195081s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000132116s elapsed
│ │ │ │ + -- .000172724s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000117308s elapsed
│ │ │ │ + -- .000153133s elapsed
│ │ │ │  2
│ │ │ │ - -- .000119734s elapsed
│ │ │ │ + -- .000171883s elapsed
│ │ │ │  1
│ │ │ │ - -- .000126405s elapsed
│ │ │ │ + -- .000167409s elapsed
│ │ │ │  1
│ │ │ │ - -- .000127438s elapsed
│ │ │ │ + -- .000163484s elapsed
│ │ │ │  1
│ │ │ │ - -- .000133549s elapsed
│ │ │ │ + -- .000151628s elapsed
│ │ │ │  1
│ │ │ │  
│ │ │ │  o2 = 3131031158784
│ │ │ │  i3 : factor d
│ │ │ │  
│ │ │ │        32 6
│ │ │ │  o3 = 2  3
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html
│ │ │ @@ -85,20 +85,20 @@
│ │ │  
│ │ │  o1 : Sequence
│ │ │              
    
│ │ │                
    i2 : computeBound 3
│ │ │ - -- .193447s elapsed
│ │ │ - -- .265378s elapsed
│ │ │ - -- .211761s elapsed
│ │ │ - -- .284175s elapsed
│ │ │ - -- .322238s elapsed
│ │ │ - -- .410165s elapsed
│ │ │ + -- .177918s elapsed
│ │ │ + -- .19592s elapsed
│ │ │ + -- .182155s elapsed
│ │ │ + -- .29462s elapsed
│ │ │ + -- .227293s elapsed
│ │ │ + -- .192633s 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
│ │ │ │ - -- .193447s elapsed
│ │ │ │ - -- .265378s elapsed
│ │ │ │ - -- .211761s elapsed
│ │ │ │ - -- .284175s elapsed
│ │ │ │ - -- .322238s elapsed
│ │ │ │ - -- .410165s elapsed
│ │ │ │ + -- .177918s elapsed
│ │ │ │ + -- .19592s elapsed
│ │ │ │ + -- .182155s elapsed
│ │ │ │ + -- .29462s elapsed
│ │ │ │ + -- .227293s elapsed
│ │ │ │ + -- .192633s 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
│ │ │ @@ -90,19 +90,19 @@
│ │ │  
│ │ │  o2 : Sequence
    
│ │ │              
    
│ │ │                
    i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .0634893s elapsed
│ │ │ - -- .00657433s elapsed
│ │ │ - -- .0254523s elapsed
│ │ │ - -- .00879327s elapsed
│ │ │ - -- .00336788s elapsed
│ │ │ + -- .0334179s elapsed
│ │ │ + -- .00737552s elapsed
│ │ │ + -- .0302602s elapsed
│ │ │ + -- .00995625s elapsed
│ │ │ + -- .00408811s 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
│ │ │ @@ -136,15 +136,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ - -- .309166s elapsed
│ │ │ + -- .257533s 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
│ │ │ @@ -178,19 +178,19 @@
│ │ │  
│ │ │  o7 : Sequence
    
│ │ │              
    
│ │ │                
    i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00250595s elapsed
│ │ │ - -- .00659358s elapsed
│ │ │ - -- .0231823s elapsed
│ │ │ - -- .0102019s elapsed
│ │ │ - -- .00377573s elapsed
│ │ │ + -- .00304241s elapsed
│ │ │ + -- .00755411s elapsed
│ │ │ + -- .0305028s elapsed
│ │ │ + -- .0162895s elapsed
│ │ │ + -- .00449834s 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)
│ │ │ │ - -- .0634893s elapsed
│ │ │ │ - -- .00657433s elapsed
│ │ │ │ - -- .0254523s elapsed
│ │ │ │ - -- .00879327s elapsed
│ │ │ │ - -- .00336788s elapsed
│ │ │ │ + -- .0334179s elapsed
│ │ │ │ + -- .00737552s elapsed
│ │ │ │ + -- .0302602s elapsed
│ │ │ │ + -- .00995625s elapsed
│ │ │ │ + -- .00408811s 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)
│ │ │ │ - -- .309166s elapsed
│ │ │ │ + -- .257533s 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)
│ │ │ │ - -- .00250595s elapsed
│ │ │ │ - -- .00659358s elapsed
│ │ │ │ - -- .0231823s elapsed
│ │ │ │ - -- .0102019s elapsed
│ │ │ │ - -- .00377573s elapsed
│ │ │ │ + -- .00304241s elapsed
│ │ │ │ + -- .00755411s elapsed
│ │ │ │ + -- .0305028s elapsed
│ │ │ │ + -- .0162895s elapsed
│ │ │ │ + -- .00449834s 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
│ │ │ @@ -78,172 +78,172 @@
│ │ │  
│ │ │  o1 = 4
    
│ │ │              
    
│ │ │                
    i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0177477s elapsed
│ │ │ + -- .0210588s 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
│ │ │ - -- .000051325s elapsed
│ │ │ + -- .000055192s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . 2
│ │ │ - -- .000095639s elapsed
│ │ │ + -- .000119865s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . .
│ │ │      9: . 2
│ │ │ - -- .000076743s elapsed
│ │ │ + -- .000083476s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │      7: 2 .
│ │ │      8: 1 .
│ │ │      9: . 1
│ │ │     10: . 2
│ │ │ - -- .000079969s elapsed
│ │ │ + -- .000106085s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (-3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      7: 1 .
│ │ │      8: 1 .
│ │ │      9: 2 2
│ │ │     10: . 1
│ │ │     11: . 1
│ │ │ - -- .000083625s elapsed
│ │ │ + -- .000107413s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      8: 1 .
│ │ │      9: 2 1
│ │ │     10: 1 2
│ │ │     11: . 1
│ │ │ - -- .000093434s elapsed
│ │ │ + -- .00010274s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      9: 1 1
│ │ │ - -- .000024326s elapsed
│ │ │ + -- .000032908s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      9: 1 1
│ │ │     10: 1 1
│ │ │ - -- .000067026s elapsed
│ │ │ + -- .000083798s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .00009138s elapsed
│ │ │ + -- .000112368s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 2 1
│ │ │     11: 1 2
│ │ │     12: . 1
│ │ │ - -- .000095969s elapsed
│ │ │ + -- .000115386s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 1 .
│ │ │     11: 2 2
│ │ │     12: . 1
│ │ │     13: . 1
│ │ │ - -- .000090488s elapsed
│ │ │ + -- .000111017s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000081492s elapsed
│ │ │ + -- .000105723s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │     10: 2 .
│ │ │     11: 1 .
│ │ │     12: . 1
│ │ │     13: . 2
│ │ │ - -- .00009612s elapsed
│ │ │ + -- .000112633s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     10: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000065421s elapsed
│ │ │ + -- .000095768s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     11: 2 .
│ │ │     12: . .
│ │ │     13: . 2
│ │ │ - -- .000077685s elapsed
│ │ │ + -- .000086287s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000024636s elapsed
│ │ │ + -- .000034508s elapsed
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     12: 2 .
│ │ │     13: . 2
│ │ │ - -- .000076392s elapsed
│ │ │ + -- .00009826s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     13: 1 1
│ │ │ - -- .000027041s elapsed
│ │ │ + -- .000032758s 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)
│ │ │ │ - -- .0177477s elapsed
│ │ │ │ + -- .0210588s 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
│ │ │ │ - -- .000051325s elapsed
│ │ │ │ + -- .000055192s elapsed
│ │ │ │  (e )(-1)
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      7: 2 .
│ │ │ │      8: . 2
│ │ │ │ - -- .000095639s elapsed
│ │ │ │ + -- .000119865s elapsed
│ │ │ │      2
│ │ │ │  (e ) (e )(-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      7: 2 .
│ │ │ │      8: . .
│ │ │ │      9: . 2
│ │ │ │ - -- .000076743s elapsed
│ │ │ │ + -- .000083476s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e )
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 3 3
│ │ │ │      7: 2 .
│ │ │ │      8: 1 .
│ │ │ │      9: . 1
│ │ │ │     10: . 2
│ │ │ │ - -- .000079969s elapsed
│ │ │ │ + -- .000106085s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (-3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      7: 1 .
│ │ │ │      8: 1 .
│ │ │ │      9: 2 2
│ │ │ │     10: . 1
│ │ │ │     11: . 1
│ │ │ │ - -- .000083625s elapsed
│ │ │ │ + -- .000107413s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      8: 1 .
│ │ │ │      9: 2 1
│ │ │ │     10: 1 2
│ │ │ │     11: . 1
│ │ │ │ - -- .000093434s elapsed
│ │ │ │ + -- .00010274s elapsed
│ │ │ │      2    3
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │      9: 1 1
│ │ │ │ - -- .000024326s elapsed
│ │ │ │ + -- .000032908s elapsed
│ │ │ │  (e )(-1)
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      9: 1 1
│ │ │ │     10: 1 1
│ │ │ │ - -- .000067026s elapsed
│ │ │ │ + -- .000083798s elapsed
│ │ │ │      2
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 2 1
│ │ │ │     10: 1 1
│ │ │ │     11: 1 2
│ │ │ │ - -- .00009138s elapsed
│ │ │ │ + -- .000112368s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e ) (-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 1 .
│ │ │ │     10: 2 1
│ │ │ │     11: 1 2
│ │ │ │     12: . 1
│ │ │ │ - -- .000095969s elapsed
│ │ │ │ + -- .000115386s elapsed
│ │ │ │      2    3
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 1 .
│ │ │ │     10: 1 .
│ │ │ │     11: 2 2
│ │ │ │     12: . 1
│ │ │ │     13: . 1
│ │ │ │ - -- .000090488s elapsed
│ │ │ │ + -- .000111017s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 2 1
│ │ │ │     10: 1 1
│ │ │ │     11: 1 2
│ │ │ │ - -- .000081492s elapsed
│ │ │ │ + -- .000105723s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e ) (-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 3 3
│ │ │ │     10: 2 .
│ │ │ │     11: 1 .
│ │ │ │     12: . 1
│ │ │ │     13: . 2
│ │ │ │ - -- .00009612s elapsed
│ │ │ │ + -- .000112633s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     10: 1 1
│ │ │ │     11: 1 1
│ │ │ │ - -- .000065421s elapsed
│ │ │ │ + -- .000095768s elapsed
│ │ │ │      2
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     11: 2 .
│ │ │ │     12: . .
│ │ │ │     13: . 2
│ │ │ │ - -- .000077685s elapsed
│ │ │ │ + -- .000086287s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e )
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │     11: 1 1
│ │ │ │ - -- .000024636s elapsed
│ │ │ │ + -- .000034508s elapsed
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     12: 2 .
│ │ │ │     13: . 2
│ │ │ │ - -- .000076392s elapsed
│ │ │ │ + -- .00009826s elapsed
│ │ │ │      2
│ │ │ │  (e ) (e )(-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │     13: 1 1
│ │ │ │ - -- .000027041s elapsed
│ │ │ │ + -- .000032758s 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 0.00904435s (cpu); 0.00904001s (thread); 0s (gc)
│ │ │ + -- used 0.0100998s (cpu); 0.0100967s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0273886s (cpu); 0.02739s (thread); 0s (gc)
│ │ │ + -- used 0.0308514s (cpu); 0.0307785s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.107453s (cpu); 0.10743s (thread); 0s (gc)
│ │ │ + -- used 0.124523s (cpu); 0.124375s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.011869s (cpu); 0.0118693s (thread); 0s (gc)
│ │ │ + -- used 0.013082s (cpu); 0.0130869s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0480703s (cpu); 0.0480742s (thread); 0s (gc)
│ │ │ + -- used 0.0620766s (cpu); 0.062084s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0590818s (cpu); 0.0590819s (thread); 0s (gc)
│ │ │ + -- used 0.0647416s (cpu); 0.0647479s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.347092s (cpu); 0.347092s (thread); 0s (gc)
│ │ │ + -- used 0.349014s (cpu); 0.349023s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.113298s (cpu); 0.113301s (thread); 0s (gc)
│ │ │ + -- used 0.153778s (cpu); 0.153794s (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
│ │ │ @@ -139,78 +139,78 @@
│ │ │                50       47
│ │ │  o2 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i3 : time LLL m;
│ │ │ - -- used 0.00904435s (cpu); 0.00904001s (thread); 0s (gc)
│ │ │ + -- used 0.0100998s (cpu); 0.0100967s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0273886s (cpu); 0.02739s (thread); 0s (gc)
│ │ │ + -- used 0.0308514s (cpu); 0.0307785s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.107453s (cpu); 0.10743s (thread); 0s (gc)
│ │ │ + -- used 0.124523s (cpu); 0.124375s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.011869s (cpu); 0.0118693s (thread); 0s (gc)
│ │ │ + -- used 0.013082s (cpu); 0.0130869s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0480703s (cpu); 0.0480742s (thread); 0s (gc)
│ │ │ + -- used 0.0620766s (cpu); 0.062084s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0590818s (cpu); 0.0590819s (thread); 0s (gc)
│ │ │ + -- used 0.0647416s (cpu); 0.0647479s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.347092s (cpu); 0.347092s (thread); 0s (gc)
│ │ │ + -- used 0.349014s (cpu); 0.349023s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.113298s (cpu); 0.113301s (thread); 0s (gc)
│ │ │ + -- used 0.153778s (cpu); 0.153794s (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 0.00904435s (cpu); 0.00904001s (thread); 0s (gc)
│ │ │ │ + -- used 0.0100998s (cpu); 0.0100967s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o3 : Matrix ZZ   <-- ZZ
│ │ │ │  i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ │ - -- used 0.0273886s (cpu); 0.02739s (thread); 0s (gc)
│ │ │ │ + -- used 0.0308514s (cpu); 0.0307785s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o4 : Matrix ZZ   <-- ZZ
│ │ │ │  i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ │ - -- used 0.107453s (cpu); 0.10743s (thread); 0s (gc)
│ │ │ │ + -- used 0.124523s (cpu); 0.124375s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o5 : Matrix ZZ   <-- ZZ
│ │ │ │  i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ │ - -- used 0.011869s (cpu); 0.0118693s (thread); 0s (gc)
│ │ │ │ + -- used 0.013082s (cpu); 0.0130869s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o6 : Matrix ZZ   <-- ZZ
│ │ │ │  i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ │ - -- used 0.0480703s (cpu); 0.0480742s (thread); 0s (gc)
│ │ │ │ + -- used 0.0620766s (cpu); 0.062084s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o7 : Matrix ZZ   <-- ZZ
│ │ │ │  i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ │ - -- used 0.0590818s (cpu); 0.0590819s (thread); 0s (gc)
│ │ │ │ + -- used 0.0647416s (cpu); 0.0647479s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o8 : Matrix ZZ   <-- ZZ
│ │ │ │  i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ │ - -- used 0.347092s (cpu); 0.347092s (thread); 0s (gc)
│ │ │ │ + -- used 0.349014s (cpu); 0.349023s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o9 : Matrix ZZ   <-- ZZ
│ │ │ │  i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ │ - -- used 0.113298s (cpu); 0.113301s (thread); 0s (gc)
│ │ │ │ + -- used 0.153778s (cpu); 0.153794s (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 0.593992s (cpu); 0.441215s (thread); 0s (gc)
│ │ │ + -- used 1.08742s (cpu); 0.59212s (thread); 0s (gc)
│ │ │  
│ │ │  i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ - -- used 0.449515s (cpu); 0.292918s (thread); 0s (gc)
│ │ │ + -- used 0.659649s (cpu); 0.364223s (thread); 0s (gc)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html
│ │ │ @@ -120,21 +120,21 @@
│ │ │              
    		
│ │ │                
    i5 : P = convexHull(M);
    
│ │ │              
    
│ │ │                
    i6 : time areIsomorphic(P,P);
│ │ │ - -- used 0.593992s (cpu); 0.441215s (thread); 0s (gc)
    
│ │ │ + -- used 1.08742s (cpu); 0.59212s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ - -- used 0.449515s (cpu); 0.292918s (thread); 0s (gc)
    
│ │ │ + -- used 0.659649s (cpu); 0.364223s (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 0.593992s (cpu); 0.441215s (thread); 0s (gc)
│ │ │ │ + -- used 1.08742s (cpu); 0.59212s (thread); 0s (gc)
│ │ │ │  i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ │ - -- used 0.449515s (cpu); 0.292918s (thread); 0s (gc)
│ │ │ │ + -- used 0.659649s (cpu); 0.364223s (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)
│ │ │ - -- .129699s elapsed
│ │ │ + -- .104153s 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}})
│ │ │ - -- .0121746s elapsed
│ │ │ + -- .0343522s 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)
│ │ │ - -- 4.16059s elapsed
│ │ │ + -- 3.40498s elapsed
│ │ │  
│ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime linearTruncationsBound M
│ │ │ - -- .0266295s elapsed
│ │ │ + -- .0292878s elapsed
│ │ │  
│ │ │  o7 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Region.html
│ │ │ @@ -125,25 +125,25 @@
│ │ │          
    
│ │ │            
    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)
│ │ │ - -- .129699s elapsed
│ │ │ + -- .104153s 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}})
│ │ │ - -- .0121746s elapsed
│ │ │ + -- .0343522s 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)
│ │ │ │ - -- .129699s elapsed
│ │ │ │ + -- .104153s 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}})
│ │ │ │ - -- .0121746s elapsed
│ │ │ │ + -- .0343522s 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
│ │ │ @@ -123,25 +123,25 @@
│ │ │          
    
│ │ │            
    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)
│ │ │ - -- 4.16059s elapsed
│ │ │ + -- 3.40498s elapsed
│ │ │  
│ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime linearTruncationsBound M
│ │ │ - -- .0266295s elapsed
│ │ │ + -- .0292878s 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)
│ │ │ │ - -- 4.16059s elapsed
│ │ │ │ + -- 3.40498s elapsed
│ │ │ │  
│ │ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : elapsedTime linearTruncationsBound M
│ │ │ │ - -- .0266295s elapsed
│ │ │ │ + -- .0292878s 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)
│ │ │ - -- .238651s elapsed
│ │ │ + -- .216742s 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
│ │ │ - -- .0754774s elapsed
│ │ │ + -- .0189068s elapsed
│ │ │  
│ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ - -- .381402s elapsed
│ │ │ + -- .306756s elapsed
│ │ │  
│ │ │  o12 = {6, 12, 18, 24, 30, 36}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/LocalRings/html/_hilbert__Samuel__Function.html
│ │ │ @@ -111,15 +111,15 @@
│ │ │                                1
│ │ │  o4 : RP-module, quotient of RP
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime hilbertSamuelFunction(M, 0, 6)
│ │ │ - -- .238651s elapsed
│ │ │ + -- .216742s elapsed
│ │ │  
│ │ │  o5 = {1, 3, 6, 7, 6, 3, 1}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -163,25 +163,25 @@
│ │ │  
│ │ │  o10 : Ideal of RP
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds
│ │ │ - -- .0754774s elapsed
│ │ │ + -- .0189068s elapsed
│ │ │  
│ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ - -- .381402s elapsed
│ │ │ + -- .306756s 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)
│ │ │ │ - -- .238651s elapsed
│ │ │ │ + -- .216742s 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
│ │ │ │ - -- .0754774s elapsed
│ │ │ │ + -- .0189068s elapsed
│ │ │ │  
│ │ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ │ - -- .381402s elapsed
│ │ │ │ + -- .306756s 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/Macaulay2Doc/example-output/___Command.out
│ │ │ @@ -5,12 +5,12 @@
│ │ │  i2 : f
│ │ │  
│ │ │  o2 = 1073741824
│ │ │  
│ │ │  i3 : (c = Command "date";)
│ │ │  
│ │ │  i4 : c
│ │ │ -Sun Dec 14 15:26:56 UTC 2025
│ │ │ +Wed Jan  7 12:22:11 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/M2-11641-0/0.dbm
│ │ │ +o1 = /tmp/M2-13231-0/0.dbm
│ │ │  
│ │ │  i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-11641-0/0.dbm
│ │ │ +o2 = /tmp/M2-13231-0/0.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" => 42969169706        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 43050590362        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 206680064
│ │ │ -               "numGCs" => 795
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 225931264
│ │ │ +               "numGCs" => 783
│ │ │ +               "numGCThreads" => 16
│ │ │  
│ │ │  o1 : HashTable
│ │ │  
│ │ │  i2 : s#"heapSize"
│ │ │  
│ │ │ -o2 = 206680064
│ │ │ +o2 = 225931264
│ │ │  
│ │ │  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 0.00906552s (cpu); 0.00905868s (thread); 0s (gc)
│ │ │ + -- used 0.00604401s (cpu); 0.00604071s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of R
│ │ │  
│ │ │  i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ - -- used 0.0794664s (cpu); 0.0794742s (thread); 0s (gc)
│ │ │ + -- used 0.0603359s (cpu); 0.0603461s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : numgens J
│ │ │  
│ │ │  o10 = 1067
│ │ ├── ./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 0.0253142s (cpu); 0.0253129s (thread); 0s (gc)
│ │ │ + -- used 0.0417848s (cpu); 0.041782s (thread); 0s (gc)
│ │ │  
│ │ │  i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0274055s (cpu); 0.0274142s (thread); 0s (gc)
│ │ │ + -- used 0.0410636s (cpu); 0.0410811s (thread); 0s (gc)
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_a_spfirst_sp__Macaulay2_spsession.out
│ │ │ @@ -351,15 +351,15 @@
│ │ │                 | 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 0.00192778s (cpu); 0.00191945s (thread); 0s (gc)
│ │ │ + -- used 0.00234372s (cpu); 0.00233367s (thread); 0s (gc)
│ │ │  
│ │ │         3      6      15      18      6
│ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R  <-- 0
│ │ │                                              
│ │ │        0      1      2       3       4      5
│ │ │  
│ │ │  o59 : ChainComplex
│ │ ├── ./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 = .000290697861367332
│ │ │ +o1 = .0003726770483759516
│ │ │  
│ │ │  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'
│ │ │ - -- .00670047s elapsed
│ │ │ + -- .00488437s elapsed
│ │ │  
│ │ │  o4 = 3
│ │ │  
│ │ │  i5 : elapsedTime pdim' M
│ │ │ - -- .000001513s elapsed
│ │ │ + -- .000002519s 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
│ │ │ @@ -18,29 +18,29 @@
│ │ │  
│ │ │  o4 = <>
│ │ │  
│ │ │  o4 : Task
│ │ │  
│ │ │  i5 : n
│ │ │  
│ │ │ -o5 = 711206
│ │ │ +o5 = 1095574
│ │ │  
│ │ │  i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
│ │ │  
│ │ │  i7 : t
│ │ │  
│ │ │  o7 = <>
│ │ │  
│ │ │  o7 : Task
│ │ │  
│ │ │  i8 : n
│ │ │  
│ │ │ -o8 = 1453533
│ │ │ +o8 = 2222205
│ │ │  
│ │ │  i9 : isReady t
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 : cancelTask t
│ │ │  
│ │ │ @@ -53,22 +53,22 @@
│ │ │  
│ │ │  o12 = <>
│ │ │  
│ │ │  o12 : Task
│ │ │  
│ │ │  i13 : n
│ │ │  
│ │ │ -o13 = 1453746
│ │ │ +o13 = 2222398
│ │ │  
│ │ │  i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
│ │ │  
│ │ │  i15 : n
│ │ │  
│ │ │ -o15 = 1453746
│ │ │ +o15 = 2222398
│ │ │  
│ │ │  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/M2-10463-0/0
│ │ │ +o1 = /tmp/M2-10833-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10463-0/0
│ │ │ +o2 = /tmp/M2-10833-0/0
│ │ │  
│ │ │  i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-10463-0/0/
│ │ │ +o3 = /tmp/M2-10833-0/0/
│ │ │  
│ │ │  i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-10463-0/0/
│ │ │ +o4 = /tmp/M2-10833-0/0/
│ │ │  
│ │ │  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")        -- .15147s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .136542s elapsed
│ │ │  
│ │ │  i3 : check FirstPackage
│ │ │ - -- capturing check(0, "FirstPackage")        -- .150181s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .150965s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .147318s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .144622s elapsed
│ │ │  
│ │ │  i4 : check_1 "FirstPackage"
│ │ │ - -- capturing check(1, "FirstPackage")        -- .152579s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .139659s elapsed
│ │ │  
│ │ │  i5 : check "FirstPackage"
│ │ │ - -- capturing check(0, "FirstPackage")        -- .152053s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .151867s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .135863s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .137943s 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")        -- .153083s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .135696s 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")        -- .152703s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .150901s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .133351s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .13031s 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")        -- .151346s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .131903s 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 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) 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 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ +o3 = "Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │       6.12.57-1 (2025-11-05) 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 0x7f67b957e540
│ │ │ +   -- registering gb 5 at 0x7f49bd44d540
│ │ │  
│ │ │     -- [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 0.303879s (cpu); 0.305901s (thread); 0s (gc)
│ │ │ + -- used 0.227867s (cpu); 0.231374s (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 0.00799525s (cpu); 0.00545052s (thread); 0s (gc)
│ │ │ + -- used 0.00398845s (cpu); 0.00376715s (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/M2-11185-0/0/
│ │ │ +o1 = /tmp/M2-12295-0/0/
│ │ │  
│ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11185-0/1/
│ │ │ +o2 = /tmp/M2-12295-0/1/
│ │ │  
│ │ │  i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11185-0/0/a/
│ │ │ +o3 = /tmp/M2-12295-0/0/a/
│ │ │  
│ │ │  i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11185-0/0/b/
│ │ │ +o4 = /tmp/M2-12295-0/0/b/
│ │ │  
│ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11185-0/0/b/c/
│ │ │ +o5 = /tmp/M2-12295-0/0/b/c/
│ │ │  
│ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11185-0/0/a/f
│ │ │ +o6 = /tmp/M2-12295-0/0/a/f
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11185-0/0/a/g
│ │ │ +o7 = /tmp/M2-12295-0/0/a/g
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11185-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12295-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-11185-0/0/
│ │ │ -     /tmp/M2-11185-0/0/b/
│ │ │ -     /tmp/M2-11185-0/0/b/c/
│ │ │ -     /tmp/M2-11185-0/0/b/c/g
│ │ │ -     /tmp/M2-11185-0/0/a/
│ │ │ -     /tmp/M2-11185-0/0/a/g
│ │ │ -     /tmp/M2-11185-0/0/a/f
│ │ │ +o9 = /tmp/M2-12295-0/0/
│ │ │ +     /tmp/M2-12295-0/0/a/
│ │ │ +     /tmp/M2-12295-0/0/a/g
│ │ │ +     /tmp/M2-12295-0/0/a/f
│ │ │ +     /tmp/M2-12295-0/0/b/
│ │ │ +     /tmp/M2-12295-0/0/b/c/
│ │ │ +     /tmp/M2-12295-0/0/b/c/g
│ │ │  
│ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11185-0/0/b/c/g -> /tmp/M2-11185-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-11185-0/0/a/g -> /tmp/M2-11185-0/1/a/g
│ │ │ - -- copying: /tmp/M2-11185-0/0/a/f -> /tmp/M2-11185-0/1/a/f
│ │ │ + -- copying: /tmp/M2-12295-0/0/a/g -> /tmp/M2-12295-0/1/a/g
│ │ │ + -- copying: /tmp/M2-12295-0/0/a/f -> /tmp/M2-12295-0/1/a/f
│ │ │ + -- copying: /tmp/M2-12295-0/0/b/c/g -> /tmp/M2-12295-0/1/b/c/g
│ │ │  
│ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11185-0/0/b/c/g not newer than /tmp/M2-11185-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-11185-0/0/a/g not newer than /tmp/M2-11185-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-11185-0/0/a/f not newer than /tmp/M2-11185-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-12295-0/0/a/g not newer than /tmp/M2-12295-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-12295-0/0/a/f not newer than /tmp/M2-12295-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-12295-0/0/b/c/g not newer than /tmp/M2-12295-0/1/b/c/g
│ │ │  
│ │ │  i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-11185-0/1/
│ │ │ -      /tmp/M2-11185-0/1/a/
│ │ │ -      /tmp/M2-11185-0/1/a/f
│ │ │ -      /tmp/M2-11185-0/1/a/g
│ │ │ -      /tmp/M2-11185-0/1/b/
│ │ │ -      /tmp/M2-11185-0/1/b/c/
│ │ │ -      /tmp/M2-11185-0/1/b/c/g
│ │ │ +o12 = /tmp/M2-12295-0/1/
│ │ │ +      /tmp/M2-12295-0/1/a/
│ │ │ +      /tmp/M2-12295-0/1/a/g
│ │ │ +      /tmp/M2-12295-0/1/a/f
│ │ │ +      /tmp/M2-12295-0/1/b/
│ │ │ +      /tmp/M2-12295-0/1/b/c/
│ │ │ +      /tmp/M2-12295-0/1/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/M2-10970-0/0
│ │ │ +o1 = /tmp/M2-11860-0/0
│ │ │  
│ │ │  i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10970-0/1
│ │ │ +o2 = /tmp/M2-11860-0/1
│ │ │  
│ │ │  i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10970-0/0
│ │ │ +o3 = /tmp/M2-11860-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-10970-0/0 -> /tmp/M2-10970-0/1
│ │ │ + -- copying: /tmp/M2-11860-0/0 -> /tmp/M2-11860-0/1
│ │ │  
│ │ │  i5 : get dst
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
│ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │  
│ │ │  i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-10970-0/0
│ │ │ +o7 = /tmp/M2-11860-0/0
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
│ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │  
│ │ │  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 = 354.029649282
│ │ │ +o1 = 383.166340652
│ │ │  
│ │ │  o1 : RR (of precision 53)
│ │ │  
│ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │  
│ │ │  i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 355.996910491
│ │ │ +o3 = 384.327232721
│ │ │  
│ │ │  o3 : RR (of precision 53)
│ │ │  
│ │ │  i4 : t2-t1
│ │ │  
│ │ │ -o4 = 1.967261209000014
│ │ │ +o4 = 1.160892068999999
│ │ │  
│ │ │  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 = 1765726091
│ │ │ +o1 = 1767788593
│ │ │  
│ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 55.95363237235333
│ │ │ +o2 = 56.01899045890111
│ │ │  
│ │ │  o2 : RR (of precision 53)
│ │ │  
│ │ │  i3 : 12 * (oo - floor oo)
│ │ │  
│ │ │ -o3 = 11.44358846823999
│ │ │ +o3 = .2278855068132941
│ │ │  
│ │ │  o3 : RR (of precision 53)
│ │ │  
│ │ │  i4 : run "date"
│ │ │ -Sun Dec 14 15:28:11 UTC 2025
│ │ │ +Wed Jan  7 12:23:13 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
│ │ │ - -- 1.00015s elapsed
│ │ │ + -- 1.00023s elapsed
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 :
│ │ ├── ./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 0.464405s (cpu); 0.277254s (thread); 0s (gc)
│ │ │ + -- used 0.1387s (cpu); 0.138699s (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 0.422625s (cpu); 0.238689s (thread); 0s (gc)
│ │ │ + -- used 0.42811s (cpu); 0.206841s (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 0.286447s (cpu); 0.110979s (thread); 0s (gc)
│ │ │ + -- used 0.0363006s (cpu); 0.0363011s (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 0.404757s (cpu); 0.227468s (thread); 0s (gc)
│ │ │ + -- used 0.1109s (cpu); 0.11091s (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 0.00191261s (cpu); 0.00191299s (thread); 0s (gc)
│ │ │ + -- used 0.00189673s (cpu); 0.00189022s (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 0.0583203s (cpu); 0.0583314s (thread); 0s (gc)
│ │ │ + -- used 0.0362551s (cpu); 0.0362736s (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/M2-10191-0/91-rundir/
│ │ │ +             source directory => /tmp/M2-10311-0/91-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/M2-10558-0/0
│ │ │ +o1 = /tmp/M2-11028-0/0
│ │ │  
│ │ │  i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
│ │ │  
│ │ │  i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10558-0/0
│ │ │ +o3 = /tmp/M2-11028-0/0
│ │ │  
│ │ │  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/M2-12150-0/0
│ │ │ +o1 = /tmp/M2-14270-0/0
│ │ │  
│ │ │  o1 : File
│ │ │  
│ │ │  i2 : fileLength f
│ │ │  
│ │ │  o2 = 8
│ │ │  
│ │ │  i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-12150-0/0
│ │ │ +o3 = /tmp/M2-14270-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-12150-0/0
│ │ │ +o4 = /tmp/M2-14270-0/0
│ │ │  
│ │ │  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/M2-11375-0/0
│ │ │ +o1 = /tmp/M2-12685-0/0
│ │ │  
│ │ │  i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11375-0/0
│ │ │ +o2 = /tmp/M2-12685-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : fileMode f
│ │ │  
│ │ │  o3 = 420
│ │ │  
│ │ │  i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-11375-0/0
│ │ │ +o4 = /tmp/M2-12685-0/0
│ │ │  
│ │ │  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/M2-10989-0/0
│ │ │ +o1 = /tmp/M2-11899-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-10989-0/0
│ │ │ +o2 = /tmp/M2-11899-0/0
│ │ │  
│ │ │  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/M2-10854-0/0
│ │ │ +o1 = /tmp/M2-11624-0/0
│ │ │  
│ │ │  i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-10854-0/0
│ │ │ +o2 = /tmp/M2-11624-0/0
│ │ │  
│ │ │  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/M2-10854-0/0
│ │ │ +o6 = /tmp/M2-11624-0/0
│ │ │  
│ │ │  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/M2-11977-0/0
│ │ │ +o1 = /tmp/M2-13917-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11977-0/0
│ │ │ +o2 = /tmp/M2-13917-0/0
│ │ │  
│ │ │  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 = 61
│ │ │ +o1 = 53
│ │ │  
│ │ │  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 0x7f5498d2ce00
│ │ │ +   -- registering gb 0 at 0x7fabc31bde00
│ │ │  
│ │ │     -- [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 = Sun Dec 14 15:27:22 UTC 2025
│ │ │ +o4 = Wed Jan  7 12:22:33 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/M2-10380-0/0
│ │ │ +o2 = /tmp/M2-10670-0/0
│ │ │  
│ │ │  i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10380-0/0
│ │ │ +o3 = /tmp/M2-10670-0/0
│ │ │  
│ │ │  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)
│ │ │ - -- 4.33674s elapsed
│ │ │ + -- 6.01582s 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
│ │ │ - -- .0569545s elapsed
│ │ │ + -- .0621642s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000114113s elapsed
│ │ │ + -- .000136165s 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/M2-12188-0/0
│ │ │ +o1 = /tmp/M2-14348-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12188-0/0
│ │ │ +o2 = /tmp/M2-14348-0/0
│ │ │  
│ │ │  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/M2-10722-0/0
│ │ │ +o1 = /tmp/M2-11352-0/0
│ │ │  
│ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-10722-0/0/a/b/c
│ │ │ +o2 = /tmp/M2-11352-0/0/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 1.26333s (cpu); 0.726987s (thread); 0s (gc)
│ │ │ + -- used 0.892603s (cpu); 0.622672s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
│ │ │  
│ │ │  i3 : fib = memoize fib
│ │ │  
│ │ │  o3 = fib
│ │ │  
│ │ │  o3 : FunctionClosure
│ │ │  
│ │ │  i4 : time fib 28
│ │ │ - -- used 7.5542e-05s (cpu); 7.4801e-05s (thread); 0s (gc)
│ │ │ + -- used 7.6e-05s (cpu); 7.0592e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
│ │ │  
│ │ │  i5 : time fib 28
│ │ │ - -- used 3.987e-06s (cpu); 3.627e-06s (thread); 0s (gc)
│ │ │ + -- used 4.41e-06s (cpu); 3.224e-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
│ │ │ @@ -18,19 +18,19 @@
│ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │       {14 => (degree, BettiTally)                                  }
│ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │       {17 => (regularity, BettiTally)                              }
│ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │       {19 => (codim, BettiTally)                                   }
│ │ │ -     {20 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ -     {21 => (dual, BettiTally)                                    }
│ │ │ +     {20 => (dual, BettiTally)                                    }
│ │ │ +     {21 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │       {22 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ -     {23 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ -     {24 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {23 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {24 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │  
│ │ │  o1 : NumberedVerticalList
│ │ │  
│ │ │  i2 : methods resolution
│ │ │  
│ │ │  o2 = {0 => (resolution, Ideal) }
│ │ │ @@ -60,20 +60,20 @@
│ │ │       {1 => (++, Module, GradedModule)}
│ │ │       {2 => (++, Module, Module)      }
│ │ │  
│ │ │  o4 : NumberedVerticalList
│ │ │  
│ │ │  i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │ -o5 = {0 => (contract, Matrix, Matrix)                            }
│ │ │ -     {1 => (diff, Matrix, Matrix)                                }
│ │ │ -     {2 => (diff', Matrix, Matrix)                               }
│ │ │ -     {3 => (-, Matrix, Matrix)                                   }
│ │ │ +o5 = {0 => (+, Matrix, Matrix)                                   }
│ │ │ +     {1 => (-, Matrix, Matrix)                                   }
│ │ │ +     {2 => (contract, Matrix, Matrix)                            }
│ │ │ +     {3 => (diff, Matrix, Matrix)                                }
│ │ │       {4 => (contract', Matrix, Matrix)                           }
│ │ │ -     {5 => (+, 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)                        }
│ │ │ @@ -88,18 +88,18 @@
│ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │       {24 => (%, Matrix, Matrix)                                  }
│ │ │       {25 => (remainder, Matrix, Matrix)                          }
│ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ -     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ -     {29 => (intersect, Matrix, Matrix)                          }
│ │ │ -     {30 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ -     {31 => (tensor, Matrix, Matrix)                             }
│ │ │ +     {28 => (intersect, Matrix, Matrix)                          }
│ │ │ +     {29 => (pullback, Matrix, Matrix)                           }
│ │ │ +     {30 => (tensor, 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
│ │ │ - -- 1.82886s elapsed
│ │ │ + -- 2.51322s 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)
│ │ │ - -- .745147s elapsed
│ │ │ + -- .977091s 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)
│ │ │ - -- .0317322s elapsed
│ │ │ + -- .0410621s 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)
│ │ │ - -- 1.20016s elapsed
│ │ │ + -- 1.68311s 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/M2-10741-0/0/
│ │ │ +o1 = /tmp/M2-11391-0/0/
│ │ │  
│ │ │  i2 : mkdir p
│ │ │  
│ │ │  i3 : isDirectory p
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10741-0/0/foo
│ │ │ +o4 = /tmp/M2-11391-0/0/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/M2-10615-0/0
│ │ │ +o1 = /tmp/M2-11145-0/0
│ │ │  
│ │ │  i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10615-0/1
│ │ │ +o2 = /tmp/M2-11145-0/1
│ │ │  
│ │ │  i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10615-0/0
│ │ │ +o3 = /tmp/M2-11145-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-10615-0/0 -> /tmp/M2-10615-0/1
│ │ │ +--moving: /tmp/M2-11145-0/0 -> /tmp/M2-11145-0/1
│ │ │  
│ │ │  i5 : get dst
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-10615-0/1.bak
│ │ │ +--backup file created: /tmp/M2-11145-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-10615-0/1.bak
│ │ │ +o6 = /tmp/M2-11145-0/1.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
│ │ │ - -- .500135s elapsed
│ │ │ + -- .500132s elapsed
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_options_lp__Function_rp.out
│ │ │ @@ -21,30 +21,30 @@
│ │ │  
│ │ │  o3 = OptionTable{Generic => false}
│ │ │  
│ │ │  o3 : OptionTable
│ │ │  
│ │ │  i4 : methods codim
│ │ │  
│ │ │ -o4 = {0 => (codim, Variety)       }
│ │ │ -     {1 => (codim, BettiTally)    }
│ │ │ -     {2 => (codim, Module)        }
│ │ │ -     {3 => (codim, QuotientRing)  }
│ │ │ -     {4 => (codim, Ideal)         }
│ │ │ -     {5 => (codim, MonomialIdeal) }
│ │ │ -     {6 => (codim, PolynomialRing)}
│ │ │ -     {7 => (codim, CoherentSheaf) }
│ │ │ +o4 = {0 => (codim, CoherentSheaf) }
│ │ │ +     {1 => (codim, Variety)       }
│ │ │ +     {2 => (codim, BettiTally)    }
│ │ │ +     {3 => (codim, Module)        }
│ │ │ +     {4 => (codim, QuotientRing)  }
│ │ │ +     {5 => (codim, Ideal)         }
│ │ │ +     {6 => (codim, MonomialIdeal) }
│ │ │ +     {7 => (codim, PolynomialRing)}
│ │ │  
│ │ │  o4 : NumberedVerticalList
│ │ │  
│ │ │  i5 : options oo
│ │ │  
│ │ │  o5 = {0 => (OptionTable{Generic => false})}
│ │ │ -     {1 => (OptionTable{})                }
│ │ │ -     {2 => (OptionTable{Generic => false})}
│ │ │ +     {1 => (OptionTable{Generic => false})}
│ │ │ +     {2 => (OptionTable{})                }
│ │ │       {3 => (OptionTable{Generic => false})}
│ │ │       {4 => (OptionTable{Generic => false})}
│ │ │       {5 => (OptionTable{Generic => false})}
│ │ │       {6 => (OptionTable{Generic => false})}
│ │ │       {7 => (OptionTable{Generic => false})}
│ │ │  
│ │ │  o5 : NumberedVerticalList
│ │ ├── ./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 = random toList (1..10000);
│ │ │  
│ │ │  i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ - -- .640674s elapsed
│ │ │ + -- .751295s elapsed
│ │ │  
│ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .307919s elapsed
│ │ │ + -- .188399s 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)
│ │ ├── ./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
│ │ │ - -- 2.07143s elapsed
│ │ │ + -- 2.45561s 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)
│ │ │ - -- 1.82839s elapsed
│ │ │ + -- 2.47016s 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)
│ │ │ - -- 1.75494s elapsed
│ │ │ + -- 2.53146s 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)
│ │ │ - -- 2.23687s elapsed
│ │ │ + -- 4.37647s 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)
│ │ │ - -- 2.55496s elapsed
│ │ │ + -- 2.92176s 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");
│ │ │ - -- 5.136s elapsed
│ │ │ + -- 4.47144s 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");
│ │ │ - -- 7.09108s elapsed
│ │ │ + -- 10.002s 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");
│ │ │ - -- 4.22028s elapsed
│ │ │ + -- 4.3477s 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);
│ │ │ - -- 1.006s elapsed
│ │ │ + -- 1.22923s 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);
│ │ │ - -- 1.19796s elapsed
│ │ │ + -- 1.73952s 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 0.00277298s (cpu); 2.153e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00267189s (cpu); 1.5885e-05s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
│ │ │  
│ │ │  i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7f3fe9e7e540
│ │ │ +   -- registering gb 19 at 0x7fc056914540
│ │ │  
│ │ │     -- [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 0.0251999s (cpu); 0.0250353s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0172706s (cpu); 0.0180053s (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 20 at 0x7f3fe9e7e380
│ │ │ +   -- registering gb 20 at 0x7fc056914380
│ │ │  
│ │ │     -- [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 21 at 0x7f3fe9e7e000
│ │ │ +   -- registering gb 21 at 0x7fc056914000
│ │ │  
│ │ │     -- [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 0.00799718s (cpu); 0.00944086s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00400687s (cpu); 0.0064099s (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 22 at 0x7f3fe9ab3e00
│ │ │ +   -- registering gb 22 at 0x7fc056a0ee00
│ │ │  
│ │ │     -- [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 0.0840427s (cpu); 0.0832054s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0599726s (cpu); 0.0597862s (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/M2-10932-0/0
│ │ │ +o3 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10932-0/0
│ │ │ +o4 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  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/M2-10932-0/0
│ │ │ +o9 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  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/M2-10932-0/0
│ │ │ +o11 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  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 = 10191
│ │ │ +o1 = 10311
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_profile.out
│ │ │ @@ -9,34 +9,34 @@
│ │ │  
│ │ │                4       5
│ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i2 : profileSummary
│ │ │  
│ │ │  o2 = #run  %time   position                         
│ │ │ -     1     94.52   ../../m2/matrix1.m2:279:4-282:58 
│ │ │ -     1     92.12   ../../m2/matrix1.m2:281:22-281:43
│ │ │ -     1     44.16   ../../m2/matrix1.m2:193:25-193:52
│ │ │ -     1     30.59   ../../m2/matrix1.m2:114:5-156:72 
│ │ │ -     1     29.47   ../../m2/matrix1.m2:140:10-155:16
│ │ │ -     1     23.83   ../../m2/matrix1.m2:181:4-181:42 
│ │ │ -     1     22.54   ../../m2/set.m2:127:5-127:61     
│ │ │ -     1     20.86   ../../m2/matrix1.m2:45:10-49:22  
│ │ │ -     1     3.30    ../../m2/matrix1.m2:112:5-112:29 
│ │ │ -     1     2.34    ../../m2/matrix1.m2:141:13-141:78
│ │ │ -     1     2.18    ../../m2/matrix1.m2:96:5-109:11  
│ │ │ -     1     1.42    ../../m2/matrix1.m2:281:7-281:16 
│ │ │ -     1     1.29    ../../m2/matrix1.m2:147:20-147:64
│ │ │ +     1     92.19   ../../m2/matrix1.m2:279:4-282:58 
│ │ │ +     1     89.59   ../../m2/matrix1.m2:281:22-281:43
│ │ │ +     1     42.99   ../../m2/matrix1.m2:193:25-193:52
│ │ │ +     1     28.77   ../../m2/matrix1.m2:114:5-156:72 
│ │ │ +     1     27.64   ../../m2/matrix1.m2:140:10-155:16
│ │ │ +     1     22.71   ../../m2/matrix1.m2:181:4-181:42 
│ │ │ +     1     21.29   ../../m2/set.m2:127:5-127:61     
│ │ │ +     1     19.23   ../../m2/matrix1.m2:45:10-49:22  
│ │ │ +     1     3.57    ../../m2/matrix1.m2:112:5-112:29 
│ │ │ +     1     2.62    ../../m2/matrix1.m2:96:5-109:11  
│ │ │ +     1     2.3     ../../m2/matrix1.m2:141:13-141:78
│ │ │ +     1     1.91    ../../m2/matrix1.m2:276:4-277:73 
│ │ │ +     1     1.72    ../../m2/matrix1.m2:147:20-147:64
│ │ │ +     1     1.63    ../../m2/matrix1.m2:281:7-281:16 
│ │ │ +     1     1.56    ../../m2/matrix1.m2:98:10-98:46  
│ │ │       1     1.29    ../../m2/matrix1.m2:111:5-111:91 
│ │ │ -     1     1.27    ../../m2/matrix1.m2:276:4-277:73 
│ │ │ -     1     1.02    ../../m2/matrix1.m2:98:10-98:46  
│ │ │ -     1     .97     ../../m2/matrix1.m2:182:4-184:74 
│ │ │ -     1     .81     ../../m2/modules.m2:279:4-279:52 
│ │ │ -     20    .64     ../../m2/matrix1.m2:191:14-192:67
│ │ │ -     20    .47     ../../m2/matrix1.m2:47:43-47:71  
│ │ │ +     1     1.13    ../../m2/matrix1.m2:182:4-184:74 
│ │ │ +     1     .96     ../../m2/modules.m2:279:4-279:52 
│ │ │ +     20    .89     ../../m2/matrix1.m2:191:14-192:67
│ │ │ +     20    .78     ../../m2/matrix1.m2:47:43-47:71  
│ │ │       1     .0038s  elapsed total                    
│ │ │  
│ │ │  i3 : coverageSummary
│ │ │  
│ │ │  o3 = covered lines:
│ │ │       ../../m2/lists.m2:145:24-145:32
│ │ │       ../../m2/lists.m2:145:34-145:58
│ │ ├── ./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 0.173474s (cpu); 0.138239s (thread); 0s (gc)
│ │ │ + -- used 0.237164s (cpu); 0.102487s (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 0.663337s (cpu); 0.35806s (thread); 0s (gc)
│ │ │ + -- used 0.843126s (cpu); 0.33904s (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 3.65398s (cpu); 1.92733s (thread); 0s (gc)
│ │ │ + -- used 4.57597s (cpu); 2.15219s (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/M2-11565-0/0
│ │ │ +o1 = /tmp/M2-13075-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11565-0/0
│ │ │ +o2 = /tmp/M2-13075-0/0
│ │ │  
│ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11565-0/0/foo
│ │ │ +o3 = /tmp/M2-13075-0/0/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/M2-11107-0/0
│ │ │ +o1 = /tmp/M2-12137-0/0
│ │ │  
│ │ │  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/M2-11107-0/0
│ │ │ +o2 = /tmp/M2-12137-0/0
│ │ │  
│ │ │  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/M2-11107-0/0
│ │ │ +o7 = /tmp/M2-12137-0/0
│ │ │  
│ │ │  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/M2-11806-0/0
│ │ │ +o1 = /tmp/M2-13556-0/0
│ │ │  
│ │ │  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/M2-10191-0/86-rundir/
│ │ │ +o1 = /tmp/M2-10311-0/86-rundir/
│ │ │  
│ │ │  i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11825-0/0
│ │ │ +o2 = /tmp/M2-13595-0/0
│ │ │  
│ │ │  i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11825-0/1
│ │ │ +o3 = /tmp/M2-13595-0/1
│ │ │  
│ │ │  i4 : symlinkFile(p,q)
│ │ │  
│ │ │  i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-11825-0/0
│ │ │ +o5 = /tmp/M2-13595-0/0
│ │ │  
│ │ │  o5 : File
│ │ │  
│ │ │  i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-11825-0/0
│ │ │ +o6 = /tmp/M2-13595-0/0
│ │ │  
│ │ │  i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-11825-0/0
│ │ │ +o7 = /tmp/M2-13595-0/0
│ │ │  
│ │ │  i8 : removeFile p
│ │ │  
│ │ │  i9 : removeFile q
│ │ │  
│ │ │  i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-10191-0/86-rundir/
│ │ │ +o10 = /tmp/M2-10311-0/86-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)[3]: -- finalizing sequence #4 --
│ │ │ ---finalization: (2)[7]: -- finalizing sequence #8 --
│ │ │ ---finalization: (3)[4]: -- finalizing sequence #5 --
│ │ │ ---finalization: (4)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (1)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (2)[8]: -- finalizing sequence #9 --
│ │ │ +--finalization: (3)[5]: -- finalizing sequence #6 --
│ │ │ +--finalization: (4)[3]: -- finalizing sequence #4 --
│ │ │  --finalization: (5)[6]: -- finalizing sequence #7 --
│ │ │ ---finalization: (6)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (7)[8]: -- finalizing sequence #9 --
│ │ │ ---finalization: (8)[2]: -- finalizing sequence #3 --
│ │ │ ---finalization: (9)[0]: -- finalizing sequence #1 --
│ │ │ +--finalization: (6)[2]: -- finalizing sequence #3 --
│ │ │ +--finalization: (7)[0]: -- finalizing sequence #1 --
│ │ │ +--finalization: (9)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (8)[7]: -- finalizing sequence #8 --
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remove__Directory.out
│ │ │ @@ -1,19 +1,19 @@
│ │ │  -- -*- M2-comint -*- hash: 8532980310097060089
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10779-0/0
│ │ │ +o1 = /tmp/M2-11469-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10779-0/0
│ │ │ +o2 = /tmp/M2-11469-0/0
│ │ │  
│ │ │  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/M2-10283-0/0
│ │ │ +o1 = /tmp/M2-10473-0/0
│ │ │  
│ │ │  i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-10283-0/0
│ │ │ +o2 = /tmp/M2-10473-0/0
│ │ │  
│ │ │  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/M2-11508-0/0
│ │ │ +o1 = /tmp/M2-12958-0/0
│ │ │  
│ │ │  i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-11508-0/0
│ │ │ +o2 = file:///tmp/M2-12958-0/0
│ │ │  
│ │ │  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/M2-11356-0/0
│ │ │ +o5 = /tmp/M2-12646-0/0
│ │ │  
│ │ │  i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11356-0/0
│ │ │ +o6 = /tmp/M2-12646-0/0
│ │ │  
│ │ │  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/_serial__Number.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 5271760183816554957
│ │ │  
│ │ │  i1 : serialNumber asdf
│ │ │  
│ │ │ -o1 = 1426273
│ │ │ +o1 = 1526273
│ │ │  
│ │ │  i2 : serialNumber foo
│ │ │  
│ │ │ -o2 = 1426275
│ │ │ +o2 = 1526275
│ │ │  
│ │ │  i3 : serialNumber ZZ
│ │ │  
│ │ │  o3 = 1000050
│ │ │  
│ │ │  i4 :
│ │ ├── ./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 0.000227156s (cpu); 0.000219682s (thread); 0s (gc)
│ │ │ + -- used 0.000221698s (cpu); 0.000206721s (thread); 0s (gc)
│ │ │  
│ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000163036s (cpu); 0.000163226s (thread); 0s (gc)
│ │ │ + -- used 0.000103953s (cpu); 0.000103242s (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 0.482514s (cpu); 0.3008s (thread); 0s (gc)
│ │ │ + -- used 0.143985s (cpu); 0.144011s (thread); 0s (gc)
│ │ │  
│ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.237275s (cpu); 0.23721s (thread); 0s (gc)
│ │ │ + -- used 0.136234s (cpu); 0.136261s (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/M2-11147-0/0/
│ │ │ +o1 = /tmp/M2-12217-0/0/
│ │ │  
│ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11147-0/1/
│ │ │ +o2 = /tmp/M2-12217-0/1/
│ │ │  
│ │ │  i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11147-0/0/a/
│ │ │ +o3 = /tmp/M2-12217-0/0/a/
│ │ │  
│ │ │  i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11147-0/0/b/
│ │ │ +o4 = /tmp/M2-12217-0/0/b/
│ │ │  
│ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11147-0/0/b/c/
│ │ │ +o5 = /tmp/M2-12217-0/0/b/c/
│ │ │  
│ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11147-0/0/a/f
│ │ │ +o6 = /tmp/M2-12217-0/0/a/f
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11147-0/0/a/g
│ │ │ +o7 = /tmp/M2-12217-0/0/a/g
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11147-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12217-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11147-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/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/M2-11147-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/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/M2-11204-0/0
│ │ │ +o1 = /tmp/M2-12334-0/0
│ │ │  
│ │ │  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/M2-12169-0/0.tex
│ │ │ +o1 = /tmp/M2-14309-0/0.tex
│ │ │  
│ │ │  i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-12169-0/1.html
│ │ │ +o2 = /tmp/M2-14309-0/1.html
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_time.out
│ │ │ @@ -1,8 +1,8 @@
│ │ │  -- -*- M2-comint -*- hash: 1332435500723
│ │ │  
│ │ │  i1 : time 3^30
│ │ │ - -- used 2.18e-05s (cpu); 1.1101e-05s (thread); 0s (gc)
│ │ │ + -- used 4.4803e-05s (cpu); 7.142e-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
│ │ │ -     -- .000018144 seconds
│ │ │ +     -- .000021827 seconds
│ │ │  
│ │ │  o1 : Time
│ │ │  
│ │ │  i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000018144, 205891132094649}
│ │ │ +o2 = Time{.000021827, 205891132094649}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_version.out
│ │ │ @@ -34,15 +34,15 @@
│ │ │                 "memtailor version" => 1.1
│ │ │                 "mpfi version" => 1.5.4
│ │ │                 "mpfr version" => 4.2.2
│ │ │                 "mpsolve version" => 3.2.2
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.11.0
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.12.57+deb13-amd64
│ │ │ +               "operating system release" => 6.12.57+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 PruneComplex 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 OldPolyhedra OldToricVectorBundles 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 NonminimalComplexes 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 MatrixFactorizations PathSignatures
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.13.11
│ │ │                 "readline version" => 8.3
│ │ │                 "scscp version" => not present
│ │ │                 "tbb version" => 2022.1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Command.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │              
│ │ │                
    i3 : (c = Command "date";)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : c
│ │ │ -Sun Dec 14 15:26:56 UTC 2025
│ │ │ +Wed Jan  7 12:22:11 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
│ │ │ │ -Sun Dec 14 15:26:56 UTC 2025
│ │ │ │ +Wed Jan  7 12:22:11 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
│ │ │ @@ -52,22 +52,22 @@
│ │ │        
    
│ │ │          
    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/M2-11641-0/0.dbm
    
│ │ │ +o1 = /tmp/M2-13231-0/0.dbm
│ │ │              
    
│ │ │                
    i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-11641-0/0.dbm
│ │ │ +o2 = /tmp/M2-13231-0/0.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/M2-11641-0/0.dbm
│ │ │ │ +o1 = /tmp/M2-13231-0/0.dbm
│ │ │ │  i2 : x = openDatabaseOut filename
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11641-0/0.dbm
│ │ │ │ +o2 = /tmp/M2-13231-0/0.dbm
│ │ │ │  
│ │ │ │  o2 : Database
│ │ │ │  i3 : x#"first" = "hi there"
│ │ │ │  
│ │ │ │  o3 = hi there
│ │ │ │  i4 : x#"first"
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__Cstats.html
│ │ │ @@ -53,33 +53,33 @@
│ │ │          
    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" => 42969169706        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 43050590362        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 206680064
│ │ │ -               "numGCs" => 795
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 225931264
│ │ │ +               "numGCs" => 783
│ │ │ +               "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 = 206680064
    
│ │ │ +o2 = 225931264
│ │ │              
    
│ │ │          
    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" => 42969169706        }
│ │ │ │ +o1 = HashTable{"bytesAlloc" => 43050590362        }
│ │ │ │                 "GC_free_space_divisor" => 3
│ │ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │ │                 "gcCpuTimeSecs" => 0
│ │ │ │ -               "heapSize" => 206680064
│ │ │ │ -               "numGCs" => 795
│ │ │ │ -               "numGCThreads" => 6
│ │ │ │ +               "heapSize" => 225931264
│ │ │ │ +               "numGCs" => 783
│ │ │ │ +               "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 = 206680064
│ │ │ │ +o2 = 225931264
│ │ │ │  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
│ │ │ @@ -128,23 +128,23 @@
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │                
    i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ - -- used 0.00906552s (cpu); 0.00905868s (thread); 0s (gc)
│ │ │ + -- used 0.00604401s (cpu); 0.00604071s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of R
    
│ │ │              
    
│ │ │                
    i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ - -- used 0.0794664s (cpu); 0.0794742s (thread); 0s (gc)
│ │ │ + -- used 0.0603359s (cpu); 0.0603461s (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 0.00906552s (cpu); 0.00905868s (thread); 0s (gc)
│ │ │ │ + -- used 0.00604401s (cpu); 0.00604071s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ │ - -- used 0.0794664s (cpu); 0.0794742s (thread); 0s (gc)
│ │ │ │ + -- used 0.0603359s (cpu); 0.0603461s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : numgens J
│ │ │ │  
│ │ │ │  o10 = 1067
│ │ │ │  i11 : numgens K
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.html
│ │ │ @@ -68,21 +68,21 @@
│ │ │  o1 : Matrix RR      <-- RR
│ │ │                53          53
    
│ │ │              
    
│ │ │                
    i2 : time SVD(M);
│ │ │ - -- used 0.0253142s (cpu); 0.0253129s (thread); 0s (gc)
    
│ │ │ + -- used 0.0417848s (cpu); 0.041782s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0274055s (cpu); 0.0274142s (thread); 0s (gc)
    
│ │ │ + -- used 0.0410636s (cpu); 0.0410811s (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 0.0253142s (cpu); 0.0253129s (thread); 0s (gc)
│ │ │ │ + -- used 0.0417848s (cpu); 0.041782s (thread); 0s (gc)
│ │ │ │  i3 : time SVD(M, DivideConquer=>true);
│ │ │ │ - -- used 0.0274055s (cpu); 0.0274142s (thread); 0s (gc)
│ │ │ │ + -- used 0.0410636s (cpu); 0.0410811s (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
│ │ │ @@ -826,15 +826,15 @@
│ │ │          
    
│ │ │            
    We may use resolution to produce a projective resolution of it, and time to report the time required.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i59 : time C = resolution M
│ │ │ - -- used 0.00192778s (cpu); 0.00191945s (thread); 0s (gc)
│ │ │ + -- used 0.00234372s (cpu); 0.00233367s (thread); 0s (gc)
│ │ │  
│ │ │         3      6      15      18      6
│ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R  <-- 0
│ │ │                                              
│ │ │        0      1      2       3       4      5
│ │ │  
│ │ │  o59 : ChainComplex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -390,15 +390,15 @@
│ │ │ │                 | 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 0.00192778s (cpu); 0.00191945s (thread); 0s (gc)
│ │ │ │ + -- used 0.00234372s (cpu); 0.00233367s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3      6      15      18      6
│ │ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R  <-- 0
│ │ │ │  
│ │ │ │        0      1      2       3       4      5
│ │ │ │  
│ │ │ │  o59 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_at__End__Of__File_lp__File_rp.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  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-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_files.m2:374:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_benchmark.html
│ │ │ @@ -68,15 +68,15 @@
│ │ │        
    
│ │ │          
    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 = .000290697861367332
│ │ │ +o1 = .0003726770483759516
│ │ │  
│ │ │  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 = .000290697861367332
│ │ │ │ +o1 = .0003726770483759516
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -69,23 +69,23 @@
│ │ │                
    i3 : M = coker vars R;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime pdim' M
│ │ │   -- computing pdim'
│ │ │ - -- .00670047s elapsed
│ │ │ + -- .00488437s elapsed
│ │ │  
│ │ │  o4 = 3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime pdim' M
│ │ │ - -- .000001513s elapsed
│ │ │ + -- .000002519s 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'
│ │ │ │ - -- .00670047s elapsed
│ │ │ │ + -- .00488437s elapsed
│ │ │ │  
│ │ │ │  o4 = 3
│ │ │ │  i5 : elapsedTime pdim' M
│ │ │ │ - -- .000001513s elapsed
│ │ │ │ + -- .000002519s 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
│ │ │ @@ -104,15 +104,15 @@
│ │ │  o4 : Task
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : n
│ │ │  
│ │ │ -o5 = 711206
    
│ │ │ +o5 = 1095574
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
    
│ │ │ @@ -127,15 +127,15 @@
│ │ │  o7 : Task
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : n
│ │ │  
│ │ │ -o8 = 1453533
    
│ │ │ +o8 = 2222205
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : isReady t
│ │ │  
│ │ │  o9 = false
    
│ │ │ @@ -163,29 +163,29 @@
│ │ │  o12 : Task
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : n
│ │ │  
│ │ │ -o13 = 1453746
    
│ │ │ +o13 = 2222398
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : n
│ │ │  
│ │ │ -o15 = 1453746
    
│ │ │ +o15 = 2222398
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : isReady t
│ │ │  
│ │ │  o16 = false
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,26 +28,26 @@
│ │ │ │  i4 : t
│ │ │ │  
│ │ │ │  o4 = <>
│ │ │ │  
│ │ │ │  o4 : Task
│ │ │ │  i5 : n
│ │ │ │  
│ │ │ │ -o5 = 711206
│ │ │ │ +o5 = 1095574
│ │ │ │  i6 : sleep 1
│ │ │ │  
│ │ │ │  o6 = 0
│ │ │ │  i7 : t
│ │ │ │  
│ │ │ │  o7 = <>
│ │ │ │  
│ │ │ │  o7 : Task
│ │ │ │  i8 : n
│ │ │ │  
│ │ │ │ -o8 = 1453533
│ │ │ │ +o8 = 2222205
│ │ │ │  i9 : isReady t
│ │ │ │  
│ │ │ │  o9 = false
│ │ │ │  i10 : cancelTask t
│ │ │ │  i11 : sleep 2
│ │ │ │  stdio:2:25:(3):[1]: error: interrupted
│ │ │ │  
│ │ │ │ @@ -55,21 +55,21 @@
│ │ │ │  i12 : t
│ │ │ │  
│ │ │ │  o12 = <>
│ │ │ │  
│ │ │ │  o12 : Task
│ │ │ │  i13 : n
│ │ │ │  
│ │ │ │ -o13 = 1453746
│ │ │ │ +o13 = 2222398
│ │ │ │  i14 : sleep 1
│ │ │ │  
│ │ │ │  o14 = 0
│ │ │ │  i15 : n
│ │ │ │  
│ │ │ │ -o15 = 1453746
│ │ │ │ +o15 = 2222398
│ │ │ │  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
│ │ │ @@ -71,36 +71,36 @@
│ │ │            
    Change the current working directory to dir.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10463-0/0
    
│ │ │ +o1 = /tmp/M2-10833-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10463-0/0
    
│ │ │ +o2 = /tmp/M2-10833-0/0
│ │ │              
    
│ │ │                
    i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-10463-0/0/
    
│ │ │ +o3 = /tmp/M2-10833-0/0/
│ │ │              
    
│ │ │                
    i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-10463-0/0/
    
│ │ │ +o4 = /tmp/M2-10833-0/0/
│ │ │              
    
│ │ │          
    
│ │ │            
    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/M2-10463-0/0
│ │ │ │ +o1 = /tmp/M2-10833-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10463-0/0
│ │ │ │ +o2 = /tmp/M2-10833-0/0
│ │ │ │  i3 : changeDirectory dir
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10463-0/0/
│ │ │ │ +o3 = /tmp/M2-10833-0/0/
│ │ │ │  i4 : currentDirectory()
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-10463-0/0/
│ │ │ │ +o4 = /tmp/M2-10833-0/0/
│ │ │ │  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
│ │ │ @@ -95,40 +95,40 @@
│ │ │  o1 : Package
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : check_1 FirstPackage
│ │ │   -- warning: reloading FirstPackage; recreate instances of types from this package
│ │ │ - -- capturing check(1, "FirstPackage")        -- .15147s elapsed
    
│ │ │ + -- capturing check(1, "FirstPackage")        -- .136542s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : check FirstPackage
│ │ │ - -- capturing check(0, "FirstPackage")        -- .150181s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .150965s elapsed
    
│ │ │ + -- capturing check(0, "FirstPackage")        -- .147318s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .144622s elapsed
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Alternatively, if the package is installed somewhere accessible, one can do the following.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : check_1 "FirstPackage"
│ │ │ - -- capturing check(1, "FirstPackage")        -- .152579s elapsed
    
│ │ │ + -- capturing check(1, "FirstPackage")        -- .139659s elapsed
│ │ │              
    
│ │ │                
    i5 : check "FirstPackage"
│ │ │ - -- capturing check(0, "FirstPackage")        -- .152053s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .151867s elapsed
    
│ │ │ + -- capturing check(0, "FirstPackage")        -- .135863s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .137943s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    A TestInput object (or a list of such objects) can also be run directly.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -140,15 +140,15 @@
│ │ │  
│ │ │  o6 : TestInput
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : check oo
│ │ │ - -- capturing check(1, "FirstPackage")        -- .153083s elapsed
    
│ │ │ + -- capturing check(1, "FirstPackage")        -- .135696s elapsed
│ │ │              
    
│ │ │                
    i8 : tests "FirstPackage"
│ │ │  
│ │ │  o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │ @@ -156,16 +156,16 @@
│ │ │  
│ │ │  o8 : NumberedVerticalList
    
│ │ │              
    
│ │ │                
    i9 : check oo
│ │ │ - -- capturing check(0, "FirstPackage")        -- .152703s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .150901s elapsed
    
│ │ │ + -- capturing check(0, "FirstPackage")        -- .133351s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .13031s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    If only an integer is passed as an argument, then the test with that index from the last call to tests is run.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -178,15 +178,15 @@
│ │ │  
│ │ │  o10 : NumberedVerticalList
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : check 1
│ │ │ - -- capturing check(1, "FirstPackage")        -- .151346s elapsed
    
│ │ │ + -- capturing check(1, "FirstPackage")        -- .131903s 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")        -- .15147s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .136542s elapsed
│ │ │ │  i3 : check FirstPackage
│ │ │ │ - -- capturing check(0, "FirstPackage")        -- .150181s elapsed
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .150965s elapsed
│ │ │ │ + -- capturing check(0, "FirstPackage")        -- .147318s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .144622s elapsed
│ │ │ │  Alternatively, if the package is installed somewhere accessible, one can do the
│ │ │ │  following.
│ │ │ │  i4 : check_1 "FirstPackage"
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .152579s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .139659s elapsed
│ │ │ │  i5 : check "FirstPackage"
│ │ │ │ - -- capturing check(0, "FirstPackage")        -- .152053s elapsed
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .151867s elapsed
│ │ │ │ + -- capturing check(0, "FirstPackage")        -- .135863s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .137943s 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")        -- .153083s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .135696s 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")        -- .152703s elapsed
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .150901s elapsed
│ │ │ │ + -- capturing check(0, "FirstPackage")        -- .133351s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .13031s 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")        -- .151346s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .131903s 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
│ │ │ @@ -50,15 +50,15 @@
│ │ │      
    
│ │ │        
    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 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) 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.        
│ │ │            
│ │ │ @@ -74,15 +74,15 @@
│ │ │            
│ │ │          
    
│ │ │  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 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ +o3 = "Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │       6.12.57-1 (2025-11-05) 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 {}
│ │ │ │ @@ -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 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-
│ │ │ │ -11-05) x86_64 GNU/Linux
│ │ │ │ +Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1
│ │ │ │ +(2025-11-05) 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 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +o3 = "Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │       6.12.57-1 (2025-11-05) 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
│ │ │ @@ -269,15 +269,15 @@
│ │ │                 1277
│ │ │              
    
│ │ │                
    i24 : gb I
│ │ │  
│ │ │ -   -- registering gb 5 at 0x7f67b957e540
│ │ │ +   -- registering gb 5 at 0x7f49bd44d540
│ │ │  
│ │ │     -- [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
│ │ │ @@ -373,15 +373,15 @@
│ │ │                1      4
│ │ │  o32 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i33 : time betti gb f
│ │ │ - -- used 0.303879s (cpu); 0.305901s (thread); 0s (gc)
│ │ │ + -- used 0.227867s (cpu); 0.231374s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o33 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ @@ -417,15 +417,15 @@
│ │ │  
│ │ │  o35 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i36 : time betti gb f
│ │ │ - -- used 0.00799525s (cpu); 0.00545052s (thread); 0s (gc)
│ │ │ + -- used 0.00398845s (cpu); 0.00376715s (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 0x7f67b957e540
│ │ │ │ +   -- registering gb 5 at 0x7f49bd44d540
│ │ │ │  
│ │ │ │     -- [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 0.303879s (cpu); 0.305901s (thread); 0s (gc)
│ │ │ │ + -- used 0.227867s (cpu); 0.231374s (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 0.00799525s (cpu); 0.00545052s (thread); 0s (gc)
│ │ │ │ + -- used 0.00398845s (cpu); 0.00376715s (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
│ │ │ @@ -80,112 +80,112 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── 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 3.0434s (cpu); 1.80018s (thread); 0s (gc)
│ │ │ │ + -- used 3.97935s (cpu); 2.18083s (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 3.14065s (cpu); 1.91214s (thread); 0s (gc)
│ │ │ │ + -- used 4.37872s (cpu); 2.44278s (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 8.07742s (cpu); 4.95873s (thread); 0s (gc)
│ │ │ │ + -- used 7.23744s (cpu); 5.14081s (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
│ │ │ @@ -89,15 +89,15 @@
│ │ │  
│ │ │  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 0.0951343s (cpu); 0.0963862s (thread); 0s (gc)
│ │ │ │ + -- used 0.189975s (cpu); 0.130013s (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 2.05743s (cpu); 1.42394s (thread); 0s (gc)
│ │ │ │ + -- used 2.4364s (cpu); 1.58346s (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
│ │ │ @@ -93,31 +93,31 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11185-0/0/
    
│ │ │ +o1 = /tmp/M2-12295-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11185-0/1/
    
│ │ │ +o2 = /tmp/M2-12295-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11185-0/0/a/
    
│ │ │ +o3 = /tmp/M2-12295-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11185-0/0/b/
    
│ │ │ +o4 = /tmp/M2-12295-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11185-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-12295-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11185-0/0/a/f
│ │ │ +o6 = /tmp/M2-12295-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11185-0/0/a/g
│ │ │ +o7 = /tmp/M2-12295-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11185-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12295-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-11185-0/0/
│ │ │ -     /tmp/M2-11185-0/0/b/
│ │ │ -     /tmp/M2-11185-0/0/b/c/
│ │ │ -     /tmp/M2-11185-0/0/b/c/g
│ │ │ -     /tmp/M2-11185-0/0/a/
│ │ │ -     /tmp/M2-11185-0/0/a/g
│ │ │ -     /tmp/M2-11185-0/0/a/f
    
│ │ │ +o9 = /tmp/M2-12295-0/0/
│ │ │ +     /tmp/M2-12295-0/0/a/
│ │ │ +     /tmp/M2-12295-0/0/a/g
│ │ │ +     /tmp/M2-12295-0/0/a/f
│ │ │ +     /tmp/M2-12295-0/0/b/
│ │ │ +     /tmp/M2-12295-0/0/b/c/
│ │ │ +     /tmp/M2-12295-0/0/b/c/g
│ │ │              
    
│ │ │                
    i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11185-0/0/b/c/g -> /tmp/M2-11185-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-11185-0/0/a/g -> /tmp/M2-11185-0/1/a/g
│ │ │ - -- copying: /tmp/M2-11185-0/0/a/f -> /tmp/M2-11185-0/1/a/f
    
│ │ │ + -- copying: /tmp/M2-12295-0/0/a/g -> /tmp/M2-12295-0/1/a/g
│ │ │ + -- copying: /tmp/M2-12295-0/0/a/f -> /tmp/M2-12295-0/1/a/f
│ │ │ + -- copying: /tmp/M2-12295-0/0/b/c/g -> /tmp/M2-12295-0/1/b/c/g
│ │ │              
    
│ │ │                
    i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11185-0/0/b/c/g not newer than /tmp/M2-11185-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-11185-0/0/a/g not newer than /tmp/M2-11185-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-11185-0/0/a/f not newer than /tmp/M2-11185-0/1/a/f
    
│ │ │ + -- skipping: /tmp/M2-12295-0/0/a/g not newer than /tmp/M2-12295-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-12295-0/0/a/f not newer than /tmp/M2-12295-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-12295-0/0/b/c/g not newer than /tmp/M2-12295-0/1/b/c/g
│ │ │              
    
│ │ │                
    i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-11185-0/1/
│ │ │ -      /tmp/M2-11185-0/1/a/
│ │ │ -      /tmp/M2-11185-0/1/a/f
│ │ │ -      /tmp/M2-11185-0/1/a/g
│ │ │ -      /tmp/M2-11185-0/1/b/
│ │ │ -      /tmp/M2-11185-0/1/b/c/
│ │ │ -      /tmp/M2-11185-0/1/b/c/g
    
│ │ │ +o12 = /tmp/M2-12295-0/1/
│ │ │ +      /tmp/M2-12295-0/1/a/
│ │ │ +      /tmp/M2-12295-0/1/a/g
│ │ │ +      /tmp/M2-12295-0/1/a/f
│ │ │ +      /tmp/M2-12295-0/1/b/
│ │ │ +      /tmp/M2-12295-0/1/b/c/
│ │ │ +      /tmp/M2-12295-0/1/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/M2-11185-0/0/
│ │ │ │ +o1 = /tmp/M2-12295-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11185-0/1/
│ │ │ │ +o2 = /tmp/M2-12295-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11185-0/0/a/
│ │ │ │ +o3 = /tmp/M2-12295-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11185-0/0/b/
│ │ │ │ +o4 = /tmp/M2-12295-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11185-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-12295-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11185-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-12295-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11185-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-12295-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-11185-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-12295-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : stack findFiles src
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-11185-0/0/
│ │ │ │ -     /tmp/M2-11185-0/0/b/
│ │ │ │ -     /tmp/M2-11185-0/0/b/c/
│ │ │ │ -     /tmp/M2-11185-0/0/b/c/g
│ │ │ │ -     /tmp/M2-11185-0/0/a/
│ │ │ │ -     /tmp/M2-11185-0/0/a/g
│ │ │ │ -     /tmp/M2-11185-0/0/a/f
│ │ │ │ +o9 = /tmp/M2-12295-0/0/
│ │ │ │ +     /tmp/M2-12295-0/0/a/
│ │ │ │ +     /tmp/M2-12295-0/0/a/g
│ │ │ │ +     /tmp/M2-12295-0/0/a/f
│ │ │ │ +     /tmp/M2-12295-0/0/b/
│ │ │ │ +     /tmp/M2-12295-0/0/b/c/
│ │ │ │ +     /tmp/M2-12295-0/0/b/c/g
│ │ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-11185-0/0/b/c/g -> /tmp/M2-11185-0/1/b/c/g
│ │ │ │ - -- copying: /tmp/M2-11185-0/0/a/g -> /tmp/M2-11185-0/1/a/g
│ │ │ │ - -- copying: /tmp/M2-11185-0/0/a/f -> /tmp/M2-11185-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-12295-0/0/a/g -> /tmp/M2-12295-0/1/a/g
│ │ │ │ + -- copying: /tmp/M2-12295-0/0/a/f -> /tmp/M2-12295-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-12295-0/0/b/c/g -> /tmp/M2-12295-0/1/b/c/g
│ │ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11185-0/0/b/c/g not newer than /tmp/M2-11185-0/1/b/c/g
│ │ │ │ - -- skipping: /tmp/M2-11185-0/0/a/g not newer than /tmp/M2-11185-0/1/a/g
│ │ │ │ - -- skipping: /tmp/M2-11185-0/0/a/f not newer than /tmp/M2-11185-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-12295-0/0/a/g not newer than /tmp/M2-12295-0/1/a/g
│ │ │ │ + -- skipping: /tmp/M2-12295-0/0/a/f not newer than /tmp/M2-12295-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-12295-0/0/b/c/g not newer than /tmp/M2-12295-0/1/b/c/g
│ │ │ │  i12 : stack findFiles dst
│ │ │ │  
│ │ │ │ -o12 = /tmp/M2-11185-0/1/
│ │ │ │ -      /tmp/M2-11185-0/1/a/
│ │ │ │ -      /tmp/M2-11185-0/1/a/f
│ │ │ │ -      /tmp/M2-11185-0/1/a/g
│ │ │ │ -      /tmp/M2-11185-0/1/b/
│ │ │ │ -      /tmp/M2-11185-0/1/b/c/
│ │ │ │ -      /tmp/M2-11185-0/1/b/c/g
│ │ │ │ +o12 = /tmp/M2-12295-0/1/
│ │ │ │ +      /tmp/M2-12295-0/1/a/
│ │ │ │ +      /tmp/M2-12295-0/1/a/g
│ │ │ │ +      /tmp/M2-12295-0/1/a/f
│ │ │ │ +      /tmp/M2-12295-0/1/b/
│ │ │ │ +      /tmp/M2-12295-0/1/b/c/
│ │ │ │ +      /tmp/M2-12295-0/1/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
│ │ │ @@ -78,65 +78,65 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10970-0/0
    
│ │ │ +o1 = /tmp/M2-11860-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10970-0/1
    
│ │ │ +o2 = /tmp/M2-11860-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10970-0/0
│ │ │ +o3 = /tmp/M2-11860-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-10970-0/0 -> /tmp/M2-10970-0/1
    
│ │ │ + -- copying: /tmp/M2-11860-0/0 -> /tmp/M2-11860-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
    
│ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │              
    
│ │ │                
    i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-10970-0/0
│ │ │ +o7 = /tmp/M2-11860-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
    
│ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │              
    
│ │ │                
    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/M2-10970-0/0
│ │ │ │ +o1 = /tmp/M2-11860-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10970-0/1
│ │ │ │ +o2 = /tmp/M2-11860-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10970-0/0
│ │ │ │ +o3 = /tmp/M2-11860-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-10970-0/0 -> /tmp/M2-10970-0/1
│ │ │ │ + -- copying: /tmp/M2-11860-0/0 -> /tmp/M2-11860-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
│ │ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │ │  i7 : src << "ho there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-10970-0/0
│ │ │ │ +o7 = /tmp/M2-11860-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
│ │ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │ │  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
│ │ │ @@ -64,38 +64,38 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : t1 = cpuTime()
│ │ │  
│ │ │ -o1 = 354.029649282
│ │ │ +o1 = 383.166340652
│ │ │  
│ │ │  o1 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i2 : for i from 0 to 1000000 do 223131321321*324234324324;
    
│ │ │              
    
│ │ │                
    i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 355.996910491
│ │ │ +o3 = 384.327232721
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i4 : t2-t1
│ │ │  
│ │ │ -o4 = 1.967261209000014
│ │ │ +o4 = 1.160892068999999
│ │ │  
│ │ │  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 = 354.029649282
│ │ │ │ +o1 = 383.166340652
│ │ │ │  
│ │ │ │  o1 : RR (of precision 53)
│ │ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │ │  i3 : t2 = cpuTime()
│ │ │ │  
│ │ │ │ -o3 = 355.996910491
│ │ │ │ +o3 = 384.327232721
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  i4 : t2-t1
│ │ │ │  
│ │ │ │ -o4 = 1.967261209000014
│ │ │ │ +o4 = 1.160892068999999
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -64,48 +64,48 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : currentTime()
│ │ │  
│ │ │ -o1 = 1765726091
    
│ │ │ +o1 = 1767788593
│ │ │              
    
│ │ │          
    We can compute, roughly, how many years ago the epoch began as follows.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 55.95363237235333
│ │ │ +o2 = 56.01899045890111
│ │ │  
│ │ │  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 = 11.44358846823999
│ │ │ +o3 = .2278855068132941
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │          
    Compare that to the current date, available from a standard Unix command.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : run "date"
│ │ │ -Sun Dec 14 15:28:11 UTC 2025
│ │ │ +Wed Jan  7 12:23:13 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 = 1765726091
│ │ │ │ +o1 = 1767788593
│ │ │ │  We can compute, roughly, how many years ago the epoch began as follows.
│ │ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │ │  
│ │ │ │ -o2 = 55.95363237235333
│ │ │ │ +o2 = 56.01899045890111
│ │ │ │  
│ │ │ │  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 = 11.44358846823999
│ │ │ │ +o3 = .2278855068132941
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  Compare that to the current date, available from a standard Unix command.
│ │ │ │  i4 : run "date"
│ │ │ │ -Sun Dec 14 15:28:11 UTC 2025
│ │ │ │ +Wed Jan  7 12:23:13 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-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:1849:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Time.html
│ │ │ @@ -59,15 +59,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │  elapsedTime e evaluates e, prints the amount of time elapsed, and returns the value of e.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime sleep 1
│ │ │ - -- 1.00015s elapsed
│ │ │ + -- 1.00023s 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
│ │ │ │ - -- 1.00015s elapsed
│ │ │ │ + -- 1.00023s 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/_elimination_spof_spvariables.html
│ │ │ @@ -65,15 +65,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
    
│ │ │                
    i3 : time leadTerm gens gb I
│ │ │ - -- used 0.464405s (cpu); 0.277254s (thread); 0s (gc)
│ │ │ + -- used 0.1387s (cpu); 0.138699s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │       ------------------------------------------------------------------------
│ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │       ------------------------------------------------------------------------
│ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -162,15 +162,15 @@
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │                
    i8 : time G = eliminate(I,{s,t})
│ │ │ - -- used 0.422625s (cpu); 0.238689s (thread); 0s (gc)
│ │ │ + -- used 0.42811s (cpu); 0.206841s (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
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -245,15 +245,15 @@
│ │ │  
│ │ │  o11 : Ideal of R1
    
│ │ │              
    
│ │ │                
    i12 : time G = eliminate(I1,{s,t})
│ │ │ - -- used 0.286447s (cpu); 0.110979s (thread); 0s (gc)
│ │ │ + -- used 0.0363006s (cpu); 0.0363011s (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  -
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -337,15 +337,15 @@
│ │ │  
│ │ │  o16 : RingMap A <-- B
    
│ │ │              
    
│ │ │                
    i17 : time G = kernel F
│ │ │ - -- used 0.404757s (cpu); 0.227468s (thread); 0s (gc)
│ │ │ + -- used 0.1109s (cpu); 0.11091s (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
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -418,26 +418,26 @@
│ │ │  
│ │ │  o19 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ - -- used 0.00191261s (cpu); 0.00191299s (thread); 0s (gc)
│ │ │ + -- used 0.00189673s (cpu); 0.00189022s (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 0.0583203s (cpu); 0.0583314s (thread); 0s (gc)
│ │ │ + -- used 0.0362551s (cpu); 0.0362736s (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 0.464405s (cpu); 0.277254s (thread); 0s (gc)
│ │ │ │ + -- used 0.1387s (cpu); 0.138699s (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 0.422625s (cpu); 0.238689s (thread); 0s (gc)
│ │ │ │ + -- used 0.42811s (cpu); 0.206841s (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 0.286447s (cpu); 0.110979s (thread); 0s (gc)
│ │ │ │ + -- used 0.0363006s (cpu); 0.0363011s (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 0.404757s (cpu); 0.227468s (thread); 0s (gc)
│ │ │ │ + -- used 0.1109s (cpu); 0.11091s (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 0.00191261s (cpu); 0.00191299s (thread); 0s (gc)
│ │ │ │ + -- used 0.00189673s (cpu); 0.00189022s (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 0.0583203s (cpu); 0.0583314s (thread); 0s (gc)
│ │ │ │ + -- used 0.0362551s (cpu); 0.0362736s (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
│ │ │ @@ -154,15 +154,15 @@
│ │ │                                      Version => 0.0
│ │ │               package prefix => /usr/
│ │ │               PackageIsLoaded => true
│ │ │               pkgname => Foo
│ │ │               private dictionary => Foo#"private dictionary"
│ │ │               processed documentation => MutableHashTable{}
│ │ │               raw documentation => MutableHashTable{}
│ │ │ -             source directory => /tmp/M2-10191-0/91-rundir/
│ │ │ +             source directory => /tmp/M2-10311-0/91-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/M2-10191-0/91-rundir/
│ │ │ │ +             source directory => /tmp/M2-10311-0/91-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
│ │ │ @@ -68,29 +68,29 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,16 +31,16 @@
│ │ │ │  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/M2-53632-0/0-in.ms -o /tmp/
│ │ │ │ -M2-53632-0/0-out.ms
│ │ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-84642-0/0-in.ms -o /tmp/
│ │ │ │ +M2-84642-0/0-out.ms
│ │ │ │  
│ │ │ │  --------------- INPUT DATA ---------------
│ │ │ │  #variables                       3
│ │ │ │  #equations                       3
│ │ │ │  #invalid equations               0
│ │ │ │  field characteristic             0
│ │ │ │  homogeneous input?               1
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  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 = 1196244169
│ │ │ │ +Initial prime = 1117130281
│ │ │ │  
│ │ │ │  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)
│ │ │ │ @@ -73,26 +73,26 @@
│ │ │ │  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
│ │ │ │ -0.02 | 0.07
│ │ │ │ +0.00 | 0.00
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  
│ │ │ │  ---------------- TIMINGS ----------------
│ │ │ │ -overall(elapsed)        0.06 sec
│ │ │ │ -overall(cpu)            0.17 sec
│ │ │ │ +overall(elapsed)        0.00 sec
│ │ │ │ +overall(cpu)            0.01 sec
│ │ │ │  select                  0.00 sec   0.0%
│ │ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ │ -update                  0.03 sec  57.3%
│ │ │ │ -convert                 0.02 sec  42.4%
│ │ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ │ +symbolic prep.          0.00 sec   0.2%
│ │ │ │ +update                  0.00 sec  87.1%
│ │ │ │ +convert                 0.00 sec   4.0%
│ │ │ │ +linear algebra          0.00 sec   0.5%
│ │ │ │  reduce gb               0.00 sec   0.0%
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │ │  size of basis                     3
│ │ │ │  #terms in basis                   3
│ │ │ │  #pairs reduced                    0
│ │ │ │ @@ -106,18 +106,18 @@
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  
│ │ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │ │  [3]
│ │ │ │  #polynomials to lift              3
│ │ │ │  -----------------------------------------
│ │ │ │ -New prime = 1107170621
│ │ │ │ +New prime = 1157291321
│ │ │ │  
│ │ │ │  ---------------- TIMINGS ----------------
│ │ │ │ -multi-mod overall(elapsed)      0.02 sec
│ │ │ │ +multi-mod overall(elapsed)      0.00 sec
│ │ │ │  learning phase                  0.00 Gops/sec
│ │ │ │  application phase               0.00 Gops/sec
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  multi-modular steps
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │ @@ -136,15 +136,15 @@
│ │ │ │  CRT     (elapsed)               0.00 sec
│ │ │ │  ratrecon(elapsed)               0.00 sec
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----
│ │ │ │ -msolve overall time           0.19 sec (elapsed) /  0.49 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
│ │ │ - -- .00186857s elapsed
│ │ │ + -- .0029236s elapsed
│ │ │  computing total degree: 2
│ │ │  number of monomials = 21
│ │ │  number of distinct multidegrees = 18
│ │ │ - -- .00833047s elapsed
│ │ │ + -- .0117245s 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
│ │ │ @@ -117,19 +117,19 @@
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── 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
│ │ │ │ - -- .0234547s elapsed
│ │ │ │ + -- .0469481s elapsed
│ │ │ │  
│ │ │ │  o3 = 2
│ │ │ │  i4 : elapsedTime monjMult I
│ │ │ │ - -- .107713s elapsed
│ │ │ │ + -- .0900121s elapsed
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  i5 : elapsedTime multiplicitySequence I
│ │ │ │ - -- .181349s elapsed
│ │ │ │ + -- .191375s 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
│ │ │ @@ -89,15 +89,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10558-0/0
    
│ │ │ +o1 = /tmp/M2-11028-0/0
│ │ │              
    
│ │ │                
    i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
    
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10558-0/0
│ │ │ +o3 = /tmp/M2-11028-0/0
│ │ │  
│ │ │  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/M2-10558-0/0
│ │ │ │ +o1 = /tmp/M2-11028-0/0
│ │ │ │  i2 : fileExists fn
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10558-0/0
│ │ │ │ +o3 = /tmp/M2-11028-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -69,15 +69,15 @@
│ │ │          
    Description
    
│ │ │          
    The length of an open output file is determined from the internal count of the number of bytes written so far.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -510,27 +510,27 @@
│ │ │  o36 = 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : f = temporaryFileName() << "hi there"
│ │ │  
│ │ │ -o1 = /tmp/M2-12150-0/0
│ │ │ +o1 = /tmp/M2-14270-0/0
│ │ │  
│ │ │  o1 : File
    
│ │ │              
    
│ │ │                
    i2 : fileLength f
│ │ │ @@ -85,24 +85,24 @@
│ │ │  o2 = 8
    
│ │ │              
    
│ │ │                
    i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-12150-0/0
│ │ │ +o3 = /tmp/M2-14270-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-12150-0/0
    
│ │ │ +o4 = /tmp/M2-14270-0/0
│ │ │              
    
│ │ │                
    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/M2-12150-0/0
│ │ │ │ +o1 = /tmp/M2-14270-0/0
│ │ │ │  
│ │ │ │  o1 : File
│ │ │ │  i2 : fileLength f
│ │ │ │  
│ │ │ │  o2 = 8
│ │ │ │  i3 : close f
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12150-0/0
│ │ │ │ +o3 = /tmp/M2-14270-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : filename = toString f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12150-0/0
│ │ │ │ +o4 = /tmp/M2-14270-0/0
│ │ │ │  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
│ │ │ @@ -69,22 +69,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11375-0/0
    
│ │ │ +o1 = /tmp/M2-12685-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11375-0/0
│ │ │ +o2 = /tmp/M2-12685-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : fileMode f
│ │ │ @@ -92,15 +92,15 @@
│ │ │  o3 = 420
    
│ │ │              
    
│ │ │                
    i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-11375-0/0
│ │ │ +o4 = /tmp/M2-12685-0/0
│ │ │  
│ │ │  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/M2-11375-0/0
│ │ │ │ +o1 = /tmp/M2-12685-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11375-0/0
│ │ │ │ +o2 = /tmp/M2-12685-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode f
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : close f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11375-0/0
│ │ │ │ +o4 = /tmp/M2-12685-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -69,22 +69,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │              
│ │ │              
│ │ │                
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -425,15 +425,15 @@
│ │ │  
│ │ │  o33 = 1
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10989-0/0
    
│ │ │ +o1 = /tmp/M2-11899-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-10989-0/0
│ │ │ +o2 = /tmp/M2-11899-0/0
│ │ │  
│ │ │  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/M2-10989-0/0
│ │ │ │ +o1 = /tmp/M2-11899-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10989-0/0
│ │ │ │ +o2 = /tmp/M2-11899-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -73,22 +73,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -245,23 +245,23 @@
│ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │        00000000000000185613
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10854-0/0
    
│ │ │ +o1 = /tmp/M2-11624-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-10854-0/0
│ │ │ +o2 = /tmp/M2-11624-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : m = 7 + 7*8 + 7*64
│ │ │ @@ -108,15 +108,15 @@
│ │ │  o5 = 511
    
│ │ │              
    
│ │ │                
    i6 : close f
│ │ │  
│ │ │ -o6 = /tmp/M2-10854-0/0
│ │ │ +o6 = /tmp/M2-11624-0/0
│ │ │  
│ │ │  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/M2-10854-0/0
│ │ │ │ +o1 = /tmp/M2-11624-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10854-0/0
│ │ │ │ +o2 = /tmp/M2-11624-0/0
│ │ │ │  
│ │ │ │  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/M2-10854-0/0
│ │ │ │ +o6 = /tmp/M2-11624-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -73,22 +73,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11977-0/0
    
│ │ │ +o1 = /tmp/M2-13917-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11977-0/0
│ │ │ +o2 = /tmp/M2-13917-0/0
│ │ │  
│ │ │  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/M2-11977-0/0
│ │ │ │ +o1 = /tmp/M2-13917-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11977-0/0
│ │ │ │ +o2 = /tmp/M2-13917-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -76,15 +76,15 @@
│ │ │        
    
│ │ │          
    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 = 61
    
│ │ │ +o1 = 53
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    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 = 61
│ │ │ │ +o1 = 53
│ │ │ │  ********** 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
│ │ │ @@ -120,15 +120,15 @@
│ │ │  o6 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i7 : syz f
│ │ │  
│ │ │ -   -- registering gb 0 at 0x7f5498d2ce00
│ │ │ +   -- registering gb 0 at 0x7fabc31bde00
│ │ │  
│ │ │     -- [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 0x7f5498d2ce00
│ │ │ │ +   -- registering gb 0 at 0x7fabc31bde00
│ │ │ │  
│ │ │ │     -- [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
│ │ │ @@ -96,15 +96,15 @@
│ │ │                
    i3 : removeFile "test-file"
    
│ │ │              
    
│ │ │                
    i4 : get "!date"
│ │ │  
│ │ │ -o4 = Sun Dec 14 15:27:22 UTC 2025
    
│ │ │ +o4 = Wed Jan  7 12:22:33 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 = Sun Dec 14 15:27:22 UTC 2025
│ │ │ │ +o4 = Wed Jan  7 12:22:33 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
│ │ │ @@ -84,15 +84,15 @@
│ │ │                 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
│ │ │ @@ -75,22 +75,22 @@
│ │ │  o1 = true
    
│ │ │              
    
│ │ │                
    i2 : fn = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10380-0/0
    
│ │ │ +o2 = /tmp/M2-10670-0/0
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10380-0/0
│ │ │ +o3 = /tmp/M2-10670-0/0
│ │ │  
│ │ │  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/M2-10380-0/0
│ │ │ │ +o2 = /tmp/M2-10670-0/0
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10380-0/0
│ │ │ │ +o3 = /tmp/M2-10670-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -211,15 +211,15 @@
│ │ │  
│ │ │  o18 = false
    
│ │ │              
    
│ │ │                
    i19 : elapsedTime facs = factor(m*m1)
│ │ │ - -- 4.33674s elapsed
│ │ │ + -- 6.01582s elapsed
│ │ │  
│ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │  
│ │ │  o19 : Expression of class Product
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime isPrime m3
│ │ │ - -- .0569545s elapsed
│ │ │ + -- .0621642s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000114113s elapsed
│ │ │ + -- .000136165s 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)
│ │ │ │ - -- 4.33674s elapsed
│ │ │ │ + -- 6.01582s 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
│ │ │ │ - -- .0569545s elapsed
│ │ │ │ + -- .0621642s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ │ - -- .000114113s elapsed
│ │ │ │ + -- .000136165s 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
│ │ │ @@ -68,22 +68,22 @@
│ │ │        
    
│ │ │          
    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/M2-12188-0/0
    
│ │ │ +o1 = /tmp/M2-14348-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12188-0/0
│ │ │ +o2 = /tmp/M2-14348-0/0
│ │ │  
│ │ │  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/M2-12188-0/0
│ │ │ │ +o1 = /tmp/M2-14348-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12188-0/0
│ │ │ │ +o2 = /tmp/M2-14348-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : isRegularFile fn
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : removeFile fn
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_make__Directory_lp__String_rp.html
│ │ │ @@ -76,22 +76,22 @@
│ │ │        
    
│ │ │          
    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/M2-10722-0/0
│ │ │ │ +o1 = /tmp/M2-11352-0/0
│ │ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10722-0/0/a/b/c
│ │ │ │ +o2 = /tmp/M2-11352-0/0/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
│ │ │ @@ -64,15 +64,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10722-0/0
    
│ │ │ +o1 = /tmp/M2-11352-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-10722-0/0/a/b/c
    
│ │ │ +o2 = /tmp/M2-11352-0/0/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-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_threads.m2:498:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_memoize.html
│ │ │ @@ -61,15 +61,15 @@
│ │ │  
│ │ │  o1 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i2 : time fib 28
│ │ │ - -- used 1.26333s (cpu); 0.726987s (thread); 0s (gc)
│ │ │ + -- used 0.892603s (cpu); 0.622672s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
    
│ │ │              
    
│ │ │                
    i3 : fib = memoize fib
│ │ │ @@ -78,23 +78,23 @@
│ │ │  
│ │ │  o3 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i4 : time fib 28
│ │ │ - -- used 7.5542e-05s (cpu); 7.4801e-05s (thread); 0s (gc)
│ │ │ + -- used 7.6e-05s (cpu); 7.0592e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
    
│ │ │              
    
│ │ │                
    i5 : time fib 28
│ │ │ - -- used 3.987e-06s (cpu); 3.627e-06s (thread); 0s (gc)
│ │ │ + -- used 4.41e-06s (cpu); 3.224e-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 1.26333s (cpu); 0.726987s (thread); 0s (gc)
│ │ │ │ + -- used 0.892603s (cpu); 0.622672s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 514229
│ │ │ │  i3 : fib = memoize fib
│ │ │ │  
│ │ │ │  o3 = fib
│ │ │ │  
│ │ │ │  o3 : FunctionClosure
│ │ │ │  i4 : time fib 28
│ │ │ │ - -- used 7.5542e-05s (cpu); 7.4801e-05s (thread); 0s (gc)
│ │ │ │ + -- used 7.6e-05s (cpu); 7.0592e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 514229
│ │ │ │  i5 : time fib 28
│ │ │ │ - -- used 3.987e-06s (cpu); 3.627e-06s (thread); 0s (gc)
│ │ │ │ + -- used 4.41e-06s (cpu); 3.224e-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
│ │ │ @@ -90,19 +90,19 @@
│ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │       {14 => (degree, BettiTally)                                  }
│ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │       {17 => (regularity, BettiTally)                              }
│ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │       {19 => (codim, BettiTally)                                   }
│ │ │ -     {20 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ -     {21 => (dual, BettiTally)                                    }
│ │ │ +     {20 => (dual, BettiTally)                                    }
│ │ │ +     {21 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │       {22 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ -     {23 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ -     {24 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {23 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {24 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │  
│ │ │  o1 : NumberedVerticalList
    
│ │ │                
    
│ │ │ @@ -188,20 +188,20 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                  
    i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │ -o5 = {0 => (contract, Matrix, Matrix)                            }
│ │ │ -     {1 => (diff, Matrix, Matrix)                                }
│ │ │ -     {2 => (diff', Matrix, Matrix)                               }
│ │ │ -     {3 => (-, Matrix, Matrix)                                   }
│ │ │ +o5 = {0 => (+, Matrix, Matrix)                                   }
│ │ │ +     {1 => (-, Matrix, Matrix)                                   }
│ │ │ +     {2 => (contract, Matrix, Matrix)                            }
│ │ │ +     {3 => (diff, Matrix, Matrix)                                }
│ │ │       {4 => (contract', Matrix, Matrix)                           }
│ │ │ -     {5 => (+, 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)                        }
│ │ │ @@ -216,18 +216,18 @@
│ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │       {24 => (%, Matrix, Matrix)                                  }
│ │ │       {25 => (remainder, Matrix, Matrix)                          }
│ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ -     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ -     {29 => (intersect, Matrix, Matrix)                          }
│ │ │ -     {30 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ -     {31 => (tensor, Matrix, Matrix)                             }
│ │ │ +     {28 => (intersect, Matrix, Matrix)                          }
│ │ │ +     {29 => (pullback, Matrix, Matrix)                           }
│ │ │ +     {30 => (tensor, 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 {}
│ │ │ │ @@ -30,19 +30,19 @@
│ │ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │ │       {14 => (degree, BettiTally)                                  }
│ │ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │ │       {17 => (regularity, BettiTally)                              }
│ │ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ │       {19 => (codim, BettiTally)                                   }
│ │ │ │ -     {20 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ │ -     {21 => (dual, BettiTally)                                    }
│ │ │ │ +     {20 => (dual, BettiTally)                                    }
│ │ │ │ +     {21 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ │       {22 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │ -     {23 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │ -     {24 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │ +     {23 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │ +     {24 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │ │  
│ │ │ │  o1 : NumberedVerticalList
│ │ │ │  i2 : methods resolution
│ │ │ │  
│ │ │ │  o2 = {0 => (resolution, Ideal) }
│ │ │ │       {1 => (resolution, Module)}
│ │ │ │ @@ -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 => (contract, Matrix, Matrix)                            }
│ │ │ │ -     {1 => (diff, Matrix, Matrix)                                }
│ │ │ │ -     {2 => (diff', Matrix, Matrix)                               }
│ │ │ │ -     {3 => (-, Matrix, Matrix)                                   }
│ │ │ │ +o5 = {0 => (+, Matrix, Matrix)                                   }
│ │ │ │ +     {1 => (-, Matrix, Matrix)                                   }
│ │ │ │ +     {2 => (contract, Matrix, Matrix)                            }
│ │ │ │ +     {3 => (diff, Matrix, Matrix)                                }
│ │ │ │       {4 => (contract', Matrix, Matrix)                           }
│ │ │ │ -     {5 => (+, 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)                        }
│ │ │ │ @@ -113,18 +113,18 @@
│ │ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │ │       {24 => (%, Matrix, Matrix)                                  }
│ │ │ │       {25 => (remainder, Matrix, Matrix)                          }
│ │ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ │ -     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ │ -     {29 => (intersect, Matrix, Matrix)                          }
│ │ │ │ -     {30 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │ -     {31 => (tensor, Matrix, Matrix)                             }
│ │ │ │ +     {28 => (intersect, Matrix, Matrix)                          }
│ │ │ │ +     {29 => (pullback, Matrix, Matrix)                           }
│ │ │ │ +     {30 => (tensor, 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
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime C = minimalBetti I
│ │ │ - -- 1.82886s elapsed
│ │ │ + -- 2.51322s 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  .
│ │ │ @@ -125,15 +125,15 @@
│ │ │  
│ │ │  o4 : Ideal of S
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ - -- .745147s elapsed
│ │ │ + -- .977091s 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
│ │ │  
│ │ │ @@ -146,15 +146,15 @@
│ │ │  
│ │ │  o6 : Ideal of S
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ - -- .0317322s elapsed
│ │ │ + -- .0410621s elapsed
│ │ │  
│ │ │              0  1   2   3  4
│ │ │  o7 = total: 1 35 140 189 84
│ │ │           0: 1  .   .   .  .
│ │ │           1: . 35 140 189 84
│ │ │  
│ │ │  o7 : BettiTally
    
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │              
    
│ │ │                
    i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ - -- 1.20016s elapsed
│ │ │ + -- 1.68311s 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
│ │ │ │ - -- 1.82886s elapsed
│ │ │ │ + -- 2.51322s 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)
│ │ │ │ - -- .745147s elapsed
│ │ │ │ + -- .977091s 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)
│ │ │ │ - -- .0317322s elapsed
│ │ │ │ + -- .0410621s 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)
│ │ │ │ - -- 1.20016s elapsed
│ │ │ │ + -- 1.68311s 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
│ │ │ @@ -72,15 +72,15 @@
│ │ │          
    Description
    
│ │ │          
    Only one directory will be made, so the components of the path p other than the last must already exist.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -91,15 +91,15 @@
│ │ │  o3 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : p = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-10741-0/0/
    
│ │ │ +o1 = /tmp/M2-11391-0/0/
│ │ │              
    
│ │ │                
    i2 : mkdir p
    
│ │ │              
    
│ │ │                
    i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10741-0/0/foo
│ │ │ +o4 = /tmp/M2-11391-0/0/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/M2-10741-0/0/
│ │ │ │ +o1 = /tmp/M2-11391-0/0/
│ │ │ │  i2 : mkdir p
│ │ │ │  i3 : isDirectory p
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-10741-0/0/foo
│ │ │ │ +o4 = /tmp/M2-11391-0/0/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
│ │ │ @@ -81,52 +81,52 @@
│ │ │        
    
│ │ │          
    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/M2-10615-0/0
│ │ │ │ +o1 = /tmp/M2-11145-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10615-0/1
│ │ │ │ +o2 = /tmp/M2-11145-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10615-0/0
│ │ │ │ +o3 = /tmp/M2-11145-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ │ ---moving: /tmp/M2-10615-0/0 -> /tmp/M2-10615-0/1
│ │ │ │ +--moving: /tmp/M2-11145-0/0 -> /tmp/M2-11145-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ │ ---backup file created: /tmp/M2-10615-0/1.bak
│ │ │ │ +--backup file created: /tmp/M2-11145-0/1.bak
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-10615-0/1.bak
│ │ │ │ +o6 = /tmp/M2-11145-0/1.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
│ │ │ @@ -51,15 +51,15 @@
│ │ │        
    nanosleep -- sleep for a given number of nanoseconds
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │  nanosleep n -- sleeps for n nanoseconds.        
    
│ │ │                
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10615-0/0
    
│ │ │ +o1 = /tmp/M2-11145-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10615-0/1
    
│ │ │ +o2 = /tmp/M2-11145-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10615-0/0
│ │ │ +o3 = /tmp/M2-11145-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-10615-0/0 -> /tmp/M2-10615-0/1
    
│ │ │ +--moving: /tmp/M2-11145-0/0 -> /tmp/M2-11145-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-10615-0/1.bak
│ │ │ +--backup file created: /tmp/M2-11145-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-10615-0/1.bak
    
│ │ │ +o6 = /tmp/M2-11145-0/1.bak
│ │ │              
    
│ │ │                
    i7 : removeFile bak
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime nanosleep 500000000
│ │ │ - -- .500135s elapsed
│ │ │ + -- .500132s 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
│ │ │ │ - -- .500135s elapsed
│ │ │ │ + -- .500132s 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
│ │ │ @@ -104,33 +104,33 @@
│ │ │  o3 : OptionTable
    
│ │ │              
    
│ │ │                
    i4 : methods codim
│ │ │  
│ │ │ -o4 = {0 => (codim, Variety)       }
│ │ │ -     {1 => (codim, BettiTally)    }
│ │ │ -     {2 => (codim, Module)        }
│ │ │ -     {3 => (codim, QuotientRing)  }
│ │ │ -     {4 => (codim, Ideal)         }
│ │ │ -     {5 => (codim, MonomialIdeal) }
│ │ │ -     {6 => (codim, PolynomialRing)}
│ │ │ -     {7 => (codim, CoherentSheaf) }
│ │ │ +o4 = {0 => (codim, CoherentSheaf) }
│ │ │ +     {1 => (codim, Variety)       }
│ │ │ +     {2 => (codim, BettiTally)    }
│ │ │ +     {3 => (codim, Module)        }
│ │ │ +     {4 => (codim, QuotientRing)  }
│ │ │ +     {5 => (codim, Ideal)         }
│ │ │ +     {6 => (codim, MonomialIdeal) }
│ │ │ +     {7 => (codim, PolynomialRing)}
│ │ │  
│ │ │  o4 : NumberedVerticalList
    
│ │ │              
    
│ │ │                
    i5 : options oo
│ │ │  
│ │ │  o5 = {0 => (OptionTable{Generic => false})}
│ │ │ -     {1 => (OptionTable{})                }
│ │ │ -     {2 => (OptionTable{Generic => false})}
│ │ │ +     {1 => (OptionTable{Generic => false})}
│ │ │ +     {2 => (OptionTable{})                }
│ │ │       {3 => (OptionTable{Generic => false})}
│ │ │       {4 => (OptionTable{Generic => false})}
│ │ │       {5 => (OptionTable{Generic => false})}
│ │ │       {6 => (OptionTable{Generic => false})}
│ │ │       {7 => (OptionTable{Generic => false})}
│ │ │  
│ │ │  o5 : NumberedVerticalList
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,29 +35,29 @@
│ │ │ │  i3 : options(codim, Ideal)
│ │ │ │  
│ │ │ │  o3 = OptionTable{Generic => false}
│ │ │ │  
│ │ │ │  o3 : OptionTable
│ │ │ │  i4 : methods codim
│ │ │ │  
│ │ │ │ -o4 = {0 => (codim, Variety)       }
│ │ │ │ -     {1 => (codim, BettiTally)    }
│ │ │ │ -     {2 => (codim, Module)        }
│ │ │ │ -     {3 => (codim, QuotientRing)  }
│ │ │ │ -     {4 => (codim, Ideal)         }
│ │ │ │ -     {5 => (codim, MonomialIdeal) }
│ │ │ │ -     {6 => (codim, PolynomialRing)}
│ │ │ │ -     {7 => (codim, CoherentSheaf) }
│ │ │ │ +o4 = {0 => (codim, CoherentSheaf) }
│ │ │ │ +     {1 => (codim, Variety)       }
│ │ │ │ +     {2 => (codim, BettiTally)    }
│ │ │ │ +     {3 => (codim, Module)        }
│ │ │ │ +     {4 => (codim, QuotientRing)  }
│ │ │ │ +     {5 => (codim, Ideal)         }
│ │ │ │ +     {6 => (codim, MonomialIdeal) }
│ │ │ │ +     {7 => (codim, PolynomialRing)}
│ │ │ │  
│ │ │ │  o4 : NumberedVerticalList
│ │ │ │  i5 : options oo
│ │ │ │  
│ │ │ │  o5 = {0 => (OptionTable{Generic => false})}
│ │ │ │ -     {1 => (OptionTable{})                }
│ │ │ │ -     {2 => (OptionTable{Generic => false})}
│ │ │ │ +     {1 => (OptionTable{Generic => false})}
│ │ │ │ +     {2 => (OptionTable{})                }
│ │ │ │       {3 => (OptionTable{Generic => false})}
│ │ │ │       {4 => (OptionTable{Generic => false})}
│ │ │ │       {5 => (OptionTable{Generic => false})}
│ │ │ │       {6 => (OptionTable{Generic => false})}
│ │ │ │       {7 => (OptionTable{Generic => false})}
│ │ │ │  
│ │ │ │  o5 : NumberedVerticalList
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallel_spprogramming_spwith_spthreads_spand_sptasks.html
│ │ │ @@ -72,21 +72,21 @@
│ │ │                  		
│ │ │                
    i2 : L = random toList (1..10000);
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ - -- .640674s elapsed
    
│ │ │ + -- .751295s elapsed
│ │ │              
    
│ │ │                
    i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .307919s elapsed
    
│ │ │ + -- .188399s 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.
    
│ │ │ @@ -100,15 +100,15 @@
│ │ │  o5 = 5
    
│ │ │              
    
│ │ │                
    i6 : allowableThreads = maxAllowableThreads
│ │ │  
│ │ │ -o6 = 7
    
│ │ │ +o6 = 17
│ │ │              
    
│ │ │          
    
│ │ │            
    To run a function in another thread use schedule, as in the following example.
    
│ │ │          
    
│ │ │          
│ │ │ ├── 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 = random toList (1..10000);
│ │ │ │  i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ │ - -- .640674s elapsed
│ │ │ │ + -- .751295s elapsed
│ │ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ │ - -- .307919s elapsed
│ │ │ │ + -- .188399s 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)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallelism_spin_spengine_spcomputations.html
│ │ │ @@ -137,15 +137,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime minimalBetti I
│ │ │ - -- 2.07143s elapsed
│ │ │ + -- 2.45561s 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  .
│ │ │ @@ -160,15 +160,15 @@
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ - -- 1.82839s elapsed
│ │ │ + -- 2.47016s 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  .
│ │ │ @@ -190,15 +190,15 @@
│ │ │  
│ │ │  o8 = 1
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime minimalBetti(I)
│ │ │ - -- 1.75494s elapsed
│ │ │ + -- 2.53146s 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  .
│ │ │ @@ -231,15 +231,15 @@
│ │ │  
│ │ │  o12 : Ideal of S
    
│ │ │              
    
│ │ │                
    i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.23687s elapsed
│ │ │ + -- 4.37647s 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
    
│ │ │ @@ -258,15 +258,15 @@
│ │ │  
│ │ │  o15 : Ideal of S
│ │ │              
    
│ │ │                
    i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.55496s elapsed
│ │ │ + -- 2.92176s 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
    
│ │ │ @@ -299,15 +299,15 @@
│ │ │  
│ │ │  o19 : Ideal of S
│ │ │              
    
│ │ │                
    i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 5.136s elapsed
│ │ │ + -- 4.47144s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o20 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -322,15 +322,15 @@
│ │ │  
│ │ │  o22 : Ideal of S
│ │ │              
    
│ │ │                
    i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 7.09108s elapsed
│ │ │ + -- 10.002s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o23 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -345,15 +345,15 @@
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │              
    
│ │ │                
    i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.22028s elapsed
│ │ │ + -- 4.3477s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o26 : Matrix S  <-- S
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -396,15 +396,15 @@
│ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │                 101
│ │ │              
    
│ │ │                
    i31 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.006s elapsed
│ │ │ + -- 1.22923s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │              
    
│ │ │                
    i34 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.19796s elapsed
│ │ │ + -- 1.73952s 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
│ │ │ │ - -- 2.07143s elapsed
│ │ │ │ + -- 2.45561s 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)
│ │ │ │ - -- 1.82839s elapsed
│ │ │ │ + -- 2.47016s 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)
│ │ │ │ - -- 1.75494s elapsed
│ │ │ │ + -- 2.53146s 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)
│ │ │ │ - -- 2.23687s elapsed
│ │ │ │ + -- 4.37647s 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)
│ │ │ │ - -- 2.55496s elapsed
│ │ │ │ + -- 2.92176s 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");
│ │ │ │ - -- 5.136s elapsed
│ │ │ │ + -- 4.47144s 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");
│ │ │ │ - -- 7.09108s elapsed
│ │ │ │ + -- 10.002s 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");
│ │ │ │ - -- 4.22028s elapsed
│ │ │ │ + -- 4.3477s 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);
│ │ │ │ - -- 1.006s elapsed
│ │ │ │ + -- 1.22923s 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);
│ │ │ │ - -- 1.19796s elapsed
│ │ │ │ + -- 1.73952s 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
│ │ │ @@ -370,36 +370,36 @@
│ │ │  
│ │ │  o27 = 3
│ │ │              
    
│ │ │                
    i28 : time poincare I
│ │ │ - -- used 0.00277298s (cpu); 2.153e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00267189s (cpu); 1.5885e-05s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7f3fe9e7e540
│ │ │ +   -- registering gb 19 at 0x7fc056914540
│ │ │  
│ │ │     -- [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 0.0251999s (cpu); 0.0250353s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0172706s (cpu); 0.0180053s (thread); 0s (gc)
│ │ │  
│ │ │                1      11
│ │ │  o29 : Matrix R  <-- R
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -411,15 +411,15 @@
│ │ │                
    i30 : R = QQ[a..d];
    
│ │ │              
    
│ │ │                
    i31 : I = ideal random(R^1, R^{3:-3});
│ │ │  
│ │ │ -   -- registering gb 20 at 0x7f3fe9e7e380
│ │ │ +   -- registering gb 20 at 0x7fc056914380
│ │ │  
│ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │     -- number of monomials                = 0
│ │ │     -- #reduction steps = 0
│ │ │     -- #spairs done = 0
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -428,24 +428,24 @@
│ │ │  o31 : Ideal of R
    
│ │ │              
    
│ │ │                
    i32 : time p = poincare I
│ │ │  
│ │ │ -   -- registering gb 21 at 0x7f3fe9e7e000
│ │ │ +   -- registering gb 21 at 0x7fc056914000
│ │ │  
│ │ │     -- [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 0.00799718s (cpu); 0.00944086s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00400687s (cpu); 0.0064099s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o32 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i37 : time gens gb J;
│ │ │  
│ │ │ -   -- registering gb 22 at 0x7f3fe9ab3e00
│ │ │ +   -- registering gb 22 at 0x7fc056a0ee00
│ │ │  
│ │ │     -- [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 0.0840427s (cpu); 0.0832054s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0599726s (cpu); 0.0597862s (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 0.00277298s (cpu); 2.153e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00267189s (cpu); 1.5885e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o28 : ZZ[T]
│ │ │ │  i29 : time gens gb I;
│ │ │ │  
│ │ │ │ -   -- registering gb 19 at 0x7f3fe9e7e540
│ │ │ │ +   -- registering gb 19 at 0x7fc056914540
│ │ │ │  
│ │ │ │     -- [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 0.0251999s (cpu); 0.0250353s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.0172706s (cpu); 0.0180053s (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 20 at 0x7f3fe9e7e380
│ │ │ │ +   -- registering gb 20 at 0x7fc056914380
│ │ │ │  
│ │ │ │     -- [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 21 at 0x7f3fe9e7e000
│ │ │ │ +   -- registering gb 21 at 0x7fc056914000
│ │ │ │  
│ │ │ │     -- [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 0.00799718s (cpu); 0.00944086s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.00400687s (cpu); 0.0064099s (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 22 at 0x7f3fe9ab3e00
│ │ │ │ +   -- registering gb 22 at 0x7fc056a0ee00
│ │ │ │  
│ │ │ │     -- [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 0.0840427s (cpu); 0.0832054s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.0599726s (cpu); 0.0597862s (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
│ │ │ @@ -97,22 +97,22 @@
│ │ │  o2 : File
│ │ │              
    
│ │ │                
    i3 : fn = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-10932-0/0
    
│ │ │ +o3 = /tmp/M2-11782-0/0
│ │ │              
    
│ │ │                
    i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10932-0/0
│ │ │ +o4 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : get fn
│ │ │ @@ -151,15 +151,15 @@
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : fn << f << close
│ │ │  
│ │ │ -o9 = /tmp/M2-10932-0/0
│ │ │ +o9 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  o9 : File
    
│ │ │              
    
│ │ │                
    i10 : get fn
│ │ │ @@ -169,15 +169,15 @@
│ │ │        + 1
    
│ │ │              
    
│ │ │                
    i11 : fn << toExternalString f << close
│ │ │  
│ │ │ -o11 = /tmp/M2-10932-0/0
│ │ │ +o11 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  o11 : File
    
│ │ │              
    
│ │ │                
    i12 : get fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,18 +36,18 @@
│ │ │ │  -- ho there --
│ │ │ │  
│ │ │ │  o2 = stdio
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10932-0/0
│ │ │ │ +o3 = /tmp/M2-11782-0/0
│ │ │ │  i4 : fn << "hi there" << endl << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-10932-0/0
│ │ │ │ +o4 = /tmp/M2-11782-0/0
│ │ │ │  
│ │ │ │  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/M2-10932-0/0
│ │ │ │ +o9 = /tmp/M2-11782-0/0
│ │ │ │  
│ │ │ │  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/M2-10932-0/0
│ │ │ │ +o11 = /tmp/M2-11782-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -64,15 +64,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : processID()
│ │ │  
│ │ │ -o1 = 10191
    
│ │ │ +o1 = 10311
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    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 = 10191
│ │ │ │ +o1 = 10311
│ │ │ │  ********** 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
│ │ │ @@ -91,34 +91,34 @@
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o9 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -173,15 +173,15 @@
│ │ │  o14 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : profileSummary
│ │ │  
│ │ │  o2 = #run  %time   position                         
│ │ │ -     1     94.52   ../../m2/matrix1.m2:279:4-282:58 
│ │ │ -     1     92.12   ../../m2/matrix1.m2:281:22-281:43
│ │ │ -     1     44.16   ../../m2/matrix1.m2:193:25-193:52
│ │ │ -     1     30.59   ../../m2/matrix1.m2:114:5-156:72 
│ │ │ -     1     29.47   ../../m2/matrix1.m2:140:10-155:16
│ │ │ -     1     23.83   ../../m2/matrix1.m2:181:4-181:42 
│ │ │ -     1     22.54   ../../m2/set.m2:127:5-127:61     
│ │ │ -     1     20.86   ../../m2/matrix1.m2:45:10-49:22  
│ │ │ -     1     3.30    ../../m2/matrix1.m2:112:5-112:29 
│ │ │ -     1     2.34    ../../m2/matrix1.m2:141:13-141:78
│ │ │ -     1     2.18    ../../m2/matrix1.m2:96:5-109:11  
│ │ │ -     1     1.42    ../../m2/matrix1.m2:281:7-281:16 
│ │ │ -     1     1.29    ../../m2/matrix1.m2:147:20-147:64
│ │ │ +     1     92.19   ../../m2/matrix1.m2:279:4-282:58 
│ │ │ +     1     89.59   ../../m2/matrix1.m2:281:22-281:43
│ │ │ +     1     42.99   ../../m2/matrix1.m2:193:25-193:52
│ │ │ +     1     28.77   ../../m2/matrix1.m2:114:5-156:72 
│ │ │ +     1     27.64   ../../m2/matrix1.m2:140:10-155:16
│ │ │ +     1     22.71   ../../m2/matrix1.m2:181:4-181:42 
│ │ │ +     1     21.29   ../../m2/set.m2:127:5-127:61     
│ │ │ +     1     19.23   ../../m2/matrix1.m2:45:10-49:22  
│ │ │ +     1     3.57    ../../m2/matrix1.m2:112:5-112:29 
│ │ │ +     1     2.62    ../../m2/matrix1.m2:96:5-109:11  
│ │ │ +     1     2.3     ../../m2/matrix1.m2:141:13-141:78
│ │ │ +     1     1.91    ../../m2/matrix1.m2:276:4-277:73 
│ │ │ +     1     1.72    ../../m2/matrix1.m2:147:20-147:64
│ │ │ +     1     1.63    ../../m2/matrix1.m2:281:7-281:16 
│ │ │ +     1     1.56    ../../m2/matrix1.m2:98:10-98:46  
│ │ │       1     1.29    ../../m2/matrix1.m2:111:5-111:91 
│ │ │ -     1     1.27    ../../m2/matrix1.m2:276:4-277:73 
│ │ │ -     1     1.02    ../../m2/matrix1.m2:98:10-98:46  
│ │ │ -     1     .97     ../../m2/matrix1.m2:182:4-184:74 
│ │ │ -     1     .81     ../../m2/modules.m2:279:4-279:52 
│ │ │ -     20    .64     ../../m2/matrix1.m2:191:14-192:67
│ │ │ -     20    .47     ../../m2/matrix1.m2:47:43-47:71  
│ │ │ +     1     1.13    ../../m2/matrix1.m2:182:4-184:74 
│ │ │ +     1     .96     ../../m2/modules.m2:279:4-279:52 
│ │ │ +     20    .89     ../../m2/matrix1.m2:191:14-192:67
│ │ │ +     20    .78     ../../m2/matrix1.m2:47:43-47:71  
│ │ │       1     .0038s  elapsed total                    
    
│ │ │              
    
│ │ │                
    i3 : coverageSummary
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,34 +25,34 @@
│ │ │ │                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     94.52   ../../m2/matrix1.m2:279:4-282:58
│ │ │ │ -     1     92.12   ../../m2/matrix1.m2:281:22-281:43
│ │ │ │ -     1     44.16   ../../m2/matrix1.m2:193:25-193:52
│ │ │ │ -     1     30.59   ../../m2/matrix1.m2:114:5-156:72
│ │ │ │ -     1     29.47   ../../m2/matrix1.m2:140:10-155:16
│ │ │ │ -     1     23.83   ../../m2/matrix1.m2:181:4-181:42
│ │ │ │ -     1     22.54   ../../m2/set.m2:127:5-127:61
│ │ │ │ -     1     20.86   ../../m2/matrix1.m2:45:10-49:22
│ │ │ │ -     1     3.30    ../../m2/matrix1.m2:112:5-112:29
│ │ │ │ -     1     2.34    ../../m2/matrix1.m2:141:13-141:78
│ │ │ │ -     1     2.18    ../../m2/matrix1.m2:96:5-109:11
│ │ │ │ -     1     1.42    ../../m2/matrix1.m2:281:7-281:16
│ │ │ │ -     1     1.29    ../../m2/matrix1.m2:147:20-147:64
│ │ │ │ +     1     92.19   ../../m2/matrix1.m2:279:4-282:58
│ │ │ │ +     1     89.59   ../../m2/matrix1.m2:281:22-281:43
│ │ │ │ +     1     42.99   ../../m2/matrix1.m2:193:25-193:52
│ │ │ │ +     1     28.77   ../../m2/matrix1.m2:114:5-156:72
│ │ │ │ +     1     27.64   ../../m2/matrix1.m2:140:10-155:16
│ │ │ │ +     1     22.71   ../../m2/matrix1.m2:181:4-181:42
│ │ │ │ +     1     21.29   ../../m2/set.m2:127:5-127:61
│ │ │ │ +     1     19.23   ../../m2/matrix1.m2:45:10-49:22
│ │ │ │ +     1     3.57    ../../m2/matrix1.m2:112:5-112:29
│ │ │ │ +     1     2.62    ../../m2/matrix1.m2:96:5-109:11
│ │ │ │ +     1     2.3     ../../m2/matrix1.m2:141:13-141:78
│ │ │ │ +     1     1.91    ../../m2/matrix1.m2:276:4-277:73
│ │ │ │ +     1     1.72    ../../m2/matrix1.m2:147:20-147:64
│ │ │ │ +     1     1.63    ../../m2/matrix1.m2:281:7-281:16
│ │ │ │ +     1     1.56    ../../m2/matrix1.m2:98:10-98:46
│ │ │ │       1     1.29    ../../m2/matrix1.m2:111:5-111:91
│ │ │ │ -     1     1.27    ../../m2/matrix1.m2:276:4-277:73
│ │ │ │ -     1     1.02    ../../m2/matrix1.m2:98:10-98:46
│ │ │ │ -     1     .97     ../../m2/matrix1.m2:182:4-184:74
│ │ │ │ -     1     .81     ../../m2/modules.m2:279:4-279:52
│ │ │ │ -     20    .64     ../../m2/matrix1.m2:191:14-192:67
│ │ │ │ -     20    .47     ../../m2/matrix1.m2:47:43-47:71
│ │ │ │ +     1     1.13    ../../m2/matrix1.m2:182:4-184:74
│ │ │ │ +     1     .96     ../../m2/modules.m2:279:4-279:52
│ │ │ │ +     20    .89     ../../m2/matrix1.m2:191:14-192:67
│ │ │ │ +     20    .78     ../../m2/matrix1.m2:47:43-47:71
│ │ │ │       1     .0038s  elapsed total
│ │ │ │  i3 : coverageSummary
│ │ │ │  
│ │ │ │  o3 = covered lines:
│ │ │ │       ../../m2/lists.m2:145:24-145:32
│ │ │ │       ../../m2/lists.m2:145:34-145:58
│ │ │ │       ../../m2/matrix.m2:12:5-12:35
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random__K__Rational__Point.html
│ │ │ @@ -99,15 +99,15 @@
│ │ │  
│ │ │  o5 : Sequence
    
│ │ │              
    
│ │ │                
    i6 : time randomKRationalPoint(I)
│ │ │ - -- used 0.173474s (cpu); 0.138239s (thread); 0s (gc)
│ │ │ + -- used 0.237164s (cpu); 0.102487s (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 0.663337s (cpu); 0.35806s (thread); 0s (gc)
│ │ │ + -- used 0.843126s (cpu); 0.33904s (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 3.65398s (cpu); 1.92733s (thread); 0s (gc)
│ │ │ + -- used 4.57597s (cpu); 2.15219s (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 0.173474s (cpu); 0.138239s (thread); 0s (gc)
│ │ │ │ + -- used 0.237164s (cpu); 0.102487s (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 0.663337s (cpu); 0.35806s (thread); 0s (gc)
│ │ │ │ + -- used 0.843126s (cpu); 0.33904s (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 3.65398s (cpu); 1.92733s (thread); 0s (gc)
│ │ │ │ + -- used 4.57597s (cpu); 2.15219s (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
│ │ │ @@ -68,38 +68,38 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11565-0/0
    
│ │ │ +o1 = /tmp/M2-13075-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11565-0/0
    
│ │ │ +o2 = /tmp/M2-13075-0/0
│ │ │              
    
│ │ │                
    i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11565-0/0/foo
│ │ │ +o3 = /tmp/M2-13075-0/0/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/M2-11565-0/0
│ │ │ │ +o1 = /tmp/M2-13075-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11565-0/0
│ │ │ │ +o2 = /tmp/M2-13075-0/0
│ │ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11565-0/0/foo
│ │ │ │ +o3 = /tmp/M2-13075-0/0/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
│ │ │ @@ -52,22 +52,22 @@
│ │ │        
    
│ │ │  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/M2-11107-0/0
    
│ │ │ +o1 = /tmp/M2-12137-0/0
│ │ │              
    
│ │ │                
    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/M2-11107-0/0
│ │ │ +o2 = /tmp/M2-12137-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │  Now we get the contents of the file, as a single string.        
│ │ │            
│ │ │ @@ -116,15 +116,15 @@
│ │ │  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/M2-11107-0/0
│ │ │ +o7 = /tmp/M2-12137-0/0
│ │ │  
│ │ │  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/M2-11107-0/0
│ │ │ │ +o1 = /tmp/M2-12137-0/0
│ │ │ │  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/M2-11107-0/0
│ │ │ │ +o2 = /tmp/M2-12137-0/0
│ │ │ │  
│ │ │ │  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/M2-11107-0/0
│ │ │ │ +o7 = /tmp/M2-12137-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -68,15 +68,15 @@
│ │ │        
    
│ │ │          
    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/M2-11806-0/0
│ │ │ │ +o1 = /tmp/M2-13556-0/0
│ │ │ │  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
│ │ │ @@ -68,57 +68,57 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │                
    i1 : p = temporaryFileName ()
│ │ │  
│ │ │ -o1 = /tmp/M2-11806-0/0
    
│ │ │ +o1 = /tmp/M2-13556-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile ("foo", p)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -130,15 +130,15 @@
│ │ │          
    
│ │ │                
    i1 : realpath "."
│ │ │  
│ │ │ -o1 = /tmp/M2-10191-0/86-rundir/
    
│ │ │ +o1 = /tmp/M2-10311-0/86-rundir/
│ │ │              
    
│ │ │                
    i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11825-0/0
    
│ │ │ +o2 = /tmp/M2-13595-0/0
│ │ │              
    
│ │ │                
    i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11825-0/1
    
│ │ │ +o3 = /tmp/M2-13595-0/1
│ │ │              
    
│ │ │                
    i4 : symlinkFile(p,q)
    
│ │ │              
    
│ │ │                
    i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-11825-0/0
│ │ │ +o5 = /tmp/M2-13595-0/0
│ │ │  
│ │ │  o5 : File
    
│ │ │              
    
│ │ │                
    i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-11825-0/0
    
│ │ │ +o6 = /tmp/M2-13595-0/0
│ │ │              
    
│ │ │                
    i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-11825-0/0
    
│ │ │ +o7 = /tmp/M2-13595-0/0
│ │ │              
    
│ │ │                
    i8 : removeFile p
    
│ │ │              
    
│ │ │          
    The empty string is interpreted as a reference to the current directory.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-10191-0/86-rundir/
    
│ │ │ +o10 = /tmp/M2-10311-0/86-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/M2-10191-0/86-rundir/
│ │ │ │ +o1 = /tmp/M2-10311-0/86-rundir/
│ │ │ │  i2 : p = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11825-0/0
│ │ │ │ +o2 = /tmp/M2-13595-0/0
│ │ │ │  i3 : q = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11825-0/1
│ │ │ │ +o3 = /tmp/M2-13595-0/1
│ │ │ │  i4 : symlinkFile(p,q)
│ │ │ │  i5 : p << close
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11825-0/0
│ │ │ │ +o5 = /tmp/M2-13595-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : readlink q
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11825-0/0
│ │ │ │ +o6 = /tmp/M2-13595-0/0
│ │ │ │  i7 : realpath q
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11825-0/0
│ │ │ │ +o7 = /tmp/M2-13595-0/0
│ │ │ │  i8 : removeFile p
│ │ │ │  i9 : removeFile q
│ │ │ │  The empty string is interpreted as a reference to the current directory.
│ │ │ │  i10 : realpath ""
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-10191-0/86-rundir/
│ │ │ │ +o10 = /tmp/M2-10311-0/86-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
│ │ │ @@ -76,23 +76,23 @@
│ │ │                  		
│ │ │                
    i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-- finalizing sequence #"|i|" --"))
    
│ │ │              
    
│ │ │                
    i2 : collectGarbage() 
│ │ │ ---finalization: (1)[3]: -- finalizing sequence #4 --
│ │ │ ---finalization: (2)[7]: -- finalizing sequence #8 --
│ │ │ ---finalization: (3)[4]: -- finalizing sequence #5 --
│ │ │ ---finalization: (4)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (1)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (2)[8]: -- finalizing sequence #9 --
│ │ │ +--finalization: (3)[5]: -- finalizing sequence #6 --
│ │ │ +--finalization: (4)[3]: -- finalizing sequence #4 --
│ │ │  --finalization: (5)[6]: -- finalizing sequence #7 --
│ │ │ ---finalization: (6)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (7)[8]: -- finalizing sequence #9 --
│ │ │ ---finalization: (8)[2]: -- finalizing sequence #3 --
│ │ │ ---finalization: (9)[0]: -- finalizing sequence #1 --
    
│ │ │ +--finalization: (6)[2]: -- finalizing sequence #3 --
│ │ │ +--finalization: (7)[0]: -- finalizing sequence #1 --
│ │ │ +--finalization: (9)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (8)[7]: -- finalizing sequence #8 --
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    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)[3]: -- finalizing sequence #4 --
│ │ │ │ ---finalization: (2)[7]: -- finalizing sequence #8 --
│ │ │ │ ---finalization: (3)[4]: -- finalizing sequence #5 --
│ │ │ │ ---finalization: (4)[1]: -- finalizing sequence #2 --
│ │ │ │ +--finalization: (1)[4]: -- finalizing sequence #5 --
│ │ │ │ +--finalization: (2)[8]: -- finalizing sequence #9 --
│ │ │ │ +--finalization: (3)[5]: -- finalizing sequence #6 --
│ │ │ │ +--finalization: (4)[3]: -- finalizing sequence #4 --
│ │ │ │  --finalization: (5)[6]: -- finalizing sequence #7 --
│ │ │ │ ---finalization: (6)[5]: -- finalizing sequence #6 --
│ │ │ │ ---finalization: (7)[8]: -- finalizing sequence #9 --
│ │ │ │ ---finalization: (8)[2]: -- finalizing sequence #3 --
│ │ │ │ ---finalization: (9)[0]: -- finalizing sequence #1 --
│ │ │ │ +--finalization: (6)[2]: -- finalizing sequence #3 --
│ │ │ │ +--finalization: (7)[0]: -- finalizing sequence #1 --
│ │ │ │ +--finalization: (9)[1]: -- finalizing sequence #2 --
│ │ │ │ +--finalization: (8)[7]: -- finalizing sequence #8 --
│ │ │ │  ********** 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
│ │ │ @@ -71,29 +71,29 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10779-0/0
    
│ │ │ +o1 = /tmp/M2-11469-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10779-0/0
    
│ │ │ +o2 = /tmp/M2-11469-0/0
│ │ │              
    
│ │ │                
    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/M2-10779-0/0
│ │ │ │ +o1 = /tmp/M2-11469-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10779-0/0
│ │ │ │ +o2 = /tmp/M2-11469-0/0
│ │ │ │  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
│ │ │ @@ -65,22 +65,22 @@
│ │ │          
    Description
    
│ │ │          
    This string may be concatenated with an absolute path to get one understandable by external programs.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10283-0/0
    
│ │ │ +o1 = /tmp/M2-10473-0/0
│ │ │              
    
│ │ │                
    i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-10283-0/0
    
│ │ │ +o2 = /tmp/M2-10473-0/0
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │          
    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/M2-10283-0/0
│ │ │ │ +o1 = /tmp/M2-10473-0/0
│ │ │ │  i2 : rootPath | fn
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10283-0/0
│ │ │ │ +o2 = /tmp/M2-10473-0/0
│ │ │ │  ********** 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-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:2025:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_root__U__R__I.html
│ │ │ @@ -65,22 +65,22 @@
│ │ │          
      Description
      
│ │ │          
      This string may be concatenated with an absolute path to get one understandable by an external browser.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11508-0/0
      
│ │ │ +o1 = /tmp/M2-12958-0/0
│ │ │              
      
│ │ │                
      i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-11508-0/0
      
│ │ │ +o2 = file:///tmp/M2-12958-0/0
│ │ │              
      
│ │ │        
    
│ │ │        
    
│ │ │          
    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/M2-11508-0/0
│ │ │ │ +o1 = /tmp/M2-12958-0/0
│ │ │ │  i2 : rootURI | fn
│ │ │ │  
│ │ │ │ -o2 = file:///tmp/M2-11508-0/0
│ │ │ │ +o2 = file:///tmp/M2-12958-0/0
│ │ │ │  ********** 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-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:2041:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_saving_sppolynomials_spand_spmatrices_spin_spfiles.html
│ │ │ @@ -90,22 +90,22 @@
│ │ │  o4 : R-module, submodule of R
│ │ │              
    
│ │ │                
    i5 : f = temporaryFileName()
│ │ │  
│ │ │ -o5 = /tmp/M2-11356-0/0
    
│ │ │ +o5 = /tmp/M2-12646-0/0
│ │ │              
    
│ │ │                
    i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11356-0/0
│ │ │ +o6 = /tmp/M2-12646-0/0
│ │ │  
│ │ │  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/M2-11356-0/0
│ │ │ │ +o5 = /tmp/M2-12646-0/0
│ │ │ │  i6 : f << toString (p,m,M) << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11356-0/0
│ │ │ │ +o6 = /tmp/M2-12646-0/0
│ │ │ │  
│ │ │ │  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/_serial__Number.html
│ │ │ @@ -68,22 +68,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : serialNumber asdf
│ │ │  
│ │ │ -o1 = 1426273
    
│ │ │ +o1 = 1526273
│ │ │              
    
│ │ │                
    i2 : serialNumber foo
│ │ │  
│ │ │ -o2 = 1426275
    
│ │ │ +o2 = 1526275
│ │ │              
    
│ │ │                
    i3 : serialNumber ZZ
│ │ │  
│ │ │  o3 = 1000050
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,18 +10,18 @@
│ │ │ │      * Inputs:
│ │ │ │            o x
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the serial number of x
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : serialNumber asdf
│ │ │ │  
│ │ │ │ -o1 = 1426273
│ │ │ │ +o1 = 1526273
│ │ │ │  i2 : serialNumber foo
│ │ │ │  
│ │ │ │ -o2 = 1426275
│ │ │ │ +o2 = 1526275
│ │ │ │  i3 : serialNumber ZZ
│ │ │ │  
│ │ │ │  o3 = 1000050
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _s_e_r_i_a_l_N_u_m_b_e_r 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/_solve.html
│ │ │ @@ -366,21 +366,21 @@
│ │ │                  		
│ │ │                
    i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
    
│ │ │              
    
│ │ │                
    i30 : time X = solve(A,B);
│ │ │ - -- used 0.000227156s (cpu); 0.000219682s (thread); 0s (gc)
    
│ │ │ + -- used 0.000221698s (cpu); 0.000206721s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000163036s (cpu); 0.000163226s (thread); 0s (gc)
    
│ │ │ + -- used 0.000103953s (cpu); 0.000103242s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i32 : norm(A*X-B)
│ │ │  
│ │ │  o32 = 5.111850690840453e-15
│ │ │ @@ -411,21 +411,21 @@
│ │ │              
    		
│ │ │                
    i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
    
│ │ │              
    
│ │ │                
    i38 : time X = solve(A,B);
│ │ │ - -- used 0.482514s (cpu); 0.3008s (thread); 0s (gc)
    
│ │ │ + -- used 0.143985s (cpu); 0.144011s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.237275s (cpu); 0.23721s (thread); 0s (gc)
    
│ │ │ + -- used 0.136234s (cpu); 0.136261s (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 0.000227156s (cpu); 0.000219682s (thread); 0s (gc)
│ │ │ │ + -- used 0.000221698s (cpu); 0.000206721s (thread); 0s (gc)
│ │ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.000163036s (cpu); 0.000163226s (thread); 0s (gc)
│ │ │ │ + -- used 0.000103953s (cpu); 0.000103242s (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 0.482514s (cpu); 0.3008s (thread); 0s (gc)
│ │ │ │ + -- used 0.143985s (cpu); 0.144011s (thread); 0s (gc)
│ │ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.237275s (cpu); 0.23721s (thread); 0s (gc)
│ │ │ │ + -- used 0.136234s (cpu); 0.136261s (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
│ │ │ @@ -80,93 +80,93 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11147-0/0/
    
│ │ │ +o1 = /tmp/M2-12217-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11147-0/1/
    
│ │ │ +o2 = /tmp/M2-12217-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11147-0/0/a/
    
│ │ │ +o3 = /tmp/M2-12217-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11147-0/0/b/
    
│ │ │ +o4 = /tmp/M2-12217-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11147-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-12217-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11147-0/0/a/f
│ │ │ +o6 = /tmp/M2-12217-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11147-0/0/a/g
│ │ │ +o7 = /tmp/M2-12217-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11147-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12217-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11147-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
    
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/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/M2-11147-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
    
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/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/M2-11147-0/0/
│ │ │ │ +o1 = /tmp/M2-12217-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11147-0/1/
│ │ │ │ +o2 = /tmp/M2-12217-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11147-0/0/a/
│ │ │ │ +o3 = /tmp/M2-12217-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11147-0/0/b/
│ │ │ │ +o4 = /tmp/M2-12217-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11147-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-12217-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11147-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-12217-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11147-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-12217-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-11147-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-12217-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11147-0/1/b/c/g
│ │ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
│ │ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/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/M2-11147-0/1/b/c/g
│ │ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
│ │ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/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
│ │ │ @@ -72,15 +72,15 @@
│ │ │        
    
│ │ │          
    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/M2-11204-0/0
│ │ │ │ +o1 = /tmp/M2-12334-0/0
│ │ │ │  i2 : symlinkFile("qwert", fn)
│ │ │ │  i3 : fileExists fn
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : readlink fn
│ │ │ │  
│ │ │ │  o4 = qwert
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_temporary__File__Name.html
│ │ │ @@ -64,22 +64,22 @@
│ │ │        
    
│ │ │          
    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/M2-11204-0/0
    
│ │ │ +o1 = /tmp/M2-12334-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile("qwert", fn)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : temporaryFileName () | ".tex"
│ │ │  
│ │ │ -o1 = /tmp/M2-12169-0/0.tex
    
│ │ │ +o1 = /tmp/M2-14309-0/0.tex
│ │ │              
    
│ │ │                
    i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-12169-0/1.html
    
│ │ │ +o2 = /tmp/M2-14309-0/1.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/M2-12169-0/0.tex
│ │ │ │ +o1 = /tmp/M2-14309-0/0.tex
│ │ │ │  i2 : temporaryFileName () | ".html"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12169-0/1.html
│ │ │ │ +o2 = /tmp/M2-14309-0/1.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
│ │ │ @@ -59,15 +59,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    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 2.18e-05s (cpu); 1.1101e-05s (thread); 0s (gc)
│ │ │ + -- used 4.4803e-05s (cpu); 7.142e-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 2.18e-05s (cpu); 1.1101e-05s (thread); 0s (gc)
│ │ │ │ + -- used 4.4803e-05s (cpu); 7.142e-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
│ │ │ @@ -54,24 +54,24 @@
│ │ │  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
│ │ │ -     -- .000018144 seconds
│ │ │ +     -- .000021827 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │              
    
│ │ │                
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000018144, 205891132094649}
    
│ │ │ +o2 = Time{.000021827, 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
│ │ │ │ -     -- .000018144 seconds
│ │ │ │ +     -- .000021827 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{.000018144, 205891132094649}
│ │ │ │ +o2 = Time{.000021827, 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
│ │ │ @@ -103,15 +103,15 @@
│ │ │                 "memtailor version" => 1.1
│ │ │                 "mpfi version" => 1.5.4
│ │ │                 "mpfr version" => 4.2.2
│ │ │                 "mpsolve version" => 3.2.2
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.11.0
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.12.57+deb13-amd64
│ │ │ +               "operating system release" => 6.12.57+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 PruneComplex 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 OldPolyhedra OldToricVectorBundles 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 NonminimalComplexes 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 MatrixFactorizations PathSignatures
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.13.11
│ │ │                 "readline version" => 8.3
│ │ │                 "scscp version" => not present
│ │ │                 "tbb version" => 2022.1
│ │ │ ├── html2text {}
│ │ │ │ @@ -64,15 +64,15 @@
│ │ │ │                 "memtailor version" => 1.1
│ │ │ │                 "mpfi version" => 1.5.4
│ │ │ │                 "mpfr version" => 4.2.2
│ │ │ │                 "mpsolve version" => 3.2.2
│ │ │ │                 "mysql version" => not present
│ │ │ │                 "normaliz version" => 3.11.0
│ │ │ │                 "ntl version" => 11.5.1
│ │ │ │ -               "operating system release" => 6.12.57+deb13-amd64
│ │ │ │ +               "operating system release" => 6.12.57+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 3.38775s (cpu); 1.60175s (thread); 0s (gc)
│ │ │ + -- used 4.59324s (cpu); 1.89244s (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
│ │ │ @@ -161,15 +161,15 @@
│ │ │          
      
│ │ │            
      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 3.38775s (cpu); 1.60175s (thread); 0s (gc)
│ │ │ + -- used 4.59324s (cpu); 1.89244s (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 3.38775s (cpu); 1.60175s (thread); 0s (gc)
│ │ │ │ + -- used 4.59324s (cpu); 1.89244s (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 0.100646s (cpu); 0.0344742s (thread); 0s (gc)
│ │ │ + -- used 0.100601s (cpu); 0.0340262s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
│ │ │  
│ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0147124s (cpu); 0.0147191s (thread); 0s (gc)
│ │ │ + -- used 0.0189303s (cpu); 0.0189429s (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 0.000285165s (cpu); 0.000279514s (thread); 0s (gc)
│ │ │ + -- used 0.000315607s (cpu); 0.000305473s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
│ │ │  
│ │ │  i17 : time regularity comodule I
│ │ │ - -- used 0.0152443s (cpu); 0.0152472s (thread); 0s (gc)
│ │ │ + -- used 0.0176733s (cpu); 0.0176848s (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 0.00455946s (cpu); 0.00455689s (thread); 0s (gc)
│ │ │ + -- used 0.00520285s (cpu); 0.00519798s (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 0.00228165s (cpu); 0.00228239s (thread); 0s (gc)
│ │ │ + -- used 0.00278733s (cpu); 0.00278889s (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
│ │ │ @@ -383,23 +383,23 @@
│ │ │          
      
│ │ │            
      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 0.100646s (cpu); 0.0344742s (thread); 0s (gc)
│ │ │ + -- used 0.100601s (cpu); 0.0340262s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
      
│ │ │              
      
│ │ │                
      i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0147124s (cpu); 0.0147191s (thread); 0s (gc)
│ │ │ + -- used 0.0189303s (cpu); 0.0189429s (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 0.100646s (cpu); 0.0344742s (thread); 0s (gc)
│ │ │ │ + -- used 0.100601s (cpu); 0.0340262s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = 1
│ │ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ │ - -- used 0.0147124s (cpu); 0.0147191s (thread); 0s (gc)
│ │ │ │ + -- used 0.0189303s (cpu); 0.0189429s (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
│ │ │ @@ -315,23 +315,23 @@
│ │ │          
      
│ │ │            
      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 0.000285165s (cpu); 0.000279514s (thread); 0s (gc)
│ │ │ + -- used 0.000315607s (cpu); 0.000305473s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
      
│ │ │              
      
│ │ │                
      i17 : time regularity comodule I
│ │ │ - -- used 0.0152443s (cpu); 0.0152472s (thread); 0s (gc)
│ │ │ + -- used 0.0176733s (cpu); 0.0176848s (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 0.000285165s (cpu); 0.000279514s (thread); 0s (gc)
│ │ │ │ + -- used 0.000315607s (cpu); 0.000305473s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 5
│ │ │ │  i17 : time regularity comodule I
│ │ │ │ - -- used 0.0152443s (cpu); 0.0152472s (thread); 0s (gc)
│ │ │ │ + -- used 0.0176733s (cpu); 0.0176848s (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
│ │ │ @@ -80,28 +80,28 @@
│ │ │  
│ │ │  o1 : List
      
│ │ │              
      
│ │ │                
      i2 : time grothendieckPolynomial w
│ │ │ - -- used 0.00455946s (cpu); 0.00455689s (thread); 0s (gc)
│ │ │ + -- used 0.00520285s (cpu); 0.00519798s (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 0.00228165s (cpu); 0.00228239s (thread); 0s (gc)
│ │ │ + -- used 0.00278733s (cpu); 0.00278889s (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 0.00455946s (cpu); 0.00455689s (thread); 0s (gc)
│ │ │ │ + -- used 0.00520285s (cpu); 0.00519798s (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 0.00228165s (cpu); 0.00228239s (thread); 0s (gc)
│ │ │ │ + -- used 0.00278733s (cpu); 0.00278889s (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 0.050878s (cpu); 0.0526062s (thread); 0s (gc)
│ │ │ + -- used 0.063962s (cpu); 0.0613665s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : time fVector R10
│ │ │ - -- used 0.195193s (cpu); 0.0769082s (thread); 0s (gc)
│ │ │ + -- used 0.266068s (cpu); 0.0949922s (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 0.000402855s (cpu); 0.000203472s (thread); 0s (gc)
│ │ │ + -- used 0.000441426s (cpu); 0.000236211s (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);
│ │ │ - -- .0861244s elapsed
│ │ │ + -- .0818794s 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 0.142029s (cpu); 0.0648897s (thread); 0s (gc)
│ │ │ + -- used 0.19236s (cpu); 0.0966993s (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 1.78938s (cpu); 1.28303s (thread); 0s (gc)
│ │ │ + -- used 2.30913s (cpu); 1.43072s (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 0.0208457s (cpu); 0.0227737s (thread); 0s (gc)
│ │ │ + -- used 0.0160114s (cpu); 0.015154s (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 4.74954s (cpu); 3.07529s (thread); 0s (gc)
│ │ │ + -- used 5.5385s (cpu); 3.37185s (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 0.000871805s (cpu); 0.000546956s (thread); 0s (gc)
│ │ │ + -- used 0.00311827s (cpu); 0.000731617s (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
│ │ │ - -- .0246909s elapsed
│ │ │ + -- .0153488s elapsed
│ │ │  
│ │ │  o6 = HashTable{0 => 1 }
│ │ │                 1 => 6
│ │ │                 2 => 15
│ │ │                 3 => 20
│ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/___Matroid.html
│ │ │ @@ -148,23 +148,23 @@
│ │ │  
│ │ │  o9 : List
│ │ │              
      
│ │ │                
      i10 : time isWellDefined R10
│ │ │ - -- used 0.050878s (cpu); 0.0526062s (thread); 0s (gc)
│ │ │ + -- used 0.063962s (cpu); 0.0613665s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
      
│ │ │              
      
│ │ │                
      i11 : time fVector R10
│ │ │ - -- used 0.195193s (cpu); 0.0769082s (thread); 0s (gc)
│ │ │ + -- used 0.266068s (cpu); 0.0949922s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ @@ -182,15 +182,15 @@
│ │ │  
│ │ │  o12 : List
      
│ │ │              
      
│ │ │                
      i13 : time fVector R10
│ │ │ - -- used 0.000402855s (cpu); 0.000203472s (thread); 0s (gc)
│ │ │ + -- used 0.000441426s (cpu); 0.000236211s (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 0.050878s (cpu); 0.0526062s (thread); 0s (gc)
│ │ │ │ + -- used 0.063962s (cpu); 0.0613665s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  i11 : time fVector R10
│ │ │ │ - -- used 0.195193s (cpu); 0.0769082s (thread); 0s (gc)
│ │ │ │ + -- used 0.266068s (cpu); 0.0949922s (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 0.000402855s (cpu); 0.000203472s (thread); 0s (gc)
│ │ │ │ + -- used 0.000441426s (cpu); 0.000236211s (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
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o2 : Matroid
      
│ │ │              
      
│ │ │                
      i3 : elapsedTime L = allMinors(V, U25);
│ │ │ - -- .0861244s elapsed
      
│ │ │ + -- .0818794s 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);
│ │ │ │ - -- .0861244s elapsed
│ │ │ │ + -- .0818794s 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
│ │ │ @@ -135,15 +135,15 @@
│ │ │  
│ │ │  o6 : Matroid
│ │ │              
      
│ │ │                
      i7 : time autF7 = getIsos(F7, F7);
│ │ │ - -- used 0.142029s (cpu); 0.0648897s (thread); 0s (gc)
      
│ │ │ + -- used 0.19236s (cpu); 0.0966993s (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 0.142029s (cpu); 0.0648897s (thread); 0s (gc)
│ │ │ │ + -- used 0.19236s (cpu); 0.0966993s (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
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o2 = false
│ │ │              
      
│ │ │                
      i3 : time hasMinor(M6, M5)
│ │ │ - -- used 1.78938s (cpu); 1.28303s (thread); 0s (gc)
│ │ │ + -- used 2.30913s (cpu); 1.43072s (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 1.78938s (cpu); 1.28303s (thread); 0s (gc)
│ │ │ │ + -- used 2.30913s (cpu); 1.43072s (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
│ │ │ @@ -118,15 +118,15 @@
│ │ │  
│ │ │  o4 : Matroid
    
│ │ │              
    
│ │ │                
    i5 : time isomorphism(M5, minorM6)
│ │ │ - -- used 0.0208457s (cpu); 0.0227737s (thread); 0s (gc)
│ │ │ + -- used 0.0160114s (cpu); 0.015154s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{0 => 1}
│ │ │                 1 => 0
│ │ │                 2 => 3
│ │ │                 3 => 2
│ │ │                 4 => 6
│ │ │                 5 => 5
│ │ │ @@ -164,15 +164,15 @@
│ │ │  
│ │ │  o7 : Matroid
    
│ │ │              
    
│ │ │                
    i8 : time phi = isomorphism(N,M6)
│ │ │ - -- used 4.74954s (cpu); 3.07529s (thread); 0s (gc)
│ │ │ + -- used 5.5385s (cpu); 3.37185s (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 0.0208457s (cpu); 0.0227737s (thread); 0s (gc)
│ │ │ │ + -- used 0.0160114s (cpu); 0.015154s (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 4.74954s (cpu); 3.07529s (thread); 0s (gc)
│ │ │ │ + -- used 5.5385s (cpu); 3.37185s (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
│ │ │ @@ -137,15 +137,15 @@
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time quickIsomorphismTest(M0, M1)
│ │ │ - -- used 0.000871805s (cpu); 0.000546956s (thread); 0s (gc)
│ │ │ + -- used 0.00311827s (cpu); 0.000731617s (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 0.000871805s (cpu); 0.000546956s (thread); 0s (gc)
│ │ │ │ + -- used 0.00311827s (cpu); 0.000731617s (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
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime fVector M
│ │ │ - -- .0246909s elapsed
│ │ │ + -- .0153488s 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
│ │ │ │ - -- .0246909s elapsed
│ │ │ │ + -- .0153488s 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 .000976812)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0138253)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time .000355486)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: DecomposeMonomials(time .000020849)  #primes = 1 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .00127335)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0358716)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .000417584)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: DecomposeMonomials(time .000020544)  #primes = 1 #prunedViaCodim = 0
│ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .000617178
│ │ │ - -- .0423566s elapsed
│ │ │ + --  Time taken : .000791597
│ │ │ + -- .0293258s 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)
│ │ │ - -- .000461911s elapsed
│ │ │ + -- .000592299s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0117995s elapsed
│ │ │ + -- .0151319s 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)
│ │ │ - -- .102182s elapsed
│ │ │ + -- .0941091s elapsed
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00174825s elapsed
│ │ │ + -- .00232212s elapsed
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00126877s elapsed
│ │ │ + -- .00166052s elapsed
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/___Hybrid.html
│ │ │ @@ -72,21 +72,21 @@
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ -  Strategy: Linear            (time .000976812)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0138253)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time .000355486)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: DecomposeMonomials(time .000020849)  #primes = 1 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .00127335)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0358716)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .000417584)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: DecomposeMonomials(time .000020544)  #primes = 1 #prunedViaCodim = 0
│ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .000617178
│ │ │ - -- .0423566s elapsed
    
│ │ │ + --  Time taken : .000791597
│ │ │ + -- .0293258s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,24 +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 .000976812)  #primes = 0 #prunedViaCodim =
│ │ │ │ +  Strategy: Linear            (time .00127335)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Birational        (time .0358716)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Factorization     (time .000417584)  #primes = 0 #prunedViaCodim =
│ │ │ │  0
│ │ │ │ -  Strategy: Birational        (time .0138253)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ -  Strategy: Factorization     (time .000355486)  #primes = 0 #prunedViaCodim =
│ │ │ │ -0
│ │ │ │ -  Strategy: DecomposeMonomials(time .000020849)  #primes = 1 #prunedViaCodim =
│ │ │ │ +  Strategy: DecomposeMonomials(time .000020544)  #primes = 1 #prunedViaCodim =
│ │ │ │  0
│ │ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ │ - --  Time taken : .000617178
│ │ │ │ - -- .0423566s elapsed
│ │ │ │ + --  Time taken : .000791597
│ │ │ │ + -- .0293258s 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
│ │ │ @@ -131,25 +131,25 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ - -- .000461911s elapsed
│ │ │ + -- .000592299s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0117995s elapsed
│ │ │ + -- .0151319s 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)
│ │ │ │ - -- .000461911s elapsed
│ │ │ │ + -- .000592299s elapsed
│ │ │ │  
│ │ │ │  o6 = ideal (a, b, c)
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ │ - -- .0117995s elapsed
│ │ │ │ + -- .0151319s 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
│ │ │ @@ -125,31 +125,31 @@
│ │ │  
│ │ │  o6 = true
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ - -- .102182s elapsed
│ │ │ + -- .0941091s elapsed
│ │ │  
│ │ │  o7 = true
    
│ │ │              
    
│ │ │                
    i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00174825s elapsed
│ │ │ + -- .00232212s elapsed
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00126877s elapsed
│ │ │ + -- .00166052s 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)
│ │ │ │ - -- .102182s elapsed
│ │ │ │ + -- .0941091s elapsed
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ │ - -- .00174825s elapsed
│ │ │ │ + -- .00232212s elapsed
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ │ - -- .00126877s elapsed
│ │ │ │ + -- .00166052s 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 0.102055s (cpu); 0.0642847s (thread); 0s (gc)
│ │ │ + -- used 0.385503s (cpu); 0.119379s (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 0.0842922s (cpu); 0.0247612s (thread); 0s (gc)
│ │ │ + -- used 0.105155s (cpu); 0.0317546s (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
│ │ │ @@ -173,15 +173,15 @@
│ │ │  
│ │ │  o9 : Ideal of U
    
│ │ │              
    
│ │ │                
    i10 : time multiReesIdeal J
│ │ │ - -- used 0.102055s (cpu); 0.0642847s (thread); 0s (gc)
│ │ │ + -- used 0.385503s (cpu); 0.119379s (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 )
│ │ │ @@ -190,15 +190,15 @@
│ │ │  o10 : Ideal of U[X ..X ]
│ │ │                    0   2
    
│ │ │              
    
│ │ │                
    i11 : time multiReesIdeal (J, a)
│ │ │ - -- used 0.0842922s (cpu); 0.0247612s (thread); 0s (gc)
│ │ │ + -- used 0.105155s (cpu); 0.0317546s (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 0.102055s (cpu); 0.0642847s (thread); 0s (gc)
│ │ │ │ + -- used 0.385503s (cpu); 0.119379s (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 0.0842922s (cpu); 0.0247612s (thread); 0s (gc)
│ │ │ │ + -- used 0.105155s (cpu); 0.0317546s (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 0.481255s (cpu); 0.333746s (thread); 0s (gc)
│ │ │ + -- used 0.63021s (cpu); 0.402966s (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 0.648667s (cpu); 0.498693s (thread); 0s (gc)
│ │ │ + -- used 0.856374s (cpu); 0.602717s (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
│ │ │ @@ -145,15 +145,15 @@
│ │ │                               1
│ │ │  o7 : R-module, submodule of R
    
│ │ │              
    
│ │ │                
    i8 : (S,N) = time deformMCMModule N0 
│ │ │ - -- used 0.481255s (cpu); 0.333746s (thread); 0s (gc)
│ │ │ + -- used 0.63021s (cpu); 0.402966s (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                                           |
│ │ │  
│ │ │ @@ -186,15 +186,15 @@
│ │ │                               2
│ │ │  o10 : R-module, quotient of R
    
│ │ │              
    
│ │ │                
    i11 : (S',N') = time deformMCMModule N0'
│ │ │ - -- used 0.648667s (cpu); 0.498693s (thread); 0s (gc)
│ │ │ + -- used 0.856374s (cpu); 0.602717s (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 0.481255s (cpu); 0.333746s (thread); 0s (gc)
│ │ │ │ + -- used 0.63021s (cpu); 0.402966s (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 0.648667s (cpu); 0.498693s (thread); 0s (gc)
│ │ │ │ + -- used 0.856374s (cpu); 0.602717s (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
│ │ │ @@ -3,27 +3,27 @@
│ │ │  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})
│ │ │ - -- .00296506s elapsed
│ │ │ - -- .00282693s elapsed
│ │ │ - -- .000335866s elapsed
│ │ │ - -- .0027534s elapsed
│ │ │ - -- .00295554s elapsed
│ │ │ - -- .000235038s elapsed
│ │ │ - -- .00283445s elapsed
│ │ │ - -- .00295772s elapsed
│ │ │ - -- .000236902s elapsed
│ │ │ - -- .00296333s elapsed
│ │ │ - -- .0029042s elapsed
│ │ │ - -- .000232784s elapsed
│ │ │ ---backup directory created: /tmp/M2-33427-0/1
│ │ │ + -- .0042466s elapsed
│ │ │ + -- .00415648s elapsed
│ │ │ + -- .000495125s elapsed
│ │ │ + -- .00413283s elapsed
│ │ │ + -- .00421453s elapsed
│ │ │ + -- .000389871s elapsed
│ │ │ + -- .00379918s elapsed
│ │ │ + -- .00392005s elapsed
│ │ │ + -- .000342196s elapsed
│ │ │ + -- .00428962s elapsed
│ │ │ + -- .00376235s elapsed
│ │ │ + -- .000332612s elapsed
│ │ │ +--backup directory created: /tmp/M2-49521-0/1
│ │ │    H01: 1
│ │ │    H10: 1
│ │ │  number of paths tracked: 2
│ │ │  found 1 points in the fiber so far
│ │ │    H01: 1
│ │ │    H10: 1
│ │ │  number of paths tracked: 4
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Group.out
│ │ │ @@ -15,128 +15,128 @@
│ │ │  
│ │ │  i7 : dLoss = diff(varMatrix, gateMatrix{{loss}});
│ │ │  
│ │ │  i8 : G = gateSystem(paramMatrix,varMatrix,transpose dLoss);
│ │ │  
│ │ │  i9 : monodromyGroup(G,"msOptions" => {NumberOfEdges=>10})
│ │ │  
│ │ │ -o9 = {{2, 0, 11, 3, 10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19,
│ │ │ +o9 = {{16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18,
│ │ │ +     4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 4, 6}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ +     17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0, 12, 10, 15, 13, 6, 17, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18,
│ │ │ +     18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 20, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 19, 15, 17, 3}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
│ │ │ +     18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 0, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 16, 17, 18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8,
│ │ │ +     3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7, 15, 8, 10, 11, 17, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 15, 12, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12,
│ │ │ +     14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 14, 15, 16, 17, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │ +     2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4, 7, 6, 11, 1, 9, 0, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16,
│ │ │ +     10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 7, 8, 15, 14, 6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15,
│ │ │ +     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6,
│ │ │ +     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4, 16, 14, 3, 5, 2, 7, 8, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 13, 11, 9, 8, 10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15,
│ │ │ +     11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20}, {12, 16, 7, 17, 10, 9, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5,
│ │ │ +     4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 11, 2, 3, 10, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 7, 10, 4, 13, 11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4,
│ │ │ +     14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18, 19, 20}, {16, 14, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 11, 14, 5, 2, 1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7,
│ │ │ +     8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {2, 0, 11, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 16, 9, 0, 4, 11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1,
│ │ │ +     10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 1, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6},
│ │ │ +     19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {17, 16, 14, 9, 8, 7, 12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18,
│ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18,
│ │ │ +     1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 4, 6}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
│ │ │ +     {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12,
│ │ │ +     20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 18, 19, 20}, {0, 9, 2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15,
│ │ │ +     19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12, 13, 14, 15, 16, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 17, 18, 19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0,
│ │ │ +     18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 19, 13, 20, 6, 9, 4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8,
│ │ │ +     16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16, 10, 7, 8, 15, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 15, 12, 17, 18, 19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17,
│ │ │ +     6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 2, 13, 20, 8, 18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10,
│ │ │ +     14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6, 2, 13, 11, 9, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 5, 13, 14, 3, 16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │ +     10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4,
│ │ │ +     3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5, 9, 7, 10, 4, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5,
│ │ │ +     11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4, 10, 11, 14, 5, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 6, 10, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1,
│ │ │ +     1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7, 17, 16, 9, 0, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4, 10, 2, 6, 11, 14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5,
│ │ │ +     11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19,
│ │ │ +     5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {17, 16, 14, 9, 8, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 11, 14, 5, 15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3,
│ │ │ +     12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18, 19}, {1, 16, 12, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2,
│ │ │ +     11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {0, 1, 2, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12,
│ │ │ +     4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {2, 0, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 3, 19, 0, 11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6},
│ │ │ +     3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4,
│ │ │ +     2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6}, {4, 1, 2, 3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18,
│ │ │ +     {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ +     4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8, 14, 15, 12, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 9, 4, 6}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │ +     19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17, 10, 2, 13, 20, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 15, 6, 9, 4}, {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10,
│ │ │ +     18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 16, 18, 9, 4, 6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16,
│ │ │ +     16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 12, 17, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17,
│ │ │ +     18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 14, 20, 16, 18, 9, 4, 6}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5,
│ │ │ +     13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5, 8, 6, 10, 7, 12,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 0, 10, 19, 13, 20, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11,
│ │ │ +     13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1, 4, 10, 2, 6, 11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 13, 14, 18, 16, 19, 15, 17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0,
│ │ │ +     14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 10, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1,
│ │ │ +     11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19, 10, 11, 14, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 0, 5, 10, 2, 13, 20, 8, 18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19,
│ │ │ +     15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3, 10, 5, 14, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 8, 10, 20, 5, 0, 14, 3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7,
│ │ │ +     8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5, 6,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 8, 10, 11, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9,
│ │ │ +     7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 1, 3, 19, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4,
│ │ │ +     11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6}, {0, 1, 3, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19,
│ │ │ +     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {4, 1, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2,
│ │ │ +     3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18, 19, 20}, {0, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4,
│ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 14, 3, 5, 2, 7, 8, 1, 11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20},
│ │ │ +     1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3, 16, 15, 6, 9, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18,
│ │ │ +     {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10, 20, 16, 18, 9, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19}, {0, 11, 2, 3, 10, 4, 14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18,
│ │ │ +     6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13,
│ │ │ +     18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 6, 9, 4}}
│ │ │ +     18, 9, 4, 6}}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html
│ │ │ @@ -96,27 +96,27 @@
│ │ │              
    		
│ │ │                
    i3 : (p0, x0) = createSeedPair polys;
    
│ │ │              
    
│ │ │                
    i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ - -- .00296506s elapsed
│ │ │ - -- .00282693s elapsed
│ │ │ - -- .000335866s elapsed
│ │ │ - -- .0027534s elapsed
│ │ │ - -- .00295554s elapsed
│ │ │ - -- .000235038s elapsed
│ │ │ - -- .00283445s elapsed
│ │ │ - -- .00295772s elapsed
│ │ │ - -- .000236902s elapsed
│ │ │ - -- .00296333s elapsed
│ │ │ - -- .0029042s elapsed
│ │ │ - -- .000232784s elapsed
│ │ │ ---backup directory created: /tmp/M2-33427-0/1
│ │ │ + -- .0042466s elapsed
│ │ │ + -- .00415648s elapsed
│ │ │ + -- .000495125s elapsed
│ │ │ + -- .00413283s elapsed
│ │ │ + -- .00421453s elapsed
│ │ │ + -- .000389871s elapsed
│ │ │ + -- .00379918s elapsed
│ │ │ + -- .00392005s elapsed
│ │ │ + -- .000342196s elapsed
│ │ │ + -- .00428962s elapsed
│ │ │ + -- .00376235s elapsed
│ │ │ + -- .000332612s elapsed
│ │ │ +--backup directory created: /tmp/M2-49521-0/1
│ │ │    H01: 1
│ │ │    H10: 1
│ │ │  number of paths tracked: 2
│ │ │  found 1 points in the fiber so far
│ │ │    H01: 1
│ │ │    H10: 1
│ │ │  number of paths tracked: 4
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,27 +22,27 @@
│ │ │ │            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})
│ │ │ │ - -- .00296506s elapsed
│ │ │ │ - -- .00282693s elapsed
│ │ │ │ - -- .000335866s elapsed
│ │ │ │ - -- .0027534s elapsed
│ │ │ │ - -- .00295554s elapsed
│ │ │ │ - -- .000235038s elapsed
│ │ │ │ - -- .00283445s elapsed
│ │ │ │ - -- .00295772s elapsed
│ │ │ │ - -- .000236902s elapsed
│ │ │ │ - -- .00296333s elapsed
│ │ │ │ - -- .0029042s elapsed
│ │ │ │ - -- .000232784s elapsed
│ │ │ │ ---backup directory created: /tmp/M2-33427-0/1
│ │ │ │ + -- .0042466s elapsed
│ │ │ │ + -- .00415648s elapsed
│ │ │ │ + -- .000495125s elapsed
│ │ │ │ + -- .00413283s elapsed
│ │ │ │ + -- .00421453s elapsed
│ │ │ │ + -- .000389871s elapsed
│ │ │ │ + -- .00379918s elapsed
│ │ │ │ + -- .00392005s elapsed
│ │ │ │ + -- .000342196s elapsed
│ │ │ │ + -- .00428962s elapsed
│ │ │ │ + -- .00376235s elapsed
│ │ │ │ + -- .000332612s elapsed
│ │ │ │ +--backup directory created: /tmp/M2-49521-0/1
│ │ │ │    H01: 1
│ │ │ │    H10: 1
│ │ │ │  number of paths tracked: 2
│ │ │ │  found 1 points in the fiber so far
│ │ │ │    H01: 1
│ │ │ │    H10: 1
│ │ │ │  number of paths tracked: 4
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/html/_monodromy__Group.html
│ │ │ @@ -118,131 +118,131 @@
│ │ │                
    i8 : G = gateSystem(paramMatrix,varMatrix,transpose dLoss);
    
│ │ │              
    
│ │ │                
    i9 : monodromyGroup(G,"msOptions" => {NumberOfEdges=>10})
│ │ │  
│ │ │ -o9 = {{2, 0, 11, 3, 10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19,
│ │ │ +o9 = {{16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18,
│ │ │ +     4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 4, 6}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ +     17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0, 12, 10, 15, 13, 6, 17, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18,
│ │ │ +     18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 20, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 19, 15, 17, 3}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
│ │ │ +     18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 0, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 16, 17, 18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8,
│ │ │ +     3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7, 15, 8, 10, 11, 17, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 15, 12, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12,
│ │ │ +     14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 14, 15, 16, 17, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │ +     2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4, 7, 6, 11, 1, 9, 0, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16,
│ │ │ +     10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 7, 8, 15, 14, 6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15,
│ │ │ +     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6,
│ │ │ +     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4, 16, 14, 3, 5, 2, 7, 8, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 13, 11, 9, 8, 10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15,
│ │ │ +     11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20}, {12, 16, 7, 17, 10, 9, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5,
│ │ │ +     4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 11, 2, 3, 10, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 7, 10, 4, 13, 11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4,
│ │ │ +     14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18, 19, 20}, {16, 14, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 11, 14, 5, 2, 1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7,
│ │ │ +     8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {2, 0, 11, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 16, 9, 0, 4, 11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1,
│ │ │ +     10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 1, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6},
│ │ │ +     19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {17, 16, 14, 9, 8, 7, 12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18,
│ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18,
│ │ │ +     1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 4, 6}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
│ │ │ +     {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12,
│ │ │ +     20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 18, 19, 20}, {0, 9, 2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15,
│ │ │ +     19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12, 13, 14, 15, 16, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 17, 18, 19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0,
│ │ │ +     18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 19, 13, 20, 6, 9, 4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8,
│ │ │ +     16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16, 10, 7, 8, 15, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 15, 12, 17, 18, 19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17,
│ │ │ +     6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 2, 13, 20, 8, 18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10,
│ │ │ +     14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6, 2, 13, 11, 9, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 5, 13, 14, 3, 16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │ +     10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4,
│ │ │ +     3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5, 9, 7, 10, 4, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5,
│ │ │ +     11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4, 10, 11, 14, 5, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 6, 10, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1,
│ │ │ +     1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7, 17, 16, 9, 0, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4, 10, 2, 6, 11, 14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5,
│ │ │ +     11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19,
│ │ │ +     5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {17, 16, 14, 9, 8, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 11, 14, 5, 15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3,
│ │ │ +     12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18, 19}, {1, 16, 12, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2,
│ │ │ +     11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {0, 1, 2, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12,
│ │ │ +     4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {2, 0, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 3, 19, 0, 11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6},
│ │ │ +     3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4,
│ │ │ +     2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6}, {4, 1, 2, 3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18,
│ │ │ +     {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ +     4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8, 14, 15, 12, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 9, 4, 6}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │ +     19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17, 10, 2, 13, 20, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 15, 6, 9, 4}, {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10,
│ │ │ +     18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 16, 18, 9, 4, 6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16,
│ │ │ +     16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 12, 17, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17,
│ │ │ +     18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 14, 20, 16, 18, 9, 4, 6}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5,
│ │ │ +     13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5, 8, 6, 10, 7, 12,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 0, 10, 19, 13, 20, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11,
│ │ │ +     13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1, 4, 10, 2, 6, 11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 13, 14, 18, 16, 19, 15, 17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0,
│ │ │ +     14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 10, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1,
│ │ │ +     11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19, 10, 11, 14, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 0, 5, 10, 2, 13, 20, 8, 18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19,
│ │ │ +     15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3, 10, 5, 14, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 8, 10, 20, 5, 0, 14, 3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7,
│ │ │ +     8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5, 6,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 8, 10, 11, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9,
│ │ │ +     7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 1, 3, 19, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4,
│ │ │ +     11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6}, {0, 1, 3, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19,
│ │ │ +     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {4, 1, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2,
│ │ │ +     3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18, 19, 20}, {0, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4,
│ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 14, 3, 5, 2, 7, 8, 1, 11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20},
│ │ │ +     1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3, 16, 15, 6, 9, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18,
│ │ │ +     {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10, 20, 16, 18, 9, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19}, {0, 11, 2, 3, 10, 4, 14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18,
│ │ │ +     6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13,
│ │ │ +     18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 6, 9, 4}}
│ │ │ +     18, 9, 4, 6}}
│ │ │  
│ │ │  o9 : List
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,131 +32,131 @@
│ │ │ │  i4 : varMatrix = gateMatrix{{t_1,t_2}};
│ │ │ │  i5 : phi = transpose gateMatrix{{t_1^3, t_1^2*t_2, t_1*t_2^2, t_2^3}};
│ │ │ │  i6 : loss = sum for i from 0 to 3 list (u_i - phi_(i,0))^2;
│ │ │ │  i7 : dLoss = diff(varMatrix, gateMatrix{{loss}});
│ │ │ │  i8 : G = gateSystem(paramMatrix,varMatrix,transpose dLoss);
│ │ │ │  i9 : monodromyGroup(G,"msOptions" => {NumberOfEdges=>10})
│ │ │ │  
│ │ │ │ -o9 = {{2, 0, 11, 3, 10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19,
│ │ │ │ +o9 = {{16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18,
│ │ │ │ +     4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     9, 4, 6}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ │ +     17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0, 12, 10, 15, 13, 6, 17, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18,
│ │ │ │ +     18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 20, 8,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 19, 15, 17, 3}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
│ │ │ │ +     18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 0, 14,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 16, 17, 18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8,
│ │ │ │ +     3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7, 15, 8, 10, 11, 17, 13,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 15, 12, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12,
│ │ │ │ +     14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     13, 14, 15, 16, 17, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │ │ +     2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4, 7, 6, 11, 1, 9, 0, 5,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16,
│ │ │ │ +     10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 7, 8, 15, 14, 6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15,
│ │ │ │ +     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6,
│ │ │ │ +     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4, 16, 14, 3, 5, 2, 7, 8, 1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     2, 13, 11, 9, 8, 10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15,
│ │ │ │ +     11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20}, {12, 16, 7, 17, 10, 9, 14,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5,
│ │ │ │ +     4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 11, 2, 3, 10, 4,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     9, 7, 10, 4, 13, 11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4,
│ │ │ │ +     14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18, 19, 20}, {16, 14, 17, 18,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 11, 14, 5, 2, 1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7,
│ │ │ │ +     8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {2, 0, 11, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     17, 16, 9, 0, 4, 11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1,
│ │ │ │ +     10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 1, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6},
│ │ │ │ +     19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {17, 16, 14, 9, 8, 7, 12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18,
│ │ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     19}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18,
│ │ │ │ +     1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     9, 4, 6}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
│ │ │ │ +     {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12,
│ │ │ │ +     20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     17, 18, 19, 20}, {0, 9, 2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15,
│ │ │ │ +     19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12, 13, 14, 15, 16, 17,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 17, 18, 19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0,
│ │ │ │ +     18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 19, 13, 20, 6, 9, 4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8,
│ │ │ │ +     16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16, 10, 7, 8, 15, 14,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     14, 15, 12, 17, 18, 19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17,
│ │ │ │ +     6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 2, 13, 20, 8, 18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10,
│ │ │ │ +     14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6, 2, 13, 11, 9, 8,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20, 5, 13, 14, 3, 16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │ │ +     10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4,
│ │ │ │ +     3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5, 9, 7, 10, 4, 13,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5,
│ │ │ │ +     11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4, 10, 11, 14, 5, 2,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     8, 6, 10, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1,
│ │ │ │ +     1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7, 17, 16, 9, 0, 4,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     4, 10, 2, 6, 11, 14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5,
│ │ │ │ +     11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19,
│ │ │ │ +     5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {17, 16, 14, 9, 8, 7,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 11, 14, 5, 15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3,
│ │ │ │ +     12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18, 19}, {1, 16, 12, 19,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2,
│ │ │ │ +     11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {0, 1, 2, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12,
│ │ │ │ +     4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {2, 0, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     1, 3, 19, 0, 11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6},
│ │ │ │ +     3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4,
│ │ │ │ +     2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     6}, {4, 1, 2, 3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18,
│ │ │ │ +     {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ │ +     4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8, 14, 15, 12, 17, 18,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     18, 9, 4, 6}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │ │ +     19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17, 10, 2, 13, 20, 8,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 15, 6, 9, 4}, {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10,
│ │ │ │ +     18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20, 16, 18, 9, 4, 6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16,
│ │ │ │ +     16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 12, 17, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17,
│ │ │ │ +     18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     13, 14, 20, 16, 18, 9, 4, 6}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5,
│ │ │ │ +     13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5, 8, 6, 10, 7, 12,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 0, 10, 19, 13, 20, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11,
│ │ │ │ +     13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1, 4, 10, 2, 6, 11,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     12, 13, 14, 18, 16, 19, 15, 17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0,
│ │ │ │ +     14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     12, 10, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1,
│ │ │ │ +     11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19, 10, 11, 14, 5,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     9, 0, 5, 10, 2, 13, 20, 8, 18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19,
│ │ │ │ +     15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3, 10, 5, 14, 7,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     11, 8, 10, 20, 5, 0, 14, 3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7,
│ │ │ │ +     8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5, 6,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 8, 10, 11, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9,
│ │ │ │ +     7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 1, 3, 19, 0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4,
│ │ │ │ +     11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6}, {0, 1, 3, 19,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19,
│ │ │ │ +     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {4, 1, 2,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2,
│ │ │ │ +     3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18, 19, 20}, {0, 1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4,
│ │ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 14, 3, 5, 2, 7, 8, 1, 11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20},
│ │ │ │ +     1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3, 16, 15, 6, 9, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18,
│ │ │ │ +     {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10, 20, 16, 18, 9, 4,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     19}, {0, 11, 2, 3, 10, 4, 14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18,
│ │ │ │ +     6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 20,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13,
│ │ │ │ +     18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20, 6, 9, 4}}
│ │ │ │ +     18, 9, 4, 6}}
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  This is still somewhat experimental.
│ │ │ │  ********** WWaayyss ttoo uussee mmoonnooddrroommyyGGrroouupp:: **********
│ │ │ │      * monodromyGroup(System)
│ │ │ │      * monodromyGroup(System,AbstractPoint,List)
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/example-output/___Msolve.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  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/M2-53632-0/0-in.ms -o /tmp/M2-53632-0/0-out.ms
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-84642-0/0-in.ms -o /tmp/M2-84642-0/0-out.ms
│ │ │  
│ │ │  --------------- INPUT DATA ---------------
│ │ │  #variables                       3
│ │ │  #equations                       3
│ │ │  #invalid equations               0
│ │ │  field characteristic             0
│ │ │  homogeneous input?               1
│ │ │ @@ -28,15 +28,15 @@
│ │ │  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 = 1196244169
│ │ │ +Initial prime = 1117130281
│ │ │  
│ │ │  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)
│ │ │ @@ -46,25 +46,25 @@
│ │ │  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         0.02 | 0.07         
│ │ │ +reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -overall(elapsed)        0.06 sec
│ │ │ -overall(cpu)            0.17 sec
│ │ │ +overall(elapsed)        0.00 sec
│ │ │ +overall(cpu)            0.01 sec
│ │ │  select                  0.00 sec   0.0%
│ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ -update                  0.03 sec  57.3%
│ │ │ -convert                 0.02 sec  42.4%
│ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ +symbolic prep.          0.00 sec   0.2%
│ │ │ +update                  0.00 sec  87.1%
│ │ │ +convert                 0.00 sec   4.0%
│ │ │ +linear algebra          0.00 sec   0.5%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │  #pairs reduced                    0
│ │ │ @@ -78,18 +78,18 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1107170621
│ │ │ +New prime = 1157291321
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -multi-mod overall(elapsed)      0.02 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%> 
│ │ │ @@ -105,15 +105,15 @@
│ │ │  ---------------- TIMINGS ----------------
│ │ │  CRT     (elapsed)               0.00 sec
│ │ │  ratrecon(elapsed)               0.00 sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.19 sec (elapsed) /  0.49 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
│ │ │ @@ -78,15 +78,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : msolveGB(I, Verbosity => 2, Threads => 6) 
│ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-53632-0/0-in.ms -o /tmp/M2-53632-0/0-out.ms
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-84642-0/0-in.ms -o /tmp/M2-84642-0/0-out.ms
│ │ │  
│ │ │  --------------- INPUT DATA ---------------
│ │ │  #variables                       3
│ │ │  #equations                       3
│ │ │  #invalid equations               0
│ │ │  field characteristic             0
│ │ │  homogeneous input?               1
│ │ │ @@ -97,15 +97,15 @@
│ │ │  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 = 1196244169
│ │ │ +Initial prime = 1117130281
│ │ │  
│ │ │  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)
│ │ │ @@ -115,25 +115,25 @@
│ │ │  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         0.02 | 0.07         
│ │ │ +reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -overall(elapsed)        0.06 sec
│ │ │ -overall(cpu)            0.17 sec
│ │ │ +overall(elapsed)        0.00 sec
│ │ │ +overall(cpu)            0.01 sec
│ │ │  select                  0.00 sec   0.0%
│ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ -update                  0.03 sec  57.3%
│ │ │ -convert                 0.02 sec  42.4%
│ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ +symbolic prep.          0.00 sec   0.2%
│ │ │ +update                  0.00 sec  87.1%
│ │ │ +convert                 0.00 sec   4.0%
│ │ │ +linear algebra          0.00 sec   0.5%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │  #pairs reduced                    0
│ │ │ @@ -147,18 +147,18 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1107170621
│ │ │ +New prime = 1157291321
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -multi-mod overall(elapsed)      0.02 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%> 
│ │ │ @@ -174,15 +174,15 @@
│ │ │  ---------------- TIMINGS ----------------
│ │ │  CRT     (elapsed)               0.00 sec
│ │ │  ratrecon(elapsed)               0.00 sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.19 sec (elapsed) /  0.49 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
│ │ │ - -- .00186857s elapsed
│ │ │ + -- .0029236s elapsed
│ │ │  computing total degree: 2
│ │ │  number of monomials = 21
│ │ │  number of distinct multidegrees = 18
│ │ │ - -- .00833047s elapsed
│ │ │ + -- .0117245s 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
│ │ │ │ - -- .00186857s elapsed
│ │ │ │ + -- .0029236s elapsed
│ │ │ │  computing total degree: 2
│ │ │ │  number of monomials = 21
│ │ │ │  number of distinct multidegrees = 18
│ │ │ │ - -- .00833047s elapsed
│ │ │ │ + -- .0117245s 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
│ │ │ - -- .0234547s elapsed
│ │ │ + -- .0469481s elapsed
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 : elapsedTime monjMult I
│ │ │ - -- .107713s elapsed
│ │ │ + -- .0900121s elapsed
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .181349s elapsed
│ │ │ + -- .191375s 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
│ │ │ - -- .171404s elapsed
│ │ │ + -- .103246s 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
│ │ │ - -- .119902s elapsed
│ │ │ + -- .122437s elapsed
│ │ │  
│ │ │  o3 = 192
│ │ │  
│ │ │  i4 : elapsedTime jMult I
│ │ │ - -- 1.54599s elapsed
│ │ │ + -- 1.50928s elapsed
│ │ │  
│ │ │  o4 = 192
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html
│ │ │ @@ -88,31 +88,31 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime jMult I
│ │ │ - -- .0234547s elapsed
│ │ │ + -- .0469481s elapsed
│ │ │  
│ │ │  o3 = 2
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime monjMult I
│ │ │ - -- .107713s elapsed
│ │ │ + -- .0900121s elapsed
│ │ │  
│ │ │  o4 = 2
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .181349s elapsed
│ │ │ + -- .191375s elapsed
│ │ │  
│ │ │  o5 = HashTable{2 => 3}
│ │ │                 3 => 2
│ │ │  
│ │ │  o5 : HashTable
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime monAnalyticSpread I
│ │ │ - -- .171404s elapsed
│ │ │ + -- .103246s 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
│ │ │ │ - -- .171404s elapsed
│ │ │ │ + -- .103246s 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
│ │ │ @@ -92,23 +92,23 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime monjMult I
│ │ │ - -- .119902s elapsed
│ │ │ + -- .122437s elapsed
│ │ │  
│ │ │  o3 = 192
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime jMult I
│ │ │ - -- 1.54599s elapsed
│ │ │ + -- 1.50928s 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
│ │ │ │ - -- .119902s elapsed
│ │ │ │ + -- .122437s elapsed
│ │ │ │  
│ │ │ │  o3 = 192
│ │ │ │  i4 : elapsedTime jMult I
│ │ │ │ - -- 1.54599s elapsed
│ │ │ │ + -- 1.50928s 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 3.0434s (cpu); 1.80018s (thread); 0s (gc)
│ │ │ + -- used 3.97935s (cpu); 2.18083s (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 3.14065s (cpu); 1.91214s (thread); 0s (gc)
│ │ │ + -- used 4.37872s (cpu); 2.44278s (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 8.07742s (cpu); 4.95873s (thread); 0s (gc)
│ │ │ + -- used 7.23744s (cpu); 5.14081s (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 5.11368s (cpu); 3.88996s (thread); 0s (gc)
│ │ │ + -- used 5.08046s (cpu); 4.1664s (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 0.0951343s (cpu); 0.0963862s (thread); 0s (gc)
│ │ │ + -- used 0.189975s (cpu); 0.130013s (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 2.05743s (cpu); 1.42394s (thread); 0s (gc)
│ │ │ + -- used 2.4364s (cpu); 1.58346s (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 3.60186s (cpu); 2.37753s (thread); 0s (gc)
│ │ │ + -- used 5.076s (cpu); 2.89314s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.300764s (cpu); 0.253978s (thread); 0s (gc)
│ │ │ + -- used 0.38506s (cpu); 0.304726s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time degree Phi
│ │ │ - -- used 0.32363s (cpu); 0.259653s (thread); 0s (gc)
│ │ │ + -- used 0.374758s (cpu); 0.297921s (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 0.39932s (cpu); 0.353204s (thread); 0s (gc)
│ │ │ + -- used 0.629243s (cpu); 0.436659s (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 0.000917731s (cpu); 0.000161313s (thread); 0s (gc)
│ │ │ + -- used 0.000686556s (cpu); 0.000185228s (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 0.00272072s (cpu); 0.000248957s (thread); 0s (gc)
│ │ │ + -- used 0.00277244s (cpu); 0.000269487s (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 1.4658s (cpu); 1.13064s (thread); 0s (gc)
│ │ │ + -- used 1.39741s (cpu); 1.10639s (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 0.000169538s (cpu); 0.000607139s (thread); 0s (gc)
│ │ │ + -- used 0.000583043s (cpu); 0.000487549s (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 0.0895117s (cpu); 0.0398444s (thread); 0s (gc)
│ │ │ + -- used 0.0403774s (cpu); 0.0285846s (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 0.032293s (cpu); 0.0340052s (thread); 0s (gc)
│ │ │ + -- used 0.107216s (cpu); 0.0478177s (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 0.194002s (cpu); 0.149271s (thread); 0s (gc)
│ │ │ + -- used 0.374728s (cpu); 0.221334s (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 0.200211s (cpu); 0.133927s (thread); 0s (gc)
│ │ │ + -- used 0.169888s (cpu); 0.138634s (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 5.791s (cpu); 2.94859s (thread); 0s (gc)
│ │ │ + -- used 10.3195s (cpu); 2.93973s (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 0.36801s (cpu); 0.291283s (thread); 0s (gc)
│ │ │ + -- used 0.463502s (cpu); 0.358226s (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 1.53154s (cpu); 1.09582s (thread); 0s (gc)
│ │ │ + -- used 1.41053s (cpu); 1.20646s (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 0.151859s (cpu); 0.0747115s (thread); 0s (gc)
│ │ │ + -- used 0.0716398s (cpu); 0.0582715s (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 0.1777s (cpu); 0.0987744s (thread); 0s (gc)
│ │ │ + -- used 0.25504s (cpu); 0.112109s (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 0.442368s (cpu); 0.288571s (thread); 0s (gc)
│ │ │ + -- used 0.633518s (cpu); 0.35739s (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 0.00299491s (cpu); 9.007e-06s (thread); 0s (gc)
│ │ │ + -- used 0.00244814s (cpu); 9.645e-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 0.331015s (cpu); 0.180919s (thread); 0s (gc)
│ │ │ + -- used 0.510622s (cpu); 0.238194s (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 1.56012s (cpu); 0.826624s (thread); 0s (gc)
│ │ │ + -- used 1.99119s (cpu); 1.01047s (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 0.369946s (cpu); 0.279856s (thread); 0s (gc)
│ │ │ + -- used 0.555355s (cpu); 0.295866s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.170151s (cpu); 0.094037s (thread); 0s (gc)
│ │ │ + -- used 0.123916s (cpu); 0.0739107s (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 4.39879s (cpu); 3.33728s (thread); 0s (gc)
│ │ │ + -- used 4.36463s (cpu); 3.58316s (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 0.010893s (cpu); 0.00980925s (thread); 0s (gc)
│ │ │ + -- used 0.124674s (cpu); 0.0353389s (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 0.564105s (cpu); 0.442475s (thread); 0s (gc)
│ │ │ + -- used 0.589895s (cpu); 0.494374s (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 0.521458s (cpu); 0.386885s (thread); 0s (gc)
│ │ │ + -- used 0.634285s (cpu); 0.398788s (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 0.00397921s (cpu); 0.00133095s (thread); 0s (gc)
│ │ │ - -- used 0.272866s (cpu); 0.142784s (thread); 0s (gc)
│ │ │ - -- used 0.237166s (cpu); 0.173525s (thread); 0s (gc)
│ │ │ - -- used 0.202997s (cpu); 0.139949s (thread); 0s (gc)
│ │ │ - -- used 0.187465s (cpu); 0.110073s (thread); 0s (gc)
│ │ │ + -- used 0.00223888s (cpu); 0.00157421s (thread); 0s (gc)
│ │ │ + -- used 0.258476s (cpu); 0.17014s (thread); 0s (gc)
│ │ │ + -- used 0.273723s (cpu); 0.190323s (thread); 0s (gc)
│ │ │ + -- used 0.262716s (cpu); 0.169153s (thread); 0s (gc)
│ │ │ + -- used 0.240155s (cpu); 0.149846s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : time assert(oo == multidegree Phi)
│ │ │ - -- used 0.127059s (cpu); 0.0818662s (thread); 0s (gc)
│ │ │ + -- used 0.21901s (cpu); 0.104875s (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 0.0161552s (cpu); 0.0167468s (thread); 0s (gc)
│ │ │ + -- used 0.0826378s (cpu); 0.0278835s (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 1.74407s (cpu); 0.993938s (thread); 0s (gc)
│ │ │ + -- used 1.85996s (cpu); 1.19095s (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 0.750404s (cpu); 0.533826s (thread); 0s (gc)
│ │ │ + -- used 1.30614s (cpu); 0.655128s (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 0.273463s (cpu); 0.167098s (thread); 0s (gc)
│ │ │ + -- used 0.267779s (cpu); 0.160765s (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
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15 
│ │ │              
    
│ │ │                
    i6 : time f = X ===> Y;
│ │ │ - -- used 3.0434s (cpu); 1.80018s (thread); 0s (gc)
│ │ │ + -- used 3.97935s (cpu); 2.18083s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (automorphism of PP^8)
    
│ │ │              
    
│ │ │                
    i7 : f X
│ │ │ @@ -143,15 +143,15 @@
│ │ │  
│ │ │  o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8
    
│ │ │              
    
│ │ │                
    i11 : time g = V ===> W;
│ │ │ - -- used 3.14065s (cpu); 1.91214s (thread); 0s (gc)
│ │ │ + -- used 4.37872s (cpu); 2.44278s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : MultirationalMap (automorphism of PP^8)
    
│ │ │              
    
│ │ │                
    i12 : g||W
│ │ │ @@ -252,15 +252,15 @@
│ │ │  
│ │ │  o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
    
│ │ │              
    
│ │ │                
    i17 : time h = Z ===> GG_K(1,4)
│ │ │ - -- used 8.07742s (cpu); 4.95873s (thread); 0s (gc)
│ │ │ + -- used 7.23744s (cpu); 5.14081s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = h
│ │ │  
│ │ │  o17 : MultirationalMap (isomorphism from PP^9 to PP^9)
    
│ │ │              
    
│ │ │                
    i4 : time X = Phi^* Y;
│ │ │ - -- used 5.11368s (cpu); 3.88996s (thread); 0s (gc)
│ │ │ + -- used 5.08046s (cpu); 4.1664s (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 5.11368s (cpu); 3.88996s (thread); 0s (gc)
│ │ │ │ + -- used 5.08046s (cpu); 4.1664s (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
│ │ │ @@ -95,15 +95,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7
    
│ │ │              
    
│ │ │                
    i5 : time Phi Z;
│ │ │ - -- used 0.0951343s (cpu); 0.0963862s (thread); 0s (gc)
│ │ │ + -- used 0.189975s (cpu); 0.130013s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7
    
│ │ │              
    
│ │ │                
    i6 : dim oo, degree oo, degrees oo
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o6 : Sequence
    
│ │ │              
    
│ │ │                
    i7 : time Phi (point Z + point Z + point Z)
│ │ │ - -- used 2.05743s (cpu); 1.42394s (thread); 0s (gc)
│ │ │ + -- used 2.4364s (cpu); 1.58346s (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 3.60186s (cpu); 2.37753s (thread); 0s (gc)
│ │ │ + -- used 5.076s (cpu); 2.89314s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
    
│ │ │              
    
│ │ │                
    i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.300764s (cpu); 0.253978s (thread); 0s (gc)
│ │ │ + -- used 0.38506s (cpu); 0.304726s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    
│ │ │              
    
│ │ │                
    i6 : time degree Phi
│ │ │ - -- used 0.32363s (cpu); 0.259653s (thread); 0s (gc)
│ │ │ + -- used 0.374758s (cpu); 0.297921s (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 3.60186s (cpu); 2.37753s (thread); 0s (gc)
│ │ │ │ + -- used 5.076s (cpu); 2.89314s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ │ - -- used 0.300764s (cpu); 0.253978s (thread); 0s (gc)
│ │ │ │ + -- used 0.38506s (cpu); 0.304726s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : time degree Phi
│ │ │ │ - -- used 0.32363s (cpu); 0.259653s (thread); 0s (gc)
│ │ │ │ + -- used 0.374758s (cpu); 0.297921s (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
│ │ │ @@ -81,15 +81,15 @@
│ │ │  
│ │ │  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 0.39932s (cpu); 0.353204s (thread); 0s (gc)
│ │ │ + -- used 0.629243s (cpu); 0.436659s (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 0.39932s (cpu); 0.353204s (thread); 0s (gc)
│ │ │ │ + -- used 0.629243s (cpu); 0.436659s (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
│ │ │ @@ -77,15 +77,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4 x PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time ? Phi
│ │ │ - -- used 0.000917731s (cpu); 0.000161313s (thread); 0s (gc)
│ │ │ + -- used 0.000686556s (cpu); 0.000185228s (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
    
│ │ │              
│ │ │ @@ -96,27 +96,27 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time ? Phi
│ │ │ - -- used 0.00272072s (cpu); 0.000248957s (thread); 0s (gc)
│ │ │ + -- used 0.00277244s (cpu); 0.000269487s (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 1.4658s (cpu); 1.13064s (thread); 0s (gc)
│ │ │ + -- used 1.39741s (cpu); 1.10639s (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 
│ │ │ @@ -126,15 +126,15 @@
│ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │       coefficient ring: ZZ/65521
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time ? Phi
│ │ │ - -- used 0.000169538s (cpu); 0.000607139s (thread); 0s (gc)
│ │ │ + -- used 0.000583043s (cpu); 0.000487549s (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 0.000917731s (cpu); 0.000161313s (thread); 0s (gc)
│ │ │ │ + -- used 0.000686556s (cpu); 0.000185228s (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 0.00272072s (cpu); 0.000248957s (thread); 0s (gc)
│ │ │ │ + -- used 0.00277244s (cpu); 0.000269487s (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 1.4658s (cpu); 1.13064s (thread); 0s (gc)
│ │ │ │ + -- used 1.39741s (cpu); 1.10639s (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 0.000169538s (cpu); 0.000607139s (thread); 0s (gc)
│ │ │ │ + -- used 0.000583043s (cpu); 0.000487549s (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
│ │ │ @@ -83,15 +83,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time (Phi1,Phi2) = graph Phi
│ │ │ - -- used 0.0895117s (cpu); 0.0398444s (thread); 0s (gc)
│ │ │ + -- used 0.0403774s (cpu); 0.0285846s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (Phi1, Phi2)
│ │ │  
│ │ │  o2 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -107,15 +107,15 @@
│ │ │  
│ │ │  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 0.032293s (cpu); 0.0340052s (thread); 0s (gc)
│ │ │ + -- used 0.107216s (cpu); 0.0478177s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = (Phi21, Phi22)
│ │ │  
│ │ │  o5 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  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 0.194002s (cpu); 0.149271s (thread); 0s (gc)
│ │ │ + -- used 0.374728s (cpu); 0.221334s (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 0.0895117s (cpu); 0.0398444s (thread); 0s (gc)
│ │ │ │ + -- used 0.0403774s (cpu); 0.0285846s (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 0.032293s (cpu); 0.0340052s (thread); 0s (gc)
│ │ │ │ + -- used 0.107216s (cpu); 0.0478177s (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 0.194002s (cpu); 0.149271s (thread); 0s (gc)
│ │ │ │ + -- used 0.374728s (cpu); 0.221334s (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
│ │ │ @@ -95,15 +95,15 @@
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time Z = image Phi;
│ │ │ - -- used 0.200211s (cpu); 0.133927s (thread); 0s (gc)
│ │ │ + -- used 0.169888s (cpu); 0.138634s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : dim Z, degree Z, degrees Z
│ │ │ @@ -115,15 +115,15 @@
│ │ │            
│ │ │          
│ │ │          
    Alternatively, the calculation can be performed using the Segre embedding as follows:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── 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 0.521458s (cpu); 0.386885s (thread); 0s (gc)
│ │ │ │ + -- used 0.634285s (cpu); 0.398788s (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
│ │ │ @@ -77,29 +77,29 @@
│ │ │  
│ │ │  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 5.791s (cpu); 2.94859s (thread); 0s (gc)
│ │ │ + -- used 10.3195s (cpu); 2.93973s (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 0.200211s (cpu); 0.133927s (thread); 0s (gc)
│ │ │ │ + -- used 0.169888s (cpu); 0.138634s (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 5.791s (cpu); 2.94859s (thread); 0s (gc)
│ │ │ │ + -- used 10.3195s (cpu); 2.93973s (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
│ │ │ @@ -83,15 +83,15 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
│ │ │              
    
│ │ │                
    i3 : time Psi = inverse2 Phi;
│ │ │ - -- used 0.36801s (cpu); 0.291283s (thread); 0s (gc)
│ │ │ + -- used 0.463502s (cpu); 0.358226s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
    
│ │ │              
    
│ │ │                
    i4 : assert(Phi * Psi == 1)
    
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
│ │ │              
    
│ │ │                
    i6 : time Psi' = inverse2 Phi';
│ │ │ - -- used 1.53154s (cpu); 1.09582s (thread); 0s (gc)
│ │ │ + -- used 1.41053s (cpu); 1.20646s (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 0.36801s (cpu); 0.291283s (thread); 0s (gc)
│ │ │ │ + -- used 0.463502s (cpu); 0.358226s (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 1.53154s (cpu); 1.09582s (thread); 0s (gc)
│ │ │ │ + -- used 1.41053s (cpu); 1.20646s (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
│ │ │ @@ -88,45 +88,45 @@
│ │ │  
│ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │              
    
│ │ │                
    i3 : time inverse Phi;
│ │ │ - -- used 0.151859s (cpu); 0.0747115s (thread); 0s (gc)
│ │ │ + -- used 0.0716398s (cpu); 0.0582715s (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 0.1777s (cpu); 0.0987744s (thread); 0s (gc)
│ │ │ + -- used 0.25504s (cpu); 0.112109s (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 0.442368s (cpu); 0.288571s (thread); 0s (gc)
│ │ │ + -- used 0.633518s (cpu); 0.35739s (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 0.151859s (cpu); 0.0747115s (thread); 0s (gc)
│ │ │ │ + -- used 0.0716398s (cpu); 0.0582715s (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 0.1777s (cpu); 0.0987744s (thread); 0s (gc)
│ │ │ │ + -- used 0.25504s (cpu); 0.112109s (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 0.442368s (cpu); 0.288571s (thread); 0s (gc)
│ │ │ │ + -- used 0.633518s (cpu); 0.35739s (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
│ │ │ @@ -83,45 +83,45 @@
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
│ │ │              
    
│ │ │                
    i4 : time isIsomorphism Phi
│ │ │ - -- used 0.00299491s (cpu); 9.007e-06s (thread); 0s (gc)
│ │ │ + -- used 0.00244814s (cpu); 9.645e-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 0.331015s (cpu); 0.180919s (thread); 0s (gc)
│ │ │ + -- used 0.510622s (cpu); 0.238194s (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 1.56012s (cpu); 0.826624s (thread); 0s (gc)
│ │ │ + -- used 1.99119s (cpu); 1.01047s (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 0.00299491s (cpu); 9.007e-06s (thread); 0s (gc)
│ │ │ │ + -- used 0.00244814s (cpu); 9.645e-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 0.331015s (cpu); 0.180919s (thread); 0s (gc)
│ │ │ │ + -- used 0.510622s (cpu); 0.238194s (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 1.56012s (cpu); 0.826624s (thread); 0s (gc)
│ │ │ │ + -- used 1.99119s (cpu); 1.01047s (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
│ │ │ @@ -80,31 +80,31 @@
│ │ │  
│ │ │  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 0.369946s (cpu); 0.279856s (thread); 0s (gc)
│ │ │ + -- used 0.555355s (cpu); 0.295866s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
    
│ │ │              
    
│ │ │                
    i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.170151s (cpu); 0.094037s (thread); 0s (gc)
│ │ │ + -- used 0.123916s (cpu); 0.0739107s (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 4.39879s (cpu); 3.33728s (thread); 0s (gc)
│ │ │ + -- used 4.36463s (cpu); 3.58316s (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 0.369946s (cpu); 0.279856s (thread); 0s (gc)
│ │ │ │ + -- used 0.555355s (cpu); 0.295866s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : time Psi = first graph Phi;
│ │ │ │ - -- used 0.170151s (cpu); 0.094037s (thread); 0s (gc)
│ │ │ │ + -- used 0.123916s (cpu); 0.0739107s (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 4.39879s (cpu); 3.33728s (thread); 0s (gc)
│ │ │ │ + -- used 4.36463s (cpu); 3.58316s (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
│ │ │ @@ -79,30 +79,30 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, curve in PP^7
│ │ │              
    
│ │ │                
    i3 : time f = linearlyNormalEmbedding X;
│ │ │ - -- used 0.010893s (cpu); 0.00980925s (thread); 0s (gc)
│ │ │ + -- used 0.124674s (cpu); 0.0353389s (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 0.564105s (cpu); 0.442475s (thread); 0s (gc)
│ │ │ + -- used 0.589895s (cpu); 0.494374s (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 0.010893s (cpu); 0.00980925s (thread); 0s (gc)
│ │ │ │ + -- used 0.124674s (cpu); 0.0353389s (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 0.564105s (cpu); 0.442475s (thread); 0s (gc)
│ │ │ │ + -- used 0.589895s (cpu); 0.494374s (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
│ │ │ @@ -81,15 +81,15 @@
│ │ │  
│ │ │  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 0.521458s (cpu); 0.386885s (thread); 0s (gc)
│ │ │ + -- used 0.634285s (cpu); 0.398788s (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 0.00397921s (cpu); 0.00133095s (thread); 0s (gc)
│ │ │ - -- used 0.272866s (cpu); 0.142784s (thread); 0s (gc)
│ │ │ - -- used 0.237166s (cpu); 0.173525s (thread); 0s (gc)
│ │ │ - -- used 0.202997s (cpu); 0.139949s (thread); 0s (gc)
│ │ │ - -- used 0.187465s (cpu); 0.110073s (thread); 0s (gc)
│ │ │ + -- used 0.00223888s (cpu); 0.00157421s (thread); 0s (gc)
│ │ │ + -- used 0.258476s (cpu); 0.17014s (thread); 0s (gc)
│ │ │ + -- used 0.273723s (cpu); 0.190323s (thread); 0s (gc)
│ │ │ + -- used 0.262716s (cpu); 0.169153s (thread); 0s (gc)
│ │ │ + -- used 0.240155s (cpu); 0.149846s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time assert(oo == multidegree Phi)
│ │ │ - -- used 0.127059s (cpu); 0.0818662s (thread); 0s (gc)
    
│ │ │ + -- used 0.21901s (cpu); 0.104875s (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 0.00397921s (cpu); 0.00133095s (thread); 0s (gc)
│ │ │ │ - -- used 0.272866s (cpu); 0.142784s (thread); 0s (gc)
│ │ │ │ - -- used 0.237166s (cpu); 0.173525s (thread); 0s (gc)
│ │ │ │ - -- used 0.202997s (cpu); 0.139949s (thread); 0s (gc)
│ │ │ │ - -- used 0.187465s (cpu); 0.110073s (thread); 0s (gc)
│ │ │ │ + -- used 0.00223888s (cpu); 0.00157421s (thread); 0s (gc)
│ │ │ │ + -- used 0.258476s (cpu); 0.17014s (thread); 0s (gc)
│ │ │ │ + -- used 0.273723s (cpu); 0.190323s (thread); 0s (gc)
│ │ │ │ + -- used 0.262716s (cpu); 0.169153s (thread); 0s (gc)
│ │ │ │ + -- used 0.240155s (cpu); 0.149846s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : time assert(oo == multidegree Phi)
│ │ │ │ - -- used 0.127059s (cpu); 0.0818662s (thread); 0s (gc)
│ │ │ │ + -- used 0.21901s (cpu); 0.104875s (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
│ │ │ @@ -80,15 +80,15 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time p := point X
│ │ │ - -- used 0.0161552s (cpu); 0.0167468s (thread); 0s (gc)
│ │ │ + -- used 0.0826378s (cpu); 0.0278835s (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
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time q = point Y
│ │ │ - -- used 1.74407s (cpu); 0.993938s (thread); 0s (gc)
│ │ │ + -- used 1.85996s (cpu); 1.19095s (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 0.0161552s (cpu); 0.0167468s (thread); 0s (gc)
│ │ │ │ + -- used 0.0826378s (cpu); 0.0278835s (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 1.74407s (cpu); 0.993938s (thread); 0s (gc)
│ │ │ │ + -- used 1.85996s (cpu); 1.19095s (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
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time segre Phi;
│ │ │ - -- used 0.750404s (cpu); 0.533826s (thread); 0s (gc)
│ │ │ + -- used 1.30614s (cpu); 0.655128s (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 0.750404s (cpu); 0.533826s (thread); 0s (gc)
│ │ │ │ + -- used 1.30614s (cpu); 0.655128s (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
│ │ │ @@ -77,15 +77,15 @@
│ │ │  
│ │ │  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 0.273463s (cpu); 0.167098s (thread); 0s (gc)
│ │ │ + -- used 0.267779s (cpu); 0.160765s (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 0.273463s (cpu); 0.167098s (thread); 0s (gc)
│ │ │ │ + -- used 0.267779s (cpu); 0.160765s (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 = (61, 77, 85, 95, 93, 96, 96, 95, 98, 96, 99, 97, 96, 96, 97, 98, 98, 96,
│ │ │ +o9 = (65, 72, 90, 94, 94, 93, 94, 95, 97, 93, 98, 96, 97, 96, 98, 98, 95, 98,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     98, 100, 97, 99, 100, 99, 100, 98, 97, 96, 99)
│ │ │ +     96, 100, 95, 97, 97, 99, 100, 97, 98, 98, 98)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (11, 6, 4, 5, 4, 2, 0, 2, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1,
│ │ │ +o10 = (17, 9, 6, 5, 3, 4, 3, 1, 3, 0, 4, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 1, 0, 0, 0, 1)
│ │ │ +      1, 1, 1, 1, 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 = {DSc, DR_, D|o, Ddw, D^k}
│ │ │ +o2 = {DT_, D`{, DZ[, DW_, DEw}
│ │ │  
│ │ │  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" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}},
│ │ │ +o2 = {Graph{"edges" => {{a, c}, {b, c}, {b, d}, {a, e}, {d, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}},
│ │ │ +     Graph{"edges" => {{b, c}, {a, d}, {c, d}, {a, e}, {b, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{b, c}, {a, d}, {b, d}, {a, e}, {c, e}}}}
│ │ │ +     Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, 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 0.000476844s (cpu); 0.000401593s (thread); 0s (gc)
│ │ │ + -- used 0.000464337s (cpu); 0.000655501s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : time (graphComplement \ G);
│ │ │ - -- used 0.147573s (cpu); 0.079233s (thread); 0s (gc)
│ │ │ + -- used 0.188207s (cpu); 0.0900525s (thread); 0s (gc)
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html
│ │ │ @@ -117,28 +117,28 @@
│ │ │                
    i8 : prob = n -> log(n)/n;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (61, 77, 85, 95, 93, 96, 96, 95, 98, 96, 99, 97, 96, 96, 97, 98, 98, 96,
│ │ │ +o9 = (65, 72, 90, 94, 94, 93, 94, 95, 97, 93, 98, 96, 97, 96, 98, 98, 95, 98,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     98, 100, 97, 99, 100, 99, 100, 98, 97, 96, 99)
│ │ │ +     96, 100, 95, 97, 97, 99, 100, 97, 98, 98, 98)
│ │ │  
│ │ │  o9 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (11, 6, 4, 5, 4, 2, 0, 2, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1,
│ │ │ +o10 = (17, 9, 6, 5, 3, 4, 3, 1, 3, 0, 4, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 1, 0, 0, 0, 1)
│ │ │ +      1, 1, 1, 1, 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 = (61, 77, 85, 95, 93, 96, 96, 95, 98, 96, 99, 97, 96, 96, 97, 98, 98, 96,
│ │ │ │ +o9 = (65, 72, 90, 94, 94, 93, 94, 95, 97, 93, 98, 96, 97, 96, 98, 98, 95, 98,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     98, 100, 97, 99, 100, 99, 100, 98, 97, 96, 99)
│ │ │ │ +     96, 100, 95, 97, 97, 99, 100, 97, 98, 98, 98)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2),
│ │ │ │  connected))
│ │ │ │  
│ │ │ │ -o10 = (11, 6, 4, 5, 4, 2, 0, 2, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1,
│ │ │ │ +o10 = (17, 9, 6, 5, 3, 4, 3, 1, 3, 0, 4, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      1, 1, 0, 0, 0, 1)
│ │ │ │ +      1, 1, 1, 1, 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/Nauty/html/_generate__Random__Graphs.html
│ │ │ @@ -100,15 +100,15 @@
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {DSc, DR_, D|o, Ddw, D^k}
│ │ │ +o2 = {DT_, D`{, DZ[, DW_, DEw}
│ │ │  
│ │ │  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 = {DSc, DR_, D|o, Ddw, D^k}
│ │ │ │ +o2 = {DT_, D`{, DZ[, DW_, DEw}
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -87,23 +87,23 @@
│ │ │                
    i1 : R = QQ[a..e];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │  
│ │ │ -o2 = {Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}},
│ │ │ +o2 = {Graph{"edges" => {{a, c}, {b, c}, {b, d}, {a, e}, {d, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}},
│ │ │ +     Graph{"edges" => {{b, c}, {a, d}, {c, d}, {a, e}, {b, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{b, c}, {a, d}, {b, d}, {a, e}, {c, e}}}}
│ │ │ +     Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, 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" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}},
│ │ │ │ +o2 = {Graph{"edges" => {{a, c}, {b, c}, {b, d}, {a, e}, {d, e}}},
│ │ │ │              "ring" => R
│ │ │ │              "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}},
│ │ │ │ +     Graph{"edges" => {{b, c}, {a, d}, {c, d}, {a, e}, {b, e}}},
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Graph{"edges" => {{b, c}, {a, d}, {b, d}, {a, e}, {c, e}}}}
│ │ │ │ +     Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, 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
│ │ │ @@ -116,21 +116,21 @@
│ │ │              
│ │ │                
    i4 : G = generateBipartiteGraphs 7;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time graphComplement G;
│ │ │ - -- used 0.000476844s (cpu); 0.000401593s (thread); 0s (gc)
    
│ │ │ + -- used 0.000464337s (cpu); 0.000655501s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time (graphComplement \ G);
│ │ │ - -- used 0.147573s (cpu); 0.079233s (thread); 0s (gc)
    
│ │ │ + -- used 0.188207s (cpu); 0.0900525s (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 0.000476844s (cpu); 0.000401593s (thread); 0s (gc)
│ │ │ │ + -- used 0.000464337s (cpu); 0.000655501s (thread); 0s (gc)
│ │ │ │  i6 : time (graphComplement \ G);
│ │ │ │ - -- used 0.147573s (cpu); 0.079233s (thread); 0s (gc)
│ │ │ │ + -- used 0.188207s (cpu); 0.0900525s (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 = (72, 79, 88, 91, 95, 95, 93, 96, 97, 99, 97, 95, 96, 97, 95, 98, 99, 99,
│ │ │ +o9 = (74, 86, 91, 91, 91, 91, 99, 97, 96, 99, 97, 98, 94, 96, 99, 100, 97,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     94, 99, 96, 98, 98, 97, 97, 98, 98, 94, 98)
│ │ │ +     98, 98, 96, 100, 99, 99, 98, 98, 98, 97, 97, 100)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (18, 11, 7, 2, 7, 3, 5, 3, 1, 1, 1, 2, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
│ │ │ +o10 = (14, 8, 5, 3, 7, 3, 1, 0, 3, 3, 2, 0, 1, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 1, 1, 1, 0, 0)
│ │ │ +      0, 1, 1, 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 = {DU?, Dt_, DFw, DTG, DxC}
│ │ │ +o2 = {Dac, DSO, DwK, DRw, Dt{}
│ │ │  
│ │ │  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 = {DMg, DLo, DLo}
│ │ │ +o1 = {DLo, Dhc, DYc}
│ │ │  
│ │ │  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 0.00101801s (cpu); 0.000811832s (thread); 0s (gc)
│ │ │ + -- used 0.000670754s (cpu); 0.000547287s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : time (graphComplement \ G);
│ │ │ - -- used 0.146022s (cpu); 0.0727874s (thread); 0s (gc)
│ │ │ + -- used 0.163163s (cpu); 0.0778834s (thread); 0s (gc)
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html
│ │ │ @@ -117,28 +117,28 @@
│ │ │                
      i8 : prob = n -> log(n)/n;
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (72, 79, 88, 91, 95, 95, 93, 96, 97, 99, 97, 95, 96, 97, 95, 98, 99, 99,
│ │ │ +o9 = (74, 86, 91, 91, 91, 91, 99, 97, 96, 99, 97, 98, 94, 96, 99, 100, 97,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     94, 99, 96, 98, 98, 97, 97, 98, 98, 94, 98)
│ │ │ +     98, 98, 96, 100, 99, 99, 98, 98, 98, 97, 97, 100)
│ │ │  
│ │ │  o9 : Sequence
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (18, 11, 7, 2, 7, 3, 5, 3, 1, 1, 1, 2, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
│ │ │ +o10 = (14, 8, 5, 3, 7, 3, 1, 0, 3, 3, 2, 0, 1, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 1, 1, 1, 0, 0)
│ │ │ +      0, 1, 1, 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 = (72, 79, 88, 91, 95, 95, 93, 96, 97, 99, 97, 95, 96, 97, 95, 98, 99, 99,
│ │ │ │ +o9 = (74, 86, 91, 91, 91, 91, 99, 97, 96, 99, 97, 98, 94, 96, 99, 100, 97,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     94, 99, 96, 98, 98, 97, 97, 98, 98, 94, 98)
│ │ │ │ +     98, 98, 96, 100, 99, 99, 98, 98, 98, 97, 97, 100)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2),
│ │ │ │  connected))
│ │ │ │  
│ │ │ │ -o10 = (18, 11, 7, 2, 7, 3, 5, 3, 1, 1, 1, 2, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
│ │ │ │ +o10 = (14, 8, 5, 3, 7, 3, 1, 0, 3, 3, 2, 0, 1, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      0, 1, 1, 1, 0, 0)
│ │ │ │ +      0, 1, 1, 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
│ │ │ @@ -93,15 +93,15 @@
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {DU?, Dt_, DFw, DTG, DxC}
│ │ │ +o2 = {Dac, DSO, DwK, DRw, Dt{}
│ │ │  
│ │ │  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 = {DU?, Dt_, DFw, DTG, DxC}
│ │ │ │ +o2 = {Dac, DSO, DwK, DRw, Dt{}
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -77,15 +77,15 @@
│ │ │            
    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 = {DMg, DLo, DLo}
│ │ │ +o1 = {DLo, Dhc, DYc}
│ │ │  
│ │ │  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 = {DMg, DLo, DLo}
│ │ │ │ +o1 = {DLo, Dhc, DYc}
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -110,21 +110,21 @@
│ │ │              
│ │ │                
    i3 : G = generateBipartiteGraphs 7;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time graphComplement G;
│ │ │ - -- used 0.00101801s (cpu); 0.000811832s (thread); 0s (gc)
    
│ │ │ + -- used 0.000670754s (cpu); 0.000547287s (thread); 0s (gc)    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time (graphComplement \ G);
│ │ │ - -- used 0.146022s (cpu); 0.0727874s (thread); 0s (gc)
    
│ │ │ + -- used 0.163163s (cpu); 0.0778834s (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 0.00101801s (cpu); 0.000811832s (thread); 0s (gc)
│ │ │ │ + -- used 0.000670754s (cpu); 0.000547287s (thread); 0s (gc)
│ │ │ │  i5 : time (graphComplement \ G);
│ │ │ │ - -- used 0.146022s (cpu); 0.0727874s (thread); 0s (gc)
│ │ │ │ + -- used 0.163163s (cpu); 0.0778834s (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")
│ │ │ - -- .113138s elapsed
│ │ │ + -- .0906126s 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
│ │ │ @@ -120,15 +120,15 @@
│ │ │  
│ │ │  o5 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot")
│ │ │ - -- .113138s elapsed
│ │ │ + -- .0906126s 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")
│ │ │ │ - -- .113138s elapsed
│ │ │ │ + -- .0906126s 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 0.195947s (cpu); 0.196132s (thread); 0s (gc)
│ │ │ + -- used 0.268141s (cpu); 0.269473s (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 0.00192857s (cpu); 0.00120085s (thread); 0s (gc)
│ │ │ - -- used 2.3484e-05s (cpu); 9.2934e-05s (thread); 0s (gc)
│ │ │ - -- used 9.298e-06s (cpu); 7.0111e-05s (thread); 0s (gc)
│ │ │ - -- used 8.586e-06s (cpu); 6.6936e-05s (thread); 0s (gc)
│ │ │ - -- used 8.175e-06s (cpu); 7.2777e-05s (thread); 0s (gc)
│ │ │ - -- used 8.416e-06s (cpu); 6.7105e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00400058s (cpu); 0.00152596s (thread); 0s (gc)
│ │ │ + -- used 3.236e-05s (cpu); 0.000134791s (thread); 0s (gc)
│ │ │ + -- used 1.216e-05s (cpu); 8.5552e-05s (thread); 0s (gc)
│ │ │ + -- used 1.0587e-05s (cpu); 8.9444e-05s (thread); 0s (gc)
│ │ │ + -- used 1.1448e-05s (cpu); 8.6472e-05s (thread); 0s (gc)
│ │ │ + -- used 1.2318e-05s (cpu); 8.3775e-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 1765726817
│ │ │ + -- setting random seed to 1767789253
│ │ │  
│ │ │  i3 : a = sort apply (3, i -> random (7))
│ │ │  
│ │ │ -o3 = {2, 4, 5}
│ │ │ +o3 = {2, 3, 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 = {5, 7, 7, 9, 9}
│ │ │ +o7 = {3, 4, 6, 8, 9}
│ │ │  
│ │ │  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
│ │ │ - -- .0534782s elapsed
│ │ │ + -- .035912s 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))
│ │ │ - -- .0017525s elapsed
│ │ │ + -- .0016787s 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
│ │ │ - -- .0763581s elapsed
│ │ │ + -- .0377816s 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))
│ │ │ - -- .00167424s elapsed
│ │ │ + -- .00136606s 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
│ │ │ - -- 40.8741s elapsed
│ │ │ + -- 32.4281s elapsed
│ │ │  
│ │ │  o11 = 7909
│ │ │  
│ │ │  i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3))
│ │ │ - -- .0293459s elapsed
│ │ │ + -- .0319871s 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 2.2001e-05s (cpu); 1.9186e-05s (thread); 0s (gc)
│ │ │ + -- used 3.5654e-05s (cpu); 2.2289e-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 0.0417375s (cpu); 0.0417407s (thread); 0s (gc)
│ │ │ + -- used 0.0543238s (cpu); 0.0543363s (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 0.0209352s (cpu); 0.020935s (thread); 0s (gc)
│ │ │ + -- used 0.0290511s (cpu); 0.0290518s (thread); 0s (gc)
│ │ │  
│ │ │  i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ - -- used 0.00216116s (cpu); 0.00216187s (thread); 0s (gc)
│ │ │ + -- used 0.00327888s (cpu); 0.0032839s (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
│ │ │ @@ -207,15 +207,15 @@
│ │ │          
    
│ │ │            
    We end with a slightly larger example.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -254,20 +254,20 @@
│ │ │  
│ │ │  o18 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i14 : Y = time smoothFanoToricVariety(5,100);
│ │ │ - -- used 0.195947s (cpu); 0.196132s (thread); 0s (gc)
    
│ │ │ + -- used 0.268141s (cpu); 0.269473s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i15 : A2 = intersectionRing Y;
    
│ │ │              
    
│ │ │                
    i19 : for i to dim Y list time hilbertFunction (i, A2)
│ │ │ - -- used 0.00192857s (cpu); 0.00120085s (thread); 0s (gc)
│ │ │ - -- used 2.3484e-05s (cpu); 9.2934e-05s (thread); 0s (gc)
│ │ │ - -- used 9.298e-06s (cpu); 7.0111e-05s (thread); 0s (gc)
│ │ │ - -- used 8.586e-06s (cpu); 6.6936e-05s (thread); 0s (gc)
│ │ │ - -- used 8.175e-06s (cpu); 7.2777e-05s (thread); 0s (gc)
│ │ │ - -- used 8.416e-06s (cpu); 6.7105e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00400058s (cpu); 0.00152596s (thread); 0s (gc)
│ │ │ + -- used 3.236e-05s (cpu); 0.000134791s (thread); 0s (gc)
│ │ │ + -- used 1.216e-05s (cpu); 8.5552e-05s (thread); 0s (gc)
│ │ │ + -- used 1.0587e-05s (cpu); 8.9444e-05s (thread); 0s (gc)
│ │ │ + -- used 1.1448e-05s (cpu); 8.6472e-05s (thread); 0s (gc)
│ │ │ + -- used 1.2318e-05s (cpu); 8.3775e-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 0.195947s (cpu); 0.196132s (thread); 0s (gc)
│ │ │ │ + -- used 0.268141s (cpu); 0.269473s (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 0.00192857s (cpu); 0.00120085s (thread); 0s (gc)
│ │ │ │ - -- used 2.3484e-05s (cpu); 9.2934e-05s (thread); 0s (gc)
│ │ │ │ - -- used 9.298e-06s (cpu); 7.0111e-05s (thread); 0s (gc)
│ │ │ │ - -- used 8.586e-06s (cpu); 6.6936e-05s (thread); 0s (gc)
│ │ │ │ - -- used 8.175e-06s (cpu); 7.2777e-05s (thread); 0s (gc)
│ │ │ │ - -- used 8.416e-06s (cpu); 6.7105e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00400058s (cpu); 0.00152596s (thread); 0s (gc)
│ │ │ │ + -- used 3.236e-05s (cpu); 0.000134791s (thread); 0s (gc)
│ │ │ │ + -- used 1.216e-05s (cpu); 8.5552e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.0587e-05s (cpu); 8.9444e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.1448e-05s (cpu); 8.6472e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.2318e-05s (cpu); 8.3775e-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
│ │ │ @@ -93,22 +93,22 @@
│ │ │          
    
│ │ │            
    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 1765726817
    
│ │ │ + -- setting random seed to 1767789253
│ │ │              
    
│ │ │                
    i3 : a = sort apply (3, i -> random (7))
│ │ │  
│ │ │ -o3 = {2, 4, 5}
│ │ │ +o3 = {2, 3, 6}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : assert isWellDefined kleinschmidt (4,a)
    
│ │ │ @@ -126,15 +126,15 @@
│ │ │                
    i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j -> random (1,9));
    
│ │ │              
    
│ │ │                
    i7 : q
│ │ │  
│ │ │ -o7 = {5, 7, 7, 9, 9}
│ │ │ +o7 = {3, 4, 6, 8, 9}
│ │ │  
│ │ │  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 1765726817
│ │ │ │ + -- setting random seed to 1767789253
│ │ │ │  i3 : a = sort apply (3, i -> random (7))
│ │ │ │  
│ │ │ │ -o3 = {2, 4, 5}
│ │ │ │ +o3 = {2, 3, 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 = {5, 7, 7, 9, 9}
│ │ │ │ +o7 = {3, 4, 6, 8, 9}
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o2 : ToricDivisor on PP2
│ │ │              
    
│ │ │                
    i3 : M1 = elapsedTime monomials D1
│ │ │ - -- .0534782s elapsed
│ │ │ + -- .035912s 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 }
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring variety D1))
│ │ │ - -- .0017525s elapsed
    
│ │ │ + -- .0016787s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    Toric varieties of Picard-rank 2 are slightly more interesting.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -138,27 +138,27 @@
│ │ │  
│ │ │  o6 : ToricDivisor on FF2
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -171,23 +171,23 @@
│ │ │  
│ │ │  o10 : ToricDivisor on X
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : M2 = elapsedTime monomials D2
│ │ │ - -- .0763581s elapsed
│ │ │ + -- .0377816s 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))
│ │ │ - -- .00167424s elapsed
    
│ │ │ + -- .00136606s elapsed
│ │ │              
    
│ │ │                
    i9 : X = kleinschmidt (5, {1,2,3});
    
│ │ │              
    
│ │ │                
    i11 : m3 = elapsedTime # monomials D3
│ │ │ - -- 40.8741s elapsed
│ │ │ + -- 32.4281s elapsed
│ │ │  
│ │ │  o11 = 7909
    
│ │ │              
    
│ │ │                
    i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3))
│ │ │ - -- .0293459s elapsed
    
│ │ │ + -- .0319871s 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
│ │ │ │ - -- .0534782s elapsed
│ │ │ │ + -- .035912s 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))
│ │ │ │ - -- .0017525s elapsed
│ │ │ │ + -- .0016787s 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
│ │ │ │ - -- .0763581s elapsed
│ │ │ │ + -- .0377816s 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))
│ │ │ │ - -- .00167424s elapsed
│ │ │ │ + -- .00136606s 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
│ │ │ │ - -- 40.8741s elapsed
│ │ │ │ + -- 32.4281s elapsed
│ │ │ │  
│ │ │ │  o11 = 7909
│ │ │ │  i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety
│ │ │ │  D3))
│ │ │ │ - -- .0293459s elapsed
│ │ │ │ + -- .0319871s 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
│ │ │ @@ -125,25 +125,25 @@
│ │ │          
    
│ │ │            
    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 2.2001e-05s (cpu); 1.9186e-05s (thread); 0s (gc)
│ │ │ │ + -- used 3.5654e-05s (cpu); 2.2289e-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 0.0417375s (cpu); 0.0417407s (thread); 0s (gc)
│ │ │ │ + -- used 0.0543238s (cpu); 0.0543363s (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
│ │ │ @@ -233,21 +233,21 @@
│ │ │                 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 0.0209352s (cpu); 0.020935s (thread); 0s (gc)
│ │ │ │ + -- used 0.0290511s (cpu); 0.0290518s (thread); 0s (gc)
│ │ │ │  i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ │ - -- used 0.00216116s (cpu); 0.00216187s (thread); 0s (gc)
│ │ │ │ + -- used 0.00327888s (cpu); 0.0032839s (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 .0884675 seconds
│ │ │ +     -- used .107765 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00923927 seconds
│ │ │ +     -- used .0280642 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00487238 seconds
│ │ │ +     -- used .00376363 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000642635 seconds
│ │ │ +     -- used .000495459 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 .00368858 seconds
│ │ │ +     -- used .00438314 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00273464 seconds
│ │ │ +     -- used .00385012 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00106818 seconds
│ │ │ +     -- used .00126874 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000315712 seconds
│ │ │ +     -- used .0003215 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 .0123198 seconds
│ │ │ +     -- used .014854 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0108776 seconds
│ │ │ +     -- used .0133614 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00719824 seconds
│ │ │ +     -- used .00772006 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000762781 seconds
│ │ │ +     -- used .000876456 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 .0026712 seconds
│ │ │ +     -- used .0035963 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00778446 seconds
│ │ │ +     -- used .0100255 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000791835 seconds
│ │ │ +     -- used .00106002 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 0.0641647s (cpu); 0.064162s (thread); 0s (gc)
│ │ │ + -- used 0.0752999s (cpu); 0.0752912s (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)
│ │ │ - -- .638062s elapsed
│ │ │ + -- .508339s elapsed
│ │ │  
│ │ │  o7 = p
│ │ │  
│ │ │  o7 : Point
│ │ │  
│ │ │  i8 : matrix pack(5, p#Coordinates)
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Convert__To__Cone.html
│ │ │ @@ -100,21 +100,21 @@
│ │ │  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 .0884675 seconds
│ │ │ │ +     -- used .107765 seconds
│ │ │ │  Creating interpolation matrix ...
│ │ │ │ -     -- used .00923927 seconds
│ │ │ │ +     -- used .0280642 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00487238 seconds
│ │ │ │ +     -- used .00376363 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │ -     -- used .000642635 seconds
│ │ │ │ +     -- used .000495459 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
│ │ │ @@ -102,21 +102,21 @@
│ │ │  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 .00368858 seconds
│ │ │ │ +     -- used .00438314 seconds
│ │ │ │  Creating interpolation matrix ...
│ │ │ │ -     -- used .00273464 seconds
│ │ │ │ +     -- used .00385012 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00106818 seconds
│ │ │ │ +     -- used .00126874 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │ -     -- used .000315712 seconds
│ │ │ │ +     -- used .0003215 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
│ │ │ @@ -107,21 +107,21 @@
│ │ │  o2 : Matrix R  <-- R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -147,19 +147,19 @@
│ │ │                
    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 .0123198 seconds
│ │ │ │ +     -- used .014854 seconds
│ │ │ │  Creating interpolation matrix ...
│ │ │ │ -     -- used .0108776 seconds
│ │ │ │ +     -- used .0133614 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00719824 seconds
│ │ │ │ +     -- used .00772006 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │ -     -- used .000762781 seconds
│ │ │ │ +     -- used .000876456 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 .0026712 seconds
│ │ │ │ +     -- used .0035963 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00778446 seconds
│ │ │ │ +     -- used .0100255 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │ -     -- used .000791835 seconds
│ │ │ │ +     -- used .00106002 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
│ │ │ @@ -140,15 +140,15 @@
│ │ │               1      70
│ │ │  o8 : Matrix R  <-- R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}})
│ │ │ - -- used 2.2001e-05s (cpu); 1.9186e-05s (thread); 0s (gc)
│ │ │ + -- used 3.5654e-05s (cpu); 2.2289e-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 0.0417375s (cpu); 0.0417407s (thread); 0s (gc)
    
│ │ │ + -- used 0.0543238s (cpu); 0.0543363s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2)
    
│ │ │              
    
│ │ │                
    i19 : X1 = time normalToricVariety convexHull (vertMatrix);
│ │ │ - -- used 0.0209352s (cpu); 0.020935s (thread); 0s (gc)
    
│ │ │ + -- used 0.0290511s (cpu); 0.0290518s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ - -- used 0.00216116s (cpu); 0.00216187s (thread); 0s (gc)
    
│ │ │ + -- used 0.00327888s (cpu); 0.0032839s (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 .0884675 seconds
│ │ │ +     -- used .107765 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00923927 seconds
│ │ │ +     -- used .0280642 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00487238 seconds
│ │ │ +     -- used .00376363 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000642635 seconds
│ │ │ +     -- used .000495459 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 .00368858 seconds
│ │ │ +     -- used .00438314 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00273464 seconds
│ │ │ +     -- used .00385012 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00106818 seconds
│ │ │ +     -- used .00126874 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000315712 seconds
│ │ │ +     -- used .0003215 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 .0123198 seconds
│ │ │ +     -- used .014854 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0108776 seconds
│ │ │ +     -- used .0133614 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00719824 seconds
│ │ │ +     -- used .00772006 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000762781 seconds
│ │ │ +     -- used .000876456 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 .0026712 seconds
│ │ │ +     -- used .0035963 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00778446 seconds
│ │ │ +     -- used .0100255 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000791835 seconds
│ │ │ +     -- used .00106002 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 0.0641647s (cpu); 0.064162s (thread); 0s (gc)
│ │ │ + -- used 0.0752999s (cpu); 0.0752912s (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 0.0641647s (cpu); 0.064162s (thread); 0s (gc)
│ │ │ │ + -- used 0.0752999s (cpu); 0.0752912s (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
│ │ │ @@ -132,15 +132,15 @@
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime p = realPoint(I, Iterations => 100)
│ │ │ - -- .638062s elapsed
│ │ │ + -- .508339s 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)
│ │ │ │ - -- .638062s elapsed
│ │ │ │ + -- .508339s elapsed
│ │ │ │  
│ │ │ │  o7 = p
│ │ │ │  
│ │ │ │  o7 : Point
│ │ │ │  i8 : matrix pack(5, p#Coordinates)
│ │ │ │  
│ │ │ │  o8 = | .722359  .289465  -.295808  .591752  -.454678 |
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/example-output/_set__Verbose__Level.out
│ │ │ @@ -52,92 +52,92 @@
│ │ │  
│ │ │  i4 : assert all(S,s->checkIncidenceSolution(s,SchPblm))
│ │ │  
│ │ │  i5 : setVerboseLevel 1; 
│ │ │  
│ │ │  i6 : S = solveSchubertProblem(SchPblm,2,4)
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00800386
│ │ │ +-- cpu time = .0119672
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00400121
│ │ │ +-- cpu time = .00403291
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00400194
│ │ │ +-- cpu time = .00414467
│ │ │  resolveNode reached node of no remaining conditions
│ │ │ --- time to make equations: .00479222
│ │ │ +-- time to make equations: .00406548
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0200437 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ --- time of performing one checker move: .01877
│ │ │ --- time of performing one checker move: .0033589
│ │ │ --- time of performing one checker move: .00399169
│ │ │ --- time to make equations: .00398021
│ │ │ + -- trackHomotopy time = .00807571 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ +-- time of performing one checker move: .01991
│ │ │ +-- time of performing one checker move: .00405521
│ │ │ +-- time of performing one checker move: 0
│ │ │ +-- time to make equations: .00799988
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0063706 sec. for [{1, 2, 3, 0}, {1, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .0199622
│ │ │ --- time to make equations: .00399873
│ │ │ + -- trackHomotopy time = .00781135 sec. for [{1, 2, 3, 0}, {1, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .0240002
│ │ │ +-- time to make equations: .00806599
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0665117 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .105096
│ │ │ --- time to make equations: .00399983
│ │ │ + -- trackHomotopy time = .0304754 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .12472
│ │ │ +-- time to make equations: .00799462
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0060564 sec. for [{2, 3, 1, 0}, {2, infinity, infinity, 1}]
│ │ │ --- time of performing one checker move: .0159996
│ │ │ --- time to make equations: .0996534
│ │ │ + -- trackHomotopy time = .00811143 sec. for [{2, 3, 1, 0}, {2, infinity, infinity, 1}]
│ │ │ +-- time of performing one checker move: .0240003
│ │ │ +-- time to make equations: .118783
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00690418 sec. for [{0, 1, 2, 3}, {infinity, 1, 2, infinity}]
│ │ │ --- time of performing one checker move: .107588
│ │ │ --- time to make equations: .011977
│ │ │ + -- trackHomotopy time = .00941138 sec. for [{0, 1, 2, 3}, {infinity, 1, 2, infinity}]
│ │ │ +-- time of performing one checker move: .134701
│ │ │ +-- time to make equations: .0159716
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0490889 sec. for [{0, 1, 3, 2}, {infinity, 1, infinity, 2}]
│ │ │ --- time of performing one checker move: .107635
│ │ │ --- time of performing one checker move: 0
│ │ │ + -- trackHomotopy time = .0311304 sec. for [{0, 1, 3, 2}, {infinity, 1, infinity, 2}]
│ │ │ +-- time of performing one checker move: .133681
│ │ │ +-- time of performing one checker move: .00400829
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time to make equations: .00799983
│ │ │ +-- time to make equations: .0159595
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0551627 sec. for [{1, 3, 2, 0}, {infinity, 3, infinity, 1}]
│ │ │ --- time of performing one checker move: .108513
│ │ │ + -- trackHomotopy time = .120965 sec. for [{1, 3, 2, 0}, {infinity, 3, infinity, 1}]
│ │ │ +-- time of performing one checker move: .23422
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00664616
│ │ │ +-- cpu time = .00805804
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = 0
│ │ │ +-- cpu time = .00397023
│ │ │  resolveNode reached node of no remaining conditions
│ │ │ --- time to make equations: .00400042
│ │ │ +-- time to make equations: .0120518
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00628713 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ --- time of performing one checker move: .0997514
│ │ │ --- time of performing one checker move: .00399816
│ │ │ --- time to make equations: .00400027
│ │ │ + -- trackHomotopy time = .00829677 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ +-- time of performing one checker move: .143704
│ │ │ +-- time of performing one checker move: 0
│ │ │ +-- time to make equations: .00398176
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00611061 sec. for [{0, 2, 3, 1}, {0, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .0159585
│ │ │ --- time of performing one checker move: .0923873
│ │ │ + -- trackHomotopy time = .00805339 sec. for [{0, 2, 3, 1}, {0, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .0239165
│ │ │ +-- time of performing one checker move: .107106
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .003972
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .00399838
│ │ │ +-- time of performing one checker move: .00400535
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .00400062
│ │ │ --- time to make equations: .0120026
│ │ │ +-- time of performing one checker move: 0
│ │ │ +-- time of performing one checker move: .00401506
│ │ │ +-- time to make equations: .0159531
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .07797 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 3}]
│ │ │ --- time of performing one checker move: .113179
│ │ │ --- time of performing one checker move: 0
│ │ │ + -- trackHomotopy time = .0341203 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 3}]
│ │ │ +-- time of performing one checker move: .141398
│ │ │ +-- time of performing one checker move: .00706141
│ │ │  
│ │ │  o6 = {| -1.65573-.600637ii .0201935+.0437095ii   |, | -.154703+.175591ii 
│ │ │        | -1.23037-1.66989ii -.0308057-.00120618ii |  | -.801221-.0354303ii
│ │ │        | 1.35971-.743988ii  -.0713133-.049047ii   |  | .325581-2.08048ii  
│ │ │        | -.397038-1.8974ii  .0102261-.024397ii    |  | -.475895-.209388ii 
│ │ │       ------------------------------------------------------------------------
│ │ │       .0376857+.0683239ii   |}
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/html/_set__Verbose__Level.html
│ │ │ @@ -147,92 +147,92 @@
│ │ │                
    i5 : setVerboseLevel 1; 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : S = solveSchubertProblem(SchPblm,2,4)
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00800386
│ │ │ +-- cpu time = .0119672
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00400121
│ │ │ +-- cpu time = .00403291
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00400194
│ │ │ +-- cpu time = .00414467
│ │ │  resolveNode reached node of no remaining conditions
│ │ │ --- time to make equations: .00479222
│ │ │ +-- time to make equations: .00406548
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0200437 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ --- time of performing one checker move: .01877
│ │ │ --- time of performing one checker move: .0033589
│ │ │ --- time of performing one checker move: .00399169
│ │ │ --- time to make equations: .00398021
│ │ │ + -- trackHomotopy time = .00807571 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ +-- time of performing one checker move: .01991
│ │ │ +-- time of performing one checker move: .00405521
│ │ │ +-- time of performing one checker move: 0
│ │ │ +-- time to make equations: .00799988
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0063706 sec. for [{1, 2, 3, 0}, {1, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .0199622
│ │ │ --- time to make equations: .00399873
│ │ │ + -- trackHomotopy time = .00781135 sec. for [{1, 2, 3, 0}, {1, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .0240002
│ │ │ +-- time to make equations: .00806599
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0665117 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .105096
│ │ │ --- time to make equations: .00399983
│ │ │ + -- trackHomotopy time = .0304754 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .12472
│ │ │ +-- time to make equations: .00799462
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0060564 sec. for [{2, 3, 1, 0}, {2, infinity, infinity, 1}]
│ │ │ --- time of performing one checker move: .0159996
│ │ │ --- time to make equations: .0996534
│ │ │ + -- trackHomotopy time = .00811143 sec. for [{2, 3, 1, 0}, {2, infinity, infinity, 1}]
│ │ │ +-- time of performing one checker move: .0240003
│ │ │ +-- time to make equations: .118783
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00690418 sec. for [{0, 1, 2, 3}, {infinity, 1, 2, infinity}]
│ │ │ --- time of performing one checker move: .107588
│ │ │ --- time to make equations: .011977
│ │ │ + -- trackHomotopy time = .00941138 sec. for [{0, 1, 2, 3}, {infinity, 1, 2, infinity}]
│ │ │ +-- time of performing one checker move: .134701
│ │ │ +-- time to make equations: .0159716
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0490889 sec. for [{0, 1, 3, 2}, {infinity, 1, infinity, 2}]
│ │ │ --- time of performing one checker move: .107635
│ │ │ --- time of performing one checker move: 0
│ │ │ + -- trackHomotopy time = .0311304 sec. for [{0, 1, 3, 2}, {infinity, 1, infinity, 2}]
│ │ │ +-- time of performing one checker move: .133681
│ │ │ +-- time of performing one checker move: .00400829
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time to make equations: .00799983
│ │ │ +-- time to make equations: .0159595
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0551627 sec. for [{1, 3, 2, 0}, {infinity, 3, infinity, 1}]
│ │ │ --- time of performing one checker move: .108513
│ │ │ + -- trackHomotopy time = .120965 sec. for [{1, 3, 2, 0}, {infinity, 3, infinity, 1}]
│ │ │ +-- time of performing one checker move: .23422
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00664616
│ │ │ +-- cpu time = .00805804
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = 0
│ │ │ +-- cpu time = .00397023
│ │ │  resolveNode reached node of no remaining conditions
│ │ │ --- time to make equations: .00400042
│ │ │ +-- time to make equations: .0120518
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00628713 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ --- time of performing one checker move: .0997514
│ │ │ --- time of performing one checker move: .00399816
│ │ │ --- time to make equations: .00400027
│ │ │ + -- trackHomotopy time = .00829677 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ +-- time of performing one checker move: .143704
│ │ │ +-- time of performing one checker move: 0
│ │ │ +-- time to make equations: .00398176
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00611061 sec. for [{0, 2, 3, 1}, {0, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .0159585
│ │ │ --- time of performing one checker move: .0923873
│ │ │ + -- trackHomotopy time = .00805339 sec. for [{0, 2, 3, 1}, {0, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .0239165
│ │ │ +-- time of performing one checker move: .107106
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .003972
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .00399838
│ │ │ +-- time of performing one checker move: .00400535
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .00400062
│ │ │ --- time to make equations: .0120026
│ │ │ +-- time of performing one checker move: 0
│ │ │ +-- time of performing one checker move: .00401506
│ │ │ +-- time to make equations: .0159531
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .07797 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 3}]
│ │ │ --- time of performing one checker move: .113179
│ │ │ --- time of performing one checker move: 0
│ │ │ + -- trackHomotopy time = .0341203 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 3}]
│ │ │ +-- time of performing one checker move: .141398
│ │ │ +-- time of performing one checker move: .00706141
│ │ │  
│ │ │  o6 = {| -1.65573-.600637ii .0201935+.0437095ii   |, | -.154703+.175591ii 
│ │ │        | -1.23037-1.66989ii -.0308057-.00120618ii |  | -.801221-.0354303ii
│ │ │        | 1.35971-.743988ii  -.0713133-.049047ii   |  | .325581-2.08048ii  
│ │ │        | -.397038-1.8974ii  .0102261-.024397ii    |  | -.475895-.209388ii 
│ │ │       ------------------------------------------------------------------------
│ │ │       .0376857+.0683239ii   |}
│ │ │ ├── html2text {}
│ │ │ │ @@ -65,102 +65,102 @@
│ │ │ │       -.0336427+.0141017ii  |
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : assert all(S,s->checkIncidenceSolution(s,SchPblm))
│ │ │ │  i5 : setVerboseLevel 1;
│ │ │ │  i6 : S = solveSchubertProblem(SchPblm,2,4)
│ │ │ │  -- playCheckers
│ │ │ │ --- cpu time = .00800386
│ │ │ │ +-- cpu time = .0119672
│ │ │ │  -- making a recursive call to resolveNode
│ │ │ │  -- playCheckers
│ │ │ │ --- cpu time = .00400121
│ │ │ │ +-- cpu time = .00403291
│ │ │ │  -- making a recursive call to resolveNode
│ │ │ │  -- playCheckers
│ │ │ │ --- cpu time = .00400194
│ │ │ │ +-- cpu time = .00414467
│ │ │ │  resolveNode reached node of no remaining conditions
│ │ │ │ --- time to make equations: .00479222
│ │ │ │ +-- time to make equations: .00406548
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0200437 sec. for [{0, 1, 2, 3}, {0, infinity, 2,
│ │ │ │ + -- trackHomotopy time = .00807571 sec. for [{0, 1, 2, 3}, {0, infinity, 2,
│ │ │ │  infinity}]
│ │ │ │ --- time of performing one checker move: .01877
│ │ │ │ --- time of performing one checker move: .0033589
│ │ │ │ --- time of performing one checker move: .00399169
│ │ │ │ --- time to make equations: .00398021
│ │ │ │ +-- time of performing one checker move: .01991
│ │ │ │ +-- time of performing one checker move: .00405521
│ │ │ │ +-- time of performing one checker move: 0
│ │ │ │ +-- time to make equations: .00799988
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0063706 sec. for [{1, 2, 3, 0}, {1, infinity,
│ │ │ │ + -- trackHomotopy time = .00781135 sec. for [{1, 2, 3, 0}, {1, infinity,
│ │ │ │  infinity, 2}]
│ │ │ │ --- time of performing one checker move: .0199622
│ │ │ │ --- time to make equations: .00399873
│ │ │ │ +-- time of performing one checker move: .0240002
│ │ │ │ +-- time to make equations: .00806599
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0665117 sec. for [{1, 3, 2, 0}, {1, infinity,
│ │ │ │ + -- trackHomotopy time = .0304754 sec. for [{1, 3, 2, 0}, {1, infinity,
│ │ │ │  infinity, 2}]
│ │ │ │ --- time of performing one checker move: .105096
│ │ │ │ --- time to make equations: .00399983
│ │ │ │ +-- time of performing one checker move: .12472
│ │ │ │ +-- time to make equations: .00799462
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0060564 sec. for [{2, 3, 1, 0}, {2, infinity,
│ │ │ │ + -- trackHomotopy time = .00811143 sec. for [{2, 3, 1, 0}, {2, infinity,
│ │ │ │  infinity, 1}]
│ │ │ │ --- time of performing one checker move: .0159996
│ │ │ │ --- time to make equations: .0996534
│ │ │ │ +-- time of performing one checker move: .0240003
│ │ │ │ +-- time to make equations: .118783
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00690418 sec. for [{0, 1, 2, 3}, {infinity, 1, 2,
│ │ │ │ + -- trackHomotopy time = .00941138 sec. for [{0, 1, 2, 3}, {infinity, 1, 2,
│ │ │ │  infinity}]
│ │ │ │ --- time of performing one checker move: .107588
│ │ │ │ --- time to make equations: .011977
│ │ │ │ +-- time of performing one checker move: .134701
│ │ │ │ +-- time to make equations: .0159716
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0490889 sec. for [{0, 1, 3, 2}, {infinity, 1,
│ │ │ │ + -- trackHomotopy time = .0311304 sec. for [{0, 1, 3, 2}, {infinity, 1,
│ │ │ │  infinity, 2}]
│ │ │ │ --- time of performing one checker move: .107635
│ │ │ │ --- time of performing one checker move: 0
│ │ │ │ +-- time of performing one checker move: .133681
│ │ │ │ +-- time of performing one checker move: .00400829
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ --- time to make equations: .00799983
│ │ │ │ +-- time to make equations: .0159595
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0551627 sec. for [{1, 3, 2, 0}, {infinity, 3,
│ │ │ │ + -- trackHomotopy time = .120965 sec. for [{1, 3, 2, 0}, {infinity, 3,
│ │ │ │  infinity, 1}]
│ │ │ │ --- time of performing one checker move: .108513
│ │ │ │ +-- time of performing one checker move: .23422
│ │ │ │  -- making a recursive call to resolveNode
│ │ │ │  -- playCheckers
│ │ │ │ --- cpu time = .00664616
│ │ │ │ +-- cpu time = .00805804
│ │ │ │  -- making a recursive call to resolveNode
│ │ │ │  -- playCheckers
│ │ │ │ --- cpu time = 0
│ │ │ │ +-- cpu time = .00397023
│ │ │ │  resolveNode reached node of no remaining conditions
│ │ │ │ --- time to make equations: .00400042
│ │ │ │ +-- time to make equations: .0120518
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00628713 sec. for [{0, 1, 2, 3}, {0, infinity, 2,
│ │ │ │ + -- trackHomotopy time = .00829677 sec. for [{0, 1, 2, 3}, {0, infinity, 2,
│ │ │ │  infinity}]
│ │ │ │ --- time of performing one checker move: .0997514
│ │ │ │ --- time of performing one checker move: .00399816
│ │ │ │ --- time to make equations: .00400027
│ │ │ │ +-- time of performing one checker move: .143704
│ │ │ │ +-- time of performing one checker move: 0
│ │ │ │ +-- time to make equations: .00398176
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00611061 sec. for [{0, 2, 3, 1}, {0, infinity,
│ │ │ │ + -- trackHomotopy time = .00805339 sec. for [{0, 2, 3, 1}, {0, infinity,
│ │ │ │  infinity, 2}]
│ │ │ │ --- time of performing one checker move: .0159585
│ │ │ │ --- time of performing one checker move: .0923873
│ │ │ │ +-- time of performing one checker move: .0239165
│ │ │ │ +-- time of performing one checker move: .107106
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ --- time of performing one checker move: .003972
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ --- time of performing one checker move: .00399838
│ │ │ │ +-- time of performing one checker move: .00400535
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ --- time of performing one checker move: .00400062
│ │ │ │ --- time to make equations: .0120026
│ │ │ │ +-- time of performing one checker move: 0
│ │ │ │ +-- time of performing one checker move: .00401506
│ │ │ │ +-- time to make equations: .0159531
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .07797 sec. for [{1, 3, 2, 0}, {1, infinity, infinity,
│ │ │ │ -3}]
│ │ │ │ --- time of performing one checker move: .113179
│ │ │ │ --- time of performing one checker move: 0
│ │ │ │ + -- trackHomotopy time = .0341203 sec. for [{1, 3, 2, 0}, {1, infinity,
│ │ │ │ +infinity, 3}]
│ │ │ │ +-- time of performing one checker move: .141398
│ │ │ │ +-- time of performing one checker move: .00706141
│ │ │ │  
│ │ │ │  o6 = {| -1.65573-.600637ii .0201935+.0437095ii   |, | -.154703+.175591ii
│ │ │ │        | -1.23037-1.66989ii -.0308057-.00120618ii |  | -.801221-.0354303ii
│ │ │ │        | 1.35971-.743988ii  -.0713133-.049047ii   |  | .325581-2.08048ii
│ │ │ │        | -.397038-1.8974ii  .0102261-.024397ii    |  | -.475895-.209388ii
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       .0376857+.0683239ii   |}
│ │ ├── ./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
│ │ │ - -- .146189s elapsed
│ │ │ + -- .135204s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .126375s elapsed
│ │ │ + -- .116809s elapsed
│ │ │  next gb
│ │ │ - -- .000828707s elapsed
│ │ │ + -- .000877162s elapsed
│ │ │  true
│ │ │  unfolding
│ │ │ - -- .190454s elapsed
│ │ │ + -- .145247s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .151259s elapsed
│ │ │ + -- .137816s elapsed
│ │ │  next gb
│ │ │ - -- .000705507s elapsed
│ │ │ + -- .00065349s elapsed
│ │ │  true
│ │ │ - -- 1.58898s elapsed
│ │ │ + -- 1.48229s elapsed
│ │ │  
│ │ │  o6 = {}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ - -- 1.5313s elapsed
│ │ │ + -- 1.31673s 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
│ │ │ - -- .207946s elapsed
│ │ │ + -- .2217s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .134913s elapsed
│ │ │ + -- .151615s elapsed
│ │ │  next gb
│ │ │ - -- .000914878s elapsed
│ │ │ + -- .00104685s elapsed
│ │ │  true
│ │ │ - -- .683163s elapsed
│ │ │ + -- .637281s elapsed
│ │ │  (5, 8,  all semigroups are smoothable)
│ │ │ - -- .716582s elapsed
│ │ │ + -- .676091s 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)
│ │ │ - -- .0484596s elapsed
│ │ │ + -- .0874803s elapsed
│ │ │  6
│ │ │  false
│ │ │  5
│ │ │  false
│ │ │  4
│ │ │  decompose
│ │ │ - -- .370099s elapsed
│ │ │ + -- .302279s 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))
│ │ │ - -- 3.62892s elapsed
│ │ │ + -- 3.38655s elapsed
│ │ │  
│ │ │  o4 = Tally{false => 6}
│ │ │             true => 4
│ │ │  
│ │ │  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)
│ │ │ - -- 1.05053s elapsed
│ │ │ + -- .931767s elapsed
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ - -- 4.34178s elapsed
│ │ │ + -- 3.82061s 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)
│ │ │ - -- 4.3542s elapsed
│ │ │ + -- 3.53196s 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)
│ │ │ - -- 1.35152s elapsed
│ │ │ + -- 1.23931s 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
│ │ │ - -- .514484s elapsed
│ │ │ + -- .439659s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .187751s elapsed
│ │ │ + -- .172486s elapsed
│ │ │  next gb
│ │ │ - -- .00166673s elapsed
│ │ │ + -- .00324603s elapsed
│ │ │  true
│ │ │ - -- 1.09437s elapsed
│ │ │ + -- .984376s elapsed
│ │ │  {6, 7, 9, 17}
│ │ │  unfolding
│ │ │ - -- .483431s elapsed
│ │ │ + -- .371182s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .204984s elapsed
│ │ │ + -- .190419s elapsed
│ │ │  next gb
│ │ │ - -- .0026011s elapsed
│ │ │ + -- .00272089s elapsed
│ │ │  decompose
│ │ │ - -- .13498s elapsed
│ │ │ + -- .161085s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │  {0, -1}
│ │ │ - -- 3.11814s elapsed
│ │ │ + -- 2.63937s elapsed
│ │ │  {6, 8, 9, 10}
│ │ │  unfolding
│ │ │ - -- .139467s elapsed
│ │ │ + -- .129154s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .154163s elapsed
│ │ │ + -- .124261s elapsed
│ │ │  next gb
│ │ │ - -- .000485587s elapsed
│ │ │ + -- .000537651s elapsed
│ │ │  true
│ │ │ - -- .687947s elapsed
│ │ │ + -- .6576s elapsed
│ │ │  {6, 8, 10, 11, 13}
│ │ │  unfolding
│ │ │ - -- .550391s elapsed
│ │ │ + -- .49442s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .233636s elapsed
│ │ │ + -- .219416s elapsed
│ │ │  next gb
│ │ │ - -- .00395566s elapsed
│ │ │ + -- .00584814s elapsed
│ │ │  decompose
│ │ │ - -- .975039s elapsed
│ │ │ + -- .851224s elapsed
│ │ │  number of components: 1
│ │ │  support c, codim c: {(5, 1)}
│ │ │  {-1}
│ │ │ - -- 2.73979s elapsed
│ │ │ - -- 7.64035s elapsed
│ │ │ + -- 2.55361s elapsed
│ │ │ + -- 6.83513s elapsed
│ │ │  0
│ │ │  
│ │ │  {}
│ │ │ - -- .000003787s elapsed
│ │ │ - -- 7.67511s elapsed
│ │ │ + -- .00000505s elapsed
│ │ │ + -- 6.88195s elapsed
│ │ │  
│ │ │  o3 = {{6, 8, 9, 11}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/___Lab__Book__Protocol.html
│ │ │ @@ -96,38 +96,38 @@
│ │ │  o5 = 2
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0,Verbose=>true))
│ │ │  unfolding
│ │ │ - -- .146189s elapsed
│ │ │ + -- .135204s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .126375s elapsed
│ │ │ + -- .116809s elapsed
│ │ │  next gb
│ │ │ - -- .000828707s elapsed
│ │ │ + -- .000877162s elapsed
│ │ │  true
│ │ │  unfolding
│ │ │ - -- .190454s elapsed
│ │ │ + -- .145247s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .151259s elapsed
│ │ │ + -- .137816s elapsed
│ │ │  next gb
│ │ │ - -- .000705507s elapsed
│ │ │ + -- .00065349s elapsed
│ │ │  true
│ │ │ - -- 1.58898s elapsed
│ │ │ + -- 1.48229s elapsed
│ │ │  
│ │ │  o6 = {}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ - -- 1.5313s elapsed
│ │ │ + -- 1.31673s elapsed
│ │ │  
│ │ │  o7 = {}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -184,23 +184,23 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true)
│ │ │  (13, 1)
│ │ │  {5, 8, 11, 12}
│ │ │  unfolding
│ │ │ - -- .207946s elapsed
│ │ │ + -- .2217s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .134913s elapsed
│ │ │ + -- .151615s elapsed
│ │ │  next gb
│ │ │ - -- .000914878s elapsed
│ │ │ + -- .00104685s elapsed
│ │ │  true
│ │ │ - -- .683163s elapsed
│ │ │ + -- .637281s elapsed
│ │ │  (5, 8,  all semigroups are smoothable)
│ │ │ - -- .716582s elapsed
│ │ │ + -- .676091s elapsed
│ │ │  
│ │ │  o11 = {}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ @@ -223,22 +223,22 @@
│ │ │  
│ │ │  o13 = 8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true)
│ │ │ - -- .0484596s elapsed
│ │ │ + -- .0874803s elapsed
│ │ │  6
│ │ │  false
│ │ │  5
│ │ │  false
│ │ │  4
│ │ │  decompose
│ │ │ - -- .370099s elapsed
│ │ │ + -- .302279s 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
│ │ │ │ - -- .146189s elapsed
│ │ │ │ + -- .135204s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .126375s elapsed
│ │ │ │ + -- .116809s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .000828707s elapsed
│ │ │ │ + -- .000877162s elapsed
│ │ │ │  true
│ │ │ │  unfolding
│ │ │ │ - -- .190454s elapsed
│ │ │ │ + -- .145247s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .151259s elapsed
│ │ │ │ + -- .137816s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .000705507s elapsed
│ │ │ │ + -- .00065349s elapsed
│ │ │ │  true
│ │ │ │ - -- 1.58898s elapsed
│ │ │ │ + -- 1.48229s elapsed
│ │ │ │  
│ │ │ │  o6 = {}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ │ - -- 1.5313s elapsed
│ │ │ │ + -- 1.31673s 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
│ │ │ │ - -- .207946s elapsed
│ │ │ │ + -- .2217s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .134913s elapsed
│ │ │ │ + -- .151615s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .000914878s elapsed
│ │ │ │ + -- .00104685s elapsed
│ │ │ │  true
│ │ │ │ - -- .683163s elapsed
│ │ │ │ + -- .637281s elapsed
│ │ │ │  (5, 8,  all semigroups are smoothable)
│ │ │ │ - -- .716582s elapsed
│ │ │ │ + -- .676091s 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)
│ │ │ │ - -- .0484596s elapsed
│ │ │ │ + -- .0874803s elapsed
│ │ │ │  6
│ │ │ │  false
│ │ │ │  5
│ │ │ │  false
│ │ │ │  4
│ │ │ │  decompose
│ │ │ │ - -- .370099s elapsed
│ │ │ │ + -- .302279s 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
│ │ │ @@ -94,15 +94,15 @@
│ │ │            
│ │ │              
│ │ │                
    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))
│ │ │ - -- 3.62892s elapsed
│ │ │ + -- 3.38655s elapsed
│ │ │  
│ │ │  o4 = Tally{false => 6}
│ │ │             true => 4
│ │ │  
│ │ │  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))
│ │ │ │ - -- 3.62892s elapsed
│ │ │ │ + -- 3.38655s elapsed
│ │ │ │  
│ │ │ │  o4 = Tally{false => 6}
│ │ │ │             true => 4
│ │ │ │  
│ │ │ │  o4 : Tally
│ │ │ │  ********** WWaayyss ttoo uussee hheeuurriissttiiccSSmmooootthhnneessss:: **********
│ │ │ │      * heuristicSmoothness(Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_is__Smoothable__Semigroup.html
│ │ │ @@ -95,23 +95,23 @@
│ │ │  
│ │ │  o2 = 8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime isSmoothableSemigroup(L,0.30,0)
│ │ │ - -- 1.05053s elapsed
│ │ │ + -- .931767s elapsed
│ │ │  
│ │ │  o3 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ - -- 4.34178s elapsed
│ │ │ + -- 3.82061s 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)
│ │ │ │ - -- 1.05053s elapsed
│ │ │ │ + -- .931767s elapsed
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ │ - -- 4.34178s elapsed
│ │ │ │ + -- 3.82061s 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
│ │ │ @@ -94,15 +94,15 @@
│ │ │  
│ │ │  o2 = 8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime isWeierstrassSemigroup(L,0.15)
│ │ │ - -- 4.3542s elapsed
│ │ │ + -- 3.53196s 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)
│ │ │ │ - -- 4.3542s elapsed
│ │ │ │ + -- 3.53196s 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
│ │ │ @@ -79,15 +79,15 @@
│ │ │            
    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.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -101,62 +101,62 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime nonWeierstrassSemigroups(6,7)
│ │ │  (6, 7,  all semigroups are smoothable)
│ │ │ - -- 1.35152s elapsed
│ │ │ + -- 1.23931s elapsed
│ │ │  
│ │ │  o1 = {}
│ │ │  
│ │ │  o1 : List
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)
│ │ │  (17, 5)
│ │ │  {6, 7, 8, 17}
│ │ │  unfolding
│ │ │ - -- .514484s elapsed
│ │ │ + -- .439659s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .187751s elapsed
│ │ │ + -- .172486s elapsed
│ │ │  next gb
│ │ │ - -- .00166673s elapsed
│ │ │ + -- .00324603s elapsed
│ │ │  true
│ │ │ - -- 1.09437s elapsed
│ │ │ + -- .984376s elapsed
│ │ │  {6, 7, 9, 17}
│ │ │  unfolding
│ │ │ - -- .483431s elapsed
│ │ │ + -- .371182s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .204984s elapsed
│ │ │ + -- .190419s elapsed
│ │ │  next gb
│ │ │ - -- .0026011s elapsed
│ │ │ + -- .00272089s elapsed
│ │ │  decompose
│ │ │ - -- .13498s elapsed
│ │ │ + -- .161085s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │  {0, -1}
│ │ │ - -- 3.11814s elapsed
│ │ │ + -- 2.63937s elapsed
│ │ │  {6, 8, 9, 10}
│ │ │  unfolding
│ │ │ - -- .139467s elapsed
│ │ │ + -- .129154s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .154163s elapsed
│ │ │ + -- .124261s elapsed
│ │ │  next gb
│ │ │ - -- .000485587s elapsed
│ │ │ + -- .000537651s elapsed
│ │ │  true
│ │ │ - -- .687947s elapsed
│ │ │ + -- .6576s elapsed
│ │ │  {6, 8, 10, 11, 13}
│ │ │  unfolding
│ │ │ - -- .550391s elapsed
│ │ │ + -- .49442s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .233636s elapsed
│ │ │ + -- .219416s elapsed
│ │ │  next gb
│ │ │ - -- .00395566s elapsed
│ │ │ + -- .00584814s elapsed
│ │ │  decompose
│ │ │ - -- .975039s elapsed
│ │ │ + -- .851224s elapsed
│ │ │  number of components: 1
│ │ │  support c, codim c: {(5, 1)}
│ │ │  {-1}
│ │ │ - -- 2.73979s elapsed
│ │ │ - -- 7.64035s elapsed
│ │ │ + -- 2.55361s elapsed
│ │ │ + -- 6.83513s elapsed
│ │ │  0
│ │ │  
│ │ │  {}
│ │ │ - -- .000003787s elapsed
│ │ │ - -- 7.67511s elapsed
│ │ │ + -- .00000505s elapsed
│ │ │ + -- 6.88195s 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)
│ │ │ │ - -- 1.35152s elapsed
│ │ │ │ + -- 1.23931s 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
│ │ │ │ - -- .514484s elapsed
│ │ │ │ + -- .439659s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .187751s elapsed
│ │ │ │ + -- .172486s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .00166673s elapsed
│ │ │ │ + -- .00324603s elapsed
│ │ │ │  true
│ │ │ │ - -- 1.09437s elapsed
│ │ │ │ + -- .984376s elapsed
│ │ │ │  {6, 7, 9, 17}
│ │ │ │  unfolding
│ │ │ │ - -- .483431s elapsed
│ │ │ │ + -- .371182s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .204984s elapsed
│ │ │ │ + -- .190419s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .0026011s elapsed
│ │ │ │ + -- .00272089s elapsed
│ │ │ │  decompose
│ │ │ │ - -- .13498s elapsed
│ │ │ │ + -- .161085s elapsed
│ │ │ │  number of components: 2
│ │ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │ │  {0, -1}
│ │ │ │ - -- 3.11814s elapsed
│ │ │ │ + -- 2.63937s elapsed
│ │ │ │  {6, 8, 9, 10}
│ │ │ │  unfolding
│ │ │ │ - -- .139467s elapsed
│ │ │ │ + -- .129154s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .154163s elapsed
│ │ │ │ + -- .124261s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .000485587s elapsed
│ │ │ │ + -- .000537651s elapsed
│ │ │ │  true
│ │ │ │ - -- .687947s elapsed
│ │ │ │ + -- .6576s elapsed
│ │ │ │  {6, 8, 10, 11, 13}
│ │ │ │  unfolding
│ │ │ │ - -- .550391s elapsed
│ │ │ │ + -- .49442s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .233636s elapsed
│ │ │ │ + -- .219416s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .00395566s elapsed
│ │ │ │ + -- .00584814s elapsed
│ │ │ │  decompose
│ │ │ │ - -- .975039s elapsed
│ │ │ │ + -- .851224s elapsed
│ │ │ │  number of components: 1
│ │ │ │  support c, codim c: {(5, 1)}
│ │ │ │  {-1}
│ │ │ │ - -- 2.73979s elapsed
│ │ │ │ - -- 7.64035s elapsed
│ │ │ │ + -- 2.55361s elapsed
│ │ │ │ + -- 6.83513s elapsed
│ │ │ │  0
│ │ │ │  
│ │ │ │  {}
│ │ │ │ - -- .000003787s elapsed
│ │ │ │ - -- 7.67511s elapsed
│ │ │ │ + -- .00000505s elapsed
│ │ │ │ + -- 6.88195s 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/___Free__O__I__Module__Map.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);
│ │ │  
│ │ │  i5 : C = oiRes({b}, 2)
│ │ │  
│ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │       1: (e1, {5, 5}, {-3, -4})
│ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-2, -5, -4, -3, -5, -4, -5, -3, -4})
│ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-5, -5, -5, -2, -4, -4, -3, -3, -4})
│ │ │  
│ │ │  o5 : OIResolution
│ │ │  
│ │ │  i6 : phi = C.dd_1
│ │ │  
│ │ │  o6 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module__Map_sp__Vector__In__Width.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);
│ │ │  
│ │ │  i5 : C = oiRes({b}, 2)
│ │ │  
│ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │       1: (e1, {5, 5}, {-4, -3})
│ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -5, -4, -3, -2, -4, -5, -5})
│ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -4, -5, -4, -5, -5, -3, -2})
│ │ │  
│ │ │  o5 : OIResolution
│ │ │  
│ │ │  i6 : phi = C.dd_1
│ │ │  
│ │ │  o6 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ ├── ./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 0.0819263s (cpu); 0.0819238s (thread); 0s (gc)
│ │ │ + -- used 0.101015s (cpu); 0.101015s (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 0.215335s (cpu); 0.118692s (thread); 0s (gc)
│ │ │ + -- used 0.237782s (cpu); 0.136776s (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 0.476094s (cpu); 0.290539s (thread); 0s (gc)
│ │ │ + -- used 0.565735s (cpu); 0.348073s (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 0.0881844s (cpu); 0.0881837s (thread); 0s (gc)
│ │ │ + -- used 0.263534s (cpu); 0.142568s (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__Free__O__I__Module__Map_rp.out
│ │ │ @@ -10,21 +10,18 @@
│ │ │  
│ │ │  i5 : C = oiRes({b}, 2);
│ │ │  
│ │ │  i6 : phi = C.dd_1;
│ │ │  
│ │ │  i7 : describe phi
│ │ │  
│ │ │ -o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ │ -     Basis element images: {-x   e0              + x   e0              +
│ │ │ -                              2,2  5,{1, 3, 5},1    2,2  5,{1, 3, 4},1  
│ │ │ +o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ │ +     Basis element images: {x   x   e0              - x   x   e0             
│ │ │ +                             2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     x   e0              - x   e0             , x   x   e0              -
│ │ │ -      2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1   2,3 1,1  5,{2, 4, 5},1  
│ │ │ +     - x   x   e0              + x   x   e0             , -x   e0        
│ │ │ +        2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3, 5},1    2,2  5,{1, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     x   x   e0              - x   x   e0              + x   x   e0        
│ │ │ -      2,4 1,1  5,{2, 3, 5},1    2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -         }
│ │ │ -     5},1
│ │ │ +          + x   e0              + x   e0              - x   e0             }
│ │ │ +     5},1    2,2  5,{1, 3, 4},1    2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1
│ │ │  
│ │ │  i8 :
│ │ ├── ./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 0.0801109s (cpu); 0.0801106s (thread); 0s (gc)
│ │ │ + -- used 0.105841s (cpu); 0.105611s (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/_get__Schreyer__Map.out
│ │ │ @@ -17,21 +17,21 @@
│ │ │        2,3 2,1 1,2 3,{1},2
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : G' = oiSyz(G, d)
│ │ │  
│ │ │  o6 = {x   d           - x   d           + 1d             , x   d             
│ │ │ -       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
│ │ │ +       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   1,2 4,{1, 3, 4},2
│ │ │       ------------------------------------------------------------------------
│ │ │ -     - x   d             , x   d              - x   d              -
│ │ │ -        2,3 4,{1, 2, 4},2   1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2  
│ │ │ +     - x   d              - x   d             , x   d              -
│ │ │ +        1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2   2,4 4,{1, 2, 3},2  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   d             }
│ │ │ -      1,3 4,{1, 2, 4},2
│ │ │ +      2,3 4,{1, 2, 4},2
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : H = getFreeOIModule G'#0
│ │ │  
│ │ │  o7 = Basis symbol: d
│ │ │       Basis element widths: {2, 3}
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_image_lp__Free__O__I__Module__Map_rp.out
│ │ │ @@ -10,19 +10,19 @@
│ │ │  
│ │ │  i5 : C = oiRes({b}, 2);
│ │ │  
│ │ │  i6 : phi = C.dd_1;
│ │ │  
│ │ │  i7 : image phi
│ │ │  
│ │ │ -o7 = {x   x   e0              - x   x   e0              - x   x   e0        
│ │ │ -       2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1    2,3 1,2  5,{1, 4,
│ │ │ +o7 = {-x   e0              + x   e0              + x   e0              -
│ │ │ +        2,2  5,{1, 3, 5},1    2,2  5,{1, 3, 4},1    2,3  5,{1, 2, 5},1  
│ │ │       ------------------------------------------------------------------------
│ │ │ -          + x   x   e0             , -x   e0              + x   e0        
│ │ │ -     5},1    2,4 1,2  5,{1, 3, 5},1    2,2  5,{1, 3, 5},1    2,2  5,{1, 3,
│ │ │ +     x   e0             , x   x   e0              - x   x   e0              -
│ │ │ +      2,3  5,{1, 2, 4},1   2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1  
│ │ │       ------------------------------------------------------------------------
│ │ │ -          + x   e0              - x   e0             }
│ │ │ -     4},1    2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1
│ │ │ +     x   x   e0              + x   x   e0             }
│ │ │ +      2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3, 5},1
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 :
│ │ ├── ./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 0.309561s (cpu); 0.253624s (thread); 0s (gc)
│ │ │ + -- used 0.405651s (cpu); 0.300532s (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 0.0233297s (cpu); 0.0233295s (thread); 0s (gc)
│ │ │ + -- used 0.0334577s (cpu); 0.0334591s (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 0.0253589s (cpu); 0.0253584s (thread); 0s (gc)
│ │ │ + -- used 0.035814s (cpu); 0.0358119s (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
│ │ │  
│ │ │ @@ -41,18 +41,17 @@
│ │ │             - x   x   e          }
│ │ │        3},3    2,1 1,2 3,{1, 3},3
│ │ │  
│ │ │  o13 : List
│ │ │  
│ │ │  i14 : minimizeOIGB C -- an element gets removed
│ │ │  
│ │ │ -                                                                        
│ │ │ -o14 = {x   x   e        + x   x   e          , x   x   x   e           -
│ │ │ -        1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3   2,3 2,2 1,1 3,{2, 3},3  
│ │ │ +                                                                   2
│ │ │ +o14 = {x   e        + x   e       , x   x   x   e           - x   x   e     
│ │ │ +        1,1 1,{1},1    2,1 1,{1},2   2,3 2,2 1,1 3,{2, 3},3    2,1 1,2 3,{1,
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2
│ │ │ -      x   x   e          , x   e        + x   e       }
│ │ │ -       2,1 1,2 3,{1, 3},3   1,1 1,{1},1    2,1 1,{1},2
│ │ │ +          , x   x   e        + x   x   e          }
│ │ │ +      3},3   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │  
│ │ │  o14 : List
│ │ │  
│ │ │  i15 :
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__Free__O__I__Module__Map_rp.out
│ │ │ @@ -10,10 +10,10 @@
│ │ │  
│ │ │  i5 : C = oiRes({b}, 2);
│ │ │  
│ │ │  i6 : phi = C.dd_1;
│ │ │  
│ │ │  i7 : net phi
│ │ │  
│ │ │ -o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ │ +o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ │  
│ │ │  i8 :
│ │ ├── ./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 0.218164s (cpu); 0.115767s (thread); 0s (gc)
│ │ │ + -- used 0.260039s (cpu); 0.136496s (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 0.0270916s (cpu); 0.0270906s (thread); 0s (gc)
│ │ │ + -- used 0.0363376s (cpu); 0.0363385s (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 0.425811s (cpu); 0.2725s (thread); 0s (gc)
│ │ │ + -- used 0.586193s (cpu); 0.367513s (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/_oi__Syz.out
│ │ │ @@ -17,18 +17,18 @@
│ │ │        2,3 2,1 1,2 3,{1},2
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : oiSyz(G, d)
│ │ │  
│ │ │  o6 = {x   d           - x   d           + 1d             , x   d             
│ │ │ -       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
│ │ │ +       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   1,2 4,{1, 3, 4},2
│ │ │       ------------------------------------------------------------------------
│ │ │ -     - x   d             , x   d              - x   d              -
│ │ │ -        2,3 4,{1, 2, 4},2   1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2  
│ │ │ +     - x   d              - x   d             , x   d              -
│ │ │ +        1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2   2,4 4,{1, 2, 3},2  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   d             }
│ │ │ -      1,3 4,{1, 2, 4},2
│ │ │ +      2,3 4,{1, 2, 4},2
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 :
│ │ ├── ./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 0.127003s (cpu); 0.127006s (thread); 0s (gc)
│ │ │ + -- used 0.220711s (cpu); 0.154099s (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/___Free__O__I__Module__Map.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : C = oiRes({b}, 2)
│ │ │  
│ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │       1: (e1, {5, 5}, {-3, -4})
│ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-2, -5, -4, -3, -5, -4, -5, -3, -4})
│ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-5, -5, -5, -2, -4, -4, -3, -3, -4})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : phi = C.dd_1
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 3);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);
│ │ │ │  i5 : C = oiRes({b}, 2)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │ │       1: (e1, {5, 5}, {-3, -4})
│ │ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-2, -5, -4, -3, -5, -4, -5, -3, -4})
│ │ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-5, -5, -5, -2, -4, -4, -3, -3, -4})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  i6 : phi = C.dd_1
│ │ │ │  
│ │ │ │  o6 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ │ │  
│ │ │ │  o6 : FreeOIModuleMap
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Free__O__I__Module__Map_sp__Vector__In__Width.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : C = oiRes({b}, 2)
│ │ │  
│ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │       1: (e1, {5, 5}, {-4, -3})
│ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -5, -4, -3, -2, -4, -5, -5})
│ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -4, -5, -4, -5, -5, -3, -2})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : phi = C.dd_1
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 3);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);
│ │ │ │  i5 : C = oiRes({b}, 2)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │ │       1: (e1, {5, 5}, {-4, -3})
│ │ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -5, -4, -3, -2, -4, -5, -5})
│ │ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -4, -5, -4, -5, -5, -3, -2})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  i6 : phi = C.dd_1
│ │ │ │  
│ │ │ │  o6 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ │ │  
│ │ │ │  o6 : FreeOIModuleMap
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Resolution.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │              
│ │ │                
    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 0.0819263s (cpu); 0.0819238s (thread); 0s (gc)
│ │ │ + -- used 0.101015s (cpu); 0.101015s (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 0.0819263s (cpu); 0.0819238s (thread); 0s (gc)
│ │ │ │ + -- used 0.101015s (cpu); 0.101015s (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
│ │ │ @@ -92,15 +92,15 @@
│ │ │              
│ │ │                
    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 0.215335s (cpu); 0.118692s (thread); 0s (gc)
    
│ │ │ + -- used 0.237782s (cpu); 0.136776s (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 0.215335s (cpu); 0.118692s (thread); 0s (gc)
│ │ │ │ + -- used 0.237782s (cpu); 0.136776s (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
│ │ │ @@ -74,15 +74,15 @@
│ │ │              
│ │ │                
    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 0.476094s (cpu); 0.290539s (thread); 0s (gc)
│ │ │ + -- used 0.565735s (cpu); 0.348073s (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 0.476094s (cpu); 0.290539s (thread); 0s (gc)
│ │ │ │ + -- used 0.565735s (cpu); 0.348073s (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
│ │ │ @@ -90,15 +90,15 @@
│ │ │              
│ │ │                
    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 0.0881844s (cpu); 0.0881837s (thread); 0s (gc)
    
│ │ │ + -- used 0.263534s (cpu); 0.142568s (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 0.0881844s (cpu); 0.0881837s (thread); 0s (gc)
│ │ │ │ + -- used 0.263534s (cpu); 0.142568s (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__Free__O__I__Module__Map_rp.html
│ │ │ @@ -102,26 +102,23 @@
│ │ │                
    i6 : phi = C.dd_1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : describe phi
│ │ │  
│ │ │ -o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ │ -     Basis element images: {-x   e0              + x   e0              +
│ │ │ -                              2,2  5,{1, 3, 5},1    2,2  5,{1, 3, 4},1  
│ │ │ +o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ │ +     Basis element images: {x   x   e0              - x   x   e0             
│ │ │ +                             2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     x   e0              - x   e0             , x   x   e0              -
│ │ │ -      2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1   2,3 1,1  5,{2, 4, 5},1  
│ │ │ +     - x   x   e0              + x   x   e0             , -x   e0        
│ │ │ +        2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3, 5},1    2,2  5,{1, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     x   x   e0              - x   x   e0              + x   x   e0        
│ │ │ -      2,4 1,1  5,{2, 3, 5},1    2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -         }
│ │ │ -     5},1
    
│ │ │ +          + x   e0              + x   e0              - x   e0             }
│ │ │ +     5},1    2,2  5,{1, 3, 4},1    2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    Ways to use this method:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,24 +18,21 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 3);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);
│ │ │ │  i5 : C = oiRes({b}, 2);
│ │ │ │  i6 : phi = C.dd_1;
│ │ │ │  i7 : describe phi
│ │ │ │  
│ │ │ │ -o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ │ │ -     Basis element images: {-x   e0              + x   e0              +
│ │ │ │ -                              2,2  5,{1, 3, 5},1    2,2  5,{1, 3, 4},1
│ │ │ │ +o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ │ │ +     Basis element images: {x   x   e0              - x   x   e0
│ │ │ │ +                             2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     x   e0              - x   e0             , x   x   e0              -
│ │ │ │ -      2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1   2,3 1,1  5,{2, 4, 5},1
│ │ │ │ +     - x   x   e0              + x   x   e0             , -x   e0
│ │ │ │ +        2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3, 5},1    2,2  5,{1, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     x   x   e0              - x   x   e0              + x   x   e0
│ │ │ │ -      2,4 1,1  5,{2, 3, 5},1    2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3,
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -         }
│ │ │ │ -     5},1
│ │ │ │ +          + x   e0              + x   e0              - x   e0             }
│ │ │ │ +     5},1    2,2  5,{1, 3, 4},1    2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _d_e_s_c_r_i_b_e_(_F_r_e_e_O_I_M_o_d_u_l_e_M_a_p_) -- display a free OI-module map
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/OIGroebnerBases.m2:1979:0.
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe_lp__O__I__Resolution_rp.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │              
│ │ │                
    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 0.0801109s (cpu); 0.0801106s (thread); 0s (gc)
    
│ │ │ + -- used 0.105841s (cpu); 0.105611s (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 0.0801109s (cpu); 0.0801106s (thread); 0s (gc)
│ │ │ │ + -- used 0.105841s (cpu); 0.105611s (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/_get__Schreyer__Map.html
│ │ │ @@ -105,21 +105,21 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : G' = oiSyz(G, d)
│ │ │  
│ │ │  o6 = {x   d           - x   d           + 1d             , x   d             
│ │ │ -       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
│ │ │ +       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   1,2 4,{1, 3, 4},2
│ │ │       ------------------------------------------------------------------------
│ │ │ -     - x   d             , x   d              - x   d              -
│ │ │ -        2,3 4,{1, 2, 4},2   1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2  
│ │ │ +     - x   d              - x   d             , x   d              -
│ │ │ +        1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2   2,4 4,{1, 2, 3},2  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   d             }
│ │ │ -      1,3 4,{1, 2, 4},2
│ │ │ +      2,3 4,{1, 2, 4},2
│ │ │  
│ │ │  o6 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : H = getFreeOIModule G'#0
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,21 +29,21 @@
│ │ │ │       x   x   x   e       }
│ │ │ │        2,3 2,1 1,2 3,{1},2
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : G' = oiSyz(G, d)
│ │ │ │  
│ │ │ │  o6 = {x   d           - x   d           + 1d             , x   d
│ │ │ │ -       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
│ │ │ │ +       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   1,2 4,{1, 3, 4},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     - x   d             , x   d              - x   d              -
│ │ │ │ -        2,3 4,{1, 2, 4},2   1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2
│ │ │ │ +     - x   d              - x   d             , x   d              -
│ │ │ │ +        1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2   2,4 4,{1, 2, 3},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x   d             }
│ │ │ │ -      1,3 4,{1, 2, 4},2
│ │ │ │ +      2,3 4,{1, 2, 4},2
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : H = getFreeOIModule G'#0
│ │ │ │  
│ │ │ │  o7 = Basis symbol: d
│ │ │ │       Basis element widths: {2, 3}
│ │ │ │       Degree shifts: {-2, -3}
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_image_lp__Free__O__I__Module__Map_rp.html
│ │ │ @@ -102,22 +102,22 @@
│ │ │                
    i6 : phi = C.dd_1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : image phi
│ │ │  
│ │ │ -o7 = {x   x   e0              - x   x   e0              - x   x   e0        
│ │ │ -       2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1    2,3 1,2  5,{1, 4,
│ │ │ +o7 = {-x   e0              + x   e0              + x   e0              -
│ │ │ +        2,2  5,{1, 3, 5},1    2,2  5,{1, 3, 4},1    2,3  5,{1, 2, 5},1  
│ │ │       ------------------------------------------------------------------------
│ │ │ -          + x   x   e0             , -x   e0              + x   e0        
│ │ │ -     5},1    2,4 1,2  5,{1, 3, 5},1    2,2  5,{1, 3, 5},1    2,2  5,{1, 3,
│ │ │ +     x   e0             , x   x   e0              - x   x   e0              -
│ │ │ +      2,3  5,{1, 2, 4},1   2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1  
│ │ │       ------------------------------------------------------------------------
│ │ │ -          + x   e0              - x   e0             }
│ │ │ -     4},1    2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1
│ │ │ +     x   x   e0              + x   x   e0             }
│ │ │ +      2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3, 5},1
│ │ │  
│ │ │  o7 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,22 +18,22 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 3);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);
│ │ │ │  i5 : C = oiRes({b}, 2);
│ │ │ │  i6 : phi = C.dd_1;
│ │ │ │  i7 : image phi
│ │ │ │  
│ │ │ │ -o7 = {x   x   e0              - x   x   e0              - x   x   e0
│ │ │ │ -       2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1    2,3 1,2  5,{1, 4,
│ │ │ │ +o7 = {-x   e0              + x   e0              + x   e0              -
│ │ │ │ +        2,2  5,{1, 3, 5},1    2,2  5,{1, 3, 4},1    2,3  5,{1, 2, 5},1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -          + x   x   e0             , -x   e0              + x   e0
│ │ │ │ -     5},1    2,4 1,2  5,{1, 3, 5},1    2,2  5,{1, 3, 5},1    2,2  5,{1, 3,
│ │ │ │ +     x   e0             , x   x   e0              - x   x   e0              -
│ │ │ │ +      2,3  5,{1, 2, 4},1   2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -          + x   e0              - x   e0             }
│ │ │ │ -     4},1    2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1
│ │ │ │ +     x   x   e0              + x   x   e0             }
│ │ │ │ +      2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3, 5},1
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _i_m_a_g_e_(_F_r_e_e_O_I_M_o_d_u_l_e_M_a_p_) -- get the basis element images of a free OI-
│ │ │ │        module map
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__Complex.html
│ │ │ @@ -94,15 +94,15 @@
│ │ │              
│ │ │                
    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 0.309561s (cpu); 0.253624s (thread); 0s (gc)
│ │ │ + -- used 0.405651s (cpu); 0.300532s (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 0.309561s (cpu); 0.253624s (thread); 0s (gc)
│ │ │ │ + -- used 0.405651s (cpu); 0.300532s (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
│ │ │ @@ -116,15 +116,15 @@
│ │ │  
│ │ │  o10 = false    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.0233297s (cpu); 0.0233295s (thread); 0s (gc)
│ │ │ + -- used 0.0334577s (cpu); 0.0334591s (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 0.0233297s (cpu); 0.0233295s (thread); 0s (gc)
│ │ │ │ + -- used 0.0334577s (cpu); 0.0334591s (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
│ │ │ @@ -109,15 +109,15 @@
│ │ │              
│ │ │                
    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 0.0253589s (cpu); 0.0253584s (thread); 0s (gc)
│ │ │ + -- used 0.035814s (cpu); 0.0358119s (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
│ │ │  
│ │ │ @@ -148,21 +148,20 @@
│ │ │  o13 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : minimizeOIGB C -- an element gets removed
│ │ │  
│ │ │ -                                                                        
│ │ │ -o14 = {x   x   e        + x   x   e          , x   x   x   e           -
│ │ │ -        1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3   2,3 2,2 1,1 3,{2, 3},3  
│ │ │ +                                                                   2
│ │ │ +o14 = {x   e        + x   e       , x   x   x   e           - x   x   e     
│ │ │ +        1,1 1,{1},1    2,1 1,{1},2   2,3 2,2 1,1 3,{2, 3},3    2,1 1,2 3,{1,
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2
│ │ │ -      x   x   e          , x   e        + x   e       }
│ │ │ -       2,1 1,2 3,{1, 3},3   1,1 1,{1},1    2,1 1,{1},2
│ │ │ +          , x   x   e        + x   x   e          }
│ │ │ +      3},3   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │  
│ │ │  o14 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── 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 0.0253589s (cpu); 0.0253584s (thread); 0s (gc)
│ │ │ │ + -- used 0.035814s (cpu); 0.0358119s (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
│ │ │ │  
│ │ │ │ @@ -49,21 +49,20 @@
│ │ │ │                    2
│ │ │ │             - x   x   e          }
│ │ │ │        3},3    2,1 1,2 3,{1, 3},3
│ │ │ │  
│ │ │ │  o13 : List
│ │ │ │  i14 : minimizeOIGB C -- an element gets removed
│ │ │ │  
│ │ │ │ -
│ │ │ │ -o14 = {x   x   e        + x   x   e          , x   x   x   e           -
│ │ │ │ -        1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3   2,3 2,2 1,1 3,{2, 3},3
│ │ │ │ +                                                                   2
│ │ │ │ +o14 = {x   e        + x   e       , x   x   x   e           - x   x   e
│ │ │ │ +        1,1 1,{1},1    2,1 1,{1},2   2,3 2,2 1,1 3,{2, 3},3    2,1 1,2 3,{1,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -           2
│ │ │ │ -      x   x   e          , x   e        + x   e       }
│ │ │ │ -       2,1 1,2 3,{1, 3},3   1,1 1,{1},1    2,1 1,{1},2
│ │ │ │ +          , x   x   e        + x   x   e          }
│ │ │ │ +      3},3   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │  
│ │ │ │  o14 : List
│ │ │ │  ********** WWaayyss ttoo uussee mmiinniimmiizzeeOOIIGGBB:: **********
│ │ │ │      * minimizeOIGB(List)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _m_i_n_i_m_i_z_e_O_I_G_B 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/OIGroebnerBases/html/_net_lp__Free__O__I__Module__Map_rp.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │                
    i6 : phi = C.dd_1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : net phi
│ │ │  
│ │ │ -o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
    
│ │ │ +o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    Ways to use this method:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,13 +18,13 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 3);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);
│ │ │ │  i5 : C = oiRes({b}, 2);
│ │ │ │  i6 : phi = C.dd_1;
│ │ │ │  i7 : net phi
│ │ │ │  
│ │ │ │ -o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ │ │ +o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _n_e_t_(_F_r_e_e_O_I_M_o_d_u_l_e_M_a_p_) -- display a free OI-module map source and target
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/OIGroebnerBases.m2:1931:0.
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__O__I__Resolution_rp.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │              
│ │ │                
    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 0.218164s (cpu); 0.115767s (thread); 0s (gc)
    
│ │ │ + -- used 0.260039s (cpu); 0.136496s (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 0.218164s (cpu); 0.115767s (thread); 0s (gc)
│ │ │ │ + -- used 0.260039s (cpu); 0.136496s (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
│ │ │ @@ -111,15 +111,15 @@
│ │ │              
│ │ │                
    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 0.0270916s (cpu); 0.0270906s (thread); 0s (gc)
│ │ │ + -- used 0.0363376s (cpu); 0.0363385s (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 {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  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 0.0270916s (cpu); 0.0270906s (thread); 0s (gc)
│ │ │ │ + -- used 0.0363376s (cpu); 0.0363385s (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
│ │ │ @@ -106,15 +106,15 @@
│ │ │              
│ │ │                
    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 0.425811s (cpu); 0.2725s (thread); 0s (gc)
│ │ │ + -- used 0.586193s (cpu); 0.367513s (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 0.425811s (cpu); 0.2725s (thread); 0s (gc)
│ │ │ │ + -- used 0.586193s (cpu); 0.367513s (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/_oi__Syz.html
│ │ │ @@ -119,21 +119,21 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : oiSyz(G, d)
│ │ │  
│ │ │  o6 = {x   d           - x   d           + 1d             , x   d             
│ │ │ -       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
│ │ │ +       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   1,2 4,{1, 3, 4},2
│ │ │       ------------------------------------------------------------------------
│ │ │ -     - x   d             , x   d              - x   d              -
│ │ │ -        2,3 4,{1, 2, 4},2   1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2  
│ │ │ +     - x   d              - x   d             , x   d              -
│ │ │ +        1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2   2,4 4,{1, 2, 3},2  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   d             }
│ │ │ -      1,3 4,{1, 2, 4},2
│ │ │ +      2,3 4,{1, 2, 4},2
│ │ │  
│ │ │  o6 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    References:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,21 +48,21 @@
│ │ │ │       x   x   x   e       }
│ │ │ │        2,3 2,1 1,2 3,{1},2
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : oiSyz(G, d)
│ │ │ │  
│ │ │ │  o6 = {x   d           - x   d           + 1d             , x   d
│ │ │ │ -       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
│ │ │ │ +       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   1,2 4,{1, 3, 4},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     - x   d             , x   d              - x   d              -
│ │ │ │ -        2,3 4,{1, 2, 4},2   1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2
│ │ │ │ +     - x   d              - x   d             , x   d              -
│ │ │ │ +        1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2   2,4 4,{1, 2, 3},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x   d             }
│ │ │ │ -      1,3 4,{1, 2, 4},2
│ │ │ │ +      2,3 4,{1, 2, 4},2
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  RReeffeerreenncceess::
│ │ │ │  [1] M. Morrow and U. Nagel, Computing Gröbner Bases and Free Resolutions of
│ │ │ │  OI-Modules, Preprint, arXiv:2303.06725, 2023.
│ │ │ │  ********** WWaayyss ttoo uussee ooiiSSyyzz:: **********
│ │ │ │      * oiSyz(List,Symbol)
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_reduce__O__I__G__B.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │              
│ │ │                
    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 0.127003s (cpu); 0.127006s (thread); 0s (gc)
│ │ │ + -- used 0.220711s (cpu); 0.154099s (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 0.127003s (cpu); 0.127006s (thread); 0s (gc)
│ │ │ │ + -- used 0.220711s (cpu); 0.154099s (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)
│ │ │ - -- 2.28836s elapsed
│ │ │ + -- 2.58568s 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_*)
│ │ │ - -- 1.58344s elapsed
│ │ │ + -- 1.58181s 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)
│ │ │ - -- 1.96896s elapsed
│ │ │ + -- 2.71421s 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
│ │ │ @@ -36,19 +36,22 @@
│ │ │            << res M << endl << endl;
│ │ │            break;
│ │ │            ) else (
│ │ │            << "-- computation interrupted" << endl;
│ │ │            status M.cache.resolution;
│ │ │            << "-- continuing the computation" << endl;
│ │ │            ))
│ │ │ - -- used 0.912289s (cpu); 0.749709s (thread); 0s (gc)
│ │ │ - -- used 0.493728s (cpu); 0.422643s (thread); 0s (gc)
│ │ │ + -- used 1.05262s (cpu); 0.977552s (thread); 0s (gc)
│ │ │ + -- used 1.07033s (cpu); 0.986918s (thread); 0s (gc)
│ │ │ + -- used 0.261263s (cpu); 0.172135s (thread); 0s (gc)
│ │ │  -- computation started: 
│ │ │  -- computation interrupted
│ │ │  -- continuing the computation
│ │ │ +-- computation interrupted
│ │ │ +-- continuing the computation
│ │ │  -- computation complete
│ │ │   4      11      89      122      40
│ │ │  R  <-- R   <-- R   <-- R    <-- R   <-- 0
│ │ │                                           
│ │ │  0      1       2       3        4       5
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Fast__Nonminimal.html
│ │ │ @@ -89,28 +89,28 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 2.28836s elapsed
│ │ │ + -- 2.58568s 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_*)
│ │ │ - -- 1.58344s elapsed
│ │ │ + -- 1.58181s 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)
│ │ │ │ - -- 2.28836s elapsed
│ │ │ │ + -- 2.58568s 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_*)
│ │ │ │ - -- 1.58344s elapsed
│ │ │ │ + -- 1.58181s 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
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 1.96896s elapsed
│ │ │ + -- 2.71421s 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)
│ │ │ │ - -- 1.96896s elapsed
│ │ │ │ + -- 2.71421s 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
│ │ │ @@ -112,19 +112,22 @@
│ │ │            << res M << endl << endl;
│ │ │            break;
│ │ │            ) else (
│ │ │            << "-- computation interrupted" << endl;
│ │ │            status M.cache.resolution;
│ │ │            << "-- continuing the computation" << endl;
│ │ │            ))
│ │ │ - -- used 0.912289s (cpu); 0.749709s (thread); 0s (gc)
│ │ │ - -- used 0.493728s (cpu); 0.422643s (thread); 0s (gc)
│ │ │ + -- used 1.05262s (cpu); 0.977552s (thread); 0s (gc)
│ │ │ + -- used 1.07033s (cpu); 0.986918s (thread); 0s (gc)
│ │ │ + -- used 0.261263s (cpu); 0.172135s (thread); 0s (gc)
│ │ │  -- computation started: 
│ │ │  -- computation interrupted
│ │ │  -- continuing the computation
│ │ │ +-- computation interrupted
│ │ │ +-- continuing the computation
│ │ │  -- computation complete
│ │ │   4      11      89      122      40
│ │ │  R  <-- R   <-- R   <-- R    <-- R   <-- 0
│ │ │                                           
│ │ │  0      1       2       3        4       5
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,19 +50,22 @@
│ │ │ │            << res M << endl << endl;
│ │ │ │            break;
│ │ │ │            ) else (
│ │ │ │            << "-- computation interrupted" << endl;
│ │ │ │            status M.cache.resolution;
│ │ │ │            << "-- continuing the computation" << endl;
│ │ │ │            ))
│ │ │ │ - -- used 0.912289s (cpu); 0.749709s (thread); 0s (gc)
│ │ │ │ - -- used 0.493728s (cpu); 0.422643s (thread); 0s (gc)
│ │ │ │ + -- used 1.05262s (cpu); 0.977552s (thread); 0s (gc)
│ │ │ │ + -- used 1.07033s (cpu); 0.986918s (thread); 0s (gc)
│ │ │ │ + -- used 0.261263s (cpu); 0.172135s (thread); 0s (gc)
│ │ │ │  -- computation started:
│ │ │ │  -- computation interrupted
│ │ │ │  -- continuing the computation
│ │ │ │ +-- computation interrupted
│ │ │ │ +-- continuing the computation
│ │ │ │  -- computation complete
│ │ │ │   4      11      89      122      40
│ │ │ │  R  <-- R   <-- R   <-- R    <-- R   <-- 0
│ │ │ │  
│ │ │ │  0      1       2       3        4       5
│ │ │ │  If the user has a chain complex in hand that is known to be a projective
│ │ │ │  resolution of M, then it can be installed with M.cache.resolution = C.
│ │ ├── ./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}
│ │ │            );
│ │ │ - -- .280854s elapsed
│ │ │ - -- .263174s elapsed
│ │ │ - -- .501439s elapsed
│ │ │ - -- .231638s elapsed
│ │ │ - -- .342923s elapsed
│ │ │ - -- .34852s elapsed
│ │ │ - -- .619954s elapsed
│ │ │ - -- .456734s elapsed
│ │ │ - -- .648967s elapsed
│ │ │ - -- .367209s elapsed
│ │ │ - -- .250328s elapsed
│ │ │ + -- .260044s elapsed
│ │ │ + -- .308781s elapsed
│ │ │ + -- .495904s elapsed
│ │ │ + -- .263177s elapsed
│ │ │ + -- .320061s elapsed
│ │ │ + -- .327781s elapsed
│ │ │ + -- .619504s elapsed
│ │ │ + -- .492191s elapsed
│ │ │ + -- .525097s elapsed
│ │ │ + -- .32743s elapsed
│ │ │ + -- .194628s 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}
│ │ │            );
│ │ │ - -- .536601s elapsed
│ │ │ - -- .816725s elapsed
│ │ │ - -- 1.00632s elapsed
│ │ │ - -- 1.37164s elapsed
│ │ │ - -- .718355s elapsed
│ │ │ - -- 1.00236s elapsed
│ │ │ - -- 1.18305s elapsed
│ │ │ - -- 1.31195s elapsed
│ │ │ - -- .727806s elapsed
│ │ │ - -- .774958s elapsed
│ │ │ - -- .333808s elapsed
│ │ │ - -- .5035s elapsed
│ │ │ - -- .490585s elapsed
│ │ │ - -- .637754s elapsed
│ │ │ - -- .880246s elapsed
│ │ │ - -- 1.2462s elapsed
│ │ │ - -- .940742s elapsed
│ │ │ - -- 1.01084s elapsed
│ │ │ - -- 1.44951s elapsed
│ │ │ - -- 1.31812s elapsed
│ │ │ - -- 1.00162s elapsed
│ │ │ - -- 1.3528s elapsed
│ │ │ - -- 1.90187s elapsed
│ │ │ - -- 1.44103s elapsed
│ │ │ - -- .432654s elapsed
│ │ │ - -- .573037s elapsed
│ │ │ - -- 1.40316s elapsed
│ │ │ - -- .640916s elapsed
│ │ │ - -- .945512s elapsed
│ │ │ - -- .952893s elapsed
│ │ │ - -- 1.08722s elapsed
│ │ │ - -- .825525s elapsed
│ │ │ - -- .672893s elapsed
│ │ │ - -- 1.13429s elapsed
│ │ │ - -- .81054s elapsed
│ │ │ - -- 1.14908s elapsed
│ │ │ - -- 1.49209s elapsed
│ │ │ - -- 1.09431s elapsed
│ │ │ - -- 1.18828s elapsed
│ │ │ - -- .875916s elapsed
│ │ │ - -- .6899s elapsed
│ │ │ - -- 1.08621s elapsed
│ │ │ - -- 1.59117s elapsed
│ │ │ - -- 1.88666s elapsed
│ │ │ - -- 1.39064s elapsed
│ │ │ - -- 1.01948s elapsed
│ │ │ - -- 1.47672s elapsed
│ │ │ - -- 1.19724s elapsed
│ │ │ - -- .910329s elapsed
│ │ │ - -- 1.02771s elapsed
│ │ │ - -- 1.06676s elapsed
│ │ │ - -- .986144s elapsed
│ │ │ - -- .739995s elapsed
│ │ │ - -- 1.05189s elapsed
│ │ │ - -- .696168s elapsed
│ │ │ - -- 1.4284s elapsed
│ │ │ - -- 1.25123s elapsed
│ │ │ - -- 1.4209s elapsed
│ │ │ - -- .786654s elapsed
│ │ │ - -- .485612s elapsed
│ │ │ - -- .40177s elapsed
│ │ │ + -- .464157s elapsed
│ │ │ + -- .463272s elapsed
│ │ │ + -- .88882s elapsed
│ │ │ + -- 1.2018s elapsed
│ │ │ + -- .719556s elapsed
│ │ │ + -- .932679s elapsed
│ │ │ + -- 1.10411s elapsed
│ │ │ + -- 1.1805s elapsed
│ │ │ + -- .765364s elapsed
│ │ │ + -- .74465s elapsed
│ │ │ + -- .439888s elapsed
│ │ │ + -- .453828s elapsed
│ │ │ + -- .487523s elapsed
│ │ │ + -- .639348s elapsed
│ │ │ + -- .919797s elapsed
│ │ │ + -- 1.25323s elapsed
│ │ │ + -- 1.09525s elapsed
│ │ │ + -- .932327s elapsed
│ │ │ + -- 1.31797s elapsed
│ │ │ + -- 1.00979s elapsed
│ │ │ + -- .80103s elapsed
│ │ │ + -- 1.0123s elapsed
│ │ │ + -- 1.5049s elapsed
│ │ │ + -- 1.27823s elapsed
│ │ │ + -- .495556s elapsed
│ │ │ + -- .68628s elapsed
│ │ │ + -- 1.3505s elapsed
│ │ │ + -- .673301s elapsed
│ │ │ + -- .577608s elapsed
│ │ │ + -- .878936s elapsed
│ │ │ + -- 1.14729s elapsed
│ │ │ + -- .842864s elapsed
│ │ │ + -- .567596s elapsed
│ │ │ + -- 1.03209s elapsed
│ │ │ + -- .884374s elapsed
│ │ │ + -- 1.0549s elapsed
│ │ │ + -- .983969s elapsed
│ │ │ + -- 1.21315s elapsed
│ │ │ + -- 1.23522s elapsed
│ │ │ + -- .748005s elapsed
│ │ │ + -- .642546s elapsed
│ │ │ + -- 1.09486s elapsed
│ │ │ + -- 1.38895s elapsed
│ │ │ + -- 1.7561s elapsed
│ │ │ + -- 1.19787s elapsed
│ │ │ + -- 1.13349s elapsed
│ │ │ + -- 1.4072s elapsed
│ │ │ + -- 1.21471s elapsed
│ │ │ + -- .999531s elapsed
│ │ │ + -- 1.14572s elapsed
│ │ │ + -- 1.10688s elapsed
│ │ │ + -- .839171s elapsed
│ │ │ + -- .874115s elapsed
│ │ │ + -- 1.02676s elapsed
│ │ │ + -- .666072s elapsed
│ │ │ + -- 1.15554s elapsed
│ │ │ + -- 1.19189s elapsed
│ │ │ + -- 1.29931s elapsed
│ │ │ + -- .766725s elapsed
│ │ │ + -- .504907s elapsed
│ │ │ + -- .434476s 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
│ │ │ - -- .662s elapsed
│ │ │ + -- 1s 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|
│ │ │  +---+---+---+---+
│ │ │  |288|216|144|72 |
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │ - -- .721s elapsed
│ │ │ + -- 1.05s elapsed
│ │ │  
│ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {.309, -.809,
│ │ │       ------------------------------------------------------------------------
│ │ │       -.809, .309, -.951, -.588, .588, .951}, {1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  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: {37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 50, 52, 53, 54, 55, 59, 60, 64, 65, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87, 88, 89}
│ │ │ - -- .763s elapsed
│ │ │ + -- .815s elapsed
│ │ │  warning: some solutions are not regular: {43, 44, 47, 49, 50, 51, 52, 53, 55, 57, 59, 61, 62, 63, 64, 65, 66, 68, 72, 74, 76, 77, 78, 79, 80, 84, 85, 87, 88, 89, 90, 91, 97}
│ │ │ - -- .659s elapsed
│ │ │ - -- .88s elapsed
│ │ │ - -- 1.02s elapsed
│ │ │ + -- .788s elapsed
│ │ │ + -- 1.03s elapsed
│ │ │ + -- 1.26s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │                                                  1 => {3, 4}
│ │ │                                                  2 => {0, 4}
│ │ │                                                  3 => {0, 1}
│ │ │                                                  4 => {2, 1}
│ │ │  +-----+----+----+-----+-----+-----+-----+-----+
│ │ │  |-.809|.309|.309|-.809|.588 |-.951|.951 |-.588|
│ │ │ @@ -69,20 +69,20 @@
│ │ │  +---+---+---+---+
│ │ │  |144|288|72 |216|
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |216|72 |288|144|
│ │ │  +---+---+---+---+
│ │ │ - -- 1.3s elapsed
│ │ │ - -- 1.31s elapsed
│ │ │ + -- 1.47s elapsed
│ │ │ + -- 1.57s elapsed
│ │ │  warning: some solutions are not regular: {27, 31, 32, 34, 37, 38, 44, 46, 47, 53, 54, 56, 57, 59, 60}
│ │ │ - -- 1.68s elapsed
│ │ │ + -- 1.91s elapsed
│ │ │  warning: some solutions are not regular: {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 34}
│ │ │ - -- 1.32s elapsed
│ │ │ - -- 1.4s elapsed
│ │ │ + -- 1.65s elapsed
│ │ │ + -- 1.67s elapsed
│ │ │  warning: some solutions are not regular: {26, 29, 30, 32, 33}
│ │ │ - -- 1.79s elapsed
│ │ │ + -- 1.71s elapsed
│ │ │  warning: some solutions are not regular: {38, 40, 42, 53, 54, 55, 62, 63, 67, 72, 77, 78}
│ │ │ - -- 1.59s elapsed
│ │ │ + -- 1.56s elapsed
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Linearly__Stable__Solutions.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 1729328129346969841
│ │ │  
│ │ │  i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}});
│ │ │  
│ │ │  i2 : getLinearlyStableSolutions(G)
│ │ │  -- warning: experimental computation over inexact field begun
│ │ │  --          results not reliable (one warning given per session)
│ │ │ - -- .250068s elapsed
│ │ │ + -- .279032s elapsed
│ │ │  warning: some solutions are not regular: {4, 5, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}
│ │ │  
│ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/_show__Exotic__Solutions.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │             4 => {0, 3}
│ │ │  
│ │ │  o1 : Graph
│ │ │  
│ │ │  i2 : showExoticSolutions G
│ │ │  -- warning: experimental computation over inexact field begun
│ │ │  --          results not reliable (one warning given per session)
│ │ │ - -- .943746s elapsed
│ │ │ + -- 1.09886s 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|
│ │ │ @@ -50,14 +50,14 @@
│ │ │             2 => {1, 3, 4}
│ │ │             3 => {2, 4}
│ │ │             4 => {0, 2, 3}
│ │ │  
│ │ │  o3 : Graph
│ │ │  
│ │ │  i4 : showExoticSolutions G
│ │ │ - -- 1.24035s elapsed
│ │ │ + -- 1.43774s 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
│ │ │ @@ -295,25 +295,25 @@
│ │ │            
│ │ │              
│ │ │                
    i16 : for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .280854s elapsed
│ │ │ - -- .263174s elapsed
│ │ │ - -- .501439s elapsed
│ │ │ - -- .231638s elapsed
│ │ │ - -- .342923s elapsed
│ │ │ - -- .34852s elapsed
│ │ │ - -- .619954s elapsed
│ │ │ - -- .456734s elapsed
│ │ │ - -- .648967s elapsed
│ │ │ - -- .367209s elapsed
│ │ │ - -- .250328s elapsed
    
│ │ │ + -- .260044s elapsed
│ │ │ + -- .308781s elapsed
│ │ │ + -- .495904s elapsed
│ │ │ + -- .263177s elapsed
│ │ │ + -- .320061s elapsed
│ │ │ + -- .327781s elapsed
│ │ │ + -- .619504s elapsed
│ │ │ + -- .492191s elapsed
│ │ │ + -- .525097s elapsed
│ │ │ + -- .32743s elapsed
│ │ │ + -- .194628s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : netList oo
│ │ │  
│ │ │        +---------------+---------------+
│ │ │ @@ -380,75 +380,75 @@
│ │ │            
│ │ │              
│ │ │                
    i23 : allcomps = for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .536601s elapsed
│ │ │ - -- .816725s elapsed
│ │ │ - -- 1.00632s elapsed
│ │ │ - -- 1.37164s elapsed
│ │ │ - -- .718355s elapsed
│ │ │ - -- 1.00236s elapsed
│ │ │ - -- 1.18305s elapsed
│ │ │ - -- 1.31195s elapsed
│ │ │ - -- .727806s elapsed
│ │ │ - -- .774958s elapsed
│ │ │ - -- .333808s elapsed
│ │ │ - -- .5035s elapsed
│ │ │ - -- .490585s elapsed
│ │ │ - -- .637754s elapsed
│ │ │ - -- .880246s elapsed
│ │ │ - -- 1.2462s elapsed
│ │ │ - -- .940742s elapsed
│ │ │ - -- 1.01084s elapsed
│ │ │ - -- 1.44951s elapsed
│ │ │ - -- 1.31812s elapsed
│ │ │ - -- 1.00162s elapsed
│ │ │ - -- 1.3528s elapsed
│ │ │ - -- 1.90187s elapsed
│ │ │ - -- 1.44103s elapsed
│ │ │ - -- .432654s elapsed
│ │ │ - -- .573037s elapsed
│ │ │ - -- 1.40316s elapsed
│ │ │ - -- .640916s elapsed
│ │ │ - -- .945512s elapsed
│ │ │ - -- .952893s elapsed
│ │ │ - -- 1.08722s elapsed
│ │ │ - -- .825525s elapsed
│ │ │ - -- .672893s elapsed
│ │ │ - -- 1.13429s elapsed
│ │ │ - -- .81054s elapsed
│ │ │ - -- 1.14908s elapsed
│ │ │ - -- 1.49209s elapsed
│ │ │ - -- 1.09431s elapsed
│ │ │ - -- 1.18828s elapsed
│ │ │ - -- .875916s elapsed
│ │ │ - -- .6899s elapsed
│ │ │ - -- 1.08621s elapsed
│ │ │ - -- 1.59117s elapsed
│ │ │ - -- 1.88666s elapsed
│ │ │ - -- 1.39064s elapsed
│ │ │ - -- 1.01948s elapsed
│ │ │ - -- 1.47672s elapsed
│ │ │ - -- 1.19724s elapsed
│ │ │ - -- .910329s elapsed
│ │ │ - -- 1.02771s elapsed
│ │ │ - -- 1.06676s elapsed
│ │ │ - -- .986144s elapsed
│ │ │ - -- .739995s elapsed
│ │ │ - -- 1.05189s elapsed
│ │ │ - -- .696168s elapsed
│ │ │ - -- 1.4284s elapsed
│ │ │ - -- 1.25123s elapsed
│ │ │ - -- 1.4209s elapsed
│ │ │ - -- .786654s elapsed
│ │ │ - -- .485612s elapsed
│ │ │ - -- .40177s elapsed
    
│ │ │ + -- .464157s elapsed
│ │ │ + -- .463272s elapsed
│ │ │ + -- .88882s elapsed
│ │ │ + -- 1.2018s elapsed
│ │ │ + -- .719556s elapsed
│ │ │ + -- .932679s elapsed
│ │ │ + -- 1.10411s elapsed
│ │ │ + -- 1.1805s elapsed
│ │ │ + -- .765364s elapsed
│ │ │ + -- .74465s elapsed
│ │ │ + -- .439888s elapsed
│ │ │ + -- .453828s elapsed
│ │ │ + -- .487523s elapsed
│ │ │ + -- .639348s elapsed
│ │ │ + -- .919797s elapsed
│ │ │ + -- 1.25323s elapsed
│ │ │ + -- 1.09525s elapsed
│ │ │ + -- .932327s elapsed
│ │ │ + -- 1.31797s elapsed
│ │ │ + -- 1.00979s elapsed
│ │ │ + -- .80103s elapsed
│ │ │ + -- 1.0123s elapsed
│ │ │ + -- 1.5049s elapsed
│ │ │ + -- 1.27823s elapsed
│ │ │ + -- .495556s elapsed
│ │ │ + -- .68628s elapsed
│ │ │ + -- 1.3505s elapsed
│ │ │ + -- .673301s elapsed
│ │ │ + -- .577608s elapsed
│ │ │ + -- .878936s elapsed
│ │ │ + -- 1.14729s elapsed
│ │ │ + -- .842864s elapsed
│ │ │ + -- .567596s elapsed
│ │ │ + -- 1.03209s elapsed
│ │ │ + -- .884374s elapsed
│ │ │ + -- 1.0549s elapsed
│ │ │ + -- .983969s elapsed
│ │ │ + -- 1.21315s elapsed
│ │ │ + -- 1.23522s elapsed
│ │ │ + -- .748005s elapsed
│ │ │ + -- .642546s elapsed
│ │ │ + -- 1.09486s elapsed
│ │ │ + -- 1.38895s elapsed
│ │ │ + -- 1.7561s elapsed
│ │ │ + -- 1.19787s elapsed
│ │ │ + -- 1.13349s elapsed
│ │ │ + -- 1.4072s elapsed
│ │ │ + -- 1.21471s elapsed
│ │ │ + -- .999531s elapsed
│ │ │ + -- 1.14572s elapsed
│ │ │ + -- 1.10688s elapsed
│ │ │ + -- .839171s elapsed
│ │ │ + -- .874115s elapsed
│ │ │ + -- 1.02676s elapsed
│ │ │ + -- .666072s elapsed
│ │ │ + -- 1.15554s elapsed
│ │ │ + -- 1.19189s elapsed
│ │ │ + -- 1.29931s elapsed
│ │ │ + -- .766725s elapsed
│ │ │ + -- .504907s elapsed
│ │ │ + -- .434476s 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}
│ │ │ │            );
│ │ │ │ - -- .280854s elapsed
│ │ │ │ - -- .263174s elapsed
│ │ │ │ - -- .501439s elapsed
│ │ │ │ - -- .231638s elapsed
│ │ │ │ - -- .342923s elapsed
│ │ │ │ - -- .34852s elapsed
│ │ │ │ - -- .619954s elapsed
│ │ │ │ - -- .456734s elapsed
│ │ │ │ - -- .648967s elapsed
│ │ │ │ - -- .367209s elapsed
│ │ │ │ - -- .250328s elapsed
│ │ │ │ + -- .260044s elapsed
│ │ │ │ + -- .308781s elapsed
│ │ │ │ + -- .495904s elapsed
│ │ │ │ + -- .263177s elapsed
│ │ │ │ + -- .320061s elapsed
│ │ │ │ + -- .327781s elapsed
│ │ │ │ + -- .619504s elapsed
│ │ │ │ + -- .492191s elapsed
│ │ │ │ + -- .525097s elapsed
│ │ │ │ + -- .32743s elapsed
│ │ │ │ + -- .194628s 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}
│ │ │ │            );
│ │ │ │ - -- .536601s elapsed
│ │ │ │ - -- .816725s elapsed
│ │ │ │ - -- 1.00632s elapsed
│ │ │ │ - -- 1.37164s elapsed
│ │ │ │ - -- .718355s elapsed
│ │ │ │ - -- 1.00236s elapsed
│ │ │ │ - -- 1.18305s elapsed
│ │ │ │ - -- 1.31195s elapsed
│ │ │ │ - -- .727806s elapsed
│ │ │ │ - -- .774958s elapsed
│ │ │ │ - -- .333808s elapsed
│ │ │ │ - -- .5035s elapsed
│ │ │ │ - -- .490585s elapsed
│ │ │ │ - -- .637754s elapsed
│ │ │ │ - -- .880246s elapsed
│ │ │ │ - -- 1.2462s elapsed
│ │ │ │ - -- .940742s elapsed
│ │ │ │ - -- 1.01084s elapsed
│ │ │ │ - -- 1.44951s elapsed
│ │ │ │ - -- 1.31812s elapsed
│ │ │ │ - -- 1.00162s elapsed
│ │ │ │ - -- 1.3528s elapsed
│ │ │ │ - -- 1.90187s elapsed
│ │ │ │ - -- 1.44103s elapsed
│ │ │ │ - -- .432654s elapsed
│ │ │ │ - -- .573037s elapsed
│ │ │ │ - -- 1.40316s elapsed
│ │ │ │ - -- .640916s elapsed
│ │ │ │ - -- .945512s elapsed
│ │ │ │ - -- .952893s elapsed
│ │ │ │ - -- 1.08722s elapsed
│ │ │ │ - -- .825525s elapsed
│ │ │ │ - -- .672893s elapsed
│ │ │ │ - -- 1.13429s elapsed
│ │ │ │ - -- .81054s elapsed
│ │ │ │ - -- 1.14908s elapsed
│ │ │ │ - -- 1.49209s elapsed
│ │ │ │ - -- 1.09431s elapsed
│ │ │ │ - -- 1.18828s elapsed
│ │ │ │ - -- .875916s elapsed
│ │ │ │ - -- .6899s elapsed
│ │ │ │ - -- 1.08621s elapsed
│ │ │ │ - -- 1.59117s elapsed
│ │ │ │ - -- 1.88666s elapsed
│ │ │ │ - -- 1.39064s elapsed
│ │ │ │ - -- 1.01948s elapsed
│ │ │ │ - -- 1.47672s elapsed
│ │ │ │ - -- 1.19724s elapsed
│ │ │ │ - -- .910329s elapsed
│ │ │ │ - -- 1.02771s elapsed
│ │ │ │ - -- 1.06676s elapsed
│ │ │ │ - -- .986144s elapsed
│ │ │ │ - -- .739995s elapsed
│ │ │ │ - -- 1.05189s elapsed
│ │ │ │ - -- .696168s elapsed
│ │ │ │ - -- 1.4284s elapsed
│ │ │ │ - -- 1.25123s elapsed
│ │ │ │ - -- 1.4209s elapsed
│ │ │ │ - -- .786654s elapsed
│ │ │ │ - -- .485612s elapsed
│ │ │ │ - -- .40177s elapsed
│ │ │ │ + -- .464157s elapsed
│ │ │ │ + -- .463272s elapsed
│ │ │ │ + -- .88882s elapsed
│ │ │ │ + -- 1.2018s elapsed
│ │ │ │ + -- .719556s elapsed
│ │ │ │ + -- .932679s elapsed
│ │ │ │ + -- 1.10411s elapsed
│ │ │ │ + -- 1.1805s elapsed
│ │ │ │ + -- .765364s elapsed
│ │ │ │ + -- .74465s elapsed
│ │ │ │ + -- .439888s elapsed
│ │ │ │ + -- .453828s elapsed
│ │ │ │ + -- .487523s elapsed
│ │ │ │ + -- .639348s elapsed
│ │ │ │ + -- .919797s elapsed
│ │ │ │ + -- 1.25323s elapsed
│ │ │ │ + -- 1.09525s elapsed
│ │ │ │ + -- .932327s elapsed
│ │ │ │ + -- 1.31797s elapsed
│ │ │ │ + -- 1.00979s elapsed
│ │ │ │ + -- .80103s elapsed
│ │ │ │ + -- 1.0123s elapsed
│ │ │ │ + -- 1.5049s elapsed
│ │ │ │ + -- 1.27823s elapsed
│ │ │ │ + -- .495556s elapsed
│ │ │ │ + -- .68628s elapsed
│ │ │ │ + -- 1.3505s elapsed
│ │ │ │ + -- .673301s elapsed
│ │ │ │ + -- .577608s elapsed
│ │ │ │ + -- .878936s elapsed
│ │ │ │ + -- 1.14729s elapsed
│ │ │ │ + -- .842864s elapsed
│ │ │ │ + -- .567596s elapsed
│ │ │ │ + -- 1.03209s elapsed
│ │ │ │ + -- .884374s elapsed
│ │ │ │ + -- 1.0549s elapsed
│ │ │ │ + -- .983969s elapsed
│ │ │ │ + -- 1.21315s elapsed
│ │ │ │ + -- 1.23522s elapsed
│ │ │ │ + -- .748005s elapsed
│ │ │ │ + -- .642546s elapsed
│ │ │ │ + -- 1.09486s elapsed
│ │ │ │ + -- 1.38895s elapsed
│ │ │ │ + -- 1.7561s elapsed
│ │ │ │ + -- 1.19787s elapsed
│ │ │ │ + -- 1.13349s elapsed
│ │ │ │ + -- 1.4072s elapsed
│ │ │ │ + -- 1.21471s elapsed
│ │ │ │ + -- .999531s elapsed
│ │ │ │ + -- 1.14572s elapsed
│ │ │ │ + -- 1.10688s elapsed
│ │ │ │ + -- .839171s elapsed
│ │ │ │ + -- .874115s elapsed
│ │ │ │ + -- 1.02676s elapsed
│ │ │ │ + -- .666072s elapsed
│ │ │ │ + -- 1.15554s elapsed
│ │ │ │ + -- 1.19189s elapsed
│ │ │ │ + -- 1.29931s elapsed
│ │ │ │ + -- .766725s elapsed
│ │ │ │ + -- .504907s elapsed
│ │ │ │ + -- .434476s 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
│ │ │ @@ -110,15 +110,15 @@
│ │ │  
│ │ │  o5 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ - -- .662s elapsed
│ │ │ + -- 1s 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|
│ │ │ @@ -131,15 +131,15 @@
│ │ │  +---+---+---+---+
│ │ │  |72 |144|216|288|
│ │ │  +---+---+---+---+
│ │ │  |288|216|144|72 |
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │ - -- .721s elapsed
│ │ │ + -- 1.05s elapsed
│ │ │  
│ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {.309, -.809,
│ │ │       ------------------------------------------------------------------------
│ │ │       -.809, .309, -.951, -.588, .588, .951}, {1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  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
│ │ │ │ - -- .662s elapsed
│ │ │ │ + -- 1s 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|
│ │ │ │  +---+---+---+---+
│ │ │ │  |288|216|144|72 |
│ │ │ │  +---+---+---+---+
│ │ │ │  |0  |0  |0  |0  |
│ │ │ │  +---+---+---+---+
│ │ │ │ - -- .721s elapsed
│ │ │ │ + -- 1.05s elapsed
│ │ │ │  
│ │ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {.309, -.809,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       -.809, .309, -.951, -.588, .588, .951}, {1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -115,19 +115,19 @@
│ │ │  o5 = 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : for G in Gs list showExoticSolutions G;
│ │ │  warning: some solutions are not regular: {37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 50, 52, 53, 54, 55, 59, 60, 64, 65, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87, 88, 89}
│ │ │ - -- .763s elapsed
│ │ │ + -- .815s elapsed
│ │ │  warning: some solutions are not regular: {43, 44, 47, 49, 50, 51, 52, 53, 55, 57, 59, 61, 62, 63, 64, 65, 66, 68, 72, 74, 76, 77, 78, 79, 80, 84, 85, 87, 88, 89, 90, 91, 97}
│ │ │ - -- .659s elapsed
│ │ │ - -- .88s elapsed
│ │ │ - -- 1.02s elapsed
│ │ │ + -- .788s elapsed
│ │ │ + -- 1.03s elapsed
│ │ │ + -- 1.26s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │                                                  1 => {3, 4}
│ │ │                                                  2 => {0, 4}
│ │ │                                                  3 => {0, 1}
│ │ │                                                  4 => {2, 1}
│ │ │  +-----+----+----+-----+-----+-----+-----+-----+
│ │ │  |-.809|.309|.309|-.809|.588 |-.951|.951 |-.588|
│ │ │ @@ -140,25 +140,25 @@
│ │ │  +---+---+---+---+
│ │ │  |144|288|72 |216|
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |216|72 |288|144|
│ │ │  +---+---+---+---+
│ │ │ - -- 1.3s elapsed
│ │ │ - -- 1.31s elapsed
│ │ │ + -- 1.47s elapsed
│ │ │ + -- 1.57s elapsed
│ │ │  warning: some solutions are not regular: {27, 31, 32, 34, 37, 38, 44, 46, 47, 53, 54, 56, 57, 59, 60}
│ │ │ - -- 1.68s elapsed
│ │ │ + -- 1.91s elapsed
│ │ │  warning: some solutions are not regular: {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 34}
│ │ │ - -- 1.32s elapsed
│ │ │ - -- 1.4s elapsed
│ │ │ + -- 1.65s elapsed
│ │ │ + -- 1.67s elapsed
│ │ │  warning: some solutions are not regular: {26, 29, 30, 32, 33}
│ │ │ - -- 1.79s elapsed
│ │ │ + -- 1.71s elapsed
│ │ │  warning: some solutions are not regular: {38, 40, 42, 53, 54, 55, 62, 63, 67, 72, 77, 78}
│ │ │ - -- 1.59s elapsed
    
│ │ │ + -- 1.56s 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: {37, 38, 40, 41, 42, 43, 44, 45, 46,
│ │ │ │  48, 50, 52, 53, 54, 55, 59, 60, 64, 65, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78,
│ │ │ │  79, 81, 83, 85, 86, 87, 88, 89}
│ │ │ │ - -- .763s elapsed
│ │ │ │ + -- .815s elapsed
│ │ │ │  warning: some solutions are not regular: {43, 44, 47, 49, 50, 51, 52, 53, 55,
│ │ │ │  57, 59, 61, 62, 63, 64, 65, 66, 68, 72, 74, 76, 77, 78, 79, 80, 84, 85, 87, 88,
│ │ │ │  89, 90, 91, 97}
│ │ │ │ - -- .659s elapsed
│ │ │ │ - -- .88s elapsed
│ │ │ │ - -- 1.02s elapsed
│ │ │ │ + -- .788s elapsed
│ │ │ │ + -- 1.03s elapsed
│ │ │ │ + -- 1.26s elapsed
│ │ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │ │                                                  1 => {3, 4}
│ │ │ │                                                  2 => {0, 4}
│ │ │ │                                                  3 => {0, 1}
│ │ │ │                                                  4 => {2, 1}
│ │ │ │  +-----+----+----+-----+-----+-----+-----+-----+
│ │ │ │  |-.809|.309|.309|-.809|.588 |-.951|.951 |-.588|
│ │ │ │ @@ -75,24 +75,24 @@
│ │ │ │  +---+---+---+---+
│ │ │ │  |144|288|72 |216|
│ │ │ │  +---+---+---+---+
│ │ │ │  |0  |0  |0  |0  |
│ │ │ │  +---+---+---+---+
│ │ │ │  |216|72 |288|144|
│ │ │ │  +---+---+---+---+
│ │ │ │ - -- 1.3s elapsed
│ │ │ │ - -- 1.31s elapsed
│ │ │ │ + -- 1.47s elapsed
│ │ │ │ + -- 1.57s elapsed
│ │ │ │  warning: some solutions are not regular: {27, 31, 32, 34, 37, 38, 44, 46, 47,
│ │ │ │  53, 54, 56, 57, 59, 60}
│ │ │ │ - -- 1.68s elapsed
│ │ │ │ + -- 1.91s elapsed
│ │ │ │  warning: some solutions are not regular: {16, 17, 18, 19, 20, 21, 22, 23, 24,
│ │ │ │  25, 26, 27, 28, 29, 31, 34}
│ │ │ │ - -- 1.32s elapsed
│ │ │ │ - -- 1.4s elapsed
│ │ │ │ + -- 1.65s elapsed
│ │ │ │ + -- 1.67s elapsed
│ │ │ │  warning: some solutions are not regular: {26, 29, 30, 32, 33}
│ │ │ │ - -- 1.79s elapsed
│ │ │ │ + -- 1.71s elapsed
│ │ │ │  warning: some solutions are not regular: {38, 40, 42, 53, 54, 55, 62, 63, 67,
│ │ │ │  72, 77, 78}
│ │ │ │ - -- 1.59s elapsed
│ │ │ │ + -- 1.56s elapsed
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:812:0.
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/_get__Linearly__Stable__Solutions.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : getLinearlyStableSolutions(G)
│ │ │  -- warning: experimental computation over inexact field begun
│ │ │  --          results not reliable (one warning given per session)
│ │ │ - -- .250068s elapsed
│ │ │ + -- .279032s elapsed
│ │ │  warning: some solutions are not regular: {4, 5, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}
│ │ │  
│ │ │  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: experimental computation over inexact field begun
│ │ │ │  --          results not reliable (one warning given per session)
│ │ │ │ - -- .250068s elapsed
│ │ │ │ + -- .279032s elapsed
│ │ │ │  warning: some solutions are not regular: {4, 5, 7, 9, 10, 12, 13, 14, 15, 16,
│ │ │ │  17, 18, 19, 20, 21}
│ │ │ │  
│ │ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/_show__Exotic__Solutions.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : showExoticSolutions G
│ │ │  -- warning: experimental computation over inexact field begun
│ │ │  --          results not reliable (one warning given per session)
│ │ │ - -- .943746s elapsed
│ │ │ + -- 1.09886s 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|
│ │ │ @@ -147,15 +147,15 @@
│ │ │  
│ │ │  o3 : Graph
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : showExoticSolutions G
│ │ │ - -- 1.24035s elapsed
│ │ │ + -- 1.43774s elapsed
│ │ │  
│ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o4 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │             3 => {2, 4}
│ │ │ │             4 => {0, 3}
│ │ │ │  
│ │ │ │  o1 : Graph
│ │ │ │  i2 : showExoticSolutions G
│ │ │ │  -- warning: experimental computation over inexact field begun
│ │ │ │  --          results not reliable (one warning given per session)
│ │ │ │ - -- .943746s elapsed
│ │ │ │ + -- 1.09886s 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|
│ │ │ │ @@ -78,15 +78,15 @@
│ │ │ │             1 => {0, 2}
│ │ │ │             2 => {1, 3, 4}
│ │ │ │             3 => {2, 4}
│ │ │ │             4 => {0, 2, 3}
│ │ │ │  
│ │ │ │  o3 : Graph
│ │ │ │  i4 : showExoticSolutions G
│ │ │ │ - -- 1.24035s elapsed
│ │ │ │ + -- 1.43774s 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
│ │ │ - -- .00892428s elapsed
│ │ │ + -- .010296s 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
│ │ │ - -- .568854s elapsed
│ │ │ + -- .611969s elapsed
│ │ │  
│ │ │  o20 : Ideal of S
│ │ │  
│ │ │  i21 : dim I
│ │ │  
│ │ │  o21 = 5
│ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html
│ │ │ @@ -208,20 +208,20 @@
│ │ │            
│ │ │              
│ │ │                
    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
│ │ │ - -- .00892428s elapsed
│ │ │ + -- .010296s 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
│ │ │ - -- .568854s elapsed
│ │ │ + -- .611969s 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
│ │ │ │ - -- .00892428s elapsed
│ │ │ │ + -- .010296s 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
│ │ │ │ - -- .568854s elapsed
│ │ │ │ + -- .611969s 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);
│ │ │ - -- 1.19915s elapsed
│ │ │ + -- 1.04914s 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 ();
│ │ │ - -- .463669s elapsed
│ │ │ + -- .366947s 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
│ │ │ - -- 20.5845s elapsed
│ │ │ + -- 14.7566s 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);
│ │ │ - -- .684809s elapsed
│ │ │ + -- .619219s 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);
│ │ │ - -- 2.41016s elapsed
│ │ │ + -- 1.93871s 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
│ │ │ @@ -128,15 +128,15 @@
│ │ │              
│ │ │                
    i4 : kk= ZZ/101;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ - -- 1.19915s elapsed
    
│ │ │ + -- 1.04914s elapsed
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o6 = CliffordModule{...6...}
│ │ │ @@ -172,15 +172,15 @@
│ │ │            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 ();
│ │ │ - -- .463669s elapsed
    
│ │ │ + -- .366947s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │  
│ │ │                 0  1          0  1
│ │ │ @@ -193,15 +193,15 @@
│ │ │  
│ │ │  o12 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the actions of generators
│ │ │ - -- 20.5845s elapsed
    
│ │ │ + -- 14.7566s 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);
│ │ │ │ - -- 1.19915s elapsed
│ │ │ │ + -- 1.04914s 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 ();
│ │ │ │ - -- .463669s elapsed
│ │ │ │ + -- .366947s 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
│ │ │ │ - -- 20.5845s elapsed
│ │ │ │ + -- 14.7566s 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
│ │ │ @@ -161,15 +161,15 @@
│ │ │  
│ │ │  o11 : CliffordModule
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ - -- .684809s elapsed
    
│ │ │ + -- .619219s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : betti freeResolution Ulr
│ │ │  
│ │ │               0  1 2
│ │ │ @@ -185,15 +185,15 @@
│ │ │  
│ │ │  o14 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ - -- 2.41016s elapsed
    
│ │ │ + -- 1.93871s 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);
│ │ │ │ - -- .684809s elapsed
│ │ │ │ + -- .619219s 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);
│ │ │ │ - -- 2.41016s elapsed
│ │ │ │ + -- 1.93871s 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);
│ │ │ - -- 2.81759s elapsed
│ │ │ + -- 2.21897s elapsed
│ │ │  
│ │ │  i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 4.60723s elapsed
│ │ │ + -- 4.39337s elapsed
│ │ │  
│ │ │  i12 : G==H
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html
│ │ │ @@ -177,21 +177,21 @@
│ │ │  
│ │ │  o9 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ - -- 2.81759s elapsed
    
│ │ │ + -- 2.21897s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 4.60723s elapsed
    
│ │ │ + -- 4.39337s 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);
│ │ │ │ - -- 2.81759s elapsed
│ │ │ │ + -- 2.21897s elapsed
│ │ │ │  i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ │ - -- 4.60723s elapsed
│ │ │ │ + -- 4.39337s 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/example-output/___Precompute.out
│ │ │ @@ -31,27 +31,27 @@
│ │ │  o5 = CacheTable{name => P}
│ │ │  
│ │ │  i6 : C == P
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : time isDistributive C
│ │ │ - -- used 1.093e-05s (cpu); 7.664e-06s (thread); 0s (gc)
│ │ │ + -- used 3.4059e-05s (cpu); 6.301e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : time isDistributive P
│ │ │ - -- used 6.40308s (cpu); 4.11716s (thread); 0s (gc)
│ │ │ + -- used 8.88289s (cpu); 5.21233s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : C' = dual C;
│ │ │  
│ │ │  i10 : time isDistributive C'
│ │ │ - -- used 6.472e-06s (cpu); 5.841e-06s (thread); 0s (gc)
│ │ │ + -- used 1.296e-05s (cpu); 2.772e-05s (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 0.479688s (cpu); 0.280179s (thread); 0s (gc)
│ │ │ + -- used 0.512123s (cpu); 0.275551s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = Partition{9, 2}
│ │ │  
│ │ │  o4 : Partition
│ │ │  
│ │ │  i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ - -- used 1.3956e-05s (cpu); 1.3846e-05s (thread); 0s (gc)
│ │ │ + -- used 1.8028e-05s (cpu); 1.5785e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = Partition{9, 2}
│ │ │  
│ │ │  o5 : Partition
│ │ │  
│ │ │  i6 : greeneKleitmanPartition chain 10
│ │ ├── ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html
│ │ │ @@ -107,23 +107,23 @@
│ │ │  
│ │ │  o6 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time isDistributive C
│ │ │ - -- used 1.093e-05s (cpu); 7.664e-06s (thread); 0s (gc)
│ │ │ + -- used 3.4059e-05s (cpu); 6.301e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time isDistributive P
│ │ │ - -- used 6.40308s (cpu); 4.11716s (thread); 0s (gc)
│ │ │ + -- used 8.88289s (cpu); 5.21233s (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.
    
│ │ │ @@ -133,15 +133,15 @@
│ │ │              
│ │ │                
    i9 : C' = dual C;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time isDistributive C'
│ │ │ - -- used 6.472e-06s (cpu); 5.841e-06s (thread); 0s (gc)
│ │ │ + -- used 1.296e-05s (cpu); 2.772e-05s (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 1.093e-05s (cpu); 7.664e-06s (thread); 0s (gc)
│ │ │ │ + -- used 3.4059e-05s (cpu); 6.301e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : time isDistributive P
│ │ │ │ - -- used 6.40308s (cpu); 4.11716s (thread); 0s (gc)
│ │ │ │ + -- used 8.88289s (cpu); 5.21233s (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 6.472e-06s (cpu); 5.841e-06s (thread); 0s (gc)
│ │ │ │ + -- used 1.296e-05s (cpu); 2.772e-05s (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
│ │ │ @@ -100,25 +100,25 @@
│ │ │              
│ │ │                
    i3 : D = dominanceLattice 6;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
│ │ │ - -- used 0.479688s (cpu); 0.280179s (thread); 0s (gc)
│ │ │ + -- used 0.512123s (cpu); 0.275551s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = Partition{9, 2}
│ │ │  
│ │ │  o4 : Partition
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ - -- used 1.3956e-05s (cpu); 1.3846e-05s (thread); 0s (gc)
│ │ │ + -- used 1.8028e-05s (cpu); 1.5785e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = Partition{9, 2}
│ │ │  
│ │ │  o5 : Partition
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,21 +28,21 @@
│ │ │ │  
│ │ │ │  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 0.479688s (cpu); 0.280179s (thread); 0s (gc)
│ │ │ │ + -- used 0.512123s (cpu); 0.275551s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = Partition{9, 2}
│ │ │ │  
│ │ │ │  o4 : Partition
│ │ │ │  i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ │ - -- used 1.3956e-05s (cpu); 1.3846e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.8028e-05s (cpu); 1.5785e-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/_associated__Primes.out
│ │ │ @@ -125,24 +125,24 @@
│ │ │        -----------------------------------------------------------------------
│ │ │        ideal (a, b, c, e), ideal (a, b, d, e), ideal (a, b, c, d, e)}
│ │ │  
│ │ │  o19 : List
│ │ │  
│ │ │  i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o20 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │ +o20 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {c, b, a}, {e, c, b, a}}
│ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │  
│ │ │  o20 : Set
│ │ │  
│ │ │  i21 : M2 = set apply(L2, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o21 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │ +o21 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {c, b, a}, {e, c, b, a}}
│ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │  
│ │ │  o21 : Set
│ │ │  
│ │ │  i22 : assert(M1 === M2)
│ │ │  
│ │ │  i23 :
│ │ ├── ./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)
│ │ │ - -- .0780135s elapsed
│ │ │ + -- .106484s 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)
│ │ │ - -- .0175249s elapsed
│ │ │ + -- .0227999s 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)
│ │ │ - -- .126458s elapsed
│ │ │ + -- .0594929s 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
│ │ │ - -- .157633s elapsed
│ │ │ + -- .0667616s 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)
│ │ │ - -- .00961892s elapsed
│ │ │ + -- .0093468s 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/_associated__Primes.html
│ │ │ @@ -286,28 +286,28 @@
│ │ │  o19 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o20 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │ +o20 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {c, b, a}, {e, c, b, a}}
│ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │  
│ │ │  o20 : Set
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : M2 = set apply(L2, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o21 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │ +o21 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {c, b, a}, {e, c, b, a}}
│ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │  
│ │ │  o21 : Set
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i22 : assert(M1 === M2)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -155,24 +155,24 @@
│ │ │ │  o19 = {ideal (a, e), ideal (a, b, c), ideal (a, b, d), ideal (a, b, c, d),
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        ideal (a, b, c, e), ideal (a, b, d, e), ideal (a, b, c, d, e)}
│ │ │ │  
│ │ │ │  o19 : List
│ │ │ │  i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │ │  
│ │ │ │ -o20 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │ │ +o20 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {c, b, a}, {e, c, b, a}}
│ │ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │ │  
│ │ │ │  o20 : Set
│ │ │ │  i21 : M2 = set apply(L2, I -> sort flatten entries gens I)
│ │ │ │  
│ │ │ │ -o21 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │ │ +o21 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {c, b, a}, {e, c, b, a}}
│ │ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │ │  
│ │ │ │  o21 : Set
│ │ │ │  i22 : assert(M1 === M2)
│ │ │ │  The method using Ext modules comes from Eisenbud-Huneke-Vasconcelos, Invent.
│ │ │ │  Math 110 (1992) 207-235.
│ │ │ │  Original author (for ideals): _C_._ _Y_a_c_k_e_l. Updated for modules by J. Chen.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html
│ │ │ @@ -107,41 +107,41 @@
│ │ │                              3
│ │ │  o3 : R-module, quotient of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ - -- .0780135s elapsed
│ │ │ + -- .106484s 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)
│ │ │ - -- .0175249s elapsed
│ │ │ + -- .0227999s 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)
│ │ │ - -- .126458s elapsed
│ │ │ + -- .0594929s 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)
│ │ │ │ - -- .0780135s elapsed
│ │ │ │ + -- .106484s 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)
│ │ │ │ - -- .0175249s elapsed
│ │ │ │ + -- .0227999s 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)
│ │ │ │ - -- .126458s elapsed
│ │ │ │ + -- .0594929s 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
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime regSeqInIdeal I
│ │ │ - -- .157633s elapsed
│ │ │ + -- .0667616s 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
    
│ │ │              
│ │ │            
│ │ │ @@ -148,15 +148,15 @@
│ │ │  
│ │ │  o6 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
│ │ │ - -- .00961892s elapsed
│ │ │ + -- .0093468s 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
│ │ │ │ - -- .157633s elapsed
│ │ │ │ + -- .0667616s 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)
│ │ │ │ - -- .00961892s elapsed
│ │ │ │ + -- .0093468s 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/M2-47275-0/0.py
│ │ │ +o1 = /tmp/M2-73596-0/0.py
│ │ │  
│ │ │  i2 : pyfile << "import math" << endl
│ │ │  
│ │ │ -o2 = /tmp/M2-47275-0/0.py
│ │ │ +o2 = /tmp/M2-73596-0/0.py
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │  
│ │ │ -o3 = /tmp/M2-47275-0/0.py
│ │ │ +o3 = /tmp/M2-73596-0/0.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 0x7fdae521ccc0>
│ │ │ +o9 =  at 0x7fdc4bc9ccc0>
│ │ │  
│ │ │  o9 : PythonObject of class function
│ │ │  
│ │ │  i10 : x
│ │ │  
│ │ │  o10 = 5
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_iterator_lp__Python__Object_rp.html
│ │ │ @@ -90,15 +90,15 @@
│ │ │  o2 : PythonObject of class range
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : i = iterator x
│ │ │  
│ │ │ -o3 = <range_iterator object at 0x7fdae5265860>
│ │ │ +o3 = <range_iterator object at 0x7fdc4bce56b0>
│ │ │  
│ │ │  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
│ │ │ @@ -86,15 +86,15 @@
│ │ │  o2 : PythonObject of class range
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : i = iterator x
│ │ │  
│ │ │ -o3 = <range_iterator object at 0x7fdae5259ec0>
│ │ │ +o3 = <range_iterator object at 0x7fdc4bcd9d70>
│ │ │  
│ │ │  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
│ │ │ @@ -76,31 +76,31 @@
│ │ │            
    The return value is a Python dictionary containing all the variables defined in the global scope.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : pyfile = temporaryFileName() | ".py"
│ │ │  
│ │ │ -o1 = /tmp/M2-47275-0/0.py
    
│ │ │ +o1 = /tmp/M2-73596-0/0.py
│ │ │              
    
│ │ │                
    i2 : pyfile << "import math" << endl
│ │ │  
│ │ │ -o2 = /tmp/M2-47275-0/0.py
│ │ │ +o2 = /tmp/M2-73596-0/0.py
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │  
│ │ │ -o3 = /tmp/M2-47275-0/0.py
│ │ │ +o3 = /tmp/M2-73596-0/0.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/M2-47275-0/0.py
│ │ │ │ +o1 = /tmp/M2-73596-0/0.py
│ │ │ │  i2 : pyfile << "import math" << endl
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-47275-0/0.py
│ │ │ │ +o2 = /tmp/M2-73596-0/0.py
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-47275-0/0.py
│ │ │ │ +o3 = /tmp/M2-73596-0/0.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
│ │ │ @@ -181,15 +181,15 @@
│ │ │  o12 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i13 : pysqrt = toPython m2sqrt
│ │ │  
│ │ │ -o13 = <built-in method m2sqrt of PyCapsule object at 0x7fdae523ac50>
│ │ │ +o13 = <built-in method m2sqrt of PyCapsule object at 0x7fdc4bcbec00>
│ │ │  
│ │ │  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
│ │ │ @@ -124,15 +124,15 @@
│ │ │                
    i8 : use ctx
    
│ │ │              
    
│ │ │                
    i9 : f
│ │ │  
│ │ │ -o9 = <function <lambda> at 0x7fdae521ccc0>
│ │ │ +o9 = <function <lambda> at 0x7fdc4bc9ccc0>
│ │ │  
│ │ │  o9 : PythonObject of class function
    
│ │ │              
    
│ │ │                
    i10 : x
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  
│ │ │ │  o7 = y
│ │ │ │  
│ │ │ │  o7 : Symbol
│ │ │ │  i8 : use ctx
│ │ │ │  i9 : f
│ │ │ │  
│ │ │ │ -o9 =  at 0x7fdae521ccc0>
│ │ │ │ +o9 =  at 0x7fdc4bc9ccc0>
│ │ │ │  
│ │ │ │  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
│ │ │ - -- 2.78979s elapsed
│ │ │ + -- 2.52317s 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;
│ │ │ - -- 2.4524s elapsed
│ │ │ + -- 2.14045s 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 = | -27 -13 45 -25 3 38 -20 -30 -41 25 -26 -44 -31 5 14 2 -45 45 21 -27
│ │ │ +o44 = | 22 -10 -8 -1 34 44 -21 -1 25 -41 6 -11 -50 -50 43 -28 -6 45 -28 22 42
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -23 -29 34 49 32 19 10 26 19 37 15 -28 -50 -10 -32 18 |
│ │ │ +      -29 -32 -28 5 -10 34 15 19 37 26 49 19 5 10 18 |
│ │ │  
│ │ │                 1       36
│ │ │  o44 : Matrix kk  <-- kk
│ │ │  
│ │ │  i45 : Fa = sub(F, (vars S) | pta)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o45 = ideal (a  + 14b*c + 25c  - 44a*d + 38b*d + 45c*d - 27d , a*b + 32b*c +
│ │ │ +              2              2                             2              
│ │ │ +o45 = ideal (a  + 43b*c - 41c  - 11a*d + 44b*d - 8c*d + 22d , a*b + 5b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │           2                              2   2              2                
│ │ │ -      21c  - 23a*d - 31b*d - 41c*d - 13d , b  - 32b*c + 26c  - 50a*d - 29b*d
│ │ │ +      28c  + 42a*d - 50b*d + 25c*d - 10d , b  + 10b*c + 15c  + 19a*d - 29b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                  2                   2                            2     2  
│ │ │ -      + 2c*d - 20d , a*c - 28b*c + 19c  + 19a*d - 27b*d + 5c*d + 3d , b*c  +
│ │ │ +                   2                   2                              2     2
│ │ │ +      - 28c*d - 21d , a*c + 49b*c - 10c  + 19a*d + 22b*d - 50c*d + 34d , b*c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3                2 
│ │ │ -      37b*c*d + 34c d + 10a*d  - 45b*d  - 26c*d  - 25d , c  + 18b*c*d + 15c d
│ │ │ +                     2         2       2       2    3   3                2   
│ │ │ +      + 37b*c*d - 32c d + 34a*d  - 6b*d  + 6c*d  - d , c  + 18b*c*d + 26c d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2        2      3
│ │ │ -      - 10a*d  + 49b*d  + 45c*d  - 30d )
│ │ │ +          2        2        2    3
│ │ │ +      5a*d  - 28b*d  + 45c*d  - d )
│ │ │  
│ │ │  o45 : Ideal of S
│ │ │  
│ │ │  i46 : betti res Fa
│ │ │  
│ │ │               0 1 2 3
│ │ │  o46 = total: 1 6 8 3
│ │ │ @@ -358,37 +358,37 @@
│ │ │            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 + 19d, b - 37d, a + 4d)                                                                                                                              |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                             2              2                      2   3                2         2        2      3     2                2         2        2 |
│ │ │ -      |ideal (a - 28b + 19c + 48d, b  - 32b*c + 26c  - 15b*d + 43c*d - 44d , c  + 18b*c*d + 15c d - 29b*d  + 33c*d  + 46d , b*c  + 37b*c*d + 34c d + 33b*d  - 14c*d )|
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +o47 = |ideal (c + 19d, b - 37d, a)                                                                                                                                        |
│ │ │ +      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                             2              2                      2   3                2         2       2     3     2                2         2        2      3 |
│ │ │ +      |ideal (a + 49b - 10c + 39d, b  + 10b*c + 15c  + 50b*d - 40c*d + 46d , c  + 18b*c*d + 26c d + 30b*d  - 6c*d  + 6d , b*c  + 37b*c*d - 32c d + 45b*d  + 43c*d  - 14d )|
│ │ │ +      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i48 : CFa = minimalPrimes Fa
│ │ │  
│ │ │ -                                                                      2  
│ │ │ -o48 = {ideal (c + 19d, b - 37d, a + 4d), ideal (a - 28b + 19c + 48d, b  -
│ │ │ +                                                                 2          
│ │ │ +o48 = {ideal (c + 19d, b - 37d, a), ideal (a + 49b - 10c + 39d, b  + 10b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                      2   3                2         2  
│ │ │ -      32b*c + 26c  - 15b*d + 43c*d - 44d , c  + 18b*c*d + 15c d - 29b*d  +
│ │ │ +         2                      2   3                2         2       2  
│ │ │ +      15c  + 50b*d - 40c*d + 46d , c  + 18b*c*d + 26c d + 30b*d  - 6c*d  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2      3     2                2         2        2
│ │ │ -      33c*d  + 46d , b*c  + 37b*c*d + 34c d + 33b*d  - 14c*d )}
│ │ │ +        3     2                2         2        2      3
│ │ │ +      6d , b*c  + 37b*c*d - 32c d + 45b*d  + 43c*d  - 14d )}
│ │ │  
│ │ │  o48 : List
│ │ │  
│ │ │  i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │  
│ │ │ -o49 = a - 28b + 19c + 48d
│ │ │ +o49 = a + 49b - 10c + 39d
│ │ │  
│ │ │  o49 : S
│ │ │  
│ │ │  i50 : CFa/degree
│ │ │  
│ │ │  o50 = {1, 5}
│ │ │  
│ │ │ @@ -402,37 +402,37 @@
│ │ │  
│ │ │  i52 : degree(Fa : (Fa : lin))  -- somewhat simpler(?) way to see the ideal of the 5 points
│ │ │  
│ │ │  o52 = 5
│ │ │  
│ │ │  i53 : ptb = randomPointOnRationalVariety compsL441_1
│ │ │  
│ │ │ -o53 = | 31 42 28 25 19 3 43 -7 -3 -42 -29 -29 14 2 50 5 36 -13 -42 47 13 31
│ │ │ +o53 = | 27 12 -34 9 -19 -43 -32 27 40 45 -13 29 -41 -13 22 -49 -4 -4 9 -23 43
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -37 -23 -24 -4 38 -29 -23 21 17 9 0 21 -9 -47 |
│ │ │ +      18 -9 -47 43 21 38 17 -20 21 -29 47 0 2 -37 9 |
│ │ │  
│ │ │                 1       36
│ │ │  o53 : Matrix kk  <-- kk
│ │ │  
│ │ │  i54 : Fb = sub(F, (vars S) | ptb)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o54 = ideal (a  + 50b*c - 42c  - 29a*d + 3b*d + 28c*d + 31d , a*b - 24b*c -
│ │ │ +              2              2                              2               
│ │ │ +o54 = ideal (a  + 22b*c + 45c  + 29a*d - 43b*d - 34c*d + 27d , a*b + 43b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2             2                 
│ │ │ -      42c  + 13a*d + 14b*d - 3c*d + 42d , b  - 9b*c - 29c  + 31b*d + 5c*d +
│ │ │ +        2                              2   2              2                  
│ │ │ +      9c  + 43a*d - 41b*d + 40c*d + 12d , b  - 37b*c + 17c  + 18b*d - 49c*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          
│ │ │ +      32d , a*c + 47b*c + 21c  - 20a*d - 23b*d - 13c*d - 19d , b*c  + 21b*c*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       2        2     3   3               2        2  
│ │ │ +      - 9c d + 38a*d  - 4b*d  - 13c*d  + 9d , c  + 9b*c*d - 29c d + 2a*d  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2     3
│ │ │ -      - 23b*d  - 13c*d  - 7d )
│ │ │ +           2       2      3
│ │ │ +      47b*d  - 4c*d  + 27d )
│ │ │  
│ │ │  o54 : Ideal of S
│ │ │  
│ │ │  i55 : betti res Fb
│ │ │  
│ │ │               0 1 2 3
│ │ │  o55 = total: 1 6 8 3
│ │ │ @@ -440,114 +440,80 @@
│ │ │            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 )|
│ │ │ -      +-------------------------------------------------------+
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                                      2              2                                                                                                                                                               |
│ │ │ +o56 = |ideal (b - 50c - 43d, a + 15c - 46d, c  + 12c*d - 37d )                                                                                                                                                              |
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |        2                             2                                 2                                   2   2                            2                                   2   2                             2 |
│ │ │ +      |ideal (c  + 46a*d - 39b*d + 2c*d - 24d , b*c - 9a*d + 16b*d + 2c*d + 27d , a*c + 43a*d + 44b*d - 48c*d + 24d , b  - 4a*d - 40b*d - 9c*d - 39d , a*b + 16a*d + 26b*d + 37c*d - 24d , a  - 25a*d + 47b*d + 34c*d + 8d )|
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │  
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                          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 )     |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +o57 = ++
│ │ │ +      ++
│ │ │  
│ │ │  i58 : pt0 = randomPointOnRationalVariety(compsL441_0)
│ │ │  
│ │ │ -o58 = | 13 17 -19 -1 -9 -15 -28 -39 -36 1 -47 29 37 -40 35 -31 12 -21 -8 -13
│ │ │ +o58 = | 32 -46 33 -7 -2 -29 -20 10 -23 -26 5 -16 1 -18 -3 46 13 -21 5 -22 17
│ │ │        -----------------------------------------------------------------------
│ │ │ -      14 15 -23 39 11 8 -24 -13 -42 -2 18 46 -18 -29 -33 -22 |
│ │ │ +      15 -33 46 -2 -29 -23 18 -42 -2 -13 39 8 -40 -24 -22 |
│ │ │  
│ │ │                 1       36
│ │ │  o58 : Matrix kk  <-- kk
│ │ │  
│ │ │  i59 : pt1 = randomPointOnRationalVariety(compsL441_1)
│ │ │  
│ │ │ -o59 = | -45 18 -9 38 21 29 50 -8 -5 45 -47 -26 37 -35 -21 28 27 46 -17 -49
│ │ │ +o59 = | -8 41 28 -44 50 33 -38 33 -23 1 -2 -47 32 46 30 -22 -2 -14 27 37 15
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -23 15 -50 37 -39 -14 21 10 -31 3 -18 32 0 3 -15 33 |
│ │ │ +      -25 -15 33 -23 3 21 -18 -9 3 10 -49 0 -35 -50 32 |
│ │ │  
│ │ │                 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  + 35b*c + c  + 29a*d - 15b*d - 19c*d + 13d , a*b + 11b*c -
│ │ │ +              2             2                              2              
│ │ │ +o60 = ideal (a  - 3b*c - 26c  - 16a*d - 29b*d + 33c*d + 32d , a*b - 2b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2   2              2                  
│ │ │ -      8c  + 14a*d + 37b*d - 36c*d + 17d , b  - 33b*c - 13c  - 18a*d + 15b*d -
│ │ │ +        2                            2   2              2                 
│ │ │ +      5c  + 17a*d + b*d - 23c*d - 46d , b  - 24b*c + 18c  + 8a*d + 15b*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                  2                             2     2  
│ │ │ -      31c*d - 28d , a*c + 46b*c + 8c  - 42a*d - 13b*d - 40c*d - 9d , b*c  -
│ │ │ +                 2                   2                             2     2  
│ │ │ +      46c*d - 20d , a*c + 39b*c - 29c  - 42a*d - 22b*d - 18c*d - 2d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                  2         2        2        2    3   3                2   
│ │ │ -      2b*c*d - 23c d - 24a*d  + 12b*d  - 47c*d  - d , c  - 22b*c*d + 18c d -
│ │ │ +                  2         2        2       2     3   3                2   
│ │ │ +      2b*c*d - 33c d - 23a*d  + 13b*d  + 5c*d  - 7d , c  - 22b*c*d - 13c d -
│ │ │        -----------------------------------------------------------------------
│ │ │             2        2        2      3
│ │ │ -      29a*d  + 39b*d  - 21c*d  - 39d )
│ │ │ +      40a*d  + 46b*d  - 21c*d  + 10d )
│ │ │  
│ │ │  o60 : Ideal of S
│ │ │  
│ │ │  i61 : I1 = sub(sub(F, (vars ring F) | sub(pt1, ring F)), S)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o61 = ideal (a  - 21b*c + 45c  - 26a*d + 29b*d - 9c*d - 45d , a*b - 39b*c -
│ │ │ +              2            2                             2               
│ │ │ +o61 = ideal (a  + 30b*c + c  - 47a*d + 33b*d + 28c*d - 8d , a*b - 23b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2              2                  
│ │ │ -      17c  - 23a*d + 37b*d - 5c*d + 18d , b  - 15b*c + 10c  + 15b*d + 28c*d +
│ │ │ +         2                              2   2              2                
│ │ │ +      27c  + 15a*d + 32b*d - 23c*d + 41d , b  - 50b*c - 18c  - 25b*d - 22c*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         
│ │ │ +      - 38d , a*c - 49b*c + 3c  - 9a*d + 37b*d + 46c*d + 50d , b*c  + 3b*c*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       2       2      3   3                2         2
│ │ │ +      - 15c d + 21a*d  - 2b*d  - 2c*d  - 44d , c  + 32b*c*d + 10c d - 35a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2     3
│ │ │ -      + 37b*d  + 46c*d  - 8d )
│ │ │ +             2        2      3
│ │ │ +      + 33b*d  - 14c*d  + 33d )
│ │ │  
│ │ │  o61 : Ideal of S
│ │ │  
│ │ │  i62 : betti res I0
│ │ │  
│ │ │               0 1 2 3
│ │ │  o62 = total: 1 6 8 3
│ │ │ @@ -565,37 +531,36 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o63 : BettiTally
│ │ │  
│ │ │  i64 : netList decompose I0
│ │ │  
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -o64 = |ideal (c - 42d, b - 6d, a + 41d)                                                             |
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -      |ideal (c + 38d, b - 41d, a - 41d)                                                            |
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -      |                                  2                      2   2      2                      2 |
│ │ │ -      |ideal (a + 46b + 8c - 7d, b*c - 7c  + 13b*d + 16c*d - 21d , b  - 42c  - 41b*d + 35c*d - 39d )|
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ +      +------------------------------------------------------------------------------------------------+
│ │ │ +o64 = |ideal (c - 42d, b + 26d, a - 30d)                                                               |
│ │ │ +      +------------------------------------------------------------------------------------------------+
│ │ │ +      |ideal (c - 47d, b + 7d, a - 44d)                                                                |
│ │ │ +      +------------------------------------------------------------------------------------------------+
│ │ │ +      |                                     2                      2   2      2                      2 |
│ │ │ +      |ideal (a + 39b - 29c - 24d, b*c - 15c  - 29b*d - 38c*d + 16d , b  - 39c  + 17b*d - 28c*d - 50d )|
│ │ │ +      +------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i65 : netList decompose I1
│ │ │  
│ │ │ -      +------------------------------------------------------+
│ │ │ -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 )|
│ │ │ -      +------------------------------------------------------+
│ │ │ +      +---------------------------------+
│ │ │ +o65 = |ideal (c - 9d, b + 15d, a + 27d) |
│ │ │ +      +---------------------------------+
│ │ │ +      |ideal (c + 48d, b + 11d, a - 37d)|
│ │ │ +      +---------------------------------+
│ │ │ +      |ideal (c + 29d, b + 46d, a + 18d)|
│ │ │ +      +---------------------------------+
│ │ │ +      |ideal (c + 24d, b + 46d, a + 33d)|
│ │ │ +      +---------------------------------+
│ │ │ +      |ideal (c + 22d, b + 38d, a - 50d)|
│ │ │ +      +---------------------------------+
│ │ │  
│ │ │  i66 : L430 = (trim minors(2, M1)) + groebnerStratum F;
│ │ │  
│ │ │  o66 : Ideal of T
│ │ │  
│ │ │  i67 : C = res(I, FastNonminimal => true)
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html
│ │ │ @@ -344,15 +344,15 @@
│ │ │              
    		
│ │ │                
    i22 : assert(dim L == 18)
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime isPrime L
│ │ │ - -- 2.78979s elapsed
│ │ │ + -- 2.52317s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The Schreyer resolution and minimal Betti numbers
    
│ │ │ @@ -556,15 +556,15 @@
│ │ │  
│ │ │  o39 : Ideal of T    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 2.4524s elapsed
    
│ │ │ + -- 2.14045s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
    
│ │ │ @@ -591,40 +591,40 @@
│ │ │            
    Both components are rational, and here are random points, one on each component:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i44 : pta = randomPointOnRationalVariety compsL441_0
│ │ │  
│ │ │ -o44 = | -27 -13 45 -25 3 38 -20 -30 -41 25 -26 -44 -31 5 14 2 -45 45 21 -27
│ │ │ +o44 = | 22 -10 -8 -1 34 44 -21 -1 25 -41 6 -11 -50 -50 43 -28 -6 45 -28 22 42
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -23 -29 34 49 32 19 10 26 19 37 15 -28 -50 -10 -32 18 |
│ │ │ +      -29 -32 -28 5 -10 34 15 19 37 26 49 19 5 10 18 |
│ │ │  
│ │ │                 1       36
│ │ │  o44 : Matrix kk  <-- kk
    
│ │ │              
    
│ │ │                
    i45 : Fa = sub(F, (vars S) | pta)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o45 = ideal (a  + 14b*c + 25c  - 44a*d + 38b*d + 45c*d - 27d , a*b + 32b*c +
│ │ │ +              2              2                             2              
│ │ │ +o45 = ideal (a  + 43b*c - 41c  - 11a*d + 44b*d - 8c*d + 22d , a*b + 5b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │           2                              2   2              2                
│ │ │ -      21c  - 23a*d - 31b*d - 41c*d - 13d , b  - 32b*c + 26c  - 50a*d - 29b*d
│ │ │ +      28c  + 42a*d - 50b*d + 25c*d - 10d , b  + 10b*c + 15c  + 19a*d - 29b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                  2                   2                            2     2  
│ │ │ -      + 2c*d - 20d , a*c - 28b*c + 19c  + 19a*d - 27b*d + 5c*d + 3d , b*c  +
│ │ │ +                   2                   2                              2     2
│ │ │ +      - 28c*d - 21d , a*c + 49b*c - 10c  + 19a*d + 22b*d - 50c*d + 34d , b*c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3                2 
│ │ │ -      37b*c*d + 34c d + 10a*d  - 45b*d  - 26c*d  - 25d , c  + 18b*c*d + 15c d
│ │ │ +                     2         2       2       2    3   3                2   
│ │ │ +      + 37b*c*d - 32c d + 34a*d  - 6b*d  + 6c*d  - d , c  + 18b*c*d + 26c d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2        2      3
│ │ │ -      - 10a*d  + 49b*d  + 45c*d  - 30d )
│ │ │ +          2        2        2    3
│ │ │ +      5a*d  - 28b*d  + 45c*d  - d )
│ │ │  
│ │ │  o45 : Ideal of S
    
│ │ │              
    
│ │ │                
    i46 : betti res Fa
│ │ │ @@ -638,43 +638,43 @@
│ │ │  o46 : BettiTally
    
│ │ │              
    
│ │ │                
    i47 : netList decompose Fa -- this one is 5 points on a plane, and another point
│ │ │  
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -o47 = |ideal (c + 19d, b - 37d, a + 4d)                                                                                                                              |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                             2              2                      2   3                2         2        2      3     2                2         2        2 |
│ │ │ -      |ideal (a - 28b + 19c + 48d, b  - 32b*c + 26c  - 15b*d + 43c*d - 44d , c  + 18b*c*d + 15c d - 29b*d  + 33c*d  + 46d , b*c  + 37b*c*d + 34c d + 33b*d  - 14c*d )|
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------+
    
│ │ │ +      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +o47 = |ideal (c + 19d, b - 37d, a)                                                                                                                                        |
│ │ │ +      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                             2              2                      2   3                2         2       2     3     2                2         2        2      3 |
│ │ │ +      |ideal (a + 49b - 10c + 39d, b  + 10b*c + 15c  + 50b*d - 40c*d + 46d , c  + 18b*c*d + 26c d + 30b*d  - 6c*d  + 6d , b*c  + 37b*c*d - 32c d + 45b*d  + 43c*d  - 14d )|
│ │ │ +      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │              
    
│ │ │                
    i48 : CFa = minimalPrimes Fa
│ │ │  
│ │ │ -                                                                      2  
│ │ │ -o48 = {ideal (c + 19d, b - 37d, a + 4d), ideal (a - 28b + 19c + 48d, b  -
│ │ │ +                                                                 2          
│ │ │ +o48 = {ideal (c + 19d, b - 37d, a), ideal (a + 49b - 10c + 39d, b  + 10b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                      2   3                2         2  
│ │ │ -      32b*c + 26c  - 15b*d + 43c*d - 44d , c  + 18b*c*d + 15c d - 29b*d  +
│ │ │ +         2                      2   3                2         2       2  
│ │ │ +      15c  + 50b*d - 40c*d + 46d , c  + 18b*c*d + 26c d + 30b*d  - 6c*d  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2      3     2                2         2        2
│ │ │ -      33c*d  + 46d , b*c  + 37b*c*d + 34c d + 33b*d  - 14c*d )}
│ │ │ +        3     2                2         2        2      3
│ │ │ +      6d , b*c  + 37b*c*d - 32c d + 45b*d  + 43c*d  - 14d )}
│ │ │  
│ │ │  o48 : List
    
│ │ │              
    
│ │ │                
    i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │  
│ │ │ -o49 = a - 28b + 19c + 48d
│ │ │ +o49 = a + 49b - 10c + 39d
│ │ │  
│ │ │  o49 : S
    
│ │ │              
    
│ │ │                
    i50 : CFa/degree
│ │ │ @@ -702,40 +702,40 @@
│ │ │            
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i53 : ptb = randomPointOnRationalVariety compsL441_1
│ │ │  
│ │ │ -o53 = | 31 42 28 25 19 3 43 -7 -3 -42 -29 -29 14 2 50 5 36 -13 -42 47 13 31
│ │ │ +o53 = | 27 12 -34 9 -19 -43 -32 27 40 45 -13 29 -41 -13 22 -49 -4 -4 9 -23 43
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -37 -23 -24 -4 38 -29 -23 21 17 9 0 21 -9 -47 |
│ │ │ +      18 -9 -47 43 21 38 17 -20 21 -29 47 0 2 -37 9 |
│ │ │  
│ │ │                 1       36
│ │ │  o53 : Matrix kk  <-- kk
    
│ │ │              
    
│ │ │                
    i54 : Fb = sub(F, (vars S) | ptb)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o54 = ideal (a  + 50b*c - 42c  - 29a*d + 3b*d + 28c*d + 31d , a*b - 24b*c -
│ │ │ +              2              2                              2               
│ │ │ +o54 = ideal (a  + 22b*c + 45c  + 29a*d - 43b*d - 34c*d + 27d , a*b + 43b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2             2                 
│ │ │ -      42c  + 13a*d + 14b*d - 3c*d + 42d , b  - 9b*c - 29c  + 31b*d + 5c*d +
│ │ │ +        2                              2   2              2                  
│ │ │ +      9c  + 43a*d - 41b*d + 40c*d + 12d , b  - 37b*c + 17c  + 18b*d - 49c*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          
│ │ │ +      32d , a*c + 47b*c + 21c  - 20a*d - 23b*d - 13c*d - 19d , b*c  + 21b*c*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       2        2     3   3               2        2  
│ │ │ +      - 9c d + 38a*d  - 4b*d  - 13c*d  + 9d , c  + 9b*c*d - 29c d + 2a*d  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2     3
│ │ │ -      - 23b*d  - 13c*d  - 7d )
│ │ │ +           2       2      3
│ │ │ +      47b*d  - 4c*d  + 27d )
│ │ │  
│ │ │  o54 : Ideal of S
    
│ │ │              
    
│ │ │                
    i55 : betti res Fb
│ │ │ @@ -749,136 +749,102 @@
│ │ │  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 )|
│ │ │ -      +-------------------------------------------------------+
    
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                                      2              2                                                                                                                                                               |
│ │ │ +o56 = |ideal (b - 50c - 43d, a + 15c - 46d, c  + 12c*d - 37d )                                                                                                                                                              |
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |        2                             2                                 2                                   2   2                            2                                   2   2                             2 |
│ │ │ +      |ideal (c  + 46a*d - 39b*d + 2c*d - 24d , b*c - 9a*d + 16b*d + 2c*d + 27d , a*c + 43a*d + 44b*d - 48c*d + 24d , b  - 4a*d - 40b*d - 9c*d - 39d , a*b + 16a*d + 26b*d + 37c*d - 24d , a  - 25a*d + 47b*d + 34c*d + 8d )|
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │              
    
│ │ │                
    i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │  
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                          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 )     |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
    
│ │ │ +o57 = ++
│ │ │ +      ++
│ │ │              
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i58 : pt0 = randomPointOnRationalVariety(compsL441_0)
│ │ │  
│ │ │ -o58 = | 13 17 -19 -1 -9 -15 -28 -39 -36 1 -47 29 37 -40 35 -31 12 -21 -8 -13
│ │ │ +o58 = | 32 -46 33 -7 -2 -29 -20 10 -23 -26 5 -16 1 -18 -3 46 13 -21 5 -22 17
│ │ │        -----------------------------------------------------------------------
│ │ │ -      14 15 -23 39 11 8 -24 -13 -42 -2 18 46 -18 -29 -33 -22 |
│ │ │ +      15 -33 46 -2 -29 -23 18 -42 -2 -13 39 8 -40 -24 -22 |
│ │ │  
│ │ │                 1       36
│ │ │  o58 : Matrix kk  <-- kk
    
│ │ │              
    
│ │ │                
    i59 : pt1 = randomPointOnRationalVariety(compsL441_1)
│ │ │  
│ │ │ -o59 = | -45 18 -9 38 21 29 50 -8 -5 45 -47 -26 37 -35 -21 28 27 46 -17 -49
│ │ │ +o59 = | -8 41 28 -44 50 33 -38 33 -23 1 -2 -47 32 46 30 -22 -2 -14 27 37 15
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -23 15 -50 37 -39 -14 21 10 -31 3 -18 32 0 3 -15 33 |
│ │ │ +      -25 -15 33 -23 3 21 -18 -9 3 10 -49 0 -35 -50 32 |
│ │ │  
│ │ │                 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  + 35b*c + c  + 29a*d - 15b*d - 19c*d + 13d , a*b + 11b*c -
│ │ │ +              2             2                              2              
│ │ │ +o60 = ideal (a  - 3b*c - 26c  - 16a*d - 29b*d + 33c*d + 32d , a*b - 2b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2   2              2                  
│ │ │ -      8c  + 14a*d + 37b*d - 36c*d + 17d , b  - 33b*c - 13c  - 18a*d + 15b*d -
│ │ │ +        2                            2   2              2                 
│ │ │ +      5c  + 17a*d + b*d - 23c*d - 46d , b  - 24b*c + 18c  + 8a*d + 15b*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                  2                             2     2  
│ │ │ -      31c*d - 28d , a*c + 46b*c + 8c  - 42a*d - 13b*d - 40c*d - 9d , b*c  -
│ │ │ +                 2                   2                             2     2  
│ │ │ +      46c*d - 20d , a*c + 39b*c - 29c  - 42a*d - 22b*d - 18c*d - 2d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                  2         2        2        2    3   3                2   
│ │ │ -      2b*c*d - 23c d - 24a*d  + 12b*d  - 47c*d  - d , c  - 22b*c*d + 18c d -
│ │ │ +                  2         2        2       2     3   3                2   
│ │ │ +      2b*c*d - 33c d - 23a*d  + 13b*d  + 5c*d  - 7d , c  - 22b*c*d - 13c d -
│ │ │        -----------------------------------------------------------------------
│ │ │             2        2        2      3
│ │ │ -      29a*d  + 39b*d  - 21c*d  - 39d )
│ │ │ +      40a*d  + 46b*d  - 21c*d  + 10d )
│ │ │  
│ │ │  o60 : Ideal of S
    
│ │ │              
    
│ │ │                
    i61 : I1 = sub(sub(F, (vars ring F) | sub(pt1, ring F)), S)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o61 = ideal (a  - 21b*c + 45c  - 26a*d + 29b*d - 9c*d - 45d , a*b - 39b*c -
│ │ │ +              2            2                             2               
│ │ │ +o61 = ideal (a  + 30b*c + c  - 47a*d + 33b*d + 28c*d - 8d , a*b - 23b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2              2                  
│ │ │ -      17c  - 23a*d + 37b*d - 5c*d + 18d , b  - 15b*c + 10c  + 15b*d + 28c*d +
│ │ │ +         2                              2   2              2                
│ │ │ +      27c  + 15a*d + 32b*d - 23c*d + 41d , b  - 50b*c - 18c  - 25b*d - 22c*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         
│ │ │ +      - 38d , a*c - 49b*c + 3c  - 9a*d + 37b*d + 46c*d + 50d , b*c  + 3b*c*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       2       2      3   3                2         2
│ │ │ +      - 15c d + 21a*d  - 2b*d  - 2c*d  - 44d , c  + 32b*c*d + 10c d - 35a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2     3
│ │ │ -      + 37b*d  + 46c*d  - 8d )
│ │ │ +             2        2      3
│ │ │ +      + 33b*d  - 14c*d  + 33d )
│ │ │  
│ │ │  o61 : Ideal of S
    
│ │ │              
    
│ │ │                
    i62 : betti res I0
│ │ │ @@ -907,40 +873,39 @@
│ │ │            
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i64 : netList decompose I0
│ │ │  
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -o64 = |ideal (c - 42d, b - 6d, a + 41d)                                                             |
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -      |ideal (c + 38d, b - 41d, a - 41d)                                                            |
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -      |                                  2                      2   2      2                      2 |
│ │ │ -      |ideal (a + 46b + 8c - 7d, b*c - 7c  + 13b*d + 16c*d - 21d , b  - 42c  - 41b*d + 35c*d - 39d )|
│ │ │ -      +---------------------------------------------------------------------------------------------+
    
│ │ │ +      +------------------------------------------------------------------------------------------------+
│ │ │ +o64 = |ideal (c - 42d, b + 26d, a - 30d)                                                               |
│ │ │ +      +------------------------------------------------------------------------------------------------+
│ │ │ +      |ideal (c - 47d, b + 7d, a - 44d)                                                                |
│ │ │ +      +------------------------------------------------------------------------------------------------+
│ │ │ +      |                                     2                      2   2      2                      2 |
│ │ │ +      |ideal (a + 39b - 29c - 24d, b*c - 15c  - 29b*d - 38c*d + 16d , b  - 39c  + 17b*d - 28c*d - 50d )|
│ │ │ +      +------------------------------------------------------------------------------------------------+
│ │ │              
    
│ │ │                
    i65 : netList decompose I1
│ │ │  
│ │ │ -      +------------------------------------------------------+
│ │ │ -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 )|
│ │ │ -      +------------------------------------------------------+
    
│ │ │ +      +---------------------------------+
│ │ │ +o65 = |ideal (c - 9d, b + 15d, a + 27d) |
│ │ │ +      +---------------------------------+
│ │ │ +      |ideal (c + 48d, b + 11d, a - 37d)|
│ │ │ +      +---------------------------------+
│ │ │ +      |ideal (c + 29d, b + 46d, a + 18d)|
│ │ │ +      +---------------------------------+
│ │ │ +      |ideal (c + 24d, b + 46d, a + 33d)|
│ │ │ +      +---------------------------------+
│ │ │ +      |ideal (c + 22d, b + 38d, a - 50d)|
│ │ │ +      +---------------------------------+
│ │ │              
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    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
│ │ │ │ - -- 2.78979s elapsed
│ │ │ │ + -- 2.52317s 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;
│ │ │ │ - -- 2.4524s elapsed
│ │ │ │ + -- 2.14045s 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 = | -27 -13 45 -25 3 38 -20 -30 -41 25 -26 -44 -31 5 14 2 -45 45 21 -27
│ │ │ │ +o44 = | 22 -10 -8 -1 34 44 -21 -1 25 -41 6 -11 -50 -50 43 -28 -6 45 -28 22 42
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -23 -29 34 49 32 19 10 26 19 37 15 -28 -50 -10 -32 18 |
│ │ │ │ +      -29 -32 -28 5 -10 34 15 19 37 26 49 19 5 10 18 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o44 : Matrix kk  <-- kk
│ │ │ │  i45 : Fa = sub(F, (vars S) | pta)
│ │ │ │  
│ │ │ │ -              2              2                              2
│ │ │ │ -o45 = ideal (a  + 14b*c + 25c  - 44a*d + 38b*d + 45c*d - 27d , a*b + 32b*c +
│ │ │ │ +              2              2                             2
│ │ │ │ +o45 = ideal (a  + 43b*c - 41c  - 11a*d + 44b*d - 8c*d + 22d , a*b + 5b*c -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │           2                              2   2              2
│ │ │ │ -      21c  - 23a*d - 31b*d - 41c*d - 13d , b  - 32b*c + 26c  - 50a*d - 29b*d
│ │ │ │ +      28c  + 42a*d - 50b*d + 25c*d - 10d , b  + 10b*c + 15c  + 19a*d - 29b*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                  2                   2                            2     2
│ │ │ │ -      + 2c*d - 20d , a*c - 28b*c + 19c  + 19a*d - 27b*d + 5c*d + 3d , b*c  +
│ │ │ │ +                   2                   2                              2     2
│ │ │ │ +      - 28c*d - 21d , a*c + 49b*c - 10c  + 19a*d + 22b*d - 50c*d + 34d , b*c
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2         2        2        2      3   3                2
│ │ │ │ -      37b*c*d + 34c d + 10a*d  - 45b*d  - 26c*d  - 25d , c  + 18b*c*d + 15c d
│ │ │ │ +                     2         2       2       2    3   3                2
│ │ │ │ +      + 37b*c*d - 32c d + 34a*d  - 6b*d  + 6c*d  - d , c  + 18b*c*d + 26c d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -             2        2        2      3
│ │ │ │ -      - 10a*d  + 49b*d  + 45c*d  - 30d )
│ │ │ │ +          2        2        2    3
│ │ │ │ +      5a*d  - 28b*d  + 45c*d  - d )
│ │ │ │  
│ │ │ │  o45 : Ideal of S
│ │ │ │  i46 : betti res Fa
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o46 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │ @@ -469,43 +469,43 @@
│ │ │ │  
│ │ │ │  o46 : BettiTally
│ │ │ │  i47 : netList decompose Fa -- this one is 5 points on a plane, and another
│ │ │ │  point
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ --------+
│ │ │ │ -o47 = |ideal (c + 19d, b - 37d, a + 4d)
│ │ │ │ +------------+
│ │ │ │ +o47 = |ideal (c + 19d, b - 37d, a)
│ │ │ │  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ --------+
│ │ │ │ +------------+
│ │ │ │        |                             2              2                      2   3
│ │ │ │ -2         2        2      3     2                2         2        2 |
│ │ │ │ -      |ideal (a - 28b + 19c + 48d, b  - 32b*c + 26c  - 15b*d + 43c*d - 44d , c
│ │ │ │ -+ 18b*c*d + 15c d - 29b*d  + 33c*d  + 46d , b*c  + 37b*c*d + 34c d + 33b*d  -
│ │ │ │ -14c*d )|
│ │ │ │ +2         2       2     3     2                2         2        2      3 |
│ │ │ │ +      |ideal (a + 49b - 10c + 39d, b  + 10b*c + 15c  + 50b*d - 40c*d + 46d , c
│ │ │ │ ++ 18b*c*d + 26c d + 30b*d  - 6c*d  + 6d , b*c  + 37b*c*d - 32c d + 45b*d  +
│ │ │ │ +43c*d  - 14d )|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ --------+
│ │ │ │ +------------+
│ │ │ │  i48 : CFa = minimalPrimes Fa
│ │ │ │  
│ │ │ │ -                                                                      2
│ │ │ │ -o48 = {ideal (c + 19d, b - 37d, a + 4d), ideal (a - 28b + 19c + 48d, b  -
│ │ │ │ +                                                                 2
│ │ │ │ +o48 = {ideal (c + 19d, b - 37d, a), ideal (a + 49b - 10c + 39d, b  + 10b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                 2                      2   3                2         2
│ │ │ │ -      32b*c + 26c  - 15b*d + 43c*d - 44d , c  + 18b*c*d + 15c d - 29b*d  +
│ │ │ │ +         2                      2   3                2         2       2
│ │ │ │ +      15c  + 50b*d - 40c*d + 46d , c  + 18b*c*d + 26c d + 30b*d  - 6c*d  +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -           2      3     2                2         2        2
│ │ │ │ -      33c*d  + 46d , b*c  + 37b*c*d + 34c d + 33b*d  - 14c*d )}
│ │ │ │ +        3     2                2         2        2      3
│ │ │ │ +      6d , b*c  + 37b*c*d - 32c d + 45b*d  + 43c*d  - 14d )}
│ │ │ │  
│ │ │ │  o48 : List
│ │ │ │  i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │ │  
│ │ │ │ -o49 = a - 28b + 19c + 48d
│ │ │ │ +o49 = a + 49b - 10c + 39d
│ │ │ │  
│ │ │ │  o49 : S
│ │ │ │  i50 : CFa/degree
│ │ │ │  
│ │ │ │  o50 = {1, 5}
│ │ │ │  
│ │ │ │  o50 : List
│ │ │ │ @@ -516,206 +516,124 @@
│ │ │ │  o51 : List
│ │ │ │  i52 : degree(Fa : (Fa : lin))  -- somewhat simpler(?) way to see the ideal of
│ │ │ │  the 5 points
│ │ │ │  
│ │ │ │  o52 = 5
│ │ │ │  i53 : ptb = randomPointOnRationalVariety compsL441_1
│ │ │ │  
│ │ │ │ -o53 = | 31 42 28 25 19 3 43 -7 -3 -42 -29 -29 14 2 50 5 36 -13 -42 47 13 31
│ │ │ │ +o53 = | 27 12 -34 9 -19 -43 -32 27 40 45 -13 29 -41 -13 22 -49 -4 -4 9 -23 43
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -37 -23 -24 -4 38 -29 -23 21 17 9 0 21 -9 -47 |
│ │ │ │ +      18 -9 -47 43 21 38 17 -20 21 -29 47 0 2 -37 9 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o53 : Matrix kk  <-- kk
│ │ │ │  i54 : Fb = sub(F, (vars S) | ptb)
│ │ │ │  
│ │ │ │ -              2              2                             2
│ │ │ │ -o54 = ideal (a  + 50b*c - 42c  - 29a*d + 3b*d + 28c*d + 31d , a*b - 24b*c -
│ │ │ │ +              2              2                              2
│ │ │ │ +o54 = ideal (a  + 22b*c + 45c  + 29a*d - 43b*d - 34c*d + 27d , a*b + 43b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                             2   2             2
│ │ │ │ -      42c  + 13a*d + 14b*d - 3c*d + 42d , b  - 9b*c - 29c  + 31b*d + 5c*d +
│ │ │ │ +        2                              2   2              2
│ │ │ │ +      9c  + 43a*d - 41b*d + 40c*d + 12d , b  - 37b*c + 17c  + 18b*d - 49c*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
│ │ │ │ +      32d , a*c + 47b*c + 21c  - 20a*d - 23b*d - 13c*d - 19d , b*c  + 21b*c*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       2        2     3   3               2        2
│ │ │ │ +      - 9c d + 38a*d  - 4b*d  - 13c*d  + 9d , c  + 9b*c*d - 29c d + 2a*d  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -             2        2     3
│ │ │ │ -      - 23b*d  - 13c*d  - 7d )
│ │ │ │ +           2       2      3
│ │ │ │ +      47b*d  - 4c*d  + 27d )
│ │ │ │  
│ │ │ │  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
│ │ │ │ -|
│ │ │ │ -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
│ │ │ │  |
│ │ │ │ -      |ideal (a - 7b + 32c + d, c  + 40b*d - 36c*d + 33d , b*c + 45b*d - 16c*d
│ │ │ │ -+ 39d , b  - 20b*d + 29c*d + 38d )
│ │ │ │ +o56 = |ideal (b - 50c - 43d, a + 15c - 46d, c  + 12c*d - 37d )
│ │ │ │  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------+
│ │ │ │ -      |                          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                            2
│ │ │ │ +2   2                             2 |
│ │ │ │ +      |ideal (c  + 46a*d - 39b*d + 2c*d - 24d , b*c - 9a*d + 16b*d + 2c*d + 27d
│ │ │ │ +, a*c + 43a*d + 44b*d - 48c*d + 24d , b  - 4a*d - 40b*d - 9c*d - 39d , a*b +
│ │ │ │ +16a*d + 26b*d + 37c*d - 24d , a  - 25a*d + 47b*d + 34c*d + 8d )|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------+
│ │ │ │ -      |                                   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 )     |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------+
│ │ │ │ +--------------------------------------------------------------+
│ │ │ │ +i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │ │ +
│ │ │ │ +o57 = ++
│ │ │ │ +      ++
│ │ │ │  i58 : pt0 = randomPointOnRationalVariety(compsL441_0)
│ │ │ │  
│ │ │ │ -o58 = | 13 17 -19 -1 -9 -15 -28 -39 -36 1 -47 29 37 -40 35 -31 12 -21 -8 -13
│ │ │ │ +o58 = | 32 -46 33 -7 -2 -29 -20 10 -23 -26 5 -16 1 -18 -3 46 13 -21 5 -22 17
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      14 15 -23 39 11 8 -24 -13 -42 -2 18 46 -18 -29 -33 -22 |
│ │ │ │ +      15 -33 46 -2 -29 -23 18 -42 -2 -13 39 8 -40 -24 -22 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o58 : Matrix kk  <-- kk
│ │ │ │  i59 : pt1 = randomPointOnRationalVariety(compsL441_1)
│ │ │ │  
│ │ │ │ -o59 = | -45 18 -9 38 21 29 50 -8 -5 45 -47 -26 37 -35 -21 28 27 46 -17 -49
│ │ │ │ +o59 = | -8 41 28 -44 50 33 -38 33 -23 1 -2 -47 32 46 30 -22 -2 -14 27 37 15
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -23 15 -50 37 -39 -14 21 10 -31 3 -18 32 0 3 -15 33 |
│ │ │ │ +      -25 -15 33 -23 3 21 -18 -9 3 10 -49 0 -35 -50 32 |
│ │ │ │  
│ │ │ │                 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  + 35b*c + c  + 29a*d - 15b*d - 19c*d + 13d , a*b + 11b*c -
│ │ │ │ +              2             2                              2
│ │ │ │ +o60 = ideal (a  - 3b*c - 26c  - 16a*d - 29b*d + 33c*d + 32d , a*b - 2b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -        2                              2   2              2
│ │ │ │ -      8c  + 14a*d + 37b*d - 36c*d + 17d , b  - 33b*c - 13c  - 18a*d + 15b*d -
│ │ │ │ +        2                            2   2              2
│ │ │ │ +      5c  + 17a*d + b*d - 23c*d - 46d , b  - 24b*c + 18c  + 8a*d + 15b*d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                 2                  2                             2     2
│ │ │ │ -      31c*d - 28d , a*c + 46b*c + 8c  - 42a*d - 13b*d - 40c*d - 9d , b*c  -
│ │ │ │ +                 2                   2                             2     2
│ │ │ │ +      46c*d - 20d , a*c + 39b*c - 29c  - 42a*d - 22b*d - 18c*d - 2d , b*c  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                  2         2        2        2    3   3                2
│ │ │ │ -      2b*c*d - 23c d - 24a*d  + 12b*d  - 47c*d  - d , c  - 22b*c*d + 18c d -
│ │ │ │ +                  2         2        2       2     3   3                2
│ │ │ │ +      2b*c*d - 33c d - 23a*d  + 13b*d  + 5c*d  - 7d , c  - 22b*c*d - 13c d -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │             2        2        2      3
│ │ │ │ -      29a*d  + 39b*d  - 21c*d  - 39d )
│ │ │ │ +      40a*d  + 46b*d  - 21c*d  + 10d )
│ │ │ │  
│ │ │ │  o60 : Ideal of S
│ │ │ │  i61 : I1 = sub(sub(F, (vars ring F) | sub(pt1, ring F)), S)
│ │ │ │  
│ │ │ │ -              2              2                             2
│ │ │ │ -o61 = ideal (a  - 21b*c + 45c  - 26a*d + 29b*d - 9c*d - 45d , a*b - 39b*c -
│ │ │ │ +              2            2                             2
│ │ │ │ +o61 = ideal (a  + 30b*c + c  - 47a*d + 33b*d + 28c*d - 8d , a*b - 23b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                             2   2              2
│ │ │ │ -      17c  - 23a*d + 37b*d - 5c*d + 18d , b  - 15b*c + 10c  + 15b*d + 28c*d +
│ │ │ │ +         2                              2   2              2
│ │ │ │ +      27c  + 15a*d + 32b*d - 23c*d + 41d , b  - 50b*c - 18c  - 25b*d - 22c*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
│ │ │ │ +      - 38d , a*c - 49b*c + 3c  - 9a*d + 37b*d + 46c*d + 50d , b*c  + 3b*c*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       2       2      3   3                2         2
│ │ │ │ +      - 15c d + 21a*d  - 2b*d  - 2c*d  - 44d , c  + 32b*c*d + 10c d - 35a*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -             2        2     3
│ │ │ │ -      + 37b*d  + 46c*d  - 8d )
│ │ │ │ +             2        2      3
│ │ │ │ +      + 33b*d  - 14c*d  + 33d )
│ │ │ │  
│ │ │ │  o61 : Ideal of S
│ │ │ │  i62 : betti res I0
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o62 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │ @@ -731,43 +649,42 @@
│ │ │ │            1: . 4 4 1
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o63 : BettiTally
│ │ │ │  i64 : netList decompose I0
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ ----------------------+
│ │ │ │ -o64 = |ideal (c - 42d, b - 6d, a + 41d)
│ │ │ │ +------------------------+
│ │ │ │ +o64 = |ideal (c - 42d, b + 26d, a - 30d)
│ │ │ │  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ ----------------------+
│ │ │ │ -      |ideal (c + 38d, b - 41d, a - 41d)
│ │ │ │ +------------------------+
│ │ │ │ +      |ideal (c - 47d, b + 7d, a - 44d)
│ │ │ │  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ ----------------------+
│ │ │ │ -      |                                  2                      2   2      2
│ │ │ │ +------------------------+
│ │ │ │ +      |                                     2                      2   2      2
│ │ │ │  2 |
│ │ │ │ -      |ideal (a + 46b + 8c - 7d, b*c - 7c  + 13b*d + 16c*d - 21d , b  - 42c  -
│ │ │ │ -41b*d + 35c*d - 39d )|
│ │ │ │ +      |ideal (a + 39b - 29c - 24d, b*c - 15c  - 29b*d - 38c*d + 16d , b  - 39c
│ │ │ │ ++ 17b*d - 28c*d - 50d )|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ ----------------------+
│ │ │ │ +------------------------+
│ │ │ │  i65 : netList decompose I1
│ │ │ │  
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ -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 )|
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ +      +---------------------------------+
│ │ │ │ +o65 = |ideal (c - 9d, b + 15d, a + 27d) |
│ │ │ │ +      +---------------------------------+
│ │ │ │ +      |ideal (c + 48d, b + 11d, a - 37d)|
│ │ │ │ +      +---------------------------------+
│ │ │ │ +      |ideal (c + 29d, b + 46d, a + 18d)|
│ │ │ │ +      +---------------------------------+
│ │ │ │ +      |ideal (c + 24d, b + 46d, a + 33d)|
│ │ │ │ +      +---------------------------------+
│ │ │ │ +      |ideal (c + 22d, b + 38d, a - 50d)|
│ │ │ │ +      +---------------------------------+
│ │ │ │  i66 : L430 = (trim minors(2, M1)) + groebnerStratum F;
│ │ │ │  
│ │ │ │  o66 : Ideal of T
│ │ │ │  i67 : C = res(I, FastNonminimal => true)
│ │ │ │  
│ │ │ │         1      4      5      2
│ │ │ │  o67 = S  <-- S  <-- S  <-- S  <-- 0
│ │ ├── ./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 8.79941s (cpu); 6.0168s (thread); 0s (gc)
│ │ │ + -- used 8.54693s (cpu); 6.72267s (thread); 0s (gc)
│ │ │  
│ │ │              0  1
│ │ │  o5 = total: 1 66
│ │ │           0: 1  .
│ │ │           1: . 66
│ │ │  
│ │ │  o5 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/_canonical__Curve.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │              
    		
│ │ │                
    i4 : R=FF[x_0..x_(g-1)];
    
│ │ │              
    
│ │ │                
    i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ - -- used 8.79941s (cpu); 6.0168s (thread); 0s (gc)
│ │ │ + -- used 8.54693s (cpu); 6.72267s (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 8.79941s (cpu); 6.0168s (thread); 0s (gc)
│ │ │ │ + -- used 8.54693s (cpu); 6.72267s (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}, .00182097, .000868731)
│ │ │ +o3 = ({5, 2.91596e52, 9}, .00210843, .000915746)
│ │ │  
│ │ │  o3 : Sequence
│ │ │  
│ │ │  i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .0051242, .0353355)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .0066944, .0483865)
│ │ │  
│ │ │  o4 : Sequence
│ │ │  
│ │ │  i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {{.00505599, .0122226}, {.00470716, .00413436}, {.00612262, .00687245},
│ │ │ +o5 = {{.00920591, .0227354}, {.00965341, .00835659}, {.00954085, .0130576},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00521976, .00990609}, {.00508104, .013251}, {.100364, .012526},
│ │ │ +     {.00912636, .0142172}, {.00608001, .0156866}, {.0975923, .0139544},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00513575, .00813997}, {.00513784, .00750516}, {.00491794, .00544569},
│ │ │ +     {.00465453, .00969963}, {.00518961, .00867869}, {.00413464, .00613861},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00753335, .00820152}}
│ │ │ +     {.00557352, .00934145}}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .01492759439999993
│ │ │ +o6 = .01607511159999984
│ │ │  
│ │ │  o6 : RR (of precision 53)
│ │ │  
│ │ │  i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .008820477599999954
│ │ │ +o7 = .01218661900000004
│ │ │  
│ │ │  o7 : RR (of precision 53)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html
│ │ │ @@ -93,57 +93,57 @@
│ │ │  o2 : Sequence
│ │ │              
    
│ │ │                
    i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │  
│ │ │ -o3 = ({5, 2.91596e52, 9}, .00182097, .000868731)
│ │ │ +o3 = ({5, 2.91596e52, 9}, .00210843, .000915746)
│ │ │  
│ │ │  o3 : Sequence
    
│ │ │              
    
│ │ │                
    i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .0051242, .0353355)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .0066944, .0483865)
│ │ │  
│ │ │  o4 : Sequence
    
│ │ │              
    
│ │ │                
    i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {{.00505599, .0122226}, {.00470716, .00413436}, {.00612262, .00687245},
│ │ │ +o5 = {{.00920591, .0227354}, {.00965341, .00835659}, {.00954085, .0130576},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00521976, .00990609}, {.00508104, .013251}, {.100364, .012526},
│ │ │ +     {.00912636, .0142172}, {.00608001, .0156866}, {.0975923, .0139544},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00513575, .00813997}, {.00513784, .00750516}, {.00491794, .00544569},
│ │ │ +     {.00465453, .00969963}, {.00518961, .00867869}, {.00413464, .00613861},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00753335, .00820152}}
│ │ │ +     {.00557352, .00934145}}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .01492759439999993
│ │ │ +o6 = .01607511159999984
│ │ │  
│ │ │  o6 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .008820477599999954
│ │ │ +o7 = .01218661900000004
│ │ │  
│ │ │  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}, .00182097, .000868731)
│ │ │ │ +o3 = ({5, 2.91596e52, 9}, .00210843, .000915746)
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │ │  
│ │ │ │ -o4 = ({50, 2.30853e454, 98}, .0051242, .0353355)
│ │ │ │ +o4 = ({50, 2.30853e454, 98}, .0066944, .0483865)
│ │ │ │  
│ │ │ │  o4 : Sequence
│ │ │ │  i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │ │  
│ │ │ │ -o5 = {{.00505599, .0122226}, {.00470716, .00413436}, {.00612262, .00687245},
│ │ │ │ +o5 = {{.00920591, .0227354}, {.00965341, .00835659}, {.00954085, .0130576},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {.00521976, .00990609}, {.00508104, .013251}, {.100364, .012526},
│ │ │ │ +     {.00912636, .0142172}, {.00608001, .0156866}, {.0975923, .0139544},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {.00513575, .00813997}, {.00513784, .00750516}, {.00491794, .00544569},
│ │ │ │ +     {.00465453, .00969963}, {.00518961, .00867869}, {.00413464, .00613861},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {.00753335, .00820152}}
│ │ │ │ +     {.00557352, .00934145}}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : 1/10*sum(L,t->t_0)
│ │ │ │  
│ │ │ │ -o6 = .01492759439999993
│ │ │ │ +o6 = .01607511159999984
│ │ │ │  
│ │ │ │  o6 : RR (of precision 53)
│ │ │ │  i7 : 1/10*sum(L,t->t_1)
│ │ │ │  
│ │ │ │ -o7 = .008820477599999954
│ │ │ │ +o7 = .01218661900000004
│ │ │ │  
│ │ │ │  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 1.56147s (cpu); 1.25457s (thread); 0s (gc)
│ │ │ + -- used 1.61722s (cpu); 1.31764s (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
│ │ │ @@ -82,15 +82,15 @@
│ │ │            
    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 1.56147s (cpu); 1.25457s (thread); 0s (gc)
│ │ │ │ + -- used 1.61722s (cpu); 1.31764s (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 1.58351s (cpu); 1.29415s (thread); 0s (gc)
│ │ │ + -- used 1.8755s (cpu); 1.55393s (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
│ │ │ @@ -93,15 +93,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── 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)
│ │ │ │ - -- 3.12043s elapsed
│ │ │ │ + -- 2.96244s 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)
│ │ │ │ - -- 2.62392s elapsed
│ │ │ │ + -- 2.16267s 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 0.612163s (cpu); 0.416347s (thread); 0s (gc)
│ │ │ + -- used 0.923715s (cpu); 0.501459s (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
│ │ │ @@ -189,15 +189,15 @@
│ │ │  
│ │ │  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 0.612163s (cpu); 0.416347s (thread); 0s (gc)
│ │ │ │ + -- used 0.923715s (cpu); 0.501459s (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 0.00322822s (cpu); 0.00322771s (thread); 0s (gc)
│ │ │ + -- used 0.00407983s (cpu); 0.00407295s (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 0.996963s (cpu); 0.801505s (thread); 0s (gc)
│ │ │ + -- used 1.21742s (cpu); 0.954415s (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 0.273936s (cpu); 0.210148s (thread); 0s (gc)
│ │ │ + -- used 0.356383s (cpu); 0.256687s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 31
│ │ │  
│ │ │  i36 :
│ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/html/_rational__Points.html
│ │ │ @@ -178,15 +178,15 @@
│ │ │  o13 : Ideal of ---[u ..u  ]
│ │ │                 101  0   10
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ - -- used 1.56147s (cpu); 1.25457s (thread); 0s (gc)
│ │ │ + -- used 1.61722s (cpu); 1.31764s (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 1.58351s (cpu); 1.29415s (thread); 0s (gc)
│ │ │ + -- used 1.8755s (cpu); 1.55393s (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 1.58351s (cpu); 1.29415s (thread); 0s (gc)
│ │ │ │ + -- used 1.8755s (cpu); 1.55393s (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 +1,24 @@
│ │ │  -- -*- M2-comint -*- hash: 9542801742429495161
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1765726966
│ │ │ + -- setting random seed to 1767789383
│ │ │  
│ │ │ -o1 = 1765726966
│ │ │ +o1 = 1767789383
│ │ │  
│ │ │  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 2.93525s (cpu); 1.5644s (thread); 0s (gc)
│ │ │ + -- used 3.87582s (cpu); 1.82874s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = Tally{4 => 50 }
│ │ │ -           5 => 198
│ │ │ -           6 => 177
│ │ │ -           7 => 62
│ │ │ -           8 => 12
│ │ │ +           5 => 217
│ │ │ +           6 => 160
│ │ │ +           7 => 58
│ │ │ +           8 => 14
│ │ │             9 => 1
│ │ │  
│ │ │  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 1765726971
│ │ │ + -- setting random seed to 1767789387
│ │ │  
│ │ │ -o1 = 1765726971
│ │ │ +o1 = 1767789387
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -15,13 +15,12 @@
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -      2
│ │ │ -o4 = a c
│ │ │ +o4 = a*b*c
│ │ │  
│ │ │  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 1765726980
│ │ │ + -- setting random seed to 1767789395
│ │ │  
│ │ │ -o1 = 1765726980
│ │ │ +o1 = 1767789395
│ │ │  
│ │ │  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*c*d)
│ │ │ +o5 = ideal(a*c*e)
│ │ │  
│ │ │  o5 : Ideal of S
│ │ │  
│ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (c*e, a*d, d*e, b*e, c*d)
│ │ │ +o6 = ideal (b*c, a*b, a*c, d*e, b*d)
│ │ │  
│ │ │  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 1765726975
│ │ │ + -- setting random seed to 1767789391
│ │ │  
│ │ │ -o1 = 1765726975
│ │ │ +o1 = 1767789391
│ │ │  
│ │ │  i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │ @@ -39,15 +39,15 @@
│ │ │  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 4.75542s (cpu); 3.39881s (thread); 0s (gc)
│ │ │ + -- used 5.17442s (cpu); 3.43358s (thread); 0s (gc)
│ │ │  
│ │ │  i9 : #T
│ │ │  
│ │ │  o9 = 168
│ │ │  
│ │ │  i10 : T
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html
│ │ │ @@ -71,17 +71,17 @@
│ │ │          
    
│ │ │            
    Chooses a random monomial.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1765726971
│ │ │ + -- setting random seed to 1767789387
│ │ │  
│ │ │ -o1 = 1765726971
    
│ │ │ +o1 = 1767789387
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -98,16 +98,15 @@
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -      2
│ │ │ -o4 = a c
│ │ │ +o4 = a*b*c
│ │ │  
│ │ │  o4 : S
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,31 +11,30 @@
│ │ │ │            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 1765726971
│ │ │ │ + -- setting random seed to 1767789387
│ │ │ │  
│ │ │ │ -o1 = 1765726971
│ │ │ │ +o1 = 1767789387
│ │ │ │  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 c
│ │ │ │ +o4 = a*b*c
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -71,17 +71,17 @@
│ │ │          
    
│ │ │            
    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 1765726980
│ │ │ + -- setting random seed to 1767789395
│ │ │  
│ │ │ -o1 = 1765726980
    
│ │ │ +o1 = 1767789395
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -108,24 +108,24 @@
│ │ │              
    
│ │ │                
    i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │  low degree gens generated everything
│ │ │  
│ │ │ -o5 = ideal(a*c*d)
│ │ │ +o5 = ideal(a*c*e)
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (c*e, a*d, d*e, b*e, c*d)
│ │ │ +o6 = ideal (b*c, a*b, a*c, d*e, b*d)
│ │ │  
│ │ │  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 1765726980
│ │ │ │ + -- setting random seed to 1767789395
│ │ │ │  
│ │ │ │ -o1 = 1765726980
│ │ │ │ +o1 = 1767789395
│ │ │ │  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*c*d)
│ │ │ │ +o5 = ideal(a*c*e)
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │ │  
│ │ │ │ -o6 = ideal (c*e, a*d, d*e, b*e, c*d)
│ │ │ │ +o6 = ideal (b*c, a*b, a*c, d*e, b*d)
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -79,17 +79,17 @@
│ │ │            
    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 1765726975
│ │ │ + -- setting random seed to 1767789391
│ │ │  
│ │ │ -o1 = 1765726975
    
│ │ │ +o1 = 1767789391
│ │ │              
    
│ │ │                
    i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │ @@ -147,15 +147,15 @@
│ │ │  
│ │ │  o7 : MonomialIdeal of S
    
│ │ │              
    
│ │ │                
    i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 4.75542s (cpu); 3.39881s (thread); 0s (gc)
    
│ │ │ + -- used 5.17442s (cpu); 3.43358s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i9 : #T
│ │ │  
│ │ │  o9 = 168
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,17 +35,17 @@
│ │ │ │  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 1765726975
│ │ │ │ + -- setting random seed to 1767789391
│ │ │ │  
│ │ │ │ -o1 = 1765726975
│ │ │ │ +o1 = 1767789391
│ │ │ │  i2 : S=ZZ/2[vars(0..3)]
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : J = monomialIdeal"ab,ad, bcd"
│ │ │ │  
│ │ │ │ @@ -74,15 +74,15 @@
│ │ │ │  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 4.75542s (cpu); 3.39881s (thread); 0s (gc)
│ │ │ │ + -- used 5.17442s (cpu); 3.43358s (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
│ │ │ @@ -54,17 +54,17 @@
│ │ │          
    
│ │ │            
    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.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -72,21 +72,21 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1765726966
│ │ │ + -- setting random seed to 1767789383
│ │ │  
│ │ │ -o1 = 1765726966
    
│ │ │ +o1 = 1767789383
│ │ │              
    
│ │ │                
    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 2.93525s (cpu); 1.5644s (thread); 0s (gc)
│ │ │ + -- used 3.87582s (cpu); 1.82874s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = Tally{4 => 50 }
│ │ │ -           5 => 198
│ │ │ -           6 => 177
│ │ │ -           7 => 62
│ │ │ -           8 => 12
│ │ │ +           5 => 217
│ │ │ +           6 => 160
│ │ │ +           7 => 58
│ │ │ +           8 => 14
│ │ │             9 => 1
│ │ │  
│ │ │  o4 : Tally
    
│ │ │              
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,27 +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 1765726966
│ │ │ │ + -- setting random seed to 1767789383
│ │ │ │  
│ │ │ │ -o1 = 1765726966
│ │ │ │ +o1 = 1767789383
│ │ │ │  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 2.93525s (cpu); 1.5644s (thread); 0s (gc)
│ │ │ │ + -- used 3.87582s (cpu); 1.82874s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = Tally{4 => 50 }
│ │ │ │ -           5 => 198
│ │ │ │ -           6 => 177
│ │ │ │ -           7 => 62
│ │ │ │ -           8 => 12
│ │ │ │ +           5 => 217
│ │ │ │ +           6 => 160
│ │ │ │ +           7 => 58
│ │ │ │ +           8 => 14
│ │ │ │             9 => 1
│ │ │ │  
│ │ │ │  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
│ │ ├── ./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)
│ │ │ - -- 1.63992s elapsed
│ │ │ + -- 1.45654s elapsed
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : elapsedTime dim I
│ │ │ - -- 3.02509s elapsed
│ │ │ + -- 3.91536s 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)) );
│ │ │ - -- 2.0239s elapsed
│ │ │ + -- 1.77502s 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)
│ │ │ - -- 3.12043s elapsed
│ │ │ + -- 2.96244s 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)
│ │ │ - -- 2.62392s elapsed
│ │ │ + -- 2.16267s 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
│ │ │ @@ -95,23 +95,23 @@
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │              
    
│ │ │                
    i4 : elapsedTime dimViaBezout(I)
│ │ │ - -- 1.63992s elapsed
│ │ │ + -- 1.45654s elapsed
│ │ │  
│ │ │  o4 = 4
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime dim I
│ │ │ - -- 3.02509s elapsed
│ │ │ + -- 3.91536s 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)
│ │ │ │ - -- 1.63992s elapsed
│ │ │ │ + -- 1.45654s elapsed
│ │ │ │  
│ │ │ │  o4 = 4
│ │ │ │  i5 : elapsedTime dim I
│ │ │ │ - -- 3.02509s elapsed
│ │ │ │ + -- 3.91536s 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
│ │ │ @@ -155,15 +155,15 @@
│ │ │  
│ │ │  o9 : Ideal of T
    
│ │ │              
    
│ │ │                
    i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J = extendIdealByNonZeroMinor(4, M, J)) );
│ │ │ - -- 2.0239s elapsed
    
│ │ │ + -- 1.77502s 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)) );
│ │ │ │ - -- 2.0239s elapsed
│ │ │ │ + -- 1.77502s 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
│ │ │ @@ -144,27 +144,27 @@
│ │ │  
│ │ │  o7 : Ideal of S
│ │ │              
    
│ │ │                
    i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable)
│ │ │ - -- 3.12043s elapsed
│ │ │ + -- 2.96244s 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)
│ │ │ - -- 2.62392s elapsed
│ │ │ + -- 2.16267s 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 0.612163s (cpu); 0.416347s (thread); 0s (gc)
│ │ │ + -- used 0.923715s (cpu); 0.501459s (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 0.00322822s (cpu); 0.00322771s (thread); 0s (gc)
│ │ │ + -- used 0.00407983s (cpu); 0.00407295s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 110462212541120451001
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Over number fields
    
│ │ │ @@ -348,15 +348,15 @@
│ │ │  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 0.996963s (cpu); 0.801505s (thread); 0s (gc)
    
│ │ │ + -- used 1.21742s (cpu); 0.954415s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i33 : #oo
│ │ │  
│ │ │  o33 = 31
    
│ │ │ @@ -373,15 +373,15 @@
│ │ │  
│ │ │  o34 : Ideal of GF 1048969271299456081[x..z, w]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true)
│ │ │ - -- used 0.273936s (cpu); 0.210148s (thread); 0s (gc)
│ │ │ + -- used 0.356383s (cpu); 0.256687s (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 0.00322822s (cpu); 0.00322771s (thread); 0s (gc)
│ │ │ │ + -- used 0.00407983s (cpu); 0.00407295s (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 0.996963s (cpu); 0.801505s (thread); 0s (gc)
│ │ │ │ + -- used 1.21742s (cpu); 0.954415s (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 0.273936s (cpu); 0.210148s (thread); 0s (gc)
│ │ │ │ + -- used 0.356383s (cpu); 0.256687s (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 0.826819s (cpu); 0.679734s (thread); 0s (gc)
│ │ │ + -- used 1.19034s (cpu); 0.951955s (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 1.03s (cpu); 0.739203s (thread); 0s (gc)
│ │ │ + -- used 1.25011s (cpu); 0.937645s (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 1.73256s (cpu); 1.40832s (thread); 0s (gc)
│ │ │ + -- used 2.07546s (cpu); 1.62957s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S[w ..w ]
│ │ │                    0   3
│ │ │  
│ │ │  i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ - -- used 1.5974s (cpu); 1.21866s (thread); 0s (gc)
│ │ │ + -- used 1.85611s (cpu); 1.53707s (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 0.0275853s (cpu); 0.0256772s (thread); 0s (gc)
│ │ │ + -- used 0.307893s (cpu); 0.0529359s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S[w ..w ]
│ │ │                   0   6
│ │ │  
│ │ │  i5 : time V2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.113109s (cpu); 0.112422s (thread); 0s (gc)
│ │ │ + -- used 0.159495s (cpu); 0.145905s (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 0.0195203s (cpu); 0.0182383s (thread); 0s (gc)
│ │ │ + -- used 0.164237s (cpu); 0.0400787s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of S[w ..w ]
│ │ │                   0   2
│ │ │  
│ │ │  i10 : time I2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.00761992s (cpu); 0.00733771s (thread); 0s (gc)
│ │ │ + -- used 0.0230433s (cpu); 0.0110679s (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
│ │ │ @@ -587,15 +587,15 @@
│ │ │          
    
│ │ │            
    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 0.826819s (cpu); 0.679734s (thread); 0s (gc)
│ │ │ │ + -- used 1.19034s (cpu); 0.951955s (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
│ │ │ @@ -151,15 +151,15 @@
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i48 : time sing2 = ideal singularLocus strictTransform2;
│ │ │ - -- used 0.826819s (cpu); 0.679734s (thread); 0s (gc)
│ │ │ + -- used 1.19034s (cpu); 0.951955s (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 1.03s (cpu); 0.739203s (thread); 0s (gc)
│ │ │ + -- used 1.25011s (cpu); 0.937645s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S[w ..w ]
│ │ │                   0   4
    
│ │ │              
    
│ │ │ @@ -185,24 +185,24 @@
│ │ │  
│ │ │  o11 : Ideal of S
│ │ │              
    
│ │ │                
    i12 : time reesIdeal (I, I_0);
│ │ │ - -- used 1.73256s (cpu); 1.40832s (thread); 0s (gc)
│ │ │ + -- used 2.07546s (cpu); 1.62957s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S[w ..w ]
│ │ │                    0   3
    
│ │ │              
    
│ │ │                
    i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ - -- used 1.5974s (cpu); 1.21866s (thread); 0s (gc)
│ │ │ + -- used 1.85611s (cpu); 1.53707s (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 1.03s (cpu); 0.739203s (thread); 0s (gc)
│ │ │ │ + -- used 1.25011s (cpu); 0.937645s (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 1.73256s (cpu); 1.40832s (thread); 0s (gc)
│ │ │ │ + -- used 2.07546s (cpu); 1.62957s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 : Ideal of S[w ..w ]
│ │ │ │                    0   3
│ │ │ │  i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ │ - -- used 1.5974s (cpu); 1.21866s (thread); 0s (gc)
│ │ │ │ + -- used 1.85611s (cpu); 1.53707s (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
│ │ │ @@ -110,24 +110,24 @@
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time V1 = reesIdeal i;
│ │ │ - -- used 0.0275853s (cpu); 0.0256772s (thread); 0s (gc)
│ │ │ + -- used 0.307893s (cpu); 0.0529359s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S[w ..w ]
│ │ │                   0   6
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time V2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.113109s (cpu); 0.112422s (thread); 0s (gc)
│ │ │ + -- used 0.159495s (cpu); 0.145905s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : Ideal of S[w ..w ]
│ │ │                   0   6
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │ @@ -164,24 +164,24 @@
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time I1 = reesIdeal i;
│ │ │ - -- used 0.0195203s (cpu); 0.0182383s (thread); 0s (gc)
│ │ │ + -- used 0.164237s (cpu); 0.0400787s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of S[w ..w ]
│ │ │                   0   2
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time I2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.00761992s (cpu); 0.00733771s (thread); 0s (gc)
│ │ │ + -- used 0.0230433s (cpu); 0.0110679s (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 0.0275853s (cpu); 0.0256772s (thread); 0s (gc)
│ │ │ │ + -- used 0.307893s (cpu); 0.0529359s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : Ideal of S[w ..w ]
│ │ │ │                   0   6
│ │ │ │  i5 : time V2 = reesIdeal(i,i_0);
│ │ │ │ - -- used 0.113109s (cpu); 0.112422s (thread); 0s (gc)
│ │ │ │ + -- used 0.159495s (cpu); 0.145905s (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 0.0195203s (cpu); 0.0182383s (thread); 0s (gc)
│ │ │ │ + -- used 0.164237s (cpu); 0.0400787s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of S[w ..w ]
│ │ │ │                   0   2
│ │ │ │  i10 : time I2 = reesIdeal(i,i_0);
│ │ │ │ - -- used 0.00761992s (cpu); 0.00733771s (thread); 0s (gc)
│ │ │ │ + -- used 0.0230433s (cpu); 0.0110679s (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 = .2590714710000002
│ │ │ +o8 = .5449336249999999
│ │ │  
│ │ │  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 = .07735652088059701
│ │ │ +o11 = .1107588754181819
│ │ │  
│ │ │  o11 : RR (of precision 53)
│ │ │  
│ │ │  i12 : time regularity I2  
│ │ │ - -- used 0.00237225s (cpu); 0.00237209s (thread); 0s (gc)
│ │ │ + -- used 0.00270185s (cpu); 0.00271108s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = 4
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Regularity/html/_m__Regularity.html
│ │ │ @@ -176,15 +176,15 @@
│ │ │  o7 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : benchmark "mRegularity I1"
│ │ │  
│ │ │ -o8 = .2590714710000002
│ │ │ +o8 = .5449336249999999
│ │ │  
│ │ │  o8 : RR (of precision 53)
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    This is an example where regularity is faster than mRegularity.
    
│ │ │          
│ │ │ @@ -204,23 +204,23 @@
│ │ │  o10 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : benchmark " mRegularity I2"
│ │ │  
│ │ │ -o11 = .07735652088059701
│ │ │ +o11 = .1107588754181819
│ │ │  
│ │ │  o11 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i12 : time regularity I2  
│ │ │ - -- used 0.00237225s (cpu); 0.00237209s (thread); 0s (gc)
│ │ │ + -- used 0.00270185s (cpu); 0.00271108s (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 = .2590714710000002
│ │ │ │ +o8 = .5449336249999999
│ │ │ │  
│ │ │ │  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 = .07735652088059701
│ │ │ │ +o11 = .1107588754181819
│ │ │ │  
│ │ │ │  o11 : RR (of precision 53)
│ │ │ │  i12 : time regularity I2
│ │ │ │ - -- used 0.00237225s (cpu); 0.00237209s (thread); 0s (gc)
│ │ │ │ + -- used 0.00270185s (cpu); 0.00271108s (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 0.127171s (cpu); 0.070403s (thread); 0s (gc)
│ │ │ + -- used 0.141126s (cpu); 0.0820603s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1)
│ │ │ - -- used 0.124664s (cpu); 0.0700762s (thread); 0s (gc)
│ │ │ + -- used 0.153371s (cpu); 0.0934494s (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 0.153844s (cpu); 0.0993609s (thread); 0s (gc)
│ │ │ + -- used 0.195631s (cpu); 0.120686s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
│ │ │  
│ │ │  i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ - -- used 0.197861s (cpu); 0.13749s (thread); 0s (gc)
│ │ │ + -- used 0.177527s (cpu); 0.118732s (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 0.127688s (cpu); 0.062s (thread); 0s (gc)
│ │ │ + -- used 0.16126s (cpu); 0.0761374s (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 0.113843s (cpu); 0.0571957s (thread); 0s (gc)
│ │ │ + -- used 0.157559s (cpu); 0.0697339s (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 0.158592s (cpu); 0.109834s (thread); 0s (gc)
│ │ │ + -- used 0.319342s (cpu); 0.165958s (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 0.150222s (cpu); 0.0893773s (thread); 0s (gc)
│ │ │ + -- used 0.179542s (cpu); 0.104182s (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 5.31048s (cpu); 4.82715s (thread); 0s (gc)
│ │ │ + -- used 5.7443s (cpu); 5.31956s (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 1.12682s (cpu); 0.99711s (thread); 0s (gc)
│ │ │ + -- used 1.26197s (cpu); 1.18582s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : -- X-resultant of V
│ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ - -- used 0.235111s (cpu); 0.179065s (thread); 0s (gc)
│ │ │ + -- used 0.347356s (cpu); 0.265667s (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 0.285058s (cpu); 0.174585s (thread); 0s (gc)
│ │ │ + -- used 0.31243s (cpu); 0.238269s (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 0.00807841s (cpu); 0.00807736s (thread); 0s (gc)
│ │ │ + -- used 0.0114216s (cpu); 0.0114235s (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 0.0087506s (cpu); 0.00875122s (thread); 0s (gc)
│ │ │ + -- used 0.0122756s (cpu); 0.0122773s (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 0.461801s (cpu); 0.427218s (thread); 0s (gc)
│ │ │ + -- used 0.508756s (cpu); 0.446603s (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 0.170711s (cpu); 0.119239s (thread); 0s (gc)
│ │ │ + -- used 0.188519s (cpu); 0.129596s (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 0.204333s (cpu); 0.146742s (thread); 0s (gc)
│ │ │ + -- used 0.243304s (cpu); 0.178114s (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 0.0566354s (cpu); 0.0566377s (thread); 0s (gc)
│ │ │ + -- used 0.0829479s (cpu); 0.082949s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o5 : Ideal of ----[a ..a ]
│ │ │                3331  0   9
│ │ │  
│ │ │  i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ - -- used 0.611958s (cpu); 0.563695s (thread); 0s (gc)
│ │ │ + -- used 0.760722s (cpu); 0.688474s (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 0.110649s (cpu); 0.0580797s (thread); 0s (gc)
│ │ │ + -- used 0.141521s (cpu); 0.0732298s (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 0.040301s (cpu); 0.0403043s (thread); 0s (gc)
│ │ │ + -- used 0.0533498s (cpu); 0.0533558s (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 0.0190728s (cpu); 0.0190737s (thread); 0s (gc)
│ │ │ + -- used 0.0254632s (cpu); 0.0254639s (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 0.0170671s (cpu); 0.0170726s (thread); 0s (gc)
│ │ │ + -- used 0.0215474s (cpu); 0.0215521s (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 0.0393717s (cpu); 0.0393716s (thread); 0s (gc)
│ │ │ + -- used 0.0537307s (cpu); 0.0535482s (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 0.00787565s (cpu); 0.00787232s (thread); 0s (gc)
│ │ │ + -- used 0.00983881s (cpu); 0.00983762s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = true
│ │ │  
│ │ │  i3 : -- random quadric in G(1,3)
│ │ │       w' = random(2,Grass(1,3))
│ │ │  
│ │ │         2     5           10 2     2                          2      3        
│ │ │ @@ -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 0.00654309s (cpu); 0.00654341s (thread); 0s (gc)
│ │ │ + -- used 0.00874447s (cpu); 0.00874582s (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 0.0197451s (cpu); 0.0197449s (thread); 0s (gc)
│ │ │ + -- used 0.0235616s (cpu); 0.0235618s (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 0.00462514s (cpu); 0.00462065s (thread); 0s (gc)
│ │ │ + -- used 0.00432433s (cpu); 0.00432479s (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 0.00237921s (cpu); 0.00238132s (thread); 0s (gc)
│ │ │ + -- used 0.00289871s (cpu); 0.00289702s (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 0.00452289s (cpu); 0.00452084s (thread); 0s (gc)
│ │ │ + -- used 0.00598204s (cpu); 0.00598267s (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 0.111022s (cpu); 0.0535001s (thread); 0s (gc)
│ │ │ + -- used 0.12953s (cpu); 0.0626089s (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 0.0361963s (cpu); 0.0361951s (thread); 0s (gc)
│ │ │ + -- used 0.0397103s (cpu); 0.0397081s (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 0.142146s (cpu); 0.142149s (thread); 0s (gc)
│ │ │ + -- used 0.179115s (cpu); 0.179121s (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 0.316083s (cpu); 0.220122s (thread); 0s (gc)
│ │ │ + -- used 0.381587s (cpu); 0.230058s (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 0.199752s (cpu); 0.138876s (thread); 0s (gc)
│ │ │ + -- used 0.181192s (cpu); 0.106558s (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 0.514703s (cpu); 0.461162s (thread); 0s (gc)
│ │ │ + -- used 0.4153s (cpu); 0.355085s (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 0.590923s (cpu); 0.537731s (thread); 0s (gc)
│ │ │ + -- used 0.648333s (cpu); 0.581225s (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 0.0229913s (cpu); 0.0229909s (thread); 0s (gc)
│ │ │ + -- used 0.0278475s (cpu); 0.0278464s (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 2.72711s (cpu); 2.06378s (thread); 0s (gc)
│ │ │ + -- used 2.6244s (cpu); 2.06772s (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 0.58204s (cpu); 0.40137s (thread); 0s (gc)
│ │ │ + -- used 0.445132s (cpu); 0.384063s (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 0.028783s (cpu); 0.0287839s (thread); 0s (gc)
│ │ │ + -- used 0.0610464s (cpu); 0.0610476s (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 0.120285s (cpu); 0.0736976s (thread); 0s (gc)
│ │ │ + -- used 0.156243s (cpu); 0.0923681s (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 0.0319774s (cpu); 0.0319788s (thread); 0s (gc)
│ │ │ + -- used 0.043272s (cpu); 0.0432777s (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 0.11267s (cpu); 0.0697575s (thread); 0s (gc)
│ │ │ + -- used 0.13669s (cpu); 0.0656883s (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 0.148995s (cpu); 0.103293s (thread); 0s (gc)
│ │ │ + -- used 0.172755s (cpu); 0.115538s (thread); 0s (gc)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html
│ │ │ @@ -86,24 +86,24 @@
│ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │                    0   3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2);
│ │ │ - -- used 0.127171s (cpu); 0.070403s (thread); 0s (gc)
    
│ │ │ + -- used 0.141126s (cpu); 0.0820603s (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 0.124664s (cpu); 0.0700762s (thread); 0s (gc)
│ │ │ + -- used 0.153371s (cpu); 0.0934494s (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   
│ │ │ @@ -130,26 +130,26 @@
│ │ │            
    
│ │ │          
    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 0.127171s (cpu); 0.070403s (thread); 0s (gc)
│ │ │ │ + -- used 0.141126s (cpu); 0.0820603s (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 0.124664s (cpu); 0.0700762s (thread); 0s (gc)
│ │ │ │ + -- used 0.153371s (cpu); 0.0934494s (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 0.153844s (cpu); 0.0993609s (thread); 0s (gc)
│ │ │ │ + -- used 0.195631s (cpu); 0.120686s (thread); 0s (gc)
│ │ │ │  i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
│ │ │ │  i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ │ - -- used 0.197861s (cpu); 0.13749s (thread); 0s (gc)
│ │ │ │ + -- used 0.177527s (cpu); 0.118732s (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
│ │ │ @@ -90,15 +90,15 @@
│ │ │  o2 : Ideal of P3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : time cayleyTrick(P1xP1,1,Duality=>true);
│ │ │ - -- used 0.153844s (cpu); 0.0993609s (thread); 0s (gc)
    
│ │ │ + -- used 0.195631s (cpu); 0.120686s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
    
│ │ │              
    
│ │ │                
    i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ - -- used 0.197861s (cpu); 0.13749s (thread); 0s (gc)
    
│ │ │ + -- used 0.177527s (cpu); 0.118732s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i8 : assert(oo == (P1xP1xP2,P1xP1xP2'))
    
│ │ │              
    
│ │ │                
    i3 : -- Chow equations of C
│ │ │       time eqsC = chowEquations chowForm C
│ │ │ - -- used 0.127688s (cpu); 0.062s (thread); 0s (gc)
│ │ │ + -- used 0.16126s (cpu); 0.0761374s (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 
│ │ │ @@ -162,15 +162,15 @@
│ │ │  o5 : Ideal of P3
    
│ │ │              
    
│ │ │                
    i6 : -- Chow equations of D
│ │ │       time eqsD = chowEquations chowForm D
│ │ │ - -- used 0.113843s (cpu); 0.0571957s (thread); 0s (gc)
│ │ │ + -- used 0.157559s (cpu); 0.0697339s (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  -
│ │ │ @@ -222,30 +222,30 @@
│ │ │  o9 : Ideal of P3
    
│ │ │              
    
│ │ │                
    i10 : -- tangential Chow forms of Q
│ │ │        time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2))
│ │ │ - -- used 0.158592s (cpu); 0.109834s (thread); 0s (gc)
│ │ │ + -- used 0.319342s (cpu); 0.165958s (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 0.150222s (cpu); 0.0893773s (thread); 0s (gc)
│ │ │ + -- used 0.179542s (cpu); 0.104182s (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 0.127688s (cpu); 0.062s (thread); 0s (gc)
│ │ │ │ + -- used 0.16126s (cpu); 0.0761374s (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 0.113843s (cpu); 0.0571957s (thread); 0s (gc)
│ │ │ │ + -- used 0.157559s (cpu); 0.0697339s (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 0.158592s (cpu); 0.109834s (thread); 0s (gc)
│ │ │ │ + -- used 0.319342s (cpu); 0.165958s (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 0.150222s (cpu); 0.0893773s (thread); 0s (gc)
│ │ │ │ + -- used 0.179542s (cpu); 0.104182s (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
│ │ │ @@ -97,15 +97,15 @@
│ │ │                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 5.31048s (cpu); 4.82715s (thread); 0s (gc)
│ │ │ + -- used 5.7443s (cpu); 5.31956s (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      +
│ │ │ @@ -227,22 +227,22 @@
│ │ │         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 1.12682s (cpu); 0.99711s (thread); 0s (gc)
    
│ │ │ + -- used 1.26197s (cpu); 1.18582s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : -- X-resultant of V
│ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ - -- used 0.235111s (cpu); 0.179065s (thread); 0s (gc)
    
│ │ │ + -- used 0.347356s (cpu); 0.265667s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : -- three generic ternary quadrics
│ │ │       F = genericPolynomials({2,2,2},ZZ/3331)
│ │ │  
│ │ │ @@ -257,15 +257,15 @@
│ │ │  o6 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : -- resultant of the three forms
│ │ │       time resF = resultant F;
│ │ │ - -- used 0.285058s (cpu); 0.174585s (thread); 0s (gc)
    
│ │ │ + -- used 0.31243s (cpu); 0.238269s (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 5.31048s (cpu); 4.82715s (thread); 0s (gc)
│ │ │ │ + -- used 5.7443s (cpu); 5.31956s (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 1.12682s (cpu); 0.99711s (thread); 0s (gc)
│ │ │ │ + -- used 1.26197s (cpu); 1.18582s (thread); 0s (gc)
│ │ │ │  i5 : -- X-resultant of V
│ │ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ │ - -- used 0.235111s (cpu); 0.179065s (thread); 0s (gc)
│ │ │ │ + -- used 0.347356s (cpu); 0.265667s (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 0.285058s (cpu); 0.174585s (thread); 0s (gc)
│ │ │ │ + -- used 0.31243s (cpu); 0.238269s (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
│ │ │ @@ -83,15 +83,15 @@
│ │ │  
│ │ │  o2 : ZZ[a..c][x..y]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time discriminant F
│ │ │ - -- used 0.00807841s (cpu); 0.00807736s (thread); 0s (gc)
│ │ │ + -- used 0.0114216s (cpu); 0.0114235s (thread); 0s (gc)
│ │ │  
│ │ │          2
│ │ │  o3 = - b  + 4a*c
│ │ │  
│ │ │  o3 : ZZ[a..c]
    
│ │ │              
│ │ │            
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o5 : ZZ[a..d][x..y]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time discriminant F
│ │ │ - -- used 0.0087506s (cpu); 0.00875122s (thread); 0s (gc)
│ │ │ + -- used 0.0122756s (cpu); 0.0122773s (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]
    
│ │ │              
│ │ │            
│ │ │ @@ -165,15 +165,15 @@
│ │ │  
│ │ │  o12 : R'
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : time D=discriminant pencil
│ │ │ - -- used 0.461801s (cpu); 0.427218s (thread); 0s (gc)
│ │ │ + -- used 0.508756s (cpu); 0.446603s (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 0.00807841s (cpu); 0.00807736s (thread); 0s (gc)
│ │ │ │ + -- used 0.0114216s (cpu); 0.0114235s (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 0.0087506s (cpu); 0.00875122s (thread); 0s (gc)
│ │ │ │ + -- used 0.0122756s (cpu); 0.0122773s (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 0.461801s (cpu); 0.427218s (thread); 0s (gc)
│ │ │ │ + -- used 0.508756s (cpu); 0.446603s (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
│ │ │ @@ -90,28 +90,28 @@
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   5
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time V' = dualVariety V
│ │ │ - -- used 0.170711s (cpu); 0.119239s (thread); 0s (gc)
│ │ │ + -- used 0.188519s (cpu); 0.129596s (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 0.204333s (cpu); 0.146742s (thread); 0s (gc)
│ │ │ + -- used 0.243304s (cpu); 0.178114s (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 $\mathbb{P}^9$
    
│ │ │          
│ │ │ @@ -131,25 +131,25 @@
│ │ │  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 0.170711s (cpu); 0.119239s (thread); 0s (gc)
│ │ │ │ + -- used 0.188519s (cpu); 0.129596s (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 0.204333s (cpu); 0.146742s (thread); 0s (gc)
│ │ │ │ + -- used 0.243304s (cpu); 0.178114s (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 0.0566354s (cpu); 0.0566377s (thread); 0s (gc)
│ │ │ │ + -- used 0.0829479s (cpu); 0.082949s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                 ZZ
│ │ │ │  o5 : Ideal of ----[a ..a ]
│ │ │ │                3331  0   9
│ │ │ │  i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ │ - -- used 0.611958s (cpu); 0.563695s (thread); 0s (gc)
│ │ │ │ + -- used 0.760722s (cpu); 0.688474s (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
│ │ │ @@ -85,15 +85,15 @@
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : time discF = ideal discriminant F;
│ │ │ - -- used 0.0566354s (cpu); 0.0566377s (thread); 0s (gc)
│ │ │ + -- used 0.0829479s (cpu); 0.082949s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o5 : Ideal of ----[a ..a ]
│ │ │                3331  0   9
    
│ │ │              
    
│ │ │                
    i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ - -- used 0.611958s (cpu); 0.563695s (thread); 0s (gc)
│ │ │ + -- used 0.760722s (cpu); 0.688474s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o6 : Ideal of ----[x ..x ]
│ │ │                3331  0   9
    
│ │ │              
    
│ │ │                
    i2 : time fromPluckerToStiefel dualize chowForm C
│ │ │ - -- used 0.110649s (cpu); 0.0580797s (thread); 0s (gc)
│ │ │ + -- used 0.141521s (cpu); 0.0732298s (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    -
│ │ │ @@ -138,15 +138,15 @@
│ │ │  o2 : QQ[x   ..x   ]
│ │ │           0,0   1,3
    
│ │ │              
    
│ │ │                
    i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1})
│ │ │ - -- used 0.040301s (cpu); 0.0403043s (thread); 0s (gc)
│ │ │ + -- used 0.0533498s (cpu); 0.0533558s (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
│ │ │ @@ -179,15 +179,15 @@
│ │ │              
    		
│ │ │                
    i5 : w = chowForm C;
    
│ │ │              
    
│ │ │                
    i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel(w,AffineChartGrass=>s))
│ │ │ - -- used 0.0190728s (cpu); 0.0190737s (thread); 0s (gc)
│ │ │ + -- used 0.0254632s (cpu); 0.0254639s (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    +
│ │ │ @@ -227,15 +227,15 @@
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time apply(U,u->dim singularLocus u)
│ │ │ - -- used 0.0170671s (cpu); 0.0170726s (thread); 0s (gc)
│ │ │ + -- used 0.0215474s (cpu); 0.0215521s (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 0.110649s (cpu); 0.0580797s (thread); 0s (gc)
│ │ │ │ + -- used 0.141521s (cpu); 0.0732298s (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 0.040301s (cpu); 0.0403043s (thread); 0s (gc)
│ │ │ │ + -- used 0.0533498s (cpu); 0.0533558s (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 0.0190728s (cpu); 0.0190737s (thread); 0s (gc)
│ │ │ │ + -- used 0.0254632s (cpu); 0.0254639s (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 0.0170671s (cpu); 0.0170726s (thread); 0s (gc)
│ │ │ │ + -- used 0.0215474s (cpu); 0.0215521s (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
│ │ │ @@ -92,15 +92,15 @@
│ │ │  o1 : Ideal of QQ[p ..p ]
│ │ │                    0   4
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time hurwitzForm Q
│ │ │ - -- used 0.0393717s (cpu); 0.0393716s (thread); 0s (gc)
│ │ │ + -- used 0.0537307s (cpu); 0.0535482s (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 0.0393717s (cpu); 0.0393716s (thread); 0s (gc)
│ │ │ │ + -- used 0.0537307s (cpu); 0.0535482s (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
│ │ │ @@ -104,15 +104,15 @@
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time isCoisotropic w
│ │ │ - -- used 0.00787565s (cpu); 0.00787232s (thread); 0s (gc)
│ │ │ + -- used 0.00983881s (cpu); 0.00983762s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : -- random quadric in G(1,3)
│ │ │ @@ -140,15 +140,15 @@
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time isCoisotropic w'
│ │ │ - -- used 0.00654309s (cpu); 0.00654341s (thread); 0s (gc)
│ │ │ + -- used 0.00874447s (cpu); 0.00874582s (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 0.00787565s (cpu); 0.00787232s (thread); 0s (gc)
│ │ │ │ + -- used 0.00983881s (cpu); 0.00983762s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = true
│ │ │ │  i3 : -- random quadric in G(1,3)
│ │ │ │       w' = random(2,Grass(1,3))
│ │ │ │  
│ │ │ │         2     5           10 2     2                          2      3
│ │ │ │  o3 = 6p    + -p   p    + --p    + -p   p    + 10p   p    + 5p    + --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 0.00654309s (cpu); 0.00654341s (thread); 0s (gc)
│ │ │ │ + -- used 0.00874447s (cpu); 0.00874582s (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
│ │ │ @@ -117,15 +117,15 @@
│ │ │  o3 : Ideal of -----[x ..x ]
│ │ │                33331  0   5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I))
│ │ │ - -- used 0.0197451s (cpu); 0.0197449s (thread); 0s (gc)
│ │ │ + -- used 0.0235616s (cpu); 0.0235618s (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 0.0197451s (cpu); 0.0197449s (thread); 0s (gc)
│ │ │ │ + -- used 0.0235616s (cpu); 0.0235618s (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
│ │ │ @@ -87,15 +87,15 @@
│ │ │  
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00462514s (cpu); 0.00462065s (thread); 0s (gc)
│ │ │ + -- used 0.00432433s (cpu); 0.00432479s (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  
│ │ │ @@ -158,15 +158,15 @@
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00237921s (cpu); 0.00238132s (thread); 0s (gc)
│ │ │ + -- used 0.00289871s (cpu); 0.00289702s (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 0.00462514s (cpu); 0.00462065s (thread); 0s (gc)
│ │ │ │ + -- used 0.00432433s (cpu); 0.00432479s (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 0.00237921s (cpu); 0.00238132s (thread); 0s (gc)
│ │ │ │ + -- used 0.00289871s (cpu); 0.00289702s (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
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o3 : Ideal of P4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time p = plucker L
│ │ │ - -- used 0.00452289s (cpu); 0.00452084s (thread); 0s (gc)
│ │ │ + -- used 0.00598204s (cpu); 0.00598267s (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  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o4 : Ideal of G'1'4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time L' = plucker p
│ │ │ - -- used 0.111022s (cpu); 0.0535001s (thread); 0s (gc)
│ │ │ + -- used 0.12953s (cpu); 0.0626089s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │               2        3         4   1        3         4   0         3  
│ │ │       ------------------------------------------------------------------------
│ │ │       664x )
│ │ │           4
│ │ │  
│ │ │ @@ -138,15 +138,15 @@
│ │ │  
│ │ │  o7 : Ideal of G'1'4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time W = plucker Y; -- surface swept out by the lines of Y
│ │ │ - -- used 0.0361963s (cpu); 0.0361951s (thread); 0s (gc)
│ │ │ + -- used 0.0397103s (cpu); 0.0397081s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of P4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : (codim W,degree W)
│ │ │ @@ -158,15 +158,15 @@
│ │ │            
│ │ │          
│ │ │          
    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 0.142146s (cpu); 0.142149s (thread); 0s (gc)
│ │ │ + -- used 0.179115s (cpu); 0.179121s (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 0.00452289s (cpu); 0.00452084s (thread); 0s (gc)
│ │ │ │ + -- used 0.00598204s (cpu); 0.00598267s (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 0.111022s (cpu); 0.0535001s (thread); 0s (gc)
│ │ │ │ + -- used 0.12953s (cpu); 0.0626089s (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 0.0361963s (cpu); 0.0361951s (thread); 0s (gc)
│ │ │ │ + -- used 0.0397103s (cpu); 0.0397081s (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 0.142146s (cpu); 0.142149s (thread); 0s (gc)
│ │ │ │ + -- used 0.179115s (cpu); 0.179121s (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
│ │ │ @@ -108,15 +108,15 @@
│ │ │  
│ │ │  o2 : List
│ │ │              
    
│ │ │                
    i3 : time resultant(F,Algorithm=>"Poisson2")
│ │ │ - -- used 0.316083s (cpu); 0.220122s (thread); 0s (gc)
│ │ │ + -- used 0.381587s (cpu); 0.230058s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -132,15 +132,15 @@
│ │ │  
│ │ │  o3 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ - -- used 0.199752s (cpu); 0.138876s (thread); 0s (gc)
│ │ │ + -- used 0.181192s (cpu); 0.106558s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -156,15 +156,15 @@
│ │ │  
│ │ │  o4 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ - -- used 0.514703s (cpu); 0.461162s (thread); 0s (gc)
│ │ │ + -- used 0.4153s (cpu); 0.355085s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -180,15 +180,15 @@
│ │ │  
│ │ │  o5 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ - -- used 0.590923s (cpu); 0.537731s (thread); 0s (gc)
│ │ │ + -- used 0.648333s (cpu); 0.581225s (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 0.316083s (cpu); 0.220122s (thread); 0s (gc)
│ │ │ │ + -- used 0.381587s (cpu); 0.230058s (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 0.199752s (cpu); 0.138876s (thread); 0s (gc)
│ │ │ │ + -- used 0.181192s (cpu); 0.106558s (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 0.514703s (cpu); 0.461162s (thread); 0s (gc)
│ │ │ │ + -- used 0.4153s (cpu); 0.355085s (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 0.590923s (cpu); 0.537731s (thread); 0s (gc)
│ │ │ │ + -- used 0.648333s (cpu); 0.581225s (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
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time resultant F
│ │ │ - -- used 0.0229913s (cpu); 0.0229909s (thread); 0s (gc)
│ │ │ + -- used 0.0278475s (cpu); 0.0278464s (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  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -155,15 +155,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : time resultant F
│ │ │ - -- used 2.72711s (cpu); 2.06378s (thread); 0s (gc)
│ │ │ + -- used 2.6244s (cpu); 2.06772s (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  -
│ │ │ @@ -1790,15 +1790,15 @@
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time # terms resultant F
│ │ │ - -- used 0.58204s (cpu); 0.40137s (thread); 0s (gc)
│ │ │ + -- used 0.445132s (cpu); 0.384063s (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 0.0229913s (cpu); 0.0229909s (thread); 0s (gc)
│ │ │ │ + -- used 0.0278475s (cpu); 0.0278464s (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 2.72711s (cpu); 2.06378s (thread); 0s (gc)
│ │ │ │ + -- used 2.6244s (cpu); 2.06772s (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 0.58204s (cpu); 0.40137s (thread); 0s (gc)
│ │ │ │ + -- used 0.445132s (cpu); 0.384063s (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
│ │ │ @@ -97,15 +97,15 @@
│ │ │                    0   4
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form)
│ │ │       time tangentialChowForm(S,0)
│ │ │ - -- used 0.028783s (cpu); 0.0287839s (thread); 0s (gc)
│ │ │ + -- used 0.0610464s (cpu); 0.0610476s (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
│ │ │ @@ -118,15 +118,15 @@
│ │ │         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 0.120285s (cpu); 0.0736976s (thread); 0s (gc)
│ │ │ + -- used 0.156243s (cpu); 0.0923681s (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      +
│ │ │ @@ -163,43 +163,43 @@
│ │ │         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 0.0319774s (cpu); 0.0319788s (thread); 0s (gc)
│ │ │ + -- used 0.043272s (cpu); 0.0432777s (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 0.11267s (cpu); 0.0697575s (thread); 0s (gc)
│ │ │ + -- used 0.13669s (cpu); 0.0656883s (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 0.148995s (cpu); 0.103293s (thread); 0s (gc)
    
│ │ │ + -- used 0.172755s (cpu); 0.115538s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │  o2 = ideal (- p p  + p p , - p p  + p p , - p  + p p )
│ │ │ │                 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 0.028783s (cpu); 0.0287839s (thread); 0s (gc)
│ │ │ │ + -- used 0.0610464s (cpu); 0.0610476s (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
│ │ │ │ @@ -88,15 +88,15 @@
│ │ │ │  - 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 0.120285s (cpu); 0.0736976s (thread); 0s (gc)
│ │ │ │ + -- used 0.156243s (cpu); 0.0923681s (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      +
│ │ │ │ @@ -138,35 +138,35 @@
│ │ │ │  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 0.0319774s (cpu); 0.0319788s (thread); 0s (gc)
│ │ │ │ + -- used 0.043272s (cpu); 0.0432777s (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 0.11267s (cpu); 0.0697575s (thread); 0s (gc)
│ │ │ │ + -- used 0.13669s (cpu); 0.0656883s (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 0.148995s (cpu); 0.103293s (thread); 0s (gc)
│ │ │ │ + -- used 0.172755s (cpu); 0.115538s (thread); 0s (gc)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_C_o_i_s_o_t_r_o_p_i_c -- whether a hypersurface of a Grassmannian is a tangential
│ │ │ │        Chow form
│ │ │ │      * _c_h_o_w_F_o_r_m -- Chow form of a projective variety
│ │ │ │  ********** WWaayyss ttoo uussee ttaannggeennttiiaallCChhoowwFFoorrmm:: **********
│ │ │ │      * tangentialChowForm(Ideal,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_resource_splimits.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  time(seconds)        700
│ │ │  file(blocks)         unlimited
│ │ │  data(kbytes)         unlimited
│ │ │  stack(kbytes)        8192
│ │ │  coredump(blocks)     unlimited
│ │ │  memory(kbytes)       850000
│ │ │  locked memory(kbytes) 8192
│ │ │ -process              63811
│ │ │ +process              63520
│ │ │  nofiles              512
│ │ │  vmemory(kbytes)      unlimited
│ │ │  locks                unlimited
│ │ │  rtprio               0
│ │ │  
│ │ │  o1 = 0
│ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out
│ │ │ @@ -1,23 +1,23 @@
│ │ │  -- -*- M2-comint -*- hash: 2927978066455787395
│ │ │  
│ │ │  i1 : fn=temporaryFileName()|".m2"
│ │ │  
│ │ │ -o1 = /tmp/M2-29954-0/0.m2
│ │ │ +o1 = /tmp/M2-43124-0/0.m2
│ │ │  
│ │ │  i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-29954-0/1.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/1.m2" >"/tmp/M2-43124-0/1.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  i7 : h
│ │ │  
│ │ │  o7 = HashTable{"answer file" => null}
│ │ │                 "exit code" => 0
│ │ │                 "output file" => null
│ │ │ @@ -33,23 +33,23 @@
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : h#"exit code"===0
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : h=runExternalM2(fn,"justexit",());
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/2.m2" >"/tmp/M2-29954-0/2.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/2.m2" >"/tmp/M2-43124-0/2.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │  i11 : h
│ │ │  
│ │ │ -o11 = HashTable{"answer file" => /tmp/M2-29954-0/2.ans}
│ │ │ +o11 = HashTable{"answer file" => /tmp/M2-43124-0/2.ans}
│ │ │                  "exit code" => 27
│ │ │ -                "output file" => /tmp/M2-29954-0/2.out
│ │ │ +                "output file" => /tmp/M2-43124-0/2.out
│ │ │                  "return code" => 6912
│ │ │                  "statistics" => null
│ │ │                  "time used" => 1
│ │ │                  value => null
│ │ │  
│ │ │  o11 : HashTable
│ │ │  
│ │ │ @@ -58,80 +58,80 @@
│ │ │  o12 = true
│ │ │  
│ │ │  i13 : fileExists(h#"answer file")
│ │ │  
│ │ │  o13 = false
│ │ │  
│ │ │  i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ -Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/3.m2" >"/tmp/M2-29954-0/3.out" 2>&1 ))
│ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/3.m2" >"/tmp/M2-43124-0/3.out" 2>&1 ))
│ │ │  Killed
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │  i15 : h
│ │ │  
│ │ │ -o15 = HashTable{"answer file" => /tmp/M2-29954-0/3.ans}
│ │ │ +o15 = HashTable{"answer file" => /tmp/M2-43124-0/3.ans}
│ │ │                  "exit code" => 0
│ │ │ -                "output file" => /tmp/M2-29954-0/3.out
│ │ │ +                "output file" => /tmp/M2-43124-0/3.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(h#"output file")
│ │ │  
│ │ │  o16 = 
│ │ │ -      i1 : -- Script /tmp/M2-29954-0/3.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-43124-0/3.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/3.ans",spin (10));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43124-0/3.ans",spin (10));
│ │ │        Spinning!!
│ │ │  
│ │ │  
│ │ │  i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get(h#"answer file")
│ │ │  
│ │ │  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/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1') >"/tmp/M2-29954-0/4.stat" 2>&1 ))
│ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1') >"/tmp/M2-43124-0/4.stat" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  i19 : h#"statistics"
│ │ │  
│ │ │ -o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1"
│ │ │ -              User time (seconds): 5.25
│ │ │ -              System time (seconds): 0.12
│ │ │ -              Percent of CPU this job got: 77%
│ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.97
│ │ │ +o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1"
│ │ │ +              User time (seconds): 5.00
│ │ │ +              System time (seconds): 0.36
│ │ │ +              Percent of CPU this job got: 119%
│ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:04.49
│ │ │                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): 251780
│ │ │ +              Maximum resident set size (kbytes): 338712
│ │ │                Average resident set size (kbytes): 0
│ │ │                Major (requiring I/O) page faults: 0
│ │ │ -              Minor (reclaiming a frame) page faults: 8554
│ │ │ -              Voluntary context switches: 1820
│ │ │ -              Involuntary context switches: 2015
│ │ │ +              Minor (reclaiming a frame) page faults: 42991
│ │ │ +              Voluntary context switches: 6143
│ │ │ +              Involuntary context switches: 1224
│ │ │                Swaps: 0
│ │ │                File system inputs: 0
│ │ │ -              File system outputs: 0
│ │ │ +              File system outputs: 16
│ │ │                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/M2-29954-0/6.m2" >"/tmp/M2-29954-0/6.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/6.m2" >"/tmp/M2-43124-0/6.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o21 = true
│ │ │  
│ │ │  i22 : R=QQ[x,y];
│ │ │  
│ │ │  i23 : v=coker random(R^2,R^{3:-1})
│ │ │ @@ -139,54 +139,54 @@
│ │ │  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/M2-29954-0/7.m2" >"/tmp/M2-29954-0/7.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/7.m2" >"/tmp/M2-43124-0/7.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │ -o24 = HashTable{"answer file" => /tmp/M2-29954-0/7.ans}
│ │ │ +o24 = HashTable{"answer file" => /tmp/M2-43124-0/7.ans}
│ │ │                  "exit code" => 1
│ │ │ -                "output file" => /tmp/M2-29954-0/7.out
│ │ │ +                "output file" => /tmp/M2-43124-0/7.out
│ │ │                  "return code" => 256
│ │ │                  "statistics" => null
│ │ │                  "time used" => 1
│ │ │                  value => null
│ │ │  
│ │ │  o24 : HashTable
│ │ │  
│ │ │  i25 : get(h#"output file")
│ │ │  
│ │ │  o25 = 
│ │ │ -      i1 : -- Script /tmp/M2-29954-0/7.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-43124-0/7.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/7.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/M2-43124-0/7.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):[1]: error: no method for binary operator ^ applied to objects:
│ │ │                    R (of class Symbol)
│ │ │              ^     2 (of class ZZ)
│ │ │  
│ │ │  
│ │ │  i26 : fn<"/tmp/M2-29954-0/8.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/8.m2" >"/tmp/M2-43124-0/8.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o27 = true
│ │ │  
│ │ │  i28 : v=R;
│ │ │  
│ │ │  i29 : h=runExternalM2(fn,identity,v);
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/9.m2" >"/tmp/M2-29954-0/9.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/9.m2" >"/tmp/M2-43124-0/9.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  i30 : h#value
│ │ │  
│ │ │  o30 = QQ[x..y]
│ │ │  
│ │ │  o30 : PolynomialRing
│ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │  time(seconds)        700
│ │ │  file(blocks)         unlimited
│ │ │  data(kbytes)         unlimited
│ │ │  stack(kbytes)        8192
│ │ │  coredump(blocks)     unlimited
│ │ │  memory(kbytes)       850000
│ │ │  locked memory(kbytes) 8192
│ │ │ -process              63811
│ │ │ +process              63520
│ │ │  nofiles              512
│ │ │  vmemory(kbytes)      unlimited
│ │ │  locks                unlimited
│ │ │  rtprio               0
│ │ │  
│ │ │  o1 = 0
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  time(seconds)        700
│ │ │ │  file(blocks)         unlimited
│ │ │ │  data(kbytes)         unlimited
│ │ │ │  stack(kbytes)        8192
│ │ │ │  coredump(blocks)     unlimited
│ │ │ │  memory(kbytes)       850000
│ │ │ │  locked memory(kbytes) 8192
│ │ │ │ -process              63811
│ │ │ │ +process              63520
│ │ │ │  nofiles              512
│ │ │ │  vmemory(kbytes)      unlimited
│ │ │ │  locks                unlimited
│ │ │ │  rtprio               0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  This starts a new shell and executes the command given, which in this case
│ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │            
      For example, we can write a few functions to a temporary file:
      
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -115,15 +115,15 @@
│ │ │          
    
│ │ │            
    and then call them:
    
│ │ │          
    
│ │ │          
    
│ │ │                
    i1 : fn=temporaryFileName()|".m2"
│ │ │  
│ │ │ -o1 = /tmp/M2-29954-0/0.m2
    
│ │ │ +o1 = /tmp/M2-43124-0/0.m2
│ │ │              
    
│ │ │                
    i2 : fn<</// square = (x) -> (stderr<<"Running"<<endl; sleep(1); x^2); ///<<endl;
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -350,20 +350,20 @@
│ │ │          
│ │ │          
    
│ │ │                
    i6 : h=runExternalM2(fn,"square",(4));
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/1.m2" >"/tmp/M2-29954-0/1.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/1.m2" >"/tmp/M2-43124-0/1.out" 2>&1 ))
│ │ │  Finished running.
    
│ │ │              
    
│ │ │                
    i7 : h
│ │ │  
│ │ │ @@ -157,26 +157,26 @@
│ │ │            
    
│ │ │            
    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.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -199,46 +199,46 @@
│ │ │          
    
│ │ │            
    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:
    
│ │ │          
    
│ │ │          
    
│ │ │                
    i10 : h=runExternalM2(fn,"justexit",());
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/2.m2" >"/tmp/M2-29954-0/2.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/2.m2" >"/tmp/M2-43124-0/2.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    
│ │ │              
    
│ │ │                
    i11 : h
│ │ │  
│ │ │ -o11 = HashTable{"answer file" => /tmp/M2-29954-0/2.ans}
│ │ │ +o11 = HashTable{"answer file" => /tmp/M2-43124-0/2.ans}
│ │ │                  "exit code" => 27
│ │ │ -                "output file" => /tmp/M2-29954-0/2.out
│ │ │ +                "output file" => /tmp/M2-43124-0/2.out
│ │ │                  "return code" => 6912
│ │ │                  "statistics" => null
│ │ │                  "time used" => 1
│ │ │                  value => null
│ │ │  
│ │ │  o11 : HashTable
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -248,40 +248,40 @@
│ │ │            
    
│ │ │            
    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/M2-29954-0/3.m2" >"/tmp/M2-29954-0/3.out" 2>&1 ))
│ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/3.m2" >"/tmp/M2-43124-0/3.out" 2>&1 ))
│ │ │  Killed
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    
│ │ │              
    
│ │ │                
    i15 : h
│ │ │  
│ │ │ -o15 = HashTable{"answer file" => /tmp/M2-29954-0/3.ans}
│ │ │ +o15 = HashTable{"answer file" => /tmp/M2-43124-0/3.ans}
│ │ │                  "exit code" => 0
│ │ │ -                "output file" => /tmp/M2-29954-0/3.out
│ │ │ +                "output file" => /tmp/M2-43124-0/3.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(h#"output file")
│ │ │  
│ │ │  o16 = 
│ │ │ -      i1 : -- Script /tmp/M2-29954-0/3.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-43124-0/3.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/3.ans",spin (10));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43124-0/3.ans",spin (10));
│ │ │        Spinning!!
    
│ │ │              
    
│ │ │                
    i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get(h#"answer file")
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -295,15 +295,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    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/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1') >"/tmp/M2-29954-0/4.stat" 2>&1 ))
│ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1') >"/tmp/M2-43124-0/4.stat" 2>&1 ))
│ │ │  Finished running.
    
│ │ │              
    
│ │ │                
    i19 : h#"statistics"
│ │ │  
│ │ │ -o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1"
│ │ │ -              User time (seconds): 5.25
│ │ │ -              System time (seconds): 0.12
│ │ │ -              Percent of CPU this job got: 77%
│ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.97
│ │ │ +o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1"
│ │ │ +              User time (seconds): 5.00
│ │ │ +              System time (seconds): 0.36
│ │ │ +              Percent of CPU this job got: 119%
│ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:04.49
│ │ │                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): 251780
│ │ │ +              Maximum resident set size (kbytes): 338712
│ │ │                Average resident set size (kbytes): 0
│ │ │                Major (requiring I/O) page faults: 0
│ │ │ -              Minor (reclaiming a frame) page faults: 8554
│ │ │ -              Voluntary context switches: 1820
│ │ │ -              Involuntary context switches: 2015
│ │ │ +              Minor (reclaiming a frame) page faults: 42991
│ │ │ +              Voluntary context switches: 6143
│ │ │ +              Involuntary context switches: 1224
│ │ │                Swaps: 0
│ │ │                File system inputs: 0
│ │ │ -              File system outputs: 0
│ │ │ +              File system outputs: 16
│ │ │                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/M2-29954-0/6.m2" >"/tmp/M2-29954-0/6.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/6.m2" >"/tmp/M2-43124-0/6.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o21 = true
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -325,21 +325,21 @@
│ │ │                               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/M2-29954-0/7.m2" >"/tmp/M2-29954-0/7.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/7.m2" >"/tmp/M2-43124-0/7.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │ -o24 = HashTable{"answer file" => /tmp/M2-29954-0/7.ans}
│ │ │ +o24 = HashTable{"answer file" => /tmp/M2-43124-0/7.ans}
│ │ │                  "exit code" => 1
│ │ │ -                "output file" => /tmp/M2-29954-0/7.out
│ │ │ +                "output file" => /tmp/M2-43124-0/7.out
│ │ │                  "return code" => 256
│ │ │                  "statistics" => null
│ │ │                  "time used" => 1
│ │ │                  value => null
│ │ │  
│ │ │  o24 : HashTable
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i25 : get(h#"output file")
│ │ │  
│ │ │  o25 = 
│ │ │ -      i1 : -- Script /tmp/M2-29954-0/7.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-43124-0/7.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/7.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/M2-43124-0/7.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):[1]: error: no method for binary operator ^ applied to objects:
│ │ │                    R (of class Symbol)
│ │ │              ^     2 (of class ZZ)
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -374,15 +374,15 @@
│ │ │              
│ │ │                
    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/M2-29954-0/8.m2" >"/tmp/M2-29954-0/8.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/8.m2" >"/tmp/M2-43124-0/8.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o27 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │ @@ -394,15 +394,15 @@
│ │ │              
│ │ │                
    i28 : v=R;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : h=runExternalM2(fn,identity,v);
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/9.m2" >"/tmp/M2-29954-0/9.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/9.m2" >"/tmp/M2-43124-0/9.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/M2-29954-0/0.m2
│ │ │ │ +o1 = /tmp/M2-43124-0/0.m2
│ │ │ │  i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-29954-0/1.out" 2>&1 ))
│ │ │ │ +M2-43124-0/1.m2" >"/tmp/M2-43124-0/1.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i7 : h
│ │ │ │  
│ │ │ │  o7 = HashTable{"answer file" => null}
│ │ │ │                 "exit code" => 0
│ │ │ │                 "output file" => null
│ │ │ │                 "return code" => 0
│ │ │ │ @@ -79,22 +79,22 @@
│ │ │ │  
│ │ │ │  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/
│ │ │ │ -M2-29954-0/2.m2" >"/tmp/M2-29954-0/2.out" 2>&1 ))
│ │ │ │ +M2-43124-0/2.m2" >"/tmp/M2-43124-0/2.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i11 : h
│ │ │ │  
│ │ │ │ -o11 = HashTable{"answer file" => /tmp/M2-29954-0/2.ans}
│ │ │ │ +o11 = HashTable{"answer file" => /tmp/M2-43124-0/2.ans}
│ │ │ │                  "exit code" => 27
│ │ │ │ -                "output file" => /tmp/M2-29954-0/2.out
│ │ │ │ +                "output file" => /tmp/M2-43124-0/2.out
│ │ │ │                  "return code" => 6912
│ │ │ │                  "statistics" => null
│ │ │ │                  "time used" => 1
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o11 : HashTable
│ │ │ │  i12 : fileExists(h#"output file")
│ │ │ │ @@ -103,143 +103,143 @@
│ │ │ │  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/M2-29954-0/3.m2" >"/tmp/M2-29954-0/3.out" 2>&1 ))
│ │ │ │ +q  <"/tmp/M2-43124-0/3.m2" >"/tmp/M2-43124-0/3.out" 2>&1 ))
│ │ │ │  Killed
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i15 : h
│ │ │ │  
│ │ │ │ -o15 = HashTable{"answer file" => /tmp/M2-29954-0/3.ans}
│ │ │ │ +o15 = HashTable{"answer file" => /tmp/M2-43124-0/3.ans}
│ │ │ │                  "exit code" => 0
│ │ │ │ -                "output file" => /tmp/M2-29954-0/3.out
│ │ │ │ +                "output file" => /tmp/M2-43124-0/3.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
│ │ │ │  (h#"output file")
│ │ │ │  
│ │ │ │  o16 =
│ │ │ │ -      i1 : -- Script /tmp/M2-29954-0/3.m2 automatically generated by
│ │ │ │ +      i1 : -- Script /tmp/M2-43124-0/3.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/3.ans",spin (10));
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43124-0/3.ans",spin (10));
│ │ │ │        Spinning!!
│ │ │ │  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/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1')
│ │ │ │ ->"/tmp/M2-29954-0/4.stat" 2>&1 ))
│ │ │ │ +-no-debug --silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1')
│ │ │ │ +>"/tmp/M2-43124-0/4.stat" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i19 : h#"statistics"
│ │ │ │  
│ │ │ │  o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug
│ │ │ │ ---silent  -q  <"/tmp/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1"
│ │ │ │ -              User time (seconds): 5.25
│ │ │ │ -              System time (seconds): 0.12
│ │ │ │ -              Percent of CPU this job got: 77%
│ │ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.97
│ │ │ │ +--silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1"
│ │ │ │ +              User time (seconds): 5.00
│ │ │ │ +              System time (seconds): 0.36
│ │ │ │ +              Percent of CPU this job got: 119%
│ │ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:04.49
│ │ │ │                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): 251780
│ │ │ │ +              Maximum resident set size (kbytes): 338712
│ │ │ │                Average resident set size (kbytes): 0
│ │ │ │                Major (requiring I/O) page faults: 0
│ │ │ │ -              Minor (reclaiming a frame) page faults: 8554
│ │ │ │ -              Voluntary context switches: 1820
│ │ │ │ -              Involuntary context switches: 2015
│ │ │ │ +              Minor (reclaiming a frame) page faults: 42991
│ │ │ │ +              Voluntary context switches: 6143
│ │ │ │ +              Involuntary context switches: 1224
│ │ │ │                Swaps: 0
│ │ │ │                File system inputs: 0
│ │ │ │ -              File system outputs: 0
│ │ │ │ +              File system outputs: 16
│ │ │ │                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/
│ │ │ │ -M2-29954-0/6.m2" >"/tmp/M2-29954-0/6.out" 2>&1 ))
│ │ │ │ +M2-43124-0/6.m2" >"/tmp/M2-43124-0/6.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/
│ │ │ │ -M2-29954-0/7.m2" >"/tmp/M2-29954-0/7.out" 2>&1 ))
│ │ │ │ +M2-43124-0/7.m2" >"/tmp/M2-43124-0/7.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  
│ │ │ │ -o24 = HashTable{"answer file" => /tmp/M2-29954-0/7.ans}
│ │ │ │ +o24 = HashTable{"answer file" => /tmp/M2-43124-0/7.ans}
│ │ │ │                  "exit code" => 1
│ │ │ │ -                "output file" => /tmp/M2-29954-0/7.out
│ │ │ │ +                "output file" => /tmp/M2-43124-0/7.out
│ │ │ │                  "return code" => 256
│ │ │ │                  "statistics" => null
│ │ │ │                  "time used" => 1
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o24 : HashTable
│ │ │ │  To view the error message:
│ │ │ │  i25 : get(h#"output file")
│ │ │ │  
│ │ │ │  o25 =
│ │ │ │ -      i1 : -- Script /tmp/M2-29954-0/7.m2 automatically generated by
│ │ │ │ +      i1 : -- Script /tmp/M2-43124-0/7.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/7.ans",identity (cokernel
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43124-0/7.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):[1]: 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/M2-29954-0/8.out" 2>&1 ))
│ │ │ │ +M2-43124-0/8.m2" >"/tmp/M2-43124-0/8.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/
│ │ │ │ -M2-29954-0/9.m2" >"/tmp/M2-29954-0/9.out" 2>&1 ))
│ │ │ │ +M2-43124-0/9.m2" >"/tmp/M2-43124-0/9.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 0.000111089s (cpu); 0.000225292s (thread); 0s (gc)
│ │ │ + -- used 0.000374163s (cpu); 0.00032314s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i7 : ZZ[y];
│ │ │  
│ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 5.14794s (cpu); 3.65518s (thread); 0s (gc)
│ │ │ + -- used 4.47422s (cpu); 3.44734s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html
│ │ │ @@ -104,29 +104,29 @@
│ │ │  
│ │ │  o5 : InterpretedSLProgram
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ - -- used 0.000111089s (cpu); 0.000225292s (thread); 0s (gc)
│ │ │ + -- used 0.000374163s (cpu); 0.00032314s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : ZZ[y];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 5.14794s (cpu); 3.65518s (thread); 0s (gc)
│ │ │ + -- used 4.47422s (cpu); 3.44734s (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 0.000111089s (cpu); 0.000225292s (thread); 0s (gc)
│ │ │ │ + -- used 0.000374163s (cpu); 0.00032314s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1       1
│ │ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │ │  i7 : ZZ[y];
│ │ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ │ - -- used 5.14794s (cpu); 3.65518s (thread); 0s (gc)
│ │ │ │ + -- used 4.47422s (cpu); 3.44734s (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)
│ │ │ - -- .00628004s elapsed
│ │ │ + -- .00744196s elapsed
│ │ │  
│ │ │         6       19       19       7       3
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │                                          
│ │ │       0       1        2        3       4
│ │ │  
│ │ │  o4 : ChainComplex
│ │ │ @@ -51,15 +51,15 @@
│ │ │         53        53         53         53        53
│ │ │                                                  
│ │ │       -1        0          1          2         3
│ │ │  
│ │ │  o6 : ChainComplex
│ │ │  
│ │ │  i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ - -- .00224378s elapsed
│ │ │ + -- .00304284s elapsed
│ │ │  
│ │ │  i8 : h
│ │ │  
│ │ │  o8 = HashTable{-1 => 1}
│ │ │                 0 => 3
│ │ │                 1 => 5
│ │ │                 2 => 2
│ │ │ @@ -95,15 +95,15 @@
│ │ │  i12 : maximalEntry chainComplex errors
│ │ │  
│ │ │  o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ - -- .00497913s elapsed
│ │ │ + -- .00636129s 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)
│ │ │ - -- .00283463s elapsed
│ │ │ + -- .00332808s elapsed
│ │ │  
│ │ │         5       10       11       5
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o4 : ChainComplex
│ │ │ @@ -47,25 +47,25 @@
│ │ │         53        53         53         53
│ │ │                                        
│ │ │       0         1          2          3
│ │ │  
│ │ │  o6 : ChainComplex
│ │ │  
│ │ │  i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ - -- .000620476s elapsed
│ │ │ + -- .000721423s 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)
│ │ │ - -- .00139661s elapsed
│ │ │ + -- .00154824s 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)
│ │ │ - -- .00359076s elapsed
│ │ │ + -- .00422273s elapsed
│ │ │  
│ │ │         6       10       13       8
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
│ │ ├── ./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)
│ │ │ - -- .0027493s elapsed
│ │ │ + -- .00320249s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
│ │ ├── ./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)
│ │ │ - -- .00303026s elapsed
│ │ │ + -- .00369821s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html
│ │ │ @@ -105,15 +105,15 @@
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
│ │ │ - -- .00628004s elapsed
│ │ │ + -- .00744196s elapsed
│ │ │  
│ │ │         6       19       19       7       3
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │                                          
│ │ │       0       1        2        3       4
│ │ │  
│ │ │  o4 : ChainComplex
    
│ │ │ @@ -150,15 +150,15 @@
│ │ │  
│ │ │  o6 : ChainComplex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ - -- .00224378s elapsed
    
│ │ │ + -- .00304284s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : h
│ │ │  
│ │ │  o8 = HashTable{-1 => 1}
│ │ │ @@ -212,15 +212,15 @@
│ │ │  
│ │ │  o12 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ - -- .00497913s elapsed
    
│ │ │ + -- .00636129s 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)
│ │ │ │ - -- .00628004s elapsed
│ │ │ │ + -- .00744196s elapsed
│ │ │ │  
│ │ │ │         6       19       19       7       3
│ │ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3       4
│ │ │ │  
│ │ │ │  o4 : ChainComplex
│ │ │ │ @@ -70,15 +70,15 @@
│ │ │ │  o6 = RR    <-- RR     <-- RR     <-- RR    <-- RR
│ │ │ │         53        53         53         53        53
│ │ │ │  
│ │ │ │       -1        0          1          2         3
│ │ │ │  
│ │ │ │  o6 : ChainComplex
│ │ │ │  i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ │ - -- .00224378s elapsed
│ │ │ │ + -- .00304284s elapsed
│ │ │ │  i8 : h
│ │ │ │  
│ │ │ │  o8 = HashTable{-1 => 1}
│ │ │ │                 0 => 3
│ │ │ │                 1 => 5
│ │ │ │                 2 => 2
│ │ │ │                 3 => 1
│ │ │ │ @@ -109,15 +109,15 @@
│ │ │ │  1)*Sigma.dd_ell*transpose U_ell);
│ │ │ │  i12 : maximalEntry chainComplex errors
│ │ │ │  
│ │ │ │  o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ │ - -- .00497913s elapsed
│ │ │ │ + -- .00636129s 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
│ │ │ @@ -107,15 +107,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00283463s elapsed
│ │ │ + -- .00332808s elapsed
│ │ │  
│ │ │         5       10       11       5
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o4 : ChainComplex
    
│ │ │ @@ -148,28 +148,28 @@
│ │ │  
│ │ │  o6 : ChainComplex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ - -- .000620476s elapsed
│ │ │ + -- .000721423s 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)
│ │ │ - -- .00139661s elapsed
│ │ │ + -- .00154824s 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)
│ │ │ │ - -- .00283463s elapsed
│ │ │ │ + -- .00332808s elapsed
│ │ │ │  
│ │ │ │         5       10       11       5
│ │ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3
│ │ │ │  
│ │ │ │  o4 : ChainComplex
│ │ │ │ @@ -69,24 +69,24 @@
│ │ │ │  o6 = RR    <-- RR     <-- RR     <-- RR
│ │ │ │         53        53         53         53
│ │ │ │  
│ │ │ │       0         1          2          3
│ │ │ │  
│ │ │ │  o6 : ChainComplex
│ │ │ │  i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ │ - -- .000620476s elapsed
│ │ │ │ + -- .000721423s 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)
│ │ │ │ - -- .00139661s elapsed
│ │ │ │ + -- .00154824s 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
│ │ │ @@ -110,15 +110,15 @@
│ │ │  
│ │ │  o4 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
│ │ │ - -- .00359076s elapsed
│ │ │ + -- .00422273s elapsed
│ │ │  
│ │ │         6       10       13       8
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
    
│ │ │ ├── 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)
│ │ │ │ - -- .00359076s elapsed
│ │ │ │ + -- .00422273s elapsed
│ │ │ │  
│ │ │ │         6       10       13       8
│ │ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3
│ │ │ │  
│ │ │ │  o5 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o4 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .0027493s elapsed
│ │ │ + -- .00320249s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
    
│ │ │ ├── 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)
│ │ │ │ - -- .0027493s elapsed
│ │ │ │ + -- .00320249s elapsed
│ │ │ │  
│ │ │ │         6       10       11       5
│ │ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3
│ │ │ │  
│ │ │ │  o5 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_project__To__Complex.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o4 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00303026s elapsed
│ │ │ + -- .00369821s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
    
│ │ │ ├── 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)
│ │ │ │ - -- .00303026s elapsed
│ │ │ │ + -- .00369821s elapsed
│ │ │ │  
│ │ │ │         6       10       11       5
│ │ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3
│ │ │ │  
│ │ │ │  o5 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/SagbiGbDetection/dump/rawdocumentation.dump
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sun Dec 14 14:09:53 2025
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sun Dec 14 14:09:54 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │  #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  d2VpZ2h0VmVjdG9yc1JlYWxpemluZ1NBR0JJ
│ │ ├── ./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 0.386864s (cpu); 0.31337s (thread); 0s (gc)
│ │ │ + -- used 0.381338s (cpu); 0.381125s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S
│ │ │  
│ │ │  i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ - -- used 0.49095s (cpu); 0.490915s (thread); 0s (gc)
│ │ │ + -- used 0.747485s (cpu); 0.646083s (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 0.0268459s (cpu); 0.0268439s (thread); 0s (gc)
│ │ │ + -- used 0.0283667s (cpu); 0.0283726s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : Ideal of S
│ │ │  
│ │ │  i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ - -- used 0.00767377s (cpu); 0.00767479s (thread); 0s (gc)
│ │ │ + -- used 0.00992101s (cpu); 0.00993252s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -125,23 +125,23 @@
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time quotient(I^3, J^2, Strategy => Iterate);
│ │ │ - -- used 0.386864s (cpu); 0.31337s (thread); 0s (gc)
│ │ │ + -- used 0.381338s (cpu); 0.381125s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ - -- used 0.49095s (cpu); 0.490915s (thread); 0s (gc)
│ │ │ + -- used 0.747485s (cpu); 0.646083s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Strategy => Quotient is faster in other cases:
    
│ │ │ @@ -158,23 +158,23 @@
│ │ │  
│ │ │  o10 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : time quotient(I^5, I^3, Strategy => Iterate);
│ │ │ - -- used 0.0268459s (cpu); 0.0268439s (thread); 0s (gc)
│ │ │ + -- used 0.0283667s (cpu); 0.0283726s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ - -- used 0.00767377s (cpu); 0.00767479s (thread); 0s (gc)
│ │ │ + -- used 0.00992101s (cpu); 0.00993252s (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 0.386864s (cpu); 0.31337s (thread); 0s (gc)
│ │ │ │ + -- used 0.381338s (cpu); 0.381125s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of S
│ │ │ │  i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ │ - -- used 0.49095s (cpu); 0.490915s (thread); 0s (gc)
│ │ │ │ + -- used 0.747485s (cpu); 0.646083s (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 0.0268459s (cpu); 0.0268439s (thread); 0s (gc)
│ │ │ │ + -- used 0.0283667s (cpu); 0.0283726s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 : Ideal of S
│ │ │ │  i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ │ - -- used 0.00767377s (cpu); 0.00767479s (thread); 0s (gc)
│ │ │ │ + -- used 0.00992101s (cpu); 0.00993252s (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 0.00503697s (cpu); 0.00503317s (thread); 0s (gc)
│ │ │ - -- used 0.00641821s (cpu); 0.0064191s (thread); 0s (gc)
│ │ │ - -- used 0.00958149s (cpu); 0.00958239s (thread); 0s (gc)
│ │ │ - -- used 0.016275s (cpu); 0.0162761s (thread); 0s (gc)
│ │ │ - -- used 0.03087s (cpu); 0.0308718s (thread); 0s (gc)
│ │ │ - -- used 0.0536082s (cpu); 0.0536137s (thread); 0s (gc)
│ │ │ - -- used 0.0913388s (cpu); 0.0913434s (thread); 0s (gc)
│ │ │ - -- used 0.278118s (cpu); 0.17744s (thread); 0s (gc)
│ │ │ - -- used 0.223118s (cpu); 0.223096s (thread); 0s (gc)
│ │ │ + -- used 0.00745073s (cpu); 0.00744723s (thread); 0s (gc)
│ │ │ + -- used 0.00860662s (cpu); 0.00861431s (thread); 0s (gc)
│ │ │ + -- used 0.012361s (cpu); 0.0123726s (thread); 0s (gc)
│ │ │ + -- used 0.0205196s (cpu); 0.0205315s (thread); 0s (gc)
│ │ │ + -- used 0.0386137s (cpu); 0.0386246s (thread); 0s (gc)
│ │ │ + -- used 0.0770707s (cpu); 0.0770816s (thread); 0s (gc)
│ │ │ + -- used 0.125028s (cpu); 0.125039s (thread); 0s (gc)
│ │ │ + -- used 0.17883s (cpu); 0.178842s (thread); 0s (gc)
│ │ │ + -- used 0.470664s (cpu); 0.326617s (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
│ │ │ @@ -126,23 +126,23 @@
│ │ │  
│ │ │  o6 : FunctionClosure
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : for n from 2 to 10 list time f n
│ │ │ - -- used 0.00503697s (cpu); 0.00503317s (thread); 0s (gc)
│ │ │ - -- used 0.00641821s (cpu); 0.0064191s (thread); 0s (gc)
│ │ │ - -- used 0.00958149s (cpu); 0.00958239s (thread); 0s (gc)
│ │ │ - -- used 0.016275s (cpu); 0.0162761s (thread); 0s (gc)
│ │ │ - -- used 0.03087s (cpu); 0.0308718s (thread); 0s (gc)
│ │ │ - -- used 0.0536082s (cpu); 0.0536137s (thread); 0s (gc)
│ │ │ - -- used 0.0913388s (cpu); 0.0913434s (thread); 0s (gc)
│ │ │ - -- used 0.278118s (cpu); 0.17744s (thread); 0s (gc)
│ │ │ - -- used 0.223118s (cpu); 0.223096s (thread); 0s (gc)
│ │ │ + -- used 0.00745073s (cpu); 0.00744723s (thread); 0s (gc)
│ │ │ + -- used 0.00860662s (cpu); 0.00861431s (thread); 0s (gc)
│ │ │ + -- used 0.012361s (cpu); 0.0123726s (thread); 0s (gc)
│ │ │ + -- used 0.0205196s (cpu); 0.0205315s (thread); 0s (gc)
│ │ │ + -- used 0.0386137s (cpu); 0.0386246s (thread); 0s (gc)
│ │ │ + -- used 0.0770707s (cpu); 0.0770816s (thread); 0s (gc)
│ │ │ + -- used 0.125028s (cpu); 0.125039s (thread); 0s (gc)
│ │ │ + -- used 0.17883s (cpu); 0.178842s (thread); 0s (gc)
│ │ │ + -- used 0.470664s (cpu); 0.326617s (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 0.00503697s (cpu); 0.00503317s (thread); 0s (gc)
│ │ │ │ - -- used 0.00641821s (cpu); 0.0064191s (thread); 0s (gc)
│ │ │ │ - -- used 0.00958149s (cpu); 0.00958239s (thread); 0s (gc)
│ │ │ │ - -- used 0.016275s (cpu); 0.0162761s (thread); 0s (gc)
│ │ │ │ - -- used 0.03087s (cpu); 0.0308718s (thread); 0s (gc)
│ │ │ │ - -- used 0.0536082s (cpu); 0.0536137s (thread); 0s (gc)
│ │ │ │ - -- used 0.0913388s (cpu); 0.0913434s (thread); 0s (gc)
│ │ │ │ - -- used 0.278118s (cpu); 0.17744s (thread); 0s (gc)
│ │ │ │ - -- used 0.223118s (cpu); 0.223096s (thread); 0s (gc)
│ │ │ │ + -- used 0.00745073s (cpu); 0.00744723s (thread); 0s (gc)
│ │ │ │ + -- used 0.00860662s (cpu); 0.00861431s (thread); 0s (gc)
│ │ │ │ + -- used 0.012361s (cpu); 0.0123726s (thread); 0s (gc)
│ │ │ │ + -- used 0.0205196s (cpu); 0.0205315s (thread); 0s (gc)
│ │ │ │ + -- used 0.0386137s (cpu); 0.0386246s (thread); 0s (gc)
│ │ │ │ + -- used 0.0770707s (cpu); 0.0770816s (thread); 0s (gc)
│ │ │ │ + -- used 0.125028s (cpu); 0.125039s (thread); 0s (gc)
│ │ │ │ + -- used 0.17883s (cpu); 0.178842s (thread); 0s (gc)
│ │ │ │ + -- used 0.470664s (cpu); 0.326617s (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/SchurFunctors/dump/rawdocumentation.dump
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sun Dec 14 14:09:54 2025
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sun Dec 14 14:09:53 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │  #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  c3BsaXRDaGFyYWN0ZXI=
│ │ ├── ./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 4.58407s (cpu); 3.45047s (thread); 0s (gc)
│ │ │ + -- used 7.70051s (cpu); 4.09959s (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 50.6991s (cpu); 46.8604s (thread); 0s (gc)
│ │ │ + -- used 66.7902s (cpu); 61.0194s (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 0.374749s (cpu); 0.242963s (thread); 0s (gc)
│ │ │ + -- used 0.638127s (cpu); 0.219781s (thread); 0s (gc)
│ │ │  
│ │ │         2             2
│ │ │  o6 = 2H  + 4H H  + 2H
│ │ │         1     1 2     2
│ │ │  
│ │ │  o6 : A
│ │ │  
│ │ │  i7 : time segre(X,Y,A)
│ │ │ - -- used 0.16565s (cpu); 0.105524s (thread); 0s (gc)
│ │ │ + -- used 0.300511s (cpu); 0.142018s (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
│ │ │ @@ -162,15 +162,15 @@
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : time isComponentContained(X,Y)
│ │ │ - -- used 4.58407s (cpu); 3.45047s (thread); 0s (gc)
│ │ │ + -- used 7.70051s (cpu); 4.09959s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : print "we could confirm this with the computation:"
│ │ │ @@ -189,15 +189,15 @@
│ │ │  
│ │ │  o13 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time isSubset(saturate(Y,B),saturate(X,B))
│ │ │ - -- used 50.6991s (cpu); 46.8604s (thread); 0s (gc)
│ │ │ + -- used 66.7902s (cpu); 61.0194s (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 4.58407s (cpu); 3.45047s (thread); 0s (gc)
│ │ │ │ + -- used 7.70051s (cpu); 4.09959s (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 50.6991s (cpu); 46.8604s (thread); 0s (gc)
│ │ │ │ + -- used 66.7902s (cpu); 61.0194s (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
│ │ │ @@ -118,27 +118,27 @@
│ │ │  
│ │ │  o5 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time s = segreDimX(X,Y,A)
│ │ │ - -- used 0.374749s (cpu); 0.242963s (thread); 0s (gc)
│ │ │ + -- used 0.638127s (cpu); 0.219781s (thread); 0s (gc)
│ │ │  
│ │ │         2             2
│ │ │  o6 = 2H  + 4H H  + 2H
│ │ │         1     1 2     2
│ │ │  
│ │ │  o6 : A
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time segre(X,Y,A)
│ │ │ - -- used 0.16565s (cpu); 0.105524s (thread); 0s (gc)
│ │ │ + -- used 0.300511s (cpu); 0.142018s (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 0.374749s (cpu); 0.242963s (thread); 0s (gc)
│ │ │ │ + -- used 0.638127s (cpu); 0.219781s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2             2
│ │ │ │  o6 = 2H  + 4H H  + 2H
│ │ │ │         1     1 2     2
│ │ │ │  
│ │ │ │  o6 : A
│ │ │ │  i7 : time segre(X,Y,A)
│ │ │ │ - -- used 0.16565s (cpu); 0.105524s (thread); 0s (gc)
│ │ │ │ + -- used 0.300511s (cpu); 0.142018s (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")           -- .226545s elapsed
│ │ │ + -- capturing check(0, "SimpleDoc")           -- .193722s elapsed
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/html/_test__Example.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │          
    
│ │ │            
    The check method executes all package tests defined this way.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : check SimpleDoc
│ │ │ - -- capturing check(0, "SimpleDoc")           -- .226545s elapsed
    
│ │ │ + -- capturing check(0, "SimpleDoc")           -- .193722s 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")           -- .226545s elapsed
│ │ │ │ + -- capturing check(0, "SimpleDoc")           -- .193722s 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/SlackIdeals/example-output/_rehomogenize__Polynomial.out
│ │ │ @@ -9,14 +9,14 @@
│ │ │  
│ │ │  i3 : (Y, T) = setOnesForest X;
│ │ │  
│ │ │  i4 : remVars := flatten entries Y - set{0_(ring Y), 1_(ring Y)};
│ │ │  
│ │ │  i5 : h = rehomogenizePolynomial(X, Y, T, remVars_0^2+remVars_0*remVars_1-1)
│ │ │  
│ │ │ -        2 2 2 2          2 2 2 2          2 2
│ │ │ -o5 = - x x x x x  x   + x x x x x  x   + x x x x x x x x
│ │ │ -        1 4 6 7 10 11    2 3 5 8 10 11    2 3 5 6 7 8 9 12
│ │ │ +      2 2 2 2          2 2 2 2                  2 2
│ │ │ +o5 = x x x x x  x   - x x x x x  x   + x x x x x x x x
│ │ │ +      1 4 6 7 10 11    2 3 5 8 10 11    1 2 3 4 6 7 9 12
│ │ │  
│ │ │  o5 : R
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/SlackIdeals/html/_rehomogenize__Polynomial.html
│ │ │ @@ -99,17 +99,17 @@
│ │ │                
      i4 : remVars := flatten entries Y - set{0_(ring Y), 1_(ring Y)};
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i5 : h = rehomogenizePolynomial(X, Y, T, remVars_0^2+remVars_0*remVars_1-1)
│ │ │  
│ │ │ -        2 2 2 2          2 2 2 2          2 2
│ │ │ -o5 = - x x x x x  x   + x x x x x  x   + x x x x x x x x
│ │ │ -        1 4 6 7 10 11    2 3 5 8 10 11    2 3 5 6 7 8 9 12
│ │ │ +      2 2 2 2          2 2 2 2                  2 2
│ │ │ +o5 = x x x x x  x   - x x x x x  x   + x x x x x x x x
│ │ │ +      1 4 6 7 10 11    2 3 5 8 10 11    1 2 3 4 6 7 9 12
│ │ │  
│ │ │  o5 : R
      
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,17 +31,17 @@
│ │ │ │  
│ │ │ │               6      5
│ │ │ │  o2 : Matrix R  <-- R
│ │ │ │  i3 : (Y, T) = setOnesForest X;
│ │ │ │  i4 : remVars := flatten entries Y - set{0_(ring Y), 1_(ring Y)};
│ │ │ │  i5 : h = rehomogenizePolynomial(X, Y, T, remVars_0^2+remVars_0*remVars_1-1)
│ │ │ │  
│ │ │ │ -        2 2 2 2          2 2 2 2          2 2
│ │ │ │ -o5 = - x x x x x  x   + x x x x x  x   + x x x x x x x x
│ │ │ │ -        1 4 6 7 10 11    2 3 5 8 10 11    2 3 5 6 7 8 9 12
│ │ │ │ +      2 2 2 2          2 2 2 2                  2 2
│ │ │ │ +o5 = x x x x x  x   - x x x x x  x   + x x x x x x x x
│ │ │ │ +      1 4 6 7 10 11    2 3 5 8 10 11    1 2 3 4 6 7 9 12
│ │ │ │  
│ │ │ │  o5 : R
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_e_t_O_n_e_s_F_o_r_e_s_t -- sets to 1 variables in a symbolic slack matrix which
│ │ │ │        corresponding to edges of a spanning forest
│ │ │ │      * _s_l_a_c_k_I_d_e_a_l -- computes the slack ideal
│ │ │ │      * _s_y_m_b_o_l_i_c_S_l_a_c_k_M_a_t_r_i_x -- computes the symbolic slack matrix
│ │ ├── ./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 0.000112722s (cpu); 0.000101711s (thread); 0s (gc)
│ │ │ + -- used 0.000102447s (cpu); 8.9785e-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 0.0350456s (cpu); 0.0342305s (thread); 0s (gc)
│ │ │ + -- used 0.171905s (cpu); 0.0551066s (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 0.408829s (cpu); 0.223024s (thread); 0s (gc)
│ │ │ + -- used 0.579826s (cpu); 0.313864s (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 0.163836s (cpu); 0.123645s (thread); 0s (gc)
│ │ │ + -- used 0.174494s (cpu); 0.115434s (thread); 0s (gc)
│ │ │  
│ │ │  i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula
│ │ │ - -- used 0.292245s (cpu); 0.237931s (thread); 0s (gc)
│ │ │ + -- used 0.379614s (cpu); 0.30724s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ - -- used 0.365133s (cpu); 0.330814s (thread); 0s (gc)
│ │ │ + -- used 0.421825s (cpu); 0.362436s (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 0.0870547s (cpu); 0.0854223s (thread); 0s (gc)
│ │ │ + -- used 0.183683s (cpu); 0.11218s (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 0.514606s (cpu); 0.440355s (thread); 0s (gc)
│ │ │ + -- used 0.480945s (cpu); 0.480945s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 912984499996938980479447727885644530753184525786986940737407301278806287
│ │ │       9257139493926586400187927813888
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_generic__Skew__Multidimensional__Matrix.out
│ │ │ @@ -34,23 +34,23 @@
│ │ │                                                     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,
│ │ │ -                              3    2        3       0        2   0           
│ │ │ +                              1    0        1       2        0   2           
│ │ │       ------------------------------------------------------------------------
│ │ │       0, b , b }, {0, 0, 0, 0}, {-b , 0, 0, -b }, {-b , 0, b , 0}}, {{0, -b ,
│ │ │ -         3   2                    3          1      2      1              3 
│ │ │ +         1   0                    1          3      0      3              1 
│ │ │       ------------------------------------------------------------------------
│ │ │       0, b }, {b , 0, 0, b }, {0, 0, 0, 0}, {-b , -b , 0, 0}}, {{0, -b , -b ,
│ │ │ -         0     3         1                    0    1                 2    0 
│ │ │ +         2     1         3                    2    3                 0    2 
│ │ │       ------------------------------------------------------------------------
│ │ │       0}, {b , 0, -b , 0}, {b , b , 0, 0}, {0, 0, 0, 0}}}
│ │ │ -           2       1        0   1
│ │ │ +           0       3        2   3
│ │ │  
│ │ │                                                     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 2.62265s (cpu); 2.22282s (thread); 0s (gc)
│ │ │ + -- used 2.97611s (cpu); 2.65251s (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 0.507781s (cpu); 0.44829s (thread); 0s (gc)
│ │ │ + -- used 0.535004s (cpu); 0.480945s (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 0.00905261s (cpu); 0.00905263s (thread); 0s (gc)
│ │ │ + -- used 0.0114584s (cpu); 0.0114583s (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 0.0304897s (cpu); 0.0292273s (thread); 0s (gc)
│ │ │ + -- used 0.251632s (cpu); 0.0743614s (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 0.00316112s (cpu); 0.00316119s (thread); 0s (gc)
│ │ │ + -- used 0.00489819s (cpu); 0.00489889s (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 0.260966s (cpu); 0.209622s (thread); 0s (gc)
│ │ │ + -- used 0.261818s (cpu); 0.206438s (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
│ │ │ @@ -76,15 +76,15 @@
│ │ │  
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time degreeDeterminant n
│ │ │ - -- used 0.000112722s (cpu); 0.000101711s (thread); 0s (gc)
│ │ │ + -- used 0.000102447s (cpu); 8.9785e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 6
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : M = genericMultidimensionalMatrix n;
│ │ │ @@ -98,15 +98,15 @@
│ │ │  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 0.0350456s (cpu); 0.0342305s (thread); 0s (gc)
│ │ │ + -- used 0.171905s (cpu); 0.0551066s (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 0.000112722s (cpu); 0.000101711s (thread); 0s (gc)
│ │ │ │ + -- used 0.000102447s (cpu); 8.9785e-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 0.0350456s (cpu); 0.0342305s (thread); 0s (gc)
│ │ │ │ + -- used 0.171905s (cpu); 0.0551066s (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
│ │ │ @@ -80,15 +80,15 @@
│ │ │              
│ │ │                
    i1 : (d,n) := (2,3);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time Disc = denseDiscriminant(d,n)
│ │ │ - -- used 0.408829s (cpu); 0.223024s (thread); 0s (gc)
│ │ │ + -- used 0.579826s (cpu); 0.313864s (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 0.408829s (cpu); 0.223024s (thread); 0s (gc)
│ │ │ │ + -- used 0.579826s (cpu); 0.313864s (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
│ │ │ @@ -90,27 +90,27 @@
│ │ │  
│ │ │  o1 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time denseResultant(f0,f1,f2); -- using Poisson formula
│ │ │ - -- used 0.163836s (cpu); 0.123645s (thread); 0s (gc)
    
│ │ │ + -- used 0.174494s (cpu); 0.115434s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula
│ │ │ - -- used 0.292245s (cpu); 0.237931s (thread); 0s (gc)
    
│ │ │ + -- used 0.379614s (cpu); 0.30724s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ - -- used 0.365133s (cpu); 0.330814s (thread); 0s (gc)
    
│ │ │ + -- used 0.421825s (cpu); 0.362436s (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 0.163836s (cpu); 0.123645s (thread); 0s (gc)
│ │ │ │ + -- used 0.174494s (cpu); 0.115434s (thread); 0s (gc)
│ │ │ │  i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay
│ │ │ │  formula
│ │ │ │ - -- used 0.292245s (cpu); 0.237931s (thread); 0s (gc)
│ │ │ │ + -- used 0.379614s (cpu); 0.30724s (thread); 0s (gc)
│ │ │ │  i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ │ - -- used 0.365133s (cpu); 0.330814s (thread); 0s (gc)
│ │ │ │ + -- used 0.421825s (cpu); 0.362436s (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
│ │ │ @@ -84,15 +84,15 @@
│ │ │  
│ │ │  o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time det M
│ │ │ - -- used 0.0870547s (cpu); 0.0854223s (thread); 0s (gc)
│ │ │ + -- used 0.183683s (cpu); 0.11218s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 9698337990421512192
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : M = randomMultidimensionalMatrix(2,2,2,2,5)
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time det M
│ │ │ - -- used 0.514606s (cpu); 0.440355s (thread); 0s (gc)
│ │ │ + -- used 0.480945s (cpu); 0.480945s (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 0.0870547s (cpu); 0.0854223s (thread); 0s (gc)
│ │ │ │ + -- used 0.183683s (cpu); 0.11218s (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 0.514606s (cpu); 0.440355s (thread); 0s (gc)
│ │ │ │ + -- used 0.480945s (cpu); 0.480945s (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
│ │ │ @@ -116,24 +116,24 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    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,
│ │ │ -                              3    2        3       0        2   0           
│ │ │ +                              1    0        1       2        0   2           
│ │ │       ------------------------------------------------------------------------
│ │ │       0, b , b }, {0, 0, 0, 0}, {-b , 0, 0, -b }, {-b , 0, b , 0}}, {{0, -b ,
│ │ │ -         3   2                    3          1      2      1              3 
│ │ │ +         1   0                    1          3      0      3              1 
│ │ │       ------------------------------------------------------------------------
│ │ │       0, b }, {b , 0, 0, b }, {0, 0, 0, 0}, {-b , -b , 0, 0}}, {{0, -b , -b ,
│ │ │ -         0     3         1                    0    1                 2    0 
│ │ │ +         2     1         3                    2    3                 0    2 
│ │ │       ------------------------------------------------------------------------
│ │ │       0}, {b , 0, -b , 0}, {b , b , 0, 0}, {0, 0, 0, 0}}}
│ │ │ -           2       1        0   1
│ │ │ +           0       3        2   3
│ │ │  
│ │ │                                                     ZZ
│ │ │  o3 : 3-dimensional matrix of shape 4 x 4 x 4 over ---[b ..b ]
│ │ │                                                    101  0   3
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,24 +51,24 @@
│ │ │ │                                                     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,
│ │ │ │ -                              3    2        3       0        2   0
│ │ │ │ +                              1    0        1       2        0   2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       0, b , b }, {0, 0, 0, 0}, {-b , 0, 0, -b }, {-b , 0, b , 0}}, {{0, -b ,
│ │ │ │ -         3   2                    3          1      2      1              3
│ │ │ │ +         1   0                    1          3      0      3              1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       0, b }, {b , 0, 0, b }, {0, 0, 0, 0}, {-b , -b , 0, 0}}, {{0, -b , -b ,
│ │ │ │ -         0     3         1                    0    1                 2    0
│ │ │ │ +         2     1         3                    2    3                 0    2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       0}, {b , 0, -b , 0}, {b , b , 0, 0}, {0, 0, 0, 0}}}
│ │ │ │ -           2       1        0   1
│ │ │ │ +           0       3        2   3
│ │ │ │  
│ │ │ │                                                     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
│ │ │ @@ -90,15 +90,15 @@
│ │ │  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 2.62265s (cpu); 2.22282s (thread); 0s (gc)
│ │ │ + -- used 2.97611s (cpu); 2.65251s (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 2.62265s (cpu); 2.22282s (thread); 0s (gc)
│ │ │ │ + -- used 2.97611s (cpu); 2.65251s (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
│ │ │ @@ -74,15 +74,15 @@
│ │ │          
    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 $f = c_{(1,1)}+c_{(1,2)} x y+c_{(1,3)} x^2 y+c_{(1,4)} x, g = c_{(2,1)} y+c_{(2,2)} x^2 y^2+c_{(2,3)} x^2 y+c_{(2,4)} x, h = c_{(3,1)}+c_{(3,2)} y+c_{(3,3)} x y+c_{(3,4)} x$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  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 0.507781s (cpu); 0.44829s (thread); 0s (gc)
│ │ │ + -- used 0.535004s (cpu); 0.480945s (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 0.00905261s (cpu); 0.00905263s (thread); 0s (gc)
│ │ │ + -- used 0.0114584s (cpu); 0.0114583s (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    +
│ │ │ @@ -825,15 +825,15 @@
│ │ │            
    
│ │ │          
    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 0.0015725s (cpu); 0.00156896s (thread); 0s (gc)
│ │ │ │ + -- used 0.00168223s (cpu); 0.00167838s (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 0.00127507s (cpu); 0.00127554s (thread); 0s (gc)
│ │ │ │ + -- used 0.00151835s (cpu); 0.00151628s (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 0.00194803s (cpu); 0.00194842s (thread); 0s (gc)
│ │ │ │ + -- used 0.00239696s (cpu); 0.00239778s (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
│ │ │ @@ -126,15 +126,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -111,15 +111,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -121,15 +121,15 @@
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : time Res = sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331);
│ │ │ - -- used 0.0304897s (cpu); 0.0292273s (thread); 0s (gc)
│ │ │ + -- used 0.251632s (cpu); 0.0743614s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over ZZ/3331)
│ │ │                                                               | 0 1 0 1 |
    
│ │ │              
    
│ │ │ @@ -849,15 +849,15 @@
│ │ │  
│ │ │  o8 : Sequence
│ │ │              
    
│ │ │                
    i9 : time Res(f,g,h)
│ │ │ - -- used 0.00316112s (cpu); 0.00316119s (thread); 0s (gc)
│ │ │ + -- used 0.00489819s (cpu); 0.00489889s (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  +
│ │ │ @@ -938,15 +938,15 @@
│ │ │  
│ │ │  o11 : Sequence
    
│ │ │              
    
│ │ │                
    i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true));
│ │ │ - -- used 0.260966s (cpu); 0.209622s (thread); 0s (gc)
    
│ │ │ + -- used 0.261818s (cpu); 0.206438s (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 0.507781s (cpu); 0.44829s (thread); 0s (gc)
│ │ │ │ + -- used 0.535004s (cpu); 0.480945s (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 0.00905261s (cpu); 0.00905263s (thread); 0s (gc)
│ │ │ │ + -- used 0.0114584s (cpu); 0.0114583s (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 0.0304897s (cpu); 0.0292273s (thread); 0s (gc)
│ │ │ │ + -- used 0.251632s (cpu); 0.0743614s (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 0.00316112s (cpu); 0.00316119s (thread); 0s (gc)
│ │ │ │ + -- used 0.00489819s (cpu); 0.00489889s (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 0.260966s (cpu); 0.209622s (thread); 0s (gc)
│ │ │ │ + -- used 0.261818s (cpu); 0.206438s (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 0.0015725s (cpu); 0.00156896s (thread); 0s (gc)
│ │ │ + -- used 0.00168223s (cpu); 0.00167838s (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 0.00127507s (cpu); 0.00127554s (thread); 0s (gc)
│ │ │ + -- used 0.00151835s (cpu); 0.00151628s (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 0.00194803s (cpu); 0.00194842s (thread); 0s (gc)
│ │ │ + -- used 0.00239696s (cpu); 0.00239778s (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 0.344281s (cpu); 0.287215s (thread); 0s (gc)
│ │ │ + -- used 0.373484s (cpu); 0.314828s (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 0.68249s (cpu); 0.493249s (thread); 0s (gc)
│ │ │ + -- used 0.842361s (cpu); 0.593415s (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
│ │ │ @@ -125,15 +125,15 @@
│ │ │  
│ │ │  o4 : YoungTableau
    
│ │ │              
    
│ │ │                
    i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ - -- used 0.0015725s (cpu); 0.00156896s (thread); 0s (gc)
│ │ │ + -- used 0.00168223s (cpu); 0.00167838s (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  -
│ │ │ @@ -149,15 +149,15 @@
│ │ │  
│ │ │  o5 : R
    
│ │ │              
    
│ │ │                
    i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ - -- used 0.00127507s (cpu); 0.00127554s (thread); 0s (gc)
│ │ │ + -- used 0.00151835s (cpu); 0.00151628s (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  -
│ │ │ @@ -173,15 +173,15 @@
│ │ │  
│ │ │  o6 : R
    
│ │ │              
    
│ │ │                
    i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ - -- used 0.00194803s (cpu); 0.00194842s (thread); 0s (gc)
│ │ │ + -- used 0.00239696s (cpu); 0.00239778s (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 0.344281s (cpu); 0.287215s (thread); 0s (gc)
│ │ │ + -- used 0.373484s (cpu); 0.314828s (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 0.344281s (cpu); 0.287215s (thread); 0s (gc)
│ │ │ │ + -- used 0.373484s (cpu); 0.314828s (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
│ │ │ @@ -109,15 +109,15 @@
│ │ │  (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 0.68249s (cpu); 0.493249s (thread); 0s (gc)
    
│ │ │ + -- used 0.842361s (cpu); 0.593415s (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 0.68249s (cpu); 0.493249s (thread); 0s (gc)
│ │ │ │ + -- used 0.842361s (cpu); 0.593415s (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
│ │ │ @@ -10,15 +10,15 @@
│ │ │       of discriminant 31 = det| 8 1 |
│ │ │                               | 1 4 |
│ │ │       containing a surface of degree 1 and sectional genus 0
│ │ │       cut out by 5 hypersurfaces of degree 1
│ │ │       (This is a classical example of rational fourfold)
│ │ │  
│ │ │  i3 : time U' = associatedCastelnuovoSurface X;
│ │ │ - -- used 2.39568s (cpu); 1.11189s (thread); 0s (gc)
│ │ │ + -- used 2.97765s (cpu); 1.24278s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │  
│ │ │  i4 : (mu,U,C,f) = building U';
│ │ │  
│ │ │  i5 : ? mu
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.out
│ │ │ @@ -7,15 +7,15 @@
│ │ │  i2 : describe X
│ │ │  
│ │ │  o2 = Special cubic fourfold of discriminant 14
│ │ │       containing a (smooth) surface of degree 4 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degree 2
│ │ │  
│ │ │  i3 : time U' = associatedK3surface X;
│ │ │ - -- used 1.99111s (cpu); 1.02887s (thread); 0s (gc)
│ │ │ + -- used 2.75827s (cpu); 1.21592s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : (mu,U,C,f) = building U';
│ │ │  
│ │ │  i5 : ? mu
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ │  
│ │ │  i3 : time U' = associatedK3surface X;
│ │ │ - -- used 7.14125s (cpu); 4.38468s (thread); 0s (gc)
│ │ │ + -- used 8.85554s (cpu); 5.02189s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : (mu,U,C,f) = building U';
│ │ │  
│ │ │  i5 : ? mu
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__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 4.03482s (cpu); 2.10467s (thread); 0s (gc)
│ │ │ + -- used 3.43085s (cpu); 2.15286s (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 0.429617s (cpu); 0.294228s (thread); 0s (gc)
│ │ │ + -- used 0.36869s (cpu); 0.301226s (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__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │       containing a surface in PP^8 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 13.4563s (cpu); 7.009s (thread); 0s (gc)
│ │ │ + -- used 24.1013s (cpu); 8.59377s (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 [7214, -1460, 7057, -2440, 15907, -14345, -5937, 13402, 1]
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │  
│ │ │  i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ - -- used 0.396s (cpu); 0.234093s (thread); 0s (gc)
│ │ │ + -- used 0.635188s (cpu); 0.325388s (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/_discriminant_lp__Special__Cubic__Fourfold_rp.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 1729890813579561111
│ │ │  
│ │ │  i1 : X = specialCubicFourfold "quintic del Pezzo surface";
│ │ │  
│ │ │  o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and sectional genus 1
│ │ │  
│ │ │  i2 : time discriminant X
│ │ │ - -- used 0.420819s (cpu); 0.147554s (thread); 0s (gc)
│ │ │ + -- used 0.433443s (cpu); 0.138104s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 14
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 1730220932418738713
│ │ │  
│ │ │  i1 : X = specialGushelMukaiFourfold "tau-quadric";
│ │ │  
│ │ │  o1 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i2 : time discriminant X
│ │ │ - -- used 1.02747s (cpu); 0.458878s (thread); 0s (gc)
│ │ │ + -- used 1.24279s (cpu); 0.527487s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
│ │ │  
│ │ │  i3 :
│ │ ├── ./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 0.296112s (cpu); 0.200839s (thread); 0s (gc)
│ │ │ + -- used 0.51251s (cpu); 0.281362s (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__Special__Cubic__Fourfold_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i3 : X = specialCubicFourfold V;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │  
│ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.641923s (cpu); 0.367693s (thread); 0s (gc)
│ │ │ + -- used 0.814949s (cpu); 0.465432s (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__Special__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in G)
│ │ │  
│ │ │  i3 : X = specialGushelMukaiFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
│ │ │  
│ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 3.48579s (cpu); 2.16969s (thread); 0s (gc)
│ │ │ + -- used 4.48101s (cpu); 3.22588s (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 1.79691s (cpu); 0.888229s (thread); 0s (gc)
│ │ │ + -- used 2.11458s (cpu); 0.87831s (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/_special__Cubic__Fourfold.out
│ │ │ @@ -7,22 +7,22 @@
│ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i4 : X = projectiveVariety ideal(x_1^2*x_3+x_0*x_2*x_3-6*x_1*x_2*x_3-x_0*x_3^2-4*x_1*x_3^2-3*x_2*x_3^2+2*x_0^2*x_4-10*x_0*x_1*x_4+13*x_1^2*x_4-x_0*x_2*x_4-3*x_1*x_2*x_4+3*x_2^2*x_4+14*x_0*x_3*x_4-8*x_1*x_3*x_4-4*x_3^2*x_4+x_0*x_4^2-7*x_1*x_4^2+4*x_2*x_4^2-2*x_3*x_4^2-2*x_4^3-x_0*x_1*x_5+x_1^2*x_5+2*x_1*x_2*x_5+3*x_0*x_3*x_5+3*x_1*x_3*x_5-x_3^2*x_5-x_0*x_4*x_5-4*x_1*x_4*x_5+3*x_2*x_4*x_5+2*x_3*x_4*x_5-x_1*x_5^2);
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^5
│ │ │  
│ │ │  i5 : time F = specialCubicFourfold(S,X,NumNodes=>3);
│ │ │ - -- used 0.0119855s (cpu); 0.00881246s (thread); 0s (gc)
│ │ │ + -- used 0.012028s (cpu); 0.0103003s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, cubic fourfold containing a surface of degree 7 and sectional genus 0
│ │ │  
│ │ │  i6 : time describe F
│ │ │  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 0.54403s (cpu); 0.233284s (thread); 0s (gc)
│ │ │ + -- used 0.880125s (cpu); 0.241297s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = 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
│ │ │  
│ │ │  i7 : assert(F == X)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold.out
│ │ │ @@ -7,22 +7,22 @@
│ │ │  o3 : ProjectiveVariety, surface in PP^8
│ │ │  
│ │ │  i4 : 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, 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);
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^8
│ │ │  
│ │ │  i5 : time F = specialGushelMukaiFourfold(S,X);
│ │ │ - -- used 2.29841s (cpu); 1.52131s (thread); 0s (gc)
│ │ │ + -- used 2.14405s (cpu); 1.73797s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i6 : time describe F
│ │ │  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 5.23566s (cpu); 3.15307s (thread); 0s (gc)
│ │ │ + -- used 7.15074s (cpu); 3.68088s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : X = specialGushelMukaiFourfold(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
│ │ │  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 4.03686s (cpu); 2.40993s (thread); 0s (gc)
│ │ │ + -- used 5.96755s (cpu); 3.21145s (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 4.71225s (cpu); 2.71013s (thread); 0s (gc)
│ │ │ + -- used 6.18495s (cpu); 3.37309s (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 = specialCubicFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │  
│ │ │  i4 : time f = unirationalParametrization X;
│ │ │ - -- used 1.16218s (cpu); 0.624472s (thread); 0s (gc)
│ │ │ + -- used 1.68429s (cpu); 0.805926s (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
│ │ │ @@ -106,15 +106,15 @@
│ │ │       cut out by 5 hypersurfaces of degree 1
│ │ │       (This is a classical example of rational fourfold)
    
│ │ │              
    
│ │ │                
    i3 : time U' = associatedCastelnuovoSurface X;
│ │ │ - -- used 2.39568s (cpu); 1.11189s (thread); 0s (gc)
│ │ │ + -- used 2.97765s (cpu); 1.24278s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
    
│ │ │              
    
│ │ │                
    i4 : (mu,U,C,f) = building U';
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  o2 = Complete intersection of 3 quadrics in PP^7
│ │ │ │       of discriminant 31 = det| 8 1 |
│ │ │ │                               | 1 4 |
│ │ │ │       containing a surface of degree 1 and sectional genus 0
│ │ │ │       cut out by 5 hypersurfaces of degree 1
│ │ │ │       (This is a classical example of rational fourfold)
│ │ │ │  i3 : time U' = associatedCastelnuovoSurface X;
│ │ │ │ - -- used 2.39568s (cpu); 1.11189s (thread); 0s (gc)
│ │ │ │ + -- used 2.97765s (cpu); 1.24278s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │ │  i4 : (mu,U,C,f) = building U';
│ │ │ │  i5 : ? mu
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 5-dimensional subvariety of PP^7 cut out by 2
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │       containing a (smooth) surface of degree 4 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degree 2
│ │ │              
    
│ │ │                
    i3 : time U' = associatedK3surface X;
│ │ │ - -- used 1.99111s (cpu); 1.02887s (thread); 0s (gc)
│ │ │ + -- used 2.75827s (cpu); 1.21592s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
    
│ │ │              
    
│ │ │                
    i4 : (mu,U,C,f) = building U';
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  sectional genus 0
│ │ │ │  i2 : describe X
│ │ │ │  
│ │ │ │  o2 = Special cubic fourfold of discriminant 14
│ │ │ │       containing a (smooth) surface of degree 4 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degree 2
│ │ │ │  i3 : time U' = associatedK3surface X;
│ │ │ │ - -- used 1.99111s (cpu); 1.02887s (thread); 0s (gc)
│ │ │ │ + -- used 2.75827s (cpu); 1.21592s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │ │  i4 : (mu,U,C,f) = building U';
│ │ │ │  i5 : ? mu
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: PP^5
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -107,15 +107,15 @@
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ │              
    
│ │ │                
    i3 : time U' = associatedK3surface X;
│ │ │ - -- used 7.14125s (cpu); 4.38468s (thread); 0s (gc)
│ │ │ + -- used 8.85554s (cpu); 5.02189s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
    
│ │ │              
    
│ │ │                
    i4 : (mu,U,C,f) = building U';
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  o2 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │ │       Type: ordinary
│ │ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ │ │  i3 : time U' = associatedK3surface X;
│ │ │ │ - -- used 7.14125s (cpu); 4.38468s (thread); 0s (gc)
│ │ │ │ + -- used 8.85554s (cpu); 5.02189s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │ │  i4 : (mu,U,C,f) = building U';
│ │ │ │  i5 : ? mu
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │       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 4.03482s (cpu); 2.10467s (thread); 0s (gc)
│ │ │ + -- used 3.43085s (cpu); 2.15286s (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 0.429617s (cpu); 0.294228s (thread); 0s (gc)
│ │ │ + -- used 0.36869s (cpu); 0.301226s (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 {}
│ │ │ │ @@ -30,28 +30,28 @@
│ │ │ │  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 4.03482s (cpu); 2.10467s (thread); 0s (gc)
│ │ │ │ + -- used 3.43085s (cpu); 2.15286s (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 0.429617s (cpu); 0.294228s (thread); 0s (gc)
│ │ │ │ + -- used 0.36869s (cpu); 0.301226s (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_(_S_p_e_c_i_a_l_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__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.html
│ │ │ @@ -94,15 +94,15 @@
│ │ │       Type: ordinary
│ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │              
    
│ │ │                
    i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 13.4563s (cpu); 7.009s (thread); 0s (gc)
│ │ │ + -- used 24.1013s (cpu); 8.59377s (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 0.396s (cpu); 0.234093s (thread); 0s (gc)
│ │ │ + -- used 0.635188s (cpu); 0.325388s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
    
│ │ │              
    
│ │ │                
    i7 : S = surface X;
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │  o2 = Special Gushel-Mukai fourfold of discriminant 20
│ │ │ │       containing a surface in PP^8 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 13.4563s (cpu); 7.009s (thread); 0s (gc)
│ │ │ │ + -- used 24.1013s (cpu); 8.59377s (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
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │  i5 : p := point Y -- random point on Y
│ │ │ │  
│ │ │ │  o5 = point of coordinates [7214, -1460, 7057, -2440, 15907, -14345, -5937,
│ │ │ │  13402, 1]
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │ │  i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ │ - -- used 0.396s (cpu); 0.234093s (thread); 0s (gc)
│ │ │ │ + -- used 0.635188s (cpu); 0.325388s (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/_discriminant_lp__Special__Cubic__Fourfold_rp.html
│ │ │ @@ -80,15 +80,15 @@
│ │ │  
│ │ │  o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and sectional genus 1
    
│ │ │              
    
│ │ │                
    i2 : time discriminant X
│ │ │ - -- used 0.420819s (cpu); 0.147554s (thread); 0s (gc)
│ │ │ + -- used 0.433443s (cpu); 0.138104s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 14
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  thanks to the functions _E_u_l_e_r_C_h_a_r_a_c_t_e_r_i_s_t_i_c and _E_u_l_e_r (the option Algorithm
│ │ │ │  allows you to select the method).
│ │ │ │  i1 : X = specialCubicFourfold "quintic del Pezzo surface";
│ │ │ │  
│ │ │ │  o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and
│ │ │ │  sectional genus 1
│ │ │ │  i2 : time discriminant X
│ │ │ │ - -- used 0.420819s (cpu); 0.147554s (thread); 0s (gc)
│ │ │ │ + -- used 0.433443s (cpu); 0.138104s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 14
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_i_s_c_r_i_m_i_n_a_n_t_(_S_p_e_c_i_a_l_G_u_s_h_e_l_M_u_k_a_i_F_o_u_r_f_o_l_d_) -- discriminant of a special
│ │ │ │        Gushel-Mukai fourfold
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * discriminant(HodgeSpecialFourfold)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -80,15 +80,15 @@
│ │ │  
│ │ │  o1 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time discriminant X
│ │ │ - -- used 1.02747s (cpu); 0.458878s (thread); 0s (gc)
│ │ │ + -- used 1.24279s (cpu); 0.527487s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  the functions _c_y_c_l_e_C_l_a_s_s, _E_u_l_e_r_C_h_a_r_a_c_t_e_r_i_s_t_i_c and _E_u_l_e_r (the option Algorithm
│ │ │ │  allows you to select the method).
│ │ │ │  i1 : X = specialGushelMukaiFourfold "tau-quadric";
│ │ │ │  
│ │ │ │  o1 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and
│ │ │ │  sectional genus 0
│ │ │ │  i2 : time discriminant X
│ │ │ │ - -- used 1.02747s (cpu); 0.458878s (thread); 0s (gc)
│ │ │ │ + -- used 1.24279s (cpu); 0.527487s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 10
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_i_s_c_r_i_m_i_n_a_n_t_(_S_p_e_c_i_a_l_C_u_b_i_c_F_o_u_r_f_o_l_d_) -- discriminant of a special cubic
│ │ │ │        fourfold
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _d_i_s_c_r_i_m_i_n_a_n_t_(_S_p_e_c_i_a_l_G_u_s_h_e_l_M_u_k_a_i_F_o_u_r_f_o_l_d_) -- discriminant of a special
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ - -- used 0.296112s (cpu); 0.200839s (thread); 0s (gc)
│ │ │ + -- used 0.51251s (cpu); 0.281362s (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 0.296112s (cpu); 0.200839s (thread); 0s (gc)
│ │ │ │ + -- used 0.51251s (cpu); 0.281362s (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__Special__Cubic__Fourfold_rp.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.641923s (cpu); 0.367693s (thread); 0s (gc)
│ │ │ + -- used 0.814949s (cpu); 0.465432s (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 = specialCubicFourfold V;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ │ - -- used 0.641923s (cpu); 0.367693s (thread); 0s (gc)
│ │ │ │ + -- used 0.814949s (cpu); 0.465432s (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__Special__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 3.48579s (cpu); 2.16969s (thread); 0s (gc)
│ │ │ + -- used 4.48101s (cpu); 3.22588s (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 = specialGushelMukaiFourfold S;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ │ - -- used 3.48579s (cpu); 2.16969s (thread); 0s (gc)
│ │ │ │ + -- used 4.48101s (cpu); 3.22588s (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
│ │ │ @@ -88,15 +88,15 @@
│ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │       1^2 2^5
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time parametrizeFanoFourfold X
│ │ │ - -- used 1.79691s (cpu); 0.888229s (thread); 0s (gc)
│ │ │ + -- used 2.11458s (cpu); 0.87831s (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 1.79691s (cpu); 0.888229s (thread); 0s (gc)
│ │ │ │ + -- used 2.11458s (cpu); 0.87831s (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/_special__Cubic__Fourfold.html
│ │ │ @@ -95,25 +95,25 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^5
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time F = specialCubicFourfold(S,X,NumNodes=>3);
│ │ │ - -- used 0.0119855s (cpu); 0.00881246s (thread); 0s (gc)
│ │ │ + -- used 0.012028s (cpu); 0.0103003s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, cubic fourfold containing a surface of degree 7 and sectional genus 0
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time describe F
│ │ │  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 0.54403s (cpu); 0.233284s (thread); 0s (gc)
│ │ │ + -- used 0.880125s (cpu); 0.241297s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = 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
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -115,24 +115,24 @@
│ │ │ │  3*x_1*x_2*x_4+3*x_2^2*x_4+14*x_0*x_3*x_4-8*x_1*x_3*x_4-4*x_3^2*x_4+x_0*x_4^2-
│ │ │ │  7*x_1*x_4^2+4*x_2*x_4^2-2*x_3*x_4^2-2*x_4^3-
│ │ │ │  x_0*x_1*x_5+x_1^2*x_5+2*x_1*x_2*x_5+3*x_0*x_3*x_5+3*x_1*x_3*x_5-x_3^2*x_5-
│ │ │ │  x_0*x_4*x_5-4*x_1*x_4*x_5+3*x_2*x_4*x_5+2*x_3*x_4*x_5-x_1*x_5^2);
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, hypersurface in PP^5
│ │ │ │  i5 : time F = specialCubicFourfold(S,X,NumNodes=>3);
│ │ │ │ - -- used 0.0119855s (cpu); 0.00881246s (thread); 0s (gc)
│ │ │ │ + -- used 0.012028s (cpu); 0.0103003s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, cubic fourfold containing a surface of degree 7 and
│ │ │ │  sectional genus 0
│ │ │ │  i6 : time describe F
│ │ │ │  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 0.54403s (cpu); 0.233284s (thread); 0s (gc)
│ │ │ │ + -- used 0.880125s (cpu); 0.241297s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = 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
│ │ │ │  i7 : assert(F == X)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_p_e_c_i_a_l_C_u_b_i_c_F_o_u_r_f_o_l_d_(_E_m_b_e_d_d_e_d_P_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_) -- random special cubic
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold.html
│ │ │ @@ -93,25 +93,25 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time F = specialGushelMukaiFourfold(S,X);
│ │ │ - -- used 2.29841s (cpu); 1.52131s (thread); 0s (gc)
│ │ │ + -- used 2.14405s (cpu); 1.73797s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time describe F
│ │ │  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 5.23566s (cpu); 3.15307s (thread); 0s (gc)
│ │ │ + -- used 7.15074s (cpu); 3.68088s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,24 +33,24 @@
│ │ │ │  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);
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^8
│ │ │ │  i5 : time F = specialGushelMukaiFourfold(S,X);
│ │ │ │ - -- used 2.29841s (cpu); 1.52131s (thread); 0s (gc)
│ │ │ │ + -- used 2.14405s (cpu); 1.73797s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and
│ │ │ │  sectional genus 0
│ │ │ │  i6 : time describe F
│ │ │ │  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 5.23566s (cpu); 3.15307s (thread); 0s (gc)
│ │ │ │ + -- used 7.15074s (cpu); 3.68088s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │ │       Type: ordinary
│ │ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html
│ │ │ @@ -81,15 +81,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    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 4.03686s (cpu); 2.40993s (thread); 0s (gc)
│ │ │ + -- used 5.96755s (cpu); 3.21145s (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))
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and
│ │ │ │  sectional genus 0
│ │ │ │  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 4.03686s (cpu); 2.40993s (thread); 0s (gc)
│ │ │ │ + -- used 5.96755s (cpu); 3.21145s (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
│ │ │ @@ -82,15 +82,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    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 4.71225s (cpu); 2.71013s (thread); 0s (gc)
│ │ │ + -- used 6.18495s (cpu); 3.37309s (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))
    
│ │ │              
│ │ │ ├── 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 4.71225s (cpu); 2.71013s (thread); 0s (gc)
│ │ │ │ + -- used 6.18495s (cpu); 3.37309s (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
│ │ │ @@ -82,15 +82,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time f = unirationalParametrization X;
│ │ │ - -- used 1.16218s (cpu); 0.624472s (thread); 0s (gc)
│ │ │ + -- used 1.68429s (cpu); 0.805926s (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 = specialCubicFourfold S;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time f = unirationalParametrization X;
│ │ │ │ - -- used 1.16218s (cpu); 0.624472s (thread); 0s (gc)
│ │ │ │ + -- used 1.68429s (cpu); 0.805926s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
│ │ │ │  i5 : degreeSequence f
│ │ │ │  
│ │ │ │  o5 = {[10]}
│ │ │ │  
│ │ │ │  o5 : List
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/example-output/_graph_lp__Mixed__Graph_rp.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │                                b => {a, c}
│ │ │                                c => {b}
│ │ │  
│ │ │  o2 : HashTable
│ │ │  
│ │ │  i3 : keys (graph G)
│ │ │  
│ │ │ -o3 = {Graph, Bigraph, Digraph}
│ │ │ +o3 = {Digraph, Graph, Bigraph}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : (graph G)#Bigraph === bigraph G
│ │ │  
│ │ │  o4 = true
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/html/_graph_lp__Mixed__Graph_rp.html
│ │ │ @@ -116,15 +116,15 @@
│ │ │  o2 : HashTable
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : keys (graph G)
│ │ │  
│ │ │ -o3 = {Graph, Bigraph, Digraph}
│ │ │ +o3 = {Digraph, Graph, Bigraph}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : (graph G)#Bigraph === bigraph G
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,15 +46,15 @@
│ │ │ │                 Graph => Graph{a => {b}   }
│ │ │ │                                b => {a, c}
│ │ │ │                                c => {b}
│ │ │ │  
│ │ │ │  o2 : HashTable
│ │ │ │  i3 : keys (graph G)
│ │ │ │  
│ │ │ │ -o3 = {Graph, Bigraph, Digraph}
│ │ │ │ +o3 = {Digraph, Graph, Bigraph}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : (graph G)#Bigraph === bigraph G
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _M_i_x_e_d_G_r_a_p_h -- a graph that has undirected, directed and bidirected edges
│ │ ├── ./usr/share/doc/Macaulay2/Style/example-output/_generate__Grammar.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 3455701143666534588
│ │ │  
│ │ │  i1 : outfile = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10069-0/0
│ │ │ +o1 = /tmp/M2-10109-0/0
│ │ │  
│ │ │  i2 : template = outfile | ".in"
│ │ │  
│ │ │ -o2 = /tmp/M2-10069-0/0.in
│ │ │ +o2 = /tmp/M2-10109-0/0.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/M2-10069-0/0
│ │ │ + -- generating /tmp/M2-10109-0/0
│ │ │  
│ │ │  i12 : get outfile
│ │ │  
│ │ │  o12 = Auto-generated for Macaulay2-1.25.11. 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
│ │ │ @@ -82,22 +82,22 @@
│ │ │            
    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.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -143,15 +143,15 @@
│ │ │            @M2KEYWORDS@
│ │ │        }
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : outfile = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10069-0/0
    
│ │ │ +o1 = /tmp/M2-10109-0/0
│ │ │              
    
│ │ │                
    i2 : template = outfile | ".in"
│ │ │  
│ │ │ -o2 = /tmp/M2-10069-0/0.in
    
│ │ │ +o2 = /tmp/M2-10109-0/0.in
│ │ │              
    
│ │ │                
    i3 : template << "@M2BANNER@" << endl << endl;
    
│ │ │              
    
│ │ │                
    i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ - -- generating /tmp/M2-10069-0/0
    
│ │ │ + -- generating /tmp/M2-10109-0/0
│ │ │              
    
│ │ │                
    i12 : get outfile
│ │ │  
│ │ │  o12 = Auto-generated for Macaulay2-1.25.11. 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/M2-10069-0/0
│ │ │ │ +o1 = /tmp/M2-10109-0/0
│ │ │ │  i2 : template = outfile | ".in"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10069-0/0.in
│ │ │ │ +o2 = /tmp/M2-10109-0/0.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/M2-10069-0/0
│ │ │ │ + -- generating /tmp/M2-10109-0/0
│ │ │ │  i12 : get outfile
│ │ │ │  
│ │ │ │  o12 = Auto-generated for Macaulay2-1.25.11. 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 0.306855s (cpu); 0.192479s (thread); 0s (gc)
│ │ │ + -- used 0.337156s (cpu); 0.220506s (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 0.0330497s (cpu); 0.0330527s (thread); 0s (gc)
│ │ │ + -- used 0.0491368s (cpu); 0.0491416s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html
│ │ │ @@ -141,15 +141,15 @@
│ │ │  
│ │ │  o6 = false
    
│ │ │              
    
│ │ │                
    i7 : time symbolicPower(P,4);
│ │ │ - -- used 0.306855s (cpu); 0.192479s (thread); 0s (gc)
│ │ │ + -- used 0.337156s (cpu); 0.220506s (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})
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time symbolicPower(Q,4);
│ │ │ - -- used 0.0330497s (cpu); 0.0330527s (thread); 0s (gc)
│ │ │ + -- used 0.0491368s (cpu); 0.0491416s (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 0.306855s (cpu); 0.192479s (thread); 0s (gc)
│ │ │ │ + -- used 0.337156s (cpu); 0.220506s (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 0.0330497s (cpu); 0.0330527s (thread); 0s (gc)
│ │ │ │ + -- used 0.0491368s (cpu); 0.0491416s (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 : ChainComplex
│ │ │  
│ │ │  i4 : time T=tateExtension W;
│ │ │ - -- used 0.116725s (cpu); 0.116725s (thread); 0s (gc)
│ │ │ + -- used 0.147515s (cpu); 0.142527s (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
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o3 : ChainComplex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time T=tateExtension W;
│ │ │ - -- used 0.116725s (cpu); 0.116725s (thread); 0s (gc)
    
│ │ │ + -- used 0.147515s (cpu); 0.142527s (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 : ChainComplex
│ │ │ │  i4 : time T=tateExtension W;
│ │ │ │ - -- used 0.116725s (cpu); 0.116725s (thread); 0s (gc)
│ │ │ │ + -- used 0.147515s (cpu); 0.142527s (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 0.854696s (cpu); 0.665482s (thread); 0s (gc)
│ │ │ + -- used 1.03689s (cpu); 0.747389s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
│ │ │  
│ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.65944s (cpu); 2.30064s (thread); 0s (gc)
│ │ │ + -- used 2.93015s (cpu); 2.40038s (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 0.00209879s (cpu); 0.00209348s (thread); 0s (gc)
│ │ │ + -- used 0.00245742s (cpu); 0.00245262s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00481737s (cpu); 0.0048183s (thread); 0s (gc)
│ │ │ + -- used 0.00585104s (cpu); 0.00585798s (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 0.0262795s (cpu); 0.0262787s (thread); 0s (gc)
│ │ │ + -- used 0.0338454s (cpu); 0.0338409s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
│ │ │  
│ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.16981s (cpu); 1.26905s (thread); 0s (gc)
│ │ │ + -- used 2.63942s (cpu); 1.5535s (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 0.141185s (cpu); 0.0902546s (thread); 0s (gc)
│ │ │ + -- used 0.179006s (cpu); 0.110652s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
│ │ │  
│ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.144332s (cpu); 0.0902735s (thread); 0s (gc)
│ │ │ + -- used 0.222088s (cpu); 0.146203s (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 0.904683s (cpu); 0.715355s (thread); 0s (gc)
│ │ │ + -- used 1.08247s (cpu); 0.838226s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
│ │ │  
│ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.412784s (cpu); 0.287986s (thread); 0s (gc)
│ │ │ + -- used 0.445748s (cpu); 0.284877s (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 0.120203s (cpu); 0.0782458s (thread); 0s (gc)
│ │ │ + -- used 0.154875s (cpu); 0.0831959s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = false
│ │ │  
│ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.228747s (cpu); 0.174944s (thread); 0s (gc)
│ │ │ + -- used 0.258187s (cpu); 0.19311s (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 0.360889s (cpu); 0.177517s (thread); 0s (gc)
│ │ │ + -- used 0.43677s (cpu); 0.230368s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.505166s (cpu); 0.327584s (thread); 0s (gc)
│ │ │ + -- used 0.527536s (cpu); 0.333067s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = ideal (y, x)
│ │ │  
│ │ │  o24 : Ideal of R
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html
│ │ │ @@ -226,23 +226,23 @@
│ │ │  
│ │ │  o16 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ - -- used 0.854696s (cpu); 0.665482s (thread); 0s (gc)
│ │ │ + -- used 1.03689s (cpu); 0.747389s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.65944s (cpu); 2.30064s (thread); 0s (gc)
│ │ │ + -- used 2.93015s (cpu); 2.40038s (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 0.854696s (cpu); 0.665482s (thread); 0s (gc)
│ │ │ │ + -- used 1.03689s (cpu); 0.747389s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 : Ideal of R
│ │ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ │ - -- used 2.65944s (cpu); 2.30064s (thread); 0s (gc)
│ │ │ │ + -- used 2.93015s (cpu); 2.40038s (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
│ │ │ @@ -96,23 +96,23 @@
│ │ │              
│ │ │                
    i4 : R = S/(ker g);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time isCohenMacaulay(R)
│ │ │ - -- used 0.00209879s (cpu); 0.00209348s (thread); 0s (gc)
│ │ │ + -- used 0.00245742s (cpu); 0.00245262s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00481737s (cpu); 0.0048183s (thread); 0s (gc)
│ │ │ + -- used 0.00585104s (cpu); 0.00585798s (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 0.00209879s (cpu); 0.00209348s (thread); 0s (gc)
│ │ │ │ + -- used 0.00245742s (cpu); 0.00245262s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ │ - -- used 0.00481737s (cpu); 0.0048183s (thread); 0s (gc)
│ │ │ │ + -- used 0.00585104s (cpu); 0.00585798s (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
│ │ │ @@ -214,23 +214,23 @@
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i20 : time isFInjective(R)
│ │ │ - -- used 0.0262795s (cpu); 0.0262787s (thread); 0s (gc)
│ │ │ + -- used 0.0338454s (cpu); 0.0338409s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
    
│ │ │              
    
│ │ │                
    i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.16981s (cpu); 1.26905s (thread); 0s (gc)
│ │ │ + -- used 2.63942s (cpu); 1.5535s (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.
    
│ │ │ @@ -240,23 +240,23 @@
│ │ │              
│ │ │                
    i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : time isFInjective(R)
│ │ │ - -- used 0.141185s (cpu); 0.0902546s (thread); 0s (gc)
│ │ │ + -- used 0.179006s (cpu); 0.110652s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.144332s (cpu); 0.0902735s (thread); 0s (gc)
│ │ │ + -- used 0.222088s (cpu); 0.146203s (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 $\mathbb{P}^1$.
    
│ │ │ @@ -283,23 +283,23 @@
│ │ │              
│ │ │                
    i28 : R = S/(ker f);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : time isFInjective(R)
│ │ │ - -- used 0.904683s (cpu); 0.715355s (thread); 0s (gc)
│ │ │ + -- used 1.08247s (cpu); 0.838226s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.412784s (cpu); 0.287986s (thread); 0s (gc)
│ │ │ + -- used 0.445748s (cpu); 0.284877s (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 0.0262795s (cpu); 0.0262787s (thread); 0s (gc)
│ │ │ │ + -- used 0.0338454s (cpu); 0.0338409s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 = true
│ │ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ │ - -- used 2.16981s (cpu); 1.26905s (thread); 0s (gc)
│ │ │ │ + -- used 2.63942s (cpu); 1.5535s (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 0.141185s (cpu); 0.0902546s (thread); 0s (gc)
│ │ │ │ + -- used 0.179006s (cpu); 0.110652s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = false
│ │ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ │ - -- used 0.144332s (cpu); 0.0902735s (thread); 0s (gc)
│ │ │ │ + -- used 0.222088s (cpu); 0.146203s (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 0.904683s (cpu); 0.715355s (thread); 0s (gc)
│ │ │ │ + -- used 1.08247s (cpu); 0.838226s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = false
│ │ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ │ - -- used 0.412784s (cpu); 0.287986s (thread); 0s (gc)
│ │ │ │ + -- used 0.445748s (cpu); 0.284877s (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
│ │ │ @@ -273,23 +273,23 @@
│ │ │                
    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 0.120203s (cpu); 0.0782458s (thread); 0s (gc)
│ │ │ + -- used 0.154875s (cpu); 0.0831959s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.228747s (cpu); 0.174944s (thread); 0s (gc)
│ │ │ + -- used 0.258187s (cpu); 0.19311s (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 0.120203s (cpu); 0.0782458s (thread); 0s (gc)
│ │ │ │ + -- used 0.154875s (cpu); 0.0831959s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = false
│ │ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ │ - -- used 0.228747s (cpu); 0.174944s (thread); 0s (gc)
│ │ │ │ + -- used 0.258187s (cpu); 0.19311s (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
│ │ │ @@ -255,25 +255,25 @@
│ │ │          
    
│ │ │            
    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 0.360889s (cpu); 0.177517s (thread); 0s (gc)
│ │ │ + -- used 0.43677s (cpu); 0.230368s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
    
│ │ │              
    
│ │ │                
    i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.505166s (cpu); 0.327584s (thread); 0s (gc)
│ │ │ + -- used 0.527536s (cpu); 0.333067s (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 0.360889s (cpu); 0.177517s (thread); 0s (gc)
│ │ │ │ + -- used 0.43677s (cpu); 0.230368s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = ideal (y, x)
│ │ │ │  
│ │ │ │  o23 : Ideal of R
│ │ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ │ - -- used 0.505166s (cpu); 0.327584s (thread); 0s (gc)
│ │ │ │ + -- used 0.527536s (cpu); 0.333067s (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
│ │ │ @@ -2,16 +2,16 @@
│ │ │  
│ │ │  i1 : S = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2", Minimal=>true)
│ │ │  
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o3 = LineageTable{((0, 1), 0) => null}
│ │ │ +                  ((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
│ │ │ @@ -2,16 +2,15 @@
│ │ │  
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o3 = 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/_minimize_lp__Lineage__Table_rp.out
│ │ │ @@ -2,35 +2,41 @@
│ │ │  
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ +                                     2
│ │ │ +o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │                                     3
│ │ │ -o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ +                  ((0, 2), 0) => -c
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │ +                             2
│ │ │ +                  (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │                                  2
│ │ │                    0 => a*b*c + c
│ │ │                            3       2
│ │ │                    1 => - b c + a*b  + a*c
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : minimize T
│ │ │  
│ │ │ -o4 = LineageTable{((0, 2), 0) => null}
│ │ │ +o4 = LineageTable{((0, 1), 0) => null}
│ │ │ +                  ((0, 2), 0) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │ +                  (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ ├── ./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"
│ │ │  
│ │ │ -                                     2
│ │ │ -o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │                                     3
│ │ │ -                  ((0, 2), 0) => -c
│ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                      2
│ │ │                    ((0, 2), 1) => a*c
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │ @@ -26,16 +24,15 @@
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{((0, 1), 0) => null}
│ │ │ -                  ((0, 2), 0) => null
│ │ │ +o4 = 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
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out
│ │ │ @@ -6,62 +6,38 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : allowableThreads  = 4;
│ │ │  
│ │ │  i4 : H = tgb I
│ │ │  
│ │ │ -                                                                   2 9
│ │ │ -o4 = LineageTable{(((((0, 1), 2), 2), ((0, 1), 2)), (0, 1)) => -22y z                               }
│ │ │ -                                                             2 9
│ │ │ -                  (((((0, 1), 2), 2), ((0, 1), 2)), 2) => 16y z
│ │ │ -                                                         2 13
│ │ │ -                  (((((0, 1), 2), 2), 1), (0, 1)) => -22y z
│ │ │ -                                                   2 13
│ │ │ -                  (((((0, 1), 2), 2), 1), 2) => 16y z
│ │ │ -                                                   2 12
│ │ │ -                  (((((0, 1), 2), 2), 2), 2) => 16y z
│ │ │ -                                                                                                 2 4
│ │ │ -                  (((((0, 1), 2), 3), (((0, 1), 2), 2)), ((((0, 1), 2), 2), ((0, 1), 2))) => -43y z
│ │ │ -                                                                  2 7
│ │ │ -                  (((((0, 1), 2), 3), (((0, 1), 2), 2)), 2) => 16y 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      4 5
│ │ │ -                  ((((0, 1), 2), 3), (((0, 1), 2), 2)) => 10y z  + 31y z
│ │ │ -                                              3 17
│ │ │ -                  ((((0, 1), 2), 3), 1) => 11y z
│ │ │ -                                                4 5     3 16
│ │ │ -                  ((((0, 1), 2), 3), 2) => - 16y z  + 9y z
│ │ │ -                                              3 17
│ │ │ -                  ((((0, 1), 2), 3), 3) => 41y 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
│ │ │ +                                   4 4     3 7
│ │ │ +o4 = LineageTable{((0, 1), 2) => 9y z  - 6y z        }
│ │ │ +                                    2 11     2 10
│ │ │ +                  ((0, 1), 3) => 44y z   + 4y z
│ │ │ +                                         2 5      2 4
│ │ │ +                  ((0, 2), (0, 1)) => 40y z  + 22y z
│ │ │ +                                    2 6
│ │ │ +                  ((0, 2), 1) => 25y z
│ │ │ +                                    2 4
│ │ │ +                  ((0, 2), 3) => 10y z
│ │ │ +                                         5       2 4
│ │ │ +                  ((0, 3), (0, 1)) => 46y z + 40y z
│ │ │ +                                     2 4
│ │ │ +                  ((1, 2), 1) => -18y 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
│ │ │ -                                3 16
│ │ │ -                  (1, 2) => -19y z
│ │ │ +                              5 3     2 4
│ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ +                              5 2      5
│ │ │ +                  (0, 3) => 5y z  + 28y z
│ │ │ +                                 4 5      3 8
│ │ │ +                  (1, 2) => - 45y z  + 30y z
│ │ │ +                               3 7      3 6
│ │ │ +                  (1, 3) => 30y z  - 34y 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
│ │ │ @@ -73,16 +73,16 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2", Minimal=>true)
│ │ │  
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o3 = LineageTable{((0, 1), 0) => null}
│ │ │ +                  ((0, 2), 0) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,16 +12,16 @@
│ │ │ │  Gr\"obner basis is minimized. Lineages of non-minimal Gr\"obner basis elements
│ │ │ │  that were added to the basis during the distributed computation are saved, with
│ │ │ │  the corresponding entry in the table being null.
│ │ │ │  i1 : S = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2", Minimal=>true)
│ │ │ │  
│ │ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │ +o3 = LineageTable{((0, 1), 0) => null}
│ │ │ │ +                  ((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
│ │ │ @@ -86,16 +86,15 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o3 = 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 {}
│ │ │ │ @@ -19,16 +19,15 @@
│ │ │ │  This simple function just returns the Gr\"obner basis computed with threaded
│ │ │ │  Gr\"obner 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, 1), 0), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │ +o3 = 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/_minimize_lp__Lineage__Table_rp.html
│ │ │ @@ -82,18 +82,22 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ +                                     2
│ │ │ +o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │                                     3
│ │ │ -o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ +                  ((0, 2), 0) => -c
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │ +                             2
│ │ │ +                  (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │                                  2
│ │ │                    0 => a*b*c + c
│ │ │                            3       2
│ │ │                    1 => - b c + a*b  + a*c
│ │ │ @@ -103,17 +107,19 @@
│ │ │  o3 : LineageTable
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : minimize T
│ │ │  
│ │ │ -o4 = LineageTable{((0, 2), 0) => null}
│ │ │ +o4 = LineageTable{((0, 1), 0) => null}
│ │ │ +                  ((0, 2), 0) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │ +                  (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,34 +19,40 @@
│ │ │ │  minimal generators of the ideal generated by the leading terms of the values of
│ │ │ │  H. If the values of H constitute a Gr\"obner basis of the ideal they generate,
│ │ │ │  this method returns a minimal Gr\"obner basis.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │ │  
│ │ │ │ +                                     2
│ │ │ │ +o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │ │                                     3
│ │ │ │ -o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ │ +                  ((0, 2), 0) => -c
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │ +                             2
│ │ │ │ +                  (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │                                  2
│ │ │ │                    0 => a*b*c + c
│ │ │ │                            3       2
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : minimize T
│ │ │ │  
│ │ │ │ -o4 = LineageTable{((0, 2), 0) => null}
│ │ │ │ +o4 = LineageTable{((0, 1), 0) => null}
│ │ │ │ +                  ((0, 2), 0) => null
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │ +                  (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html
│ │ │ @@ -82,18 +82,16 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │ -                                     2
│ │ │ -o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │                                     3
│ │ │ -                  ((0, 2), 0) => -c
│ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                      2
│ │ │                    ((0, 2), 1) => a*c
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │ @@ -109,16 +107,15 @@
│ │ │  o3 : LineageTable
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{((0, 1), 0) => null}
│ │ │ -                  ((0, 2), 0) => null
│ │ │ +o4 = 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
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,18 +20,16 @@
│ │ │ │  remainder on the division by the remaining values H.
│ │ │ │  If values H constitute a Gr\"obner basis of the ideal they generate, this
│ │ │ │  method returns a reduced Gr\"obner basis.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │ │  
│ │ │ │ -                                     2
│ │ │ │ -o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │ │                                     3
│ │ │ │ -                  ((0, 2), 0) => -c
│ │ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ │                                      2
│ │ │ │                    ((0, 2), 1) => a*c
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │                               2
│ │ │ │                    (0, 1) => a c
│ │ │ │                                 2
│ │ │ │ @@ -43,16 +41,15 @@
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : reduce T
│ │ │ │  
│ │ │ │ -o4 = LineageTable{((0, 1), 0) => null}
│ │ │ │ -                  ((0, 2), 0) => null
│ │ │ │ +o4 = 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
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html
│ │ │ @@ -95,62 +95,38 @@
│ │ │                
    i3 : allowableThreads  = 4;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : H = tgb I
│ │ │  
│ │ │ -                                                                   2 9
│ │ │ -o4 = LineageTable{(((((0, 1), 2), 2), ((0, 1), 2)), (0, 1)) => -22y z                               }
│ │ │ -                                                             2 9
│ │ │ -                  (((((0, 1), 2), 2), ((0, 1), 2)), 2) => 16y z
│ │ │ -                                                         2 13
│ │ │ -                  (((((0, 1), 2), 2), 1), (0, 1)) => -22y z
│ │ │ -                                                   2 13
│ │ │ -                  (((((0, 1), 2), 2), 1), 2) => 16y z
│ │ │ -                                                   2 12
│ │ │ -                  (((((0, 1), 2), 2), 2), 2) => 16y z
│ │ │ -                                                                                                 2 4
│ │ │ -                  (((((0, 1), 2), 3), (((0, 1), 2), 2)), ((((0, 1), 2), 2), ((0, 1), 2))) => -43y z
│ │ │ -                                                                  2 7
│ │ │ -                  (((((0, 1), 2), 3), (((0, 1), 2), 2)), 2) => 16y 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      4 5
│ │ │ -                  ((((0, 1), 2), 3), (((0, 1), 2), 2)) => 10y z  + 31y z
│ │ │ -                                              3 17
│ │ │ -                  ((((0, 1), 2), 3), 1) => 11y z
│ │ │ -                                                4 5     3 16
│ │ │ -                  ((((0, 1), 2), 3), 2) => - 16y z  + 9y z
│ │ │ -                                              3 17
│ │ │ -                  ((((0, 1), 2), 3), 3) => 41y 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
│ │ │ +                                   4 4     3 7
│ │ │ +o4 = LineageTable{((0, 1), 2) => 9y z  - 6y z        }
│ │ │ +                                    2 11     2 10
│ │ │ +                  ((0, 1), 3) => 44y z   + 4y z
│ │ │ +                                         2 5      2 4
│ │ │ +                  ((0, 2), (0, 1)) => 40y z  + 22y z
│ │ │ +                                    2 6
│ │ │ +                  ((0, 2), 1) => 25y z
│ │ │ +                                    2 4
│ │ │ +                  ((0, 2), 3) => 10y z
│ │ │ +                                         5       2 4
│ │ │ +                  ((0, 3), (0, 1)) => 46y z + 40y z
│ │ │ +                                     2 4
│ │ │ +                  ((1, 2), 1) => -18y 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
│ │ │ -                                3 16
│ │ │ -                  (1, 2) => -19y z
│ │ │ +                              5 3     2 4
│ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ +                              5 2      5
│ │ │ +                  (0, 3) => 5y z  + 28y z
│ │ │ +                                 4 5      3 8
│ │ │ +                  (1, 2) => - 45y z  + 30y z
│ │ │ +                               3 7      3 6
│ │ │ +                  (1, 3) => 30y z  - 34y 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 {}
│ │ │ │ @@ -26,65 +26,38 @@
│ │ │ │  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 9
│ │ │ │ -o4 = LineageTable{(((((0, 1), 2), 2), ((0, 1), 2)), (0, 1)) => -22y z
│ │ │ │ -}
│ │ │ │ -                                                             2 9
│ │ │ │ -                  (((((0, 1), 2), 2), ((0, 1), 2)), 2) => 16y z
│ │ │ │ -                                                         2 13
│ │ │ │ -                  (((((0, 1), 2), 2), 1), (0, 1)) => -22y z
│ │ │ │ -                                                   2 13
│ │ │ │ -                  (((((0, 1), 2), 2), 1), 2) => 16y z
│ │ │ │ -                                                   2 12
│ │ │ │ -                  (((((0, 1), 2), 2), 2), 2) => 16y z
│ │ │ │ -
│ │ │ │ -2 4
│ │ │ │ -                  (((((0, 1), 2), 3), (((0, 1), 2), 2)), ((((0, 1), 2), 2), (
│ │ │ │ -(0, 1), 2))) => -43y z
│ │ │ │ -                                                                  2 7
│ │ │ │ -                  (((((0, 1), 2), 3), (((0, 1), 2), 2)), 2) => 16y 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      4 5
│ │ │ │ -                  ((((0, 1), 2), 3), (((0, 1), 2), 2)) => 10y z  + 31y z
│ │ │ │ -                                              3 17
│ │ │ │ -                  ((((0, 1), 2), 3), 1) => 11y z
│ │ │ │ -                                                4 5     3 16
│ │ │ │ -                  ((((0, 1), 2), 3), 2) => - 16y z  + 9y z
│ │ │ │ -                                              3 17
│ │ │ │ -                  ((((0, 1), 2), 3), 3) => 41y 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
│ │ │ │ +                                   4 4     3 7
│ │ │ │ +o4 = LineageTable{((0, 1), 2) => 9y z  - 6y z        }
│ │ │ │ +                                    2 11     2 10
│ │ │ │ +                  ((0, 1), 3) => 44y z   + 4y z
│ │ │ │ +                                         2 5      2 4
│ │ │ │ +                  ((0, 2), (0, 1)) => 40y z  + 22y z
│ │ │ │ +                                    2 6
│ │ │ │ +                  ((0, 2), 1) => 25y z
│ │ │ │ +                                    2 4
│ │ │ │ +                  ((0, 2), 3) => 10y z
│ │ │ │ +                                         5       2 4
│ │ │ │ +                  ((0, 3), (0, 1)) => 46y z + 40y z
│ │ │ │ +                                     2 4
│ │ │ │ +                  ((1, 2), 1) => -18y 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
│ │ │ │ -                                3 16
│ │ │ │ -                  (1, 2) => -19y z
│ │ │ │ +                              5 3     2 4
│ │ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ │ +                              5 2      5
│ │ │ │ +                  (0, 3) => 5y z  + 28y z
│ │ │ │ +                                 4 5      3 8
│ │ │ │ +                  (1, 2) => - 45y z  + 30y z
│ │ │ │ +                               3 7      3 6
│ │ │ │ +                  (1, 3) => 30y z  - 34y 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 1.13366s (cpu); 0.779417s (thread); 0s (gc)
│ │ │ + -- used 1.52408s (cpu); 0.987592s (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 4.61337s (cpu); 3.0038s (thread); 0s (gc)
│ │ │ + -- used 6.05469s (cpu); 3.86889s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 252
│ │ │  
│ │ │  o5 : QQ
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html
│ │ │ @@ -131,15 +131,15 @@
│ │ │  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 1.13366s (cpu); 0.779417s (thread); 0s (gc)
│ │ │ + -- used 1.52408s (cpu); 0.987592s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 252
│ │ │  
│ │ │  o4 : QQ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -150,15 +150,15 @@
│ │ │  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 4.61337s (cpu); 3.0038s (thread); 0s (gc)
│ │ │ + -- used 6.05469s (cpu); 3.86889s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 252
│ │ │  
│ │ │  o5 : 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 1.13366s (cpu); 0.779417s (thread); 0s (gc)
│ │ │ │ + -- used 1.52408s (cpu); 0.987592s (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 4.61337s (cpu); 3.0038s (thread); 0s (gc)
│ │ │ │ + -- used 6.05469s (cpu); 3.86889s (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);
│ │ │ - -- .0931162s elapsed
│ │ │ + -- .116782s 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;
│ │ │ - -- 1.28881s elapsed
│ │ │ + -- 1.09395s elapsed
│ │ │  
│ │ │  i9 : #Ts2 == #Ts
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/example-output/_generate__Triangulations.out
│ │ │ @@ -21,87 +21,15 @@
│ │ │  
│ │ │  o3 = triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}
│ │ │  
│ │ │  o3 : Triangulation
│ │ │  
│ │ │  i4 : Ts1 = generateTriangulations A -- list of Triangulation's.
│ │ │  
│ │ │ -o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -227,275 +155,275 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o4 : List
│ │ │ -
│ │ │ -i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ -
│ │ │ -o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7},
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 5, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5},
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7},
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6},
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 7},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │ +     {2, 4, 6, 7}}}
│ │ │ +
│ │ │ +o4 : List
│ │ │ +
│ │ │ +i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ +
│ │ │ +o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 6},
│ │ │ +     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 2, 7},
│ │ │ +     {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ +     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ +     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {0, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 6},
│ │ │ +     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 6, 7},
│ │ │ +     {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {0, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 6},
│ │ │ +     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 6},
│ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ +     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o5 : List
│ │ │ -
│ │ │ -i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ -
│ │ │ -o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ +     {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}}
│ │ │ +
│ │ │ +o5 : List
│ │ │ +
│ │ │ +i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ +
│ │ │ +o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -621,93 +549,93 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o6 : List
│ │ │ -
│ │ │ -i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ -
│ │ │ -o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {2, 4, 6, 7}}}
│ │ │ +
│ │ │ +o6 : List
│ │ │ +
│ │ │ +i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ +
│ │ │ +o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -833,15 +761,87 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 6, 7}}}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : all(Ts4, isFine)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │ @@ -858,191 +858,193 @@
│ │ │  o11 = Tally{false => 66}
│ │ │              true => 8
│ │ │  
│ │ │  o11 : Tally
│ │ │  
│ │ │  i12 : Ts4/gkzVector
│ │ │  
│ │ │ -        16     16  4     8  4       20     8  8     4  8          4  8     8 
│ │ │ -o12 = {{--, 4, --, -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ -         3      3  3     3  3        3     3  3     3  3          3  3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8  20       16  16  4  16  4  4       8        4     16  4  16      
│ │ │ -      4, -, --}, {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4,
│ │ │ -         3   3        3   3  3   3  3  3       3        3      3  3   3      
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20  4  4  20          16        8  16  4  4          20  4        4 
│ │ │ -      4, --, -, -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -,
│ │ │ -          3  3  3   3           3        3   3  3  3           3  3        3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      20       16  16  4     4        8       4  4  16  8        16    16  4 
│ │ │ -      --, 4}, {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -,
│ │ │ -       3        3   3  3     3        3       3  3   3  3         3     3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16     4        8    4  16     4     16  8       8  8     8  8     8 
│ │ │ -      --, 4, -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -,
│ │ │ -       3     3        3    3   3     3      3  3       3  3     3  3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8       8  4     20  8     8    16  4  4        16  16  4       4  16 
│ │ │ -      -}, {4, -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --,
│ │ │ -      3       3  3      3  3     3     3  3  3         3   3  3       3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16  8        4    4  20              20  4    4        20  20       
│ │ │ -      --, -, 8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4,
│ │ │ -       3  3        3    3   3               3  3    3         3   3       
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4    4        8  16  16  4          4     20  20     4             4 
│ │ │ -      -}, {-, 4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -,
│ │ │ -      3    3        3   3   3  3          3      3   3     3             3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      20  20  4          8        16     4  4  16    8        4  8  8  20 
│ │ │ -      --, --, -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --,
│ │ │ -       3   3  3          3         3     3  3   3    3        3  3  3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -           8     8  8     4  20       20  4  4  20  4  20  20  4    4  20 
│ │ │ -      4}, {-, 8, -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --,
│ │ │ -           3     3  3     3   3        3  3  3   3  3   3   3  3    3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      20  4  20  4  4  20    20        4  4        20       8  4     4    
│ │ │ -      --, -, --, -, -, --}, {--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4,
│ │ │ -       3  3   3  3  3   3     3        3  3         3       3  3     3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16  16    8        4     4  16  16    4  16     16     4  8       4    
│ │ │ -      --, --}, {-, 8, 4, -, 4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4,
│ │ │ -       3   3    3        3     3   3   3    3   3      3     3  3       3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20     8  8  8    8  8  8     20        4    4        8     8  20 
│ │ │ -      4, --, 8, -, -, -}, {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --,
│ │ │ -          3     3  3  3    3  3  3      3        3    3        3     3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8    4     16  4     8  16          4  20     8     8  8    16  4  4 
│ │ │ -      -}, {-, 8, --, -, 4, -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -,
│ │ │ -      3    3      3  3     3   3          3   3     3     3  3     3  3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         16        8    16     4  16  4  16     4       8  20  8  4       
│ │ │ -      8, --, 4, 4, -}, {--, 4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4,
│ │ │ -          3        3     3     3   3  3   3     3       3   3  3  3       
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8    8  8     8     20  4       4  16     4  16     4  16       20  8 
│ │ │ -      -}, {-, -, 8, -, 4, --, -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -,
│ │ │ -      3    3  3     3      3  3       3   3     3   3     3   3        3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8  4        8    20        4  8  8  8          8  8  8  4        20  
│ │ │ -      -, -, 4, 8, -}, {--, 4, 4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --},
│ │ │ -      3  3        3     3        3  3  3  3          3  3  3  3         3  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -       16  16     4     4  8          8  4     8  8     20       20     4  4 
│ │ │ -      {--, --, 4, -, 4, -, -, 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -,
│ │ │ -        3   3     3     3  3          3  3     3  3      3        3     3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20       20  4              4  20       4  20        20  4       8 
│ │ │ -      4, --, 4}, {--, -, 4, 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-,
│ │ │ -          3        3  3              3   3       3   3         3  3       3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8  20     4  8          16  4  16  8        4    4     16  16     8 
│ │ │ -      4, -, --, 8, -, -, 4}, {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -,
│ │ │ -         3   3     3  3           3  3   3  3        3    3      3   3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4          4  8     16  16     4    4     20        20     4    8  8 
│ │ │ -      -, 4}, {4, -, -, 8, --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -,
│ │ │ -      3          3  3      3   3     3    3      3         3     3    3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8        8  8  8    8  20  8     8        4       16  8     4  16    
│ │ │ -      -, 8, 8, -, -, -}, {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8,
│ │ │ -      3        3  3  3    3   3  3     3        3        3  3     3   3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4    4        8  16  4  16       16  4     16  4     16  4    4     16 
│ │ │ -      -}, {-, 8, 4, -, --, -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --,
│ │ │ -      3    3        3   3  3   3        3  3      3  3      3  3    3      3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  16  4     16       8  8  8  8  8  8       20  8     8     8  4     
│ │ │ -      -, --, -, 4, --}, {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8},
│ │ │ -      3   3  3      3       3  3  3  3  3  3        3  3     3     3  3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          4  8     4  16     16    20     4        4     20       4  4  16 
│ │ │ -      {8, -, -, 4, -, --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --,
│ │ │ -          3  3     3   3      3     3     3        3      3       3  3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  16  16          8  16     4     16  4    8  8     20     8  4     
│ │ │ -      -, --, --, 4}, {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4},
│ │ │ -      3   3   3          3   3     3      3  3    3  3      3     3  3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          4  8     20     8  8       8  4     16     16  4    8  8  20     8 
│ │ │ -      {4, -, -, 8, --, 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -,
│ │ │ -          3  3      3     3  3       3  3      3      3  3    3  3   3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -            4    4        8     20  8  8    8     8  8  8  8     8    4  16 
│ │ │ -      8, 4, -}, {-, 4, 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --,
│ │ │ -            3    3        3      3  3  3    3     3  3  3  3     3    3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16        4  4  16       20  4     8  8     8    8        4  8  20  8
│ │ │ -      --, 4, 8, -, -, --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -,
│ │ │ -       3        3  3   3        3  3     3  3     3    3        3  3   3  3
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4}}
│ │ │ +        20        4  4        20       8  4     4     16  16    8        4 
│ │ │ +o12 = {{--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4, --, --}, {-, 8, 4, -,
│ │ │ +         3        3  3         3       3  3     3      3   3    3        3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  16  16    4  16     16     4  8       4        20     8  8  8  
│ │ │ +      4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4, 4, --, 8, -, -, -},
│ │ │ +         3   3   3    3   3      3     3  3       3         3     3  3  3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       8  8  8     20        4    4        8     8  20  8    4     16  4    
│ │ │ +      {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --, -}, {-, 8, --, -, 4,
│ │ │ +       3  3  3      3        3    3        3     3   3  3    3      3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  16          4  20     8     8  8    16  4  4     16        8    16 
│ │ │ +      -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -, 8, --, 4, 4, -}, {--,
│ │ │ +      3   3          3   3     3     3  3     3  3  3      3        3     3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  16  4  16     4       8  20  8  4        8    8  8     8     20 
│ │ │ +      4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4, -}, {-, -, 8, -, 4, --,
│ │ │ +         3   3  3   3     3       3   3  3  3        3    3  3     3      3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4       4  16     4  16     4  16       20  8  8  4        8    20    
│ │ │ +      -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -, -, -, 4, 8, -}, {--, 4,
│ │ │ +      3       3   3     3   3     3   3        3  3  3  3        3     3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  8  8  8          8  8  8  4        20    16  16     4     4  8 
│ │ │ +      4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, {--, --, 4, -, 4, -, -,
│ │ │ +         3  3  3  3          3  3  3  3         3     3   3     3     3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +              8  4     8  8     20       20     4  4     20       20  4    
│ │ │ +      8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, 4, --, 4}, {--, -, 4,
│ │ │ +              3  3     3  3      3        3     3  3      3        3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +               4  20       4  20        20  4       8     8  20     4  8     
│ │ │ +      4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, 4, -, --, 8, -, -, 4},
│ │ │ +               3   3       3   3         3  3       3     3   3     3  3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          16  4  16  8        4    4     16  16     8  4          4  8    
│ │ │ +      {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, -, 4}, {4, -, -, 8,
│ │ │ +           3  3   3  3        3    3      3   3     3  3          3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16  16     4    4     20        20     4    8  8  8        8  8  8  
│ │ │ +      --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, -, 8, 8, -, -, -},
│ │ │ +       3   3     3    3      3         3     3    3  3  3        3  3  3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       8  20  8     8        4       16  8     4  16     4    4        8  16 
│ │ │ +      {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, -}, {-, 8, 4, -, --,
│ │ │ +       3   3  3     3        3        3  3     3   3     3    3        3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  16       16  4     16  4     16  4    4     16  4  16  4     16  
│ │ │ +      -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, -, --, -, 4, --},
│ │ │ +      3   3        3  3      3  3      3  3    3      3  3   3  3      3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          8  8  8  8  8  8       20  8     8     8  4          4  8     4 
│ │ │ +      {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ +          3  3  3  3  3  3        3  3     3     3  3          3  3     3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16     16    20     4        4     20       4  4  16  4  16  16     
│ │ │ +      --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, -, --, --, 4},
│ │ │ +       3      3     3     3        3      3       3  3   3  3   3   3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          8  16     4     16  4    8  8     20     8  4          4  8     20 
│ │ │ +      {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, {4, -, -, 8, --,
│ │ │ +          3   3     3      3  3    3  3      3     3  3          3  3      3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         8  8       8  4     16     16  4    8  8  20     8        4    4    
│ │ │ +      4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, 8, 4, -}, {-, 4,
│ │ │ +         3  3       3  3      3      3  3    3  3   3     3        3    3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         8     20  8  8    8     8  8  8  8     8    4  16  16        4  4 
│ │ │ +      8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, --, 4, 8, -, -,
│ │ │ +         3      3  3  3    3     3  3  3  3     3    3   3   3        3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16       20  4     8  8     8    8        4  8  20  8       16     16 
│ │ │ +      --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, 4}, {--, 4, --,
│ │ │ +       3        3  3     3  3     3    3        3  3   3  3        3      3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4     8  4       20     8  8     4  8          4  8     8     8  20  
│ │ │ +      -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, 4, -, --},
│ │ │ +      3     3  3        3     3  3     3  3          3  3     3     3   3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          16  16  4  16  4  4       8        4     16  4  16          20  4 
│ │ │ +      {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, 4, --, -,
│ │ │ +           3   3  3   3  3  3       3        3      3  3   3           3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  20          16        8  16  4  4          20  4        4  20     
│ │ │ +      -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -, --, 4},
│ │ │ +      3   3           3        3   3  3  3           3  3        3   3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       16  16  4     4        8       4  4  16  8        16    16  4  16    
│ │ │ +      {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -, --, 4,
│ │ │ +        3   3  3     3        3       3  3   3  3         3     3  3   3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4        8    4  16     4     16  8       8  8     8  8     8  8      
│ │ │ +      -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -, -}, {4,
│ │ │ +      3        3    3   3     3      3  3       3  3     3  3     3  3      
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  4     20  8     8    16  4  4        16  16  4       4  16  16  8 
│ │ │ +      -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --, --, -,
│ │ │ +      3  3      3  3     3     3  3  3         3   3  3       3   3   3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +            4    4  20              20  4    4        20  20        4    4 
│ │ │ +      8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4, -}, {-,
│ │ │ +            3    3   3               3  3    3         3   3        3    3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +            8  16  16  4          4     20  20     4             4  20  20 
│ │ │ +      4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -, --, --,
│ │ │ +            3   3   3  3          3      3   3     3             3   3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4          8        16     4  4  16    8        4  8  8  20       8    
│ │ │ +      -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --, 4}, {-, 8,
│ │ │ +      3          3         3     3  3   3    3        3  3  3   3       3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  8     4  20       20  4  4  20  4  20  20  4    4  20  20  4  20  4 
│ │ │ +      -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --, --, -, --, -,
│ │ │ +      3  3     3   3        3  3  3   3  3   3   3  3    3   3   3  3   3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  20
│ │ │ +      -, --}}
│ │ │ +      3   3
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : volume convexHull A -- 8
│ │ │  
│ │ │  o13 = 8
│ │ │  
│ │ │  o13 : QQ
│ │ │  
│ │ │  i14 : stars1 = select(Ts4, t -> (gkzVector t)#-1 == 8)
│ │ │  
│ │ │ -o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ +      {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │        {2, 4, 6, 7}}}
│ │ │  
│ │ │  o14 : List
│ │ │  
│ │ │  i15 : stars2 = select(Ts4, isStar)
│ │ │  
│ │ │ -o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ +      {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │        {2, 4, 6, 7}}}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : stars1 == stars2
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/html/_generate__Triangulations.html
│ │ │ @@ -117,87 +117,15 @@
│ │ │  o3 : Triangulation
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : Ts1 = generateTriangulations A -- list of Triangulation's.
│ │ │  
│ │ │ -o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -323,281 +251,281 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o4 : List
    
│ │ │ -            
│ │ │ -          
│ │ │ -          
│ │ │ -            
│ │ │ -              
    i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ -
│ │ │ -o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7},
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 5, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5},
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7},
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6},
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 7},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │ +     {2, 4, 6, 7}}}
│ │ │ +
│ │ │ +o4 : List
    
│ │ │ +            
│ │ │ +          
│ │ │ +          
│ │ │ +            
│ │ │ +              
    i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ +
│ │ │ +o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 6},
│ │ │ +     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 2, 7},
│ │ │ +     {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ +     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ +     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {0, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 6},
│ │ │ +     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 6, 7},
│ │ │ +     {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {0, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 6},
│ │ │ +     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 6},
│ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ +     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o5 : List
    
│ │ │ -            
│ │ │ -          
│ │ │ -          
│ │ │ -            
│ │ │ -              
    i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ -
│ │ │ -o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ +     {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}}
│ │ │ +
│ │ │ +o5 : List
    
│ │ │ +            
│ │ │ +          
│ │ │ +          
│ │ │ +            
│ │ │ +              
    i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ +
│ │ │ +o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -723,96 +651,96 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o6 : List
    
│ │ │ -            
│ │ │ -          
│ │ │ -          
│ │ │ -            
│ │ │ -              
    i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ -
│ │ │ -o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {2, 4, 6, 7}}}
│ │ │ +
│ │ │ +o6 : List
    
│ │ │ +            
│ │ │ +          
│ │ │ +          
│ │ │ +            
│ │ │ +              
    i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ +
│ │ │ +o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -938,15 +866,87 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 6, 7}}}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : all(Ts4, isFine)
│ │ │ @@ -978,131 +978,133 @@
│ │ │  o11 : Tally
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : Ts4/gkzVector
│ │ │  
│ │ │ -        16     16  4     8  4       20     8  8     4  8          4  8     8 
│ │ │ -o12 = {{--, 4, --, -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ -         3      3  3     3  3        3     3  3     3  3          3  3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8  20       16  16  4  16  4  4       8        4     16  4  16      
│ │ │ -      4, -, --}, {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4,
│ │ │ -         3   3        3   3  3   3  3  3       3        3      3  3   3      
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20  4  4  20          16        8  16  4  4          20  4        4 
│ │ │ -      4, --, -, -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -,
│ │ │ -          3  3  3   3           3        3   3  3  3           3  3        3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      20       16  16  4     4        8       4  4  16  8        16    16  4 
│ │ │ -      --, 4}, {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -,
│ │ │ -       3        3   3  3     3        3       3  3   3  3         3     3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16     4        8    4  16     4     16  8       8  8     8  8     8 
│ │ │ -      --, 4, -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -,
│ │ │ -       3     3        3    3   3     3      3  3       3  3     3  3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8       8  4     20  8     8    16  4  4        16  16  4       4  16 
│ │ │ -      -}, {4, -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --,
│ │ │ -      3       3  3      3  3     3     3  3  3         3   3  3       3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16  8        4    4  20              20  4    4        20  20       
│ │ │ -      --, -, 8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4,
│ │ │ -       3  3        3    3   3               3  3    3         3   3       
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4    4        8  16  16  4          4     20  20     4             4 
│ │ │ -      -}, {-, 4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -,
│ │ │ -      3    3        3   3   3  3          3      3   3     3             3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      20  20  4          8        16     4  4  16    8        4  8  8  20 
│ │ │ -      --, --, -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --,
│ │ │ -       3   3  3          3         3     3  3   3    3        3  3  3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -           8     8  8     4  20       20  4  4  20  4  20  20  4    4  20 
│ │ │ -      4}, {-, 8, -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --,
│ │ │ -           3     3  3     3   3        3  3  3   3  3   3   3  3    3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      20  4  20  4  4  20    20        4  4        20       8  4     4    
│ │ │ -      --, -, --, -, -, --}, {--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4,
│ │ │ -       3  3   3  3  3   3     3        3  3         3       3  3     3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16  16    8        4     4  16  16    4  16     16     4  8       4    
│ │ │ -      --, --}, {-, 8, 4, -, 4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4,
│ │ │ -       3   3    3        3     3   3   3    3   3      3     3  3       3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20     8  8  8    8  8  8     20        4    4        8     8  20 
│ │ │ -      4, --, 8, -, -, -}, {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --,
│ │ │ -          3     3  3  3    3  3  3      3        3    3        3     3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8    4     16  4     8  16          4  20     8     8  8    16  4  4 
│ │ │ -      -}, {-, 8, --, -, 4, -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -,
│ │ │ -      3    3      3  3     3   3          3   3     3     3  3     3  3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         16        8    16     4  16  4  16     4       8  20  8  4       
│ │ │ -      8, --, 4, 4, -}, {--, 4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4,
│ │ │ -          3        3     3     3   3  3   3     3       3   3  3  3       
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8    8  8     8     20  4       4  16     4  16     4  16       20  8 
│ │ │ -      -}, {-, -, 8, -, 4, --, -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -,
│ │ │ -      3    3  3     3      3  3       3   3     3   3     3   3        3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8  4        8    20        4  8  8  8          8  8  8  4        20  
│ │ │ -      -, -, 4, 8, -}, {--, 4, 4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --},
│ │ │ -      3  3        3     3        3  3  3  3          3  3  3  3         3  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -       16  16     4     4  8          8  4     8  8     20       20     4  4 
│ │ │ -      {--, --, 4, -, 4, -, -, 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -,
│ │ │ -        3   3     3     3  3          3  3     3  3      3        3     3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20       20  4              4  20       4  20        20  4       8 
│ │ │ -      4, --, 4}, {--, -, 4, 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-,
│ │ │ -          3        3  3              3   3       3   3         3  3       3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8  20     4  8          16  4  16  8        4    4     16  16     8 
│ │ │ -      4, -, --, 8, -, -, 4}, {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -,
│ │ │ -         3   3     3  3           3  3   3  3        3    3      3   3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4          4  8     16  16     4    4     20        20     4    8  8 
│ │ │ -      -, 4}, {4, -, -, 8, --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -,
│ │ │ -      3          3  3      3   3     3    3      3         3     3    3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8        8  8  8    8  20  8     8        4       16  8     4  16    
│ │ │ -      -, 8, 8, -, -, -}, {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8,
│ │ │ -      3        3  3  3    3   3  3     3        3        3  3     3   3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4    4        8  16  4  16       16  4     16  4     16  4    4     16 
│ │ │ -      -}, {-, 8, 4, -, --, -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --,
│ │ │ -      3    3        3   3  3   3        3  3      3  3      3  3    3      3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  16  4     16       8  8  8  8  8  8       20  8     8     8  4     
│ │ │ -      -, --, -, 4, --}, {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8},
│ │ │ -      3   3  3      3       3  3  3  3  3  3        3  3     3     3  3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          4  8     4  16     16    20     4        4     20       4  4  16 
│ │ │ -      {8, -, -, 4, -, --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --,
│ │ │ -          3  3     3   3      3     3     3        3      3       3  3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  16  16          8  16     4     16  4    8  8     20     8  4     
│ │ │ -      -, --, --, 4}, {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4},
│ │ │ -      3   3   3          3   3     3      3  3    3  3      3     3  3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          4  8     20     8  8       8  4     16     16  4    8  8  20     8 
│ │ │ -      {4, -, -, 8, --, 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -,
│ │ │ -          3  3      3     3  3       3  3      3      3  3    3  3   3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -            4    4        8     20  8  8    8     8  8  8  8     8    4  16 
│ │ │ -      8, 4, -}, {-, 4, 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --,
│ │ │ -            3    3        3      3  3  3    3     3  3  3  3     3    3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16        4  4  16       20  4     8  8     8    8        4  8  20  8
│ │ │ -      --, 4, 8, -, -, --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -,
│ │ │ -       3        3  3   3        3  3     3  3     3    3        3  3   3  3
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4}}
│ │ │ +        20        4  4        20       8  4     4     16  16    8        4 
│ │ │ +o12 = {{--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4, --, --}, {-, 8, 4, -,
│ │ │ +         3        3  3         3       3  3     3      3   3    3        3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  16  16    4  16     16     4  8       4        20     8  8  8  
│ │ │ +      4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4, 4, --, 8, -, -, -},
│ │ │ +         3   3   3    3   3      3     3  3       3         3     3  3  3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       8  8  8     20        4    4        8     8  20  8    4     16  4    
│ │ │ +      {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --, -}, {-, 8, --, -, 4,
│ │ │ +       3  3  3      3        3    3        3     3   3  3    3      3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  16          4  20     8     8  8    16  4  4     16        8    16 
│ │ │ +      -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -, 8, --, 4, 4, -}, {--,
│ │ │ +      3   3          3   3     3     3  3     3  3  3      3        3     3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  16  4  16     4       8  20  8  4        8    8  8     8     20 
│ │ │ +      4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4, -}, {-, -, 8, -, 4, --,
│ │ │ +         3   3  3   3     3       3   3  3  3        3    3  3     3      3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4       4  16     4  16     4  16       20  8  8  4        8    20    
│ │ │ +      -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -, -, -, 4, 8, -}, {--, 4,
│ │ │ +      3       3   3     3   3     3   3        3  3  3  3        3     3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  8  8  8          8  8  8  4        20    16  16     4     4  8 
│ │ │ +      4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, {--, --, 4, -, 4, -, -,
│ │ │ +         3  3  3  3          3  3  3  3         3     3   3     3     3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +              8  4     8  8     20       20     4  4     20       20  4    
│ │ │ +      8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, 4, --, 4}, {--, -, 4,
│ │ │ +              3  3     3  3      3        3     3  3      3        3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +               4  20       4  20        20  4       8     8  20     4  8     
│ │ │ +      4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, 4, -, --, 8, -, -, 4},
│ │ │ +               3   3       3   3         3  3       3     3   3     3  3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          16  4  16  8        4    4     16  16     8  4          4  8    
│ │ │ +      {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, -, 4}, {4, -, -, 8,
│ │ │ +           3  3   3  3        3    3      3   3     3  3          3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16  16     4    4     20        20     4    8  8  8        8  8  8  
│ │ │ +      --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, -, 8, 8, -, -, -},
│ │ │ +       3   3     3    3      3         3     3    3  3  3        3  3  3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       8  20  8     8        4       16  8     4  16     4    4        8  16 
│ │ │ +      {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, -}, {-, 8, 4, -, --,
│ │ │ +       3   3  3     3        3        3  3     3   3     3    3        3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  16       16  4     16  4     16  4    4     16  4  16  4     16  
│ │ │ +      -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, -, --, -, 4, --},
│ │ │ +      3   3        3  3      3  3      3  3    3      3  3   3  3      3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          8  8  8  8  8  8       20  8     8     8  4          4  8     4 
│ │ │ +      {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ +          3  3  3  3  3  3        3  3     3     3  3          3  3     3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16     16    20     4        4     20       4  4  16  4  16  16     
│ │ │ +      --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, -, --, --, 4},
│ │ │ +       3      3     3     3        3      3       3  3   3  3   3   3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          8  16     4     16  4    8  8     20     8  4          4  8     20 
│ │ │ +      {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, {4, -, -, 8, --,
│ │ │ +          3   3     3      3  3    3  3      3     3  3          3  3      3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         8  8       8  4     16     16  4    8  8  20     8        4    4    
│ │ │ +      4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, 8, 4, -}, {-, 4,
│ │ │ +         3  3       3  3      3      3  3    3  3   3     3        3    3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         8     20  8  8    8     8  8  8  8     8    4  16  16        4  4 
│ │ │ +      8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, --, 4, 8, -, -,
│ │ │ +         3      3  3  3    3     3  3  3  3     3    3   3   3        3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16       20  4     8  8     8    8        4  8  20  8       16     16 
│ │ │ +      --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, 4}, {--, 4, --,
│ │ │ +       3        3  3     3  3     3    3        3  3   3  3        3      3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4     8  4       20     8  8     4  8          4  8     8     8  20  
│ │ │ +      -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, 4, -, --},
│ │ │ +      3     3  3        3     3  3     3  3          3  3     3     3   3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          16  16  4  16  4  4       8        4     16  4  16          20  4 
│ │ │ +      {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, 4, --, -,
│ │ │ +           3   3  3   3  3  3       3        3      3  3   3           3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  20          16        8  16  4  4          20  4        4  20     
│ │ │ +      -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -, --, 4},
│ │ │ +      3   3           3        3   3  3  3           3  3        3   3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       16  16  4     4        8       4  4  16  8        16    16  4  16    
│ │ │ +      {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -, --, 4,
│ │ │ +        3   3  3     3        3       3  3   3  3         3     3  3   3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4        8    4  16     4     16  8       8  8     8  8     8  8      
│ │ │ +      -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -, -}, {4,
│ │ │ +      3        3    3   3     3      3  3       3  3     3  3     3  3      
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  4     20  8     8    16  4  4        16  16  4       4  16  16  8 
│ │ │ +      -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --, --, -,
│ │ │ +      3  3      3  3     3     3  3  3         3   3  3       3   3   3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +            4    4  20              20  4    4        20  20        4    4 
│ │ │ +      8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4, -}, {-,
│ │ │ +            3    3   3               3  3    3         3   3        3    3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +            8  16  16  4          4     20  20     4             4  20  20 
│ │ │ +      4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -, --, --,
│ │ │ +            3   3   3  3          3      3   3     3             3   3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4          8        16     4  4  16    8        4  8  8  20       8    
│ │ │ +      -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --, 4}, {-, 8,
│ │ │ +      3          3         3     3  3   3    3        3  3  3   3       3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  8     4  20       20  4  4  20  4  20  20  4    4  20  20  4  20  4 
│ │ │ +      -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --, --, -, --, -,
│ │ │ +      3  3     3   3        3  3  3   3  3   3   3  3    3   3   3  3   3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  20
│ │ │ +      -, --}}
│ │ │ +      3   3
│ │ │  
│ │ │  o12 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : volume convexHull A -- 8
│ │ │ @@ -1112,66 +1114,66 @@
│ │ │  o13 : QQ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : stars1 = select(Ts4, t -> (gkzVector t)#-1 == 8)
│ │ │  
│ │ │ -o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ +      {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │        {2, 4, 6, 7}}}
│ │ │  
│ │ │  o14 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : stars2 = select(Ts4, isStar)
│ │ │  
│ │ │ -o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ +      {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │        {2, 4, 6, 7}}}
│ │ │  
│ │ │  o15 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -57,87 +57,15 @@
│ │ │ │  
│ │ │ │  o3 = triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3,
│ │ │ │  4, 5, 6}, {3, 5, 6, 7}}
│ │ │ │  
│ │ │ │  o3 : Triangulation
│ │ │ │  i4 : Ts1 = generateTriangulations A -- list of Triangulation's.
│ │ │ │  
│ │ │ │ -o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │ +o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -263,273 +191,273 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}}
│ │ │ │ -
│ │ │ │ -o4 : List
│ │ │ │ -i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ │ -
│ │ │ │ -o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7},
│ │ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 3, 5, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5},
│ │ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7},
│ │ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7},
│ │ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6},
│ │ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 4, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 7},
│ │ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}},
│ │ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6},
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │ │ +     {2, 4, 6, 7}}}
│ │ │ │ +
│ │ │ │ +o4 : List
│ │ │ │ +i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ │ +
│ │ │ │ +o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6},
│ │ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 6},
│ │ │ │ +     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ │ +     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 2, 7},
│ │ │ │ +     {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │ +     {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ │ +     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6},
│ │ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ │ +     {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ │ +     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ │ +     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ │ +     {0, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}},
│ │ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 6},
│ │ │ │ +     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │ │ +     {1, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 6, 7},
│ │ │ │ +     {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7},
│ │ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │ +     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │ +     {0, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 6},
│ │ │ │ +     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 6},
│ │ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}}
│ │ │ │ -
│ │ │ │ -o5 : List
│ │ │ │ -i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ │ -
│ │ │ │ -o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +     {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │ +     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ │ +     {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ │ +     {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ │ +     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ │ +     {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ │ +     {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ │ +     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}}
│ │ │ │ +
│ │ │ │ +o5 : List
│ │ │ │ +i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ │ +
│ │ │ │ +o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -655,92 +583,92 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}}
│ │ │ │ -
│ │ │ │ -o6 : List
│ │ │ │ -i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ │ -
│ │ │ │ -o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │ +     {2, 4, 6, 7}}}
│ │ │ │ +
│ │ │ │ +o6 : List
│ │ │ │ +i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ │ +
│ │ │ │ +o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -866,15 +794,87 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}}
│ │ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {2, 4, 6, 7}}}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : all(Ts4, isFine)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : all(Ts4, isStar)
│ │ │ │  
│ │ │ │ @@ -886,188 +886,190 @@
│ │ │ │  
│ │ │ │  o11 = Tally{false => 66}
│ │ │ │              true => 8
│ │ │ │  
│ │ │ │  o11 : Tally
│ │ │ │  i12 : Ts4/gkzVector
│ │ │ │  
│ │ │ │ -        16     16  4     8  4       20     8  8     4  8          4  8     8
│ │ │ │ -o12 = {{--, 4, --, -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ │ -         3      3  3     3  3        3     3  3     3  3          3  3     3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         8  20       16  16  4  16  4  4       8        4     16  4  16
│ │ │ │ -      4, -, --}, {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4,
│ │ │ │ -         3   3        3   3  3   3  3  3       3        3      3  3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         20  4  4  20          16        8  16  4  4          20  4        4
│ │ │ │ -      4, --, -, -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -,
│ │ │ │ -          3  3  3   3           3        3   3  3  3           3  3        3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      20       16  16  4     4        8       4  4  16  8        16    16  4
│ │ │ │ -      --, 4}, {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -,
│ │ │ │ -       3        3   3  3     3        3       3  3   3  3         3     3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      16     4        8    4  16     4     16  8       8  8     8  8     8
│ │ │ │ -      --, 4, -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -,
│ │ │ │ -       3     3        3    3   3     3      3  3       3  3     3  3     3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      8       8  4     20  8     8    16  4  4        16  16  4       4  16
│ │ │ │ -      -}, {4, -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --,
│ │ │ │ -      3       3  3      3  3     3     3  3  3         3   3  3       3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      16  8        4    4  20              20  4    4        20  20
│ │ │ │ -      --, -, 8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4,
│ │ │ │ -       3  3        3    3   3               3  3    3         3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4    4        8  16  16  4          4     20  20     4             4
│ │ │ │ -      -}, {-, 4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -,
│ │ │ │ -      3    3        3   3   3  3          3      3   3     3             3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      20  20  4          8        16     4  4  16    8        4  8  8  20
│ │ │ │ -      --, --, -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --,
│ │ │ │ -       3   3  3          3         3     3  3   3    3        3  3  3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -           8     8  8     4  20       20  4  4  20  4  20  20  4    4  20
│ │ │ │ -      4}, {-, 8, -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --,
│ │ │ │ -           3     3  3     3   3        3  3  3   3  3   3   3  3    3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      20  4  20  4  4  20    20        4  4        20       8  4     4
│ │ │ │ -      --, -, --, -, -, --}, {--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4,
│ │ │ │ -       3  3   3  3  3   3     3        3  3         3       3  3     3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      16  16    8        4     4  16  16    4  16     16     4  8       4
│ │ │ │ -      --, --}, {-, 8, 4, -, 4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4,
│ │ │ │ -       3   3    3        3     3   3   3    3   3      3     3  3       3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         20     8  8  8    8  8  8     20        4    4        8     8  20
│ │ │ │ -      4, --, 8, -, -, -}, {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --,
│ │ │ │ -          3     3  3  3    3  3  3      3        3    3        3     3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      8    4     16  4     8  16          4  20     8     8  8    16  4  4
│ │ │ │ -      -}, {-, 8, --, -, 4, -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -,
│ │ │ │ -      3    3      3  3     3   3          3   3     3     3  3     3  3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         16        8    16     4  16  4  16     4       8  20  8  4
│ │ │ │ -      8, --, 4, 4, -}, {--, 4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4,
│ │ │ │ -          3        3     3     3   3  3   3     3       3   3  3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      8    8  8     8     20  4       4  16     4  16     4  16       20  8
│ │ │ │ -      -}, {-, -, 8, -, 4, --, -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -,
│ │ │ │ -      3    3  3     3      3  3       3   3     3   3     3   3        3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      8  4        8    20        4  8  8  8          8  8  8  4        20
│ │ │ │ -      -, -, 4, 8, -}, {--, 4, 4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --},
│ │ │ │ -      3  3        3     3        3  3  3  3          3  3  3  3         3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -       16  16     4     4  8          8  4     8  8     20       20     4  4
│ │ │ │ -      {--, --, 4, -, 4, -, -, 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -,
│ │ │ │ -        3   3     3     3  3          3  3     3  3      3        3     3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         20       20  4              4  20       4  20        20  4       8
│ │ │ │ -      4, --, 4}, {--, -, 4, 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-,
│ │ │ │ -          3        3  3              3   3       3   3         3  3       3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         8  20     4  8          16  4  16  8        4    4     16  16     8
│ │ │ │ -      4, -, --, 8, -, -, 4}, {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -,
│ │ │ │ -         3   3     3  3           3  3   3  3        3    3      3   3     3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4          4  8     16  16     4    4     20        20     4    8  8
│ │ │ │ -      -, 4}, {4, -, -, 8, --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -,
│ │ │ │ -      3          3  3      3   3     3    3      3         3     3    3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      8        8  8  8    8  20  8     8        4       16  8     4  16
│ │ │ │ -      -, 8, 8, -, -, -}, {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8,
│ │ │ │ -      3        3  3  3    3   3  3     3        3        3  3     3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4    4        8  16  4  16       16  4     16  4     16  4    4     16
│ │ │ │ -      -}, {-, 8, 4, -, --, -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --,
│ │ │ │ -      3    3        3   3  3   3        3  3      3  3      3  3    3      3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4  16  4     16       8  8  8  8  8  8       20  8     8     8  4
│ │ │ │ -      -, --, -, 4, --}, {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8},
│ │ │ │ -      3   3  3      3       3  3  3  3  3  3        3  3     3     3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -          4  8     4  16     16    20     4        4     20       4  4  16
│ │ │ │ -      {8, -, -, 4, -, --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --,
│ │ │ │ -          3  3     3   3      3     3     3        3      3       3  3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4  16  16          8  16     4     16  4    8  8     20     8  4
│ │ │ │ -      -, --, --, 4}, {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4},
│ │ │ │ -      3   3   3          3   3     3      3  3    3  3      3     3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -          4  8     20     8  8       8  4     16     16  4    8  8  20     8
│ │ │ │ -      {4, -, -, 8, --, 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -,
│ │ │ │ -          3  3      3     3  3       3  3      3      3  3    3  3   3     3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -            4    4        8     20  8  8    8     8  8  8  8     8    4  16
│ │ │ │ -      8, 4, -}, {-, 4, 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --,
│ │ │ │ -            3    3        3      3  3  3    3     3  3  3  3     3    3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      16        4  4  16       20  4     8  8     8    8        4  8  20  8
│ │ │ │ -      --, 4, 8, -, -, --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -,
│ │ │ │ -       3        3  3   3        3  3     3  3     3    3        3  3   3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4}}
│ │ │ │ +        20        4  4        20       8  4     4     16  16    8        4
│ │ │ │ +o12 = {{--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4, --, --}, {-, 8, 4, -,
│ │ │ │ +         3        3  3         3       3  3     3      3   3    3        3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         4  16  16    4  16     16     4  8       4        20     8  8  8
│ │ │ │ +      4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4, 4, --, 8, -, -, -},
│ │ │ │ +         3   3   3    3   3      3     3  3       3         3     3  3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +       8  8  8     20        4    4        8     8  20  8    4     16  4
│ │ │ │ +      {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --, -}, {-, 8, --, -, 4,
│ │ │ │ +       3  3  3      3        3    3        3     3   3  3    3      3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      8  16          4  20     8     8  8    16  4  4     16        8    16
│ │ │ │ +      -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -, 8, --, 4, 4, -}, {--,
│ │ │ │ +      3   3          3   3     3     3  3     3  3  3      3        3     3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         4  16  4  16     4       8  20  8  4        8    8  8     8     20
│ │ │ │ +      4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4, -}, {-, -, 8, -, 4, --,
│ │ │ │ +         3   3  3   3     3       3   3  3  3        3    3  3     3      3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      4       4  16     4  16     4  16       20  8  8  4        8    20
│ │ │ │ +      -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -, -, -, 4, 8, -}, {--, 4,
│ │ │ │ +      3       3   3     3   3     3   3        3  3  3  3        3     3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         4  8  8  8          8  8  8  4        20    16  16     4     4  8
│ │ │ │ +      4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, {--, --, 4, -, 4, -, -,
│ │ │ │ +         3  3  3  3          3  3  3  3         3     3   3     3     3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +              8  4     8  8     20       20     4  4     20       20  4
│ │ │ │ +      8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, 4, --, 4}, {--, -, 4,
│ │ │ │ +              3  3     3  3      3        3     3  3      3        3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +               4  20       4  20        20  4       8     8  20     4  8
│ │ │ │ +      4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, 4, -, --, 8, -, -, 4},
│ │ │ │ +               3   3       3   3         3  3       3     3   3     3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +          16  4  16  8        4    4     16  16     8  4          4  8
│ │ │ │ +      {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, -, 4}, {4, -, -, 8,
│ │ │ │ +           3  3   3  3        3    3      3   3     3  3          3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      16  16     4    4     20        20     4    8  8  8        8  8  8
│ │ │ │ +      --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, -, 8, 8, -, -, -},
│ │ │ │ +       3   3     3    3      3         3     3    3  3  3        3  3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +       8  20  8     8        4       16  8     4  16     4    4        8  16
│ │ │ │ +      {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, -}, {-, 8, 4, -, --,
│ │ │ │ +       3   3  3     3        3        3  3     3   3     3    3        3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      4  16       16  4     16  4     16  4    4     16  4  16  4     16
│ │ │ │ +      -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, -, --, -, 4, --},
│ │ │ │ +      3   3        3  3      3  3      3  3    3      3  3   3  3      3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +          8  8  8  8  8  8       20  8     8     8  4          4  8     4
│ │ │ │ +      {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ │ +          3  3  3  3  3  3        3  3     3     3  3          3  3     3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      16     16    20     4        4     20       4  4  16  4  16  16
│ │ │ │ +      --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, -, --, --, 4},
│ │ │ │ +       3      3     3     3        3      3       3  3   3  3   3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +          8  16     4     16  4    8  8     20     8  4          4  8     20
│ │ │ │ +      {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, {4, -, -, 8, --,
│ │ │ │ +          3   3     3      3  3    3  3      3     3  3          3  3      3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         8  8       8  4     16     16  4    8  8  20     8        4    4
│ │ │ │ +      4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, 8, 4, -}, {-, 4,
│ │ │ │ +         3  3       3  3      3      3  3    3  3   3     3        3    3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         8     20  8  8    8     8  8  8  8     8    4  16  16        4  4
│ │ │ │ +      8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, --, 4, 8, -, -,
│ │ │ │ +         3      3  3  3    3     3  3  3  3     3    3   3   3        3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      16       20  4     8  8     8    8        4  8  20  8       16     16
│ │ │ │ +      --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, 4}, {--, 4, --,
│ │ │ │ +       3        3  3     3  3     3    3        3  3   3  3        3      3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      4     8  4       20     8  8     4  8          4  8     8     8  20
│ │ │ │ +      -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, 4, -, --},
│ │ │ │ +      3     3  3        3     3  3     3  3          3  3     3     3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +          16  16  4  16  4  4       8        4     16  4  16          20  4
│ │ │ │ +      {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, 4, --, -,
│ │ │ │ +           3   3  3   3  3  3       3        3      3  3   3           3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      4  20          16        8  16  4  4          20  4        4  20
│ │ │ │ +      -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -, --, 4},
│ │ │ │ +      3   3           3        3   3  3  3           3  3        3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +       16  16  4     4        8       4  4  16  8        16    16  4  16
│ │ │ │ +      {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -, --, 4,
│ │ │ │ +        3   3  3     3        3       3  3   3  3         3     3  3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      4        8    4  16     4     16  8       8  8     8  8     8  8
│ │ │ │ +      -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -, -}, {4,
│ │ │ │ +      3        3    3   3     3      3  3       3  3     3  3     3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      8  4     20  8     8    16  4  4        16  16  4       4  16  16  8
│ │ │ │ +      -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --, --, -,
│ │ │ │ +      3  3      3  3     3     3  3  3         3   3  3       3   3   3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +            4    4  20              20  4    4        20  20        4    4
│ │ │ │ +      8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4, -}, {-,
│ │ │ │ +            3    3   3               3  3    3         3   3        3    3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +            8  16  16  4          4     20  20     4             4  20  20
│ │ │ │ +      4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -, --, --,
│ │ │ │ +            3   3   3  3          3      3   3     3             3   3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      4          8        16     4  4  16    8        4  8  8  20       8
│ │ │ │ +      -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --, 4}, {-, 8,
│ │ │ │ +      3          3         3     3  3   3    3        3  3  3   3       3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      8  8     4  20       20  4  4  20  4  20  20  4    4  20  20  4  20  4
│ │ │ │ +      -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --, --, -, --, -,
│ │ │ │ +      3  3     3   3        3  3  3   3  3   3   3  3    3   3   3  3   3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      4  20
│ │ │ │ +      -, --}}
│ │ │ │ +      3   3
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : volume convexHull A -- 8
│ │ │ │  
│ │ │ │  o13 = 8
│ │ │ │  
│ │ │ │  o13 : QQ
│ │ │ │  i14 : stars1 = select(Ts4, t -> (gkzVector t)#-1 == 8)
│ │ │ │  
│ │ │ │ -o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ │ +      {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {2, 4, 6, 7}}}
│ │ │ │  
│ │ │ │  o14 : List
│ │ │ │  i15 : stars2 = select(Ts4, isStar)
│ │ │ │  
│ │ │ │ -o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ │ +      {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {2, 4, 6, 7}}}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : stars1 == stars2
│ │ │ │  
│ │ │ │  o16 = true
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/html/index.html
│ │ │ @@ -150,15 +150,15 @@
│ │ │                4       10
│ │ │  o2 : Matrix ZZ  <-- ZZ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ - -- .0931162s elapsed
    
│ │ │ + -- .116782s 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,
│ │ │ @@ -198,15 +198,15 @@
│ │ │  
│ │ │  o7 : Triangulation
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ - -- 1.28881s elapsed
    
│ │ │ + -- 1.09395s 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);
│ │ │ │ - -- .0931162s elapsed
│ │ │ │ + -- .116782s 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;
│ │ │ │ - -- 1.28881s elapsed
│ │ │ │ + -- 1.09395s 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/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 0.91597s (cpu); 0.578592s (thread); 0s (gc)
│ │ │ + -- used 1.16791s (cpu); 0.705476s (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 0.654992s (cpu); 0.406513s (thread); 0s (gc)
│ │ │ + -- used 0.897132s (cpu); 0.508072s (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
│ │ │ @@ -71,15 +71,15 @@
│ │ │            
│ │ │          
│ │ │          
    With the default setting SmartLift=>true we get very nice equations for the base space:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -98,15 +98,15 @@
│ │ │            
│ │ │          
    
│ │ │                
    i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ - -- used 0.91597s (cpu); 0.578592s (thread); 0s (gc)
    
│ │ │ + -- used 1.16791s (cpu); 0.705476s (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 0.654992s (cpu); 0.406513s (thread); 0s (gc)
    
│ │ │ + -- used 0.897132s (cpu); 0.508072s (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 0.91597s (cpu); 0.578592s (thread); 0s (gc)
│ │ │ │ + -- used 1.16791s (cpu); 0.705476s (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 0.654992s (cpu); 0.406513s (thread); 0s (gc)
│ │ │ │ + -- used 0.897132s (cpu); 0.508072s (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/___Basic__Divisor.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 18380296066161043289
│ │ │  
│ │ │  i1 : R = QQ[x,y,z];
│ │ │  
│ │ │  i2 : D = divisor(x*y^2*z^3)
│ │ │  
│ │ │ -o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ +o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : H = new HashTable from D
│ │ │  
│ │ │  o3 = HashTable{{x} => {1}                  }
│ │ │                 {y} => {2}
│ │ │ @@ -16,18 +16,18 @@
│ │ │                 cache => CacheTable{...1...}
│ │ │                 ring => R
│ │ │  
│ │ │  o3 : HashTable
│ │ │  
│ │ │  i4 : (2/3)*D
│ │ │  
│ │ │ -o4 = 2/3*Div(x) + 4/3*Div(y) + 2*Div(z)
│ │ │ +o4 = 2*Div(z) + 4/3*Div(y) + 2/3*Div(x)
│ │ │  
│ │ │  o4 : QWeilDivisor on R
│ │ │  
│ │ │  i5 : 0.6*D
│ │ │  
│ │ │ -o5 = .6*Div(x) + 1.2*Div(y) + 1.8*Div(z)
│ │ │ +o5 = 1.8*Div(z) + 1.2*Div(y) + .6*Div(x)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/___Number_sp_st_sp__Basic__Divisor.out
│ │ │ @@ -16,21 +16,21 @@
│ │ │  
│ │ │  o4 = -3.2*Div(-y^3+x^2) + 1.5*Div(x) + 0*Div(y)
│ │ │  
│ │ │  o4 : RWeilDivisor on R
│ │ │  
│ │ │  i5 : 8*D
│ │ │  
│ │ │ -o5 = -8*Div(x+y) + 16*Div(x) + 8*Div(y)
│ │ │ +o5 = -8*Div(x+y) + 8*Div(y) + 16*Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : (-2/3)*D
│ │ │  
│ │ │ -o6 = 2/3*Div(x+y) + -4/3*Div(x) + -2/3*Div(y)
│ │ │ +o6 = 2/3*Div(x+y) + -2/3*Div(y) + -4/3*Div(x)
│ │ │  
│ │ │  o6 : QWeilDivisor on R
│ │ │  
│ │ │  i7 : 0.0*D
│ │ │  
│ │ │  o7 = 0, the zero divisor
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ceiling_lp__R__Weil__Divisor_rp.out
│ │ │ @@ -1,32 +1,32 @@
│ │ │  -- -*- M2-comint -*- hash: 992133077988949640
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x *y - z^2);
│ │ │  
│ │ │  i2 : D = divisor({1/2, 4/3}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o2 = 4/3*Div(y, z) + 1/2*Div(x, z)
│ │ │ +o2 = 1/2*Div(x, z) + 4/3*Div(y, z)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
│ │ │  
│ │ │  i3 : ceiling( D )
│ │ │  
│ │ │ -o3 = 2*Div(y, z) + Div(x, z)
│ │ │ +o3 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 : floor( D )
│ │ │  
│ │ │  o4 = Div(y, z)
│ │ │  
│ │ │  o4 : WeilDivisor on R
│ │ │  
│ │ │  i5 : E = divisor({0.3, -0.7}, {ideal(x, z), ideal(y,z)}, CoefficientType => RR)
│ │ │  
│ │ │ -o5 = -.7*Div(y, z) + .3*Div(x, z)
│ │ │ +o5 = .3*Div(x, z) + -.7*Div(y, z)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
│ │ │  
│ │ │  i6 : ceiling( E )
│ │ │  
│ │ │  o6 = Div(x, z)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_divisor.out
│ │ │ @@ -1,38 +1,38 @@
│ │ │  -- -*- M2-comint -*- hash: 16935688116980988371
│ │ │  
│ │ │  i1 : R = QQ[x,y,z];
│ │ │  
│ │ │  i2 : D = divisor({1,2,3}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : E = divisor(x*y^2*z^3)
│ │ │  
│ │ │ -o3 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o3 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 : F = divisor(ideal(x*y^2*z^3))
│ │ │  
│ │ │ -o4 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o4 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o4 : WeilDivisor on R
│ │ │  
│ │ │  i5 : G = divisor({{1, ideal(x)}, {2, ideal(y)}, {3, ideal(z)}})
│ │ │  
│ │ │ -o5 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o5 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : H = divisor(x) + 2*divisor(y) + 3*divisor(z)
│ │ │  
│ │ │ -o6 = 3*Div(z) + 2*Div(y) + Div(x)
│ │ │ +o6 = Div(x) + 3*Div(z) + 2*Div(y)
│ │ │  
│ │ │  o6 : WeilDivisor on R
│ │ │  
│ │ │  i7 : R = QQ[x,y,z]/ideal(x^2-y*z);
│ │ │  
│ │ │  i8 : D = divisor({2}, {ideal(x,y)})
│ │ │  
│ │ │ @@ -60,29 +60,29 @@
│ │ │  
│ │ │  o14 = 3*Div(xz2, xyz, xy2, x2z, x2y, x3)
│ │ │  
│ │ │  o14 : WeilDivisor on A
│ │ │  
│ │ │  i15 : E = divisor(y2z)
│ │ │  
│ │ │ -o15 = 2*Div(yz2, y2z, y3, xyz, xy2, x2y) + Div(z3, yz2, y2z, xz2, xyz, x2z)
│ │ │ +o15 = Div(z3, yz2, y2z, xz2, xyz, x2z) + 2*Div(yz2, y2z, y3, xyz, xy2, x2y)
│ │ │  
│ │ │  o15 : WeilDivisor on A
│ │ │  
│ │ │  i16 : R = ZZ/7[x,y];
│ │ │  
│ │ │  i17 : D = divisor({-1/2, 2/1}, {ideal(y^2-x^3), ideal(x)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o17 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ +o17 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │  
│ │ │  o17 : QWeilDivisor on R
│ │ │  
│ │ │  i18 : D = (-1/2)*divisor(y^2-x^3) + (2/1)*divisor(x)
│ │ │  
│ │ │ -o18 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ +o18 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │  
│ │ │  o18 : QWeilDivisor on R
│ │ │  
│ │ │  i19 : R = ZZ/11[x,y,u,v]/ideal(x*y-u*v);
│ │ │  
│ │ │  i20 : D = divisor({1.1, -3.14159}, {ideal(x,u), ideal(x, v)}, CoefficientType=>RR)
│ │ ├── ./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 0.135013s (cpu); 0.0782845s (thread); 0s (gc)
│ │ │ + -- used 0.15575s (cpu); 0.0816305s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
│ │ │  
│ │ │  i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.455344s (cpu); 0.455348s (thread); 0s (gc)
│ │ │ + -- used 0.548671s (cpu); 0.548681s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.57369s (cpu); 0.498594s (thread); 0s (gc)
│ │ │ + -- used 0.691125s (cpu); 0.608267s (thread); 0s (gc)
│ │ │  
│ │ │  i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00285362s (cpu); 0.00285446s (thread); 0s (gc)
│ │ │ + -- used 0.00346799s (cpu); 0.00347585s (thread); 0s (gc)
│ │ │  
│ │ │  i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00222665s (cpu); 0.0022275s (thread); 0s (gc)
│ │ │ + -- used 0.00285906s (cpu); 0.00286407s (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 0.279007s (cpu); 0.150681s (thread); 0s (gc)
│ │ │ + -- used 0.321724s (cpu); 0.151468s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
│ │ │  
│ │ │  i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00604243s (cpu); 0.00604328s (thread); 0s (gc)
│ │ │ + -- used 0.00756043s (cpu); 0.00756775s (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/_is__Cartier.out
│ │ │ @@ -12,15 +12,15 @@
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i5 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o5 = Div(x, z) + 2*Div(y, z)
│ │ │ +o5 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : isCartier( D )
│ │ │  
│ │ │  o6 = false
│ │ │  
│ │ │ @@ -48,15 +48,15 @@
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │  
│ │ │  i14 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o14 = Div(x, z) + 2*Div(y, z)
│ │ │ +o14 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o14 : WeilDivisor on R
│ │ │  
│ │ │  i15 : isCartier(D, IsGraded => true)
│ │ │  
│ │ │  o15 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Homogeneous_lp__Basic__Divisor_rp.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 18048197335381839324
│ │ │  
│ │ │  i1 : R = QQ[x, y, z];
│ │ │  
│ │ │  i2 : D = divisor({1, 2, 3}, {ideal(x * y - z^2), ideal(y * z - x^2), ideal(x * z - y^2)})
│ │ │  
│ │ │ -o2 = 3*Div(-y^2+x*z) + Div(x*y-z^2) + 2*Div(-x^2+y*z)
│ │ │ +o2 = Div(x*y-z^2) + 2*Div(-x^2+y*z) + 3*Div(-y^2+x*z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : isHomogeneous( D )
│ │ │  
│ │ │  o3 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Linear__Equivalent.out
│ │ │ @@ -1,38 +1,38 @@
│ │ │  -- -*- M2-comint -*- hash: 6019119347082811396
│ │ │  
│ │ │  i1 : R = QQ[x, y, z]/ ideal(x * y - z^2);
│ │ │  
│ │ │  i2 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 8*Div(y, z) + 3*Div(x, z)
│ │ │ +o2 = 3*Div(x, z) + 8*Div(y, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │  
│ │ │ -o3 = 8*Div(y, z) + Div(x, z)
│ │ │ +o3 = Div(x, z) + 8*Div(y, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 : isLinearEquivalent(D1, D2)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : R = QQ[x, y, z]/ ideal(x * y - z^2);
│ │ │  
│ │ │  i6 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o6 = 3*Div(x, z) + 8*Div(y, z)
│ │ │ +o6 = 8*Div(y, z) + 3*Div(x, z)
│ │ │  
│ │ │  o6 : WeilDivisor on R
│ │ │  
│ │ │  i7 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │  
│ │ │ -o7 = Div(x, z) + 8*Div(y, z)
│ │ │ +o7 = 8*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o7 : WeilDivisor on R
│ │ │  
│ │ │  i8 : isLinearEquivalent(D1, D2, IsGraded => true)
│ │ │  
│ │ │  o8 = false
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Q__Cartier.out
│ │ │ @@ -1,20 +1,20 @@
│ │ │  -- -*- M2-comint -*- hash: 13719144060491348416
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i2 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 2*Div(y, z) + Div(x, z)
│ │ │ +o2 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o3 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ +o3 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
│ │ │  
│ │ │  i4 : isQCartier(10, D1)
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │ @@ -44,21 +44,21 @@
│ │ │  
│ │ │  o10 = 0
│ │ │  
│ │ │  i11 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i12 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o12 = 2*Div(y, z) + Div(x, z)
│ │ │ +o12 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o12 : WeilDivisor on R
│ │ │  
│ │ │  i13 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o13 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ +o13 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │  
│ │ │  o13 : QWeilDivisor on R
│ │ │  
│ │ │  i14 : isQCartier(10, D1, IsGraded => true)
│ │ │  
│ │ │  o14 = 1
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Q__Linear__Equivalent.out
│ │ │ @@ -36,21 +36,21 @@
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │  
│ │ │  i11 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o11 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │ +o11 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │  
│ │ │  o11 : QWeilDivisor on R
│ │ │  
│ │ │  i12 : E = divisor({3/2, -1/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o12 = 3/2*Div(y, z) + -1/4*Div(x, z)
│ │ │ +o12 = -1/4*Div(x, z) + 3/2*Div(y, z)
│ │ │  
│ │ │  o12 : QWeilDivisor on R
│ │ │  
│ │ │  i13 : isQLinearEquivalent(10, D, E, IsGraded => true)
│ │ │  
│ │ │  o13 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__S__N__C.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 2360371518304120718
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i2 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = Div(x, z) + -2*Div(y, z)
│ │ │ +o2 = -2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : isSNC( D )
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │ @@ -36,15 +36,15 @@
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i11 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o11 = -2*Div(y, z) + Div(x, z)
│ │ │ +o11 = Div(x, z) + -2*Div(y, z)
│ │ │  
│ │ │  o11 : WeilDivisor on R
│ │ │  
│ │ │  i12 : isSNC( D, IsGraded => true )
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │ @@ -60,15 +60,15 @@
│ │ │  
│ │ │  o15 = true
│ │ │  
│ │ │  i16 : R = QQ[x,y,z];
│ │ │  
│ │ │  i17 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o17 = Div(x) + Div(x+y) + Div(y)
│ │ │ +o17 = Div(x+y) + Div(y) + Div(x)
│ │ │  
│ │ │  o17 : WeilDivisor on R
│ │ │  
│ │ │  i18 : isSNC( D, IsGraded => true)
│ │ │  
│ │ │  o18 = false
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_map__To__Projective__Space.out
│ │ │ @@ -16,15 +16,15 @@
│ │ │  o3 : RingMap R <-- QQ[YY ..YY ]
│ │ │                          1    2
│ │ │  
│ │ │  i4 : R = ZZ/7[x,y,z];
│ │ │  
│ │ │  i5 : D = divisor(x*y)
│ │ │  
│ │ │ -o5 = Div(y) + Div(x)
│ │ │ +o5 = Div(x) + Div(y)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : mapToProjectiveSpace(D, Variable=>"Z")
│ │ │  
│ │ │               ZZ            2             2        2
│ │ │  o6 = map (R, --[Z ..Z ], {x , x*y, x*z, y , y*z, z })
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.out
│ │ │ @@ -6,21 +6,21 @@
│ │ │  
│ │ │  i3 : f = map(T, R, {a^3, a^2*b, a*b^2, b^3});
│ │ │  
│ │ │  o3 : RingMap T <-- R
│ │ │  
│ │ │  i4 : D = divisor(y*z)
│ │ │  
│ │ │ -o4 = 3*Div(w, z, y) + 3*Div(z, y, x)
│ │ │ +o4 = 3*Div(z, y, x) + 3*Div(w, z, y)
│ │ │  
│ │ │  o4 : WeilDivisor on R
│ │ │  
│ │ │  i5 : pullback(f, D, Strategy=>Primes)
│ │ │  
│ │ │ -o5 = 3*Div(b) + 3*Div(a)
│ │ │ +o5 = 3*Div(a) + 3*Div(b)
│ │ │  
│ │ │  o5 : WeilDivisor on T
│ │ │  
│ │ │  i6 : pullback(f, D, Strategy=>Sheaves)
│ │ │  
│ │ │  o6 = 3*Div(b) + 3*Div(a)
│ │ │  
│ │ │ @@ -36,18 +36,18 @@
│ │ │  
│ │ │  i10 : D = divisor(x*y*(x+y));
│ │ │  
│ │ │  o10 : WeilDivisor on R
│ │ │  
│ │ │  i11 : D1 = pullback(f, D)
│ │ │  
│ │ │ -o11 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │ +o11 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │  
│ │ │  o11 : WeilDivisor on S
│ │ │  
│ │ │  i12 : f^* D
│ │ │  
│ │ │ -o12 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │ +o12 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │  
│ │ │  o12 : WeilDivisor on S
│ │ │  
│ │ │  i13 :
│ │ ├── ./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 0.260716s (cpu); 0.201186s (thread); 0s (gc)
│ │ │ + -- used 0.305298s (cpu); 0.234436s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.527272s (cpu); 0.394308s (thread); 0s (gc)
│ │ │ + -- used 0.484222s (cpu); 0.39818s (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 0.273432s (cpu); 0.158699s (thread); 0s (gc)
│ │ │ + -- used 0.328963s (cpu); 0.156132s (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 6.95165s (cpu); 4.6997s (thread); 0s (gc)
│ │ │ + -- used 6.33337s (cpu); 4.98751s (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 5.67659s (cpu); 4.46311s (thread); 0s (gc)
│ │ │ + -- used 6.71641s (cpu); 5.39144s (thread); 0s (gc)
│ │ │  
│ │ │  i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.775911s (cpu); 0.448237s (thread); 0s (gc)
│ │ │ + -- used 0.663711s (cpu); 0.45656s (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 1.05331s (cpu); 0.384322s (thread); 0s (gc)
│ │ │ + -- used 1.28401s (cpu); 0.46856s (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 0.238734s (cpu); 0.0653637s (thread); 0s (gc)
│ │ │ + -- used 0.259904s (cpu); 0.0660955s (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 0.276556s (cpu); 0.103744s (thread); 0s (gc)
│ │ │ + -- used 0.323261s (cpu); 0.106148s (thread); 0s (gc)
│ │ │  
│ │ │  i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00801402s (cpu); 0.00801143s (thread); 0s (gc)
│ │ │ + -- used 0.00832269s (cpu); 0.00833187s (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 0.0323817s (cpu); 0.0323825s (thread); 0s (gc)
│ │ │ + -- used 0.0477471s (cpu); 0.0477492s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
│ │ │  
│ │ │  i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.204644s (cpu); 0.153031s (thread); 0s (gc)
│ │ │ + -- used 0.234538s (cpu); 0.159666s (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 0.029315s (cpu); 0.0293196s (thread); 0s (gc)
│ │ │ + -- used 0.0376128s (cpu); 0.0376168s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.143657s (cpu); 0.0818326s (thread); 0s (gc)
│ │ │ + -- used 0.188293s (cpu); 0.101737s (thread); 0s (gc)
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : J1 == J2
│ │ │  
│ │ │  o15 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ring_lp__Basic__Divisor_rp.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 5006859181202351713
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i2 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 2*Div(y, z) + Div(x, z)
│ │ │ +o2 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : ring( D )
│ │ │  
│ │ │  o3 = R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_to__R__Weil__Divisor.out
│ │ │ @@ -1,32 +1,32 @@
│ │ │  -- -*- M2-comint -*- hash: 12819564349892123361
│ │ │  
│ │ │  i1 : R = ZZ/5[x,y];
│ │ │  
│ │ │  i2 : D = divisor({2, 0, -4}, {ideal(x), ideal(y), ideal(x-y)})
│ │ │  
│ │ │ -o2 = -4*Div(x-y) + 2*Div(x) + 0*Div(y)
│ │ │ +o2 = 2*Div(x) + 0*Div(y) + -4*Div(x-y)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : E = (1/2)*D
│ │ │  
│ │ │ -o3 = -2*Div(x-y) + Div(x)
│ │ │ +o3 = Div(x) + -2*Div(x-y)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
│ │ │  
│ │ │  i4 : F = toRWeilDivisor(D)
│ │ │  
│ │ │ -o4 = -4*Div(x-y) + 2*Div(x)
│ │ │ +o4 = 2*Div(x) + -4*Div(x-y)
│ │ │  
│ │ │  o4 : RWeilDivisor on R
│ │ │  
│ │ │  i5 : G = toRWeilDivisor(E)
│ │ │  
│ │ │ -o5 = -2*Div(x-y) + Div(x)
│ │ │ +o5 = Div(x) + -2*Div(x-y)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
│ │ │  
│ │ │  i6 : F == 2*G
│ │ │  
│ │ │  o6 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/___Basic__Divisor.html
│ │ │ @@ -72,15 +72,15 @@
│ │ │                
    i1 : R = QQ[x,y,z];
    
│ │ │              
    
│ │ │                
    i2 : D = divisor(x*y^2*z^3)
│ │ │  
│ │ │ -o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ +o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
    
│ │ │                
    i3 : H = new HashTable from D
│ │ │ @@ -94,24 +94,24 @@
│ │ │  o3 : HashTable
    
│ │ │              
    
│ │ │                
    i4 : (2/3)*D
│ │ │  
│ │ │ -o4 = 2/3*Div(x) + 4/3*Div(y) + 2*Div(z)
│ │ │ +o4 = 2*Div(z) + 4/3*Div(y) + 2/3*Div(x)
│ │ │  
│ │ │  o4 : QWeilDivisor on R
    
│ │ │              
    
│ │ │                
    i5 : 0.6*D
│ │ │  
│ │ │ -o5 = .6*Div(x) + 1.2*Div(y) + 1.8*Div(z)
│ │ │ +o5 = 1.8*Div(z) + 1.2*Div(y) + .6*Div(x)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,34 +15,34 @@
│ │ │ │  specifies the ambient ring. Another key is cache which points to a CacheTable.
│ │ │ │  The remaining keys are a Groebner basis $L$ for each prime ideal $P$ in the
│ │ │ │  support with corresponding value a list with one entry {$n$} where $n$ is the
│ │ │ │  coefficient of the height one prime.
│ │ │ │  i1 : R = QQ[x,y,z];
│ │ │ │  i2 : D = divisor(x*y^2*z^3)
│ │ │ │  
│ │ │ │ -o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ │ +o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : H = new HashTable from D
│ │ │ │  
│ │ │ │  o3 = HashTable{{x} => {1}                  }
│ │ │ │                 {y} => {2}
│ │ │ │                 {z} => {3}
│ │ │ │                 cache => CacheTable{...1...}
│ │ │ │                 ring => R
│ │ │ │  
│ │ │ │  o3 : HashTable
│ │ │ │  i4 : (2/3)*D
│ │ │ │  
│ │ │ │ -o4 = 2/3*Div(x) + 4/3*Div(y) + 2*Div(z)
│ │ │ │ +o4 = 2*Div(z) + 4/3*Div(y) + 2/3*Div(x)
│ │ │ │  
│ │ │ │  o4 : QWeilDivisor on R
│ │ │ │  i5 : 0.6*D
│ │ │ │  
│ │ │ │ -o5 = .6*Div(x) + 1.2*Div(y) + 1.8*Div(z)
│ │ │ │ +o5 = 1.8*Div(z) + 1.2*Div(y) + .6*Div(x)
│ │ │ │  
│ │ │ │  o5 : RWeilDivisor on R
│ │ │ │  ********** TTyyppeess ooff BBaassiiccDDiivviissoorr:: **********
│ │ │ │      * RWeilDivisor
│ │ │ │  ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aann oobbjjeecctt ooff ccllaassss BBaassiiccDDiivviissoorr:: **********
│ │ │ │      * applyToCoefficients(BasicDivisor,Function) -- see _a_p_p_l_y_T_o_C_o_e_f_f_i_c_i_e_n_t_s -
│ │ │ │        - apply a function to the coefficients of a divisor
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/___Number_sp_st_sp__Basic__Divisor.html
│ │ │ @@ -103,24 +103,24 @@
│ │ │  o4 : RWeilDivisor on R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : 8*D
│ │ │  
│ │ │ -o5 = -8*Div(x+y) + 16*Div(x) + 8*Div(y)
│ │ │ +o5 = -8*Div(x+y) + 8*Div(y) + 16*Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : (-2/3)*D
│ │ │  
│ │ │ -o6 = 2/3*Div(x+y) + -4/3*Div(x) + -2/3*Div(y)
│ │ │ +o6 = 2/3*Div(x+y) + -2/3*Div(y) + -4/3*Div(x)
│ │ │  
│ │ │  o6 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : 0.0*D
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,20 +27,20 @@
│ │ │ │  CoefficientType=>RR)
│ │ │ │  
│ │ │ │  o4 = -3.2*Div(-y^3+x^2) + 1.5*Div(x) + 0*Div(y)
│ │ │ │  
│ │ │ │  o4 : RWeilDivisor on R
│ │ │ │  i5 : 8*D
│ │ │ │  
│ │ │ │ -o5 = -8*Div(x+y) + 16*Div(x) + 8*Div(y)
│ │ │ │ +o5 = -8*Div(x+y) + 8*Div(y) + 16*Div(x)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  i6 : (-2/3)*D
│ │ │ │  
│ │ │ │ -o6 = 2/3*Div(x+y) + -4/3*Div(x) + -2/3*Div(y)
│ │ │ │ +o6 = 2/3*Div(x+y) + -2/3*Div(y) + -4/3*Div(x)
│ │ │ │  
│ │ │ │  o6 : QWeilDivisor on R
│ │ │ │  i7 : 0.0*D
│ │ │ │  
│ │ │ │  o7 = 0, the zero divisor
│ │ │ │  
│ │ │ │  o7 : RWeilDivisor on R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_ceiling_lp__R__Weil__Divisor_rp.html
│ │ │ @@ -78,24 +78,24 @@
│ │ │                
    i1 : R = QQ[x, y, z] / ideal(x *y - z^2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({1/2, 4/3}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o2 = 4/3*Div(y, z) + 1/2*Div(x, z)
│ │ │ +o2 = 1/2*Div(x, z) + 4/3*Div(y, z)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : ceiling( D )
│ │ │  
│ │ │ -o3 = 2*Div(y, z) + Div(x, z)
│ │ │ +o3 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : floor( D )
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o4 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : E = divisor({0.3, -0.7}, {ideal(x, z), ideal(y,z)}, CoefficientType => RR)
│ │ │  
│ │ │ -o5 = -.7*Div(y, z) + .3*Div(x, z)
│ │ │ +o5 = .3*Div(x, z) + -.7*Div(y, z)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : ceiling( E )
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,30 +15,30 @@
│ │ │ │            o an instance of the type _W_e_i_l_D_i_v_i_s_o_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Start with a rational or real Weil divisor. We form a new divisor whose
│ │ │ │  coefficients are obtained by applying the ceiling or floor function to them.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x *y - z^2);
│ │ │ │  i2 : D = divisor({1/2, 4/3}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │ │  
│ │ │ │ -o2 = 4/3*Div(y, z) + 1/2*Div(x, z)
│ │ │ │ +o2 = 1/2*Div(x, z) + 4/3*Div(y, z)
│ │ │ │  
│ │ │ │  o2 : QWeilDivisor on R
│ │ │ │  i3 : ceiling( D )
│ │ │ │  
│ │ │ │ -o3 = 2*Div(y, z) + Div(x, z)
│ │ │ │ +o3 = Div(x, z) + 2*Div(y, z)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ │ │  i4 : floor( D )
│ │ │ │  
│ │ │ │  o4 = Div(y, z)
│ │ │ │  
│ │ │ │  o4 : WeilDivisor on R
│ │ │ │  i5 : E = divisor({0.3, -0.7}, {ideal(x, z), ideal(y,z)}, CoefficientType => RR)
│ │ │ │  
│ │ │ │ -o5 = -.7*Div(y, z) + .3*Div(x, z)
│ │ │ │ +o5 = .3*Div(x, z) + -.7*Div(y, z)
│ │ │ │  
│ │ │ │  o5 : RWeilDivisor on R
│ │ │ │  i6 : ceiling( E )
│ │ │ │  
│ │ │ │  o6 = Div(x, z)
│ │ │ │  
│ │ │ │  o6 : WeilDivisor on R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_divisor.html
│ │ │ @@ -95,51 +95,51 @@
│ │ │                
    i1 : R = QQ[x,y,z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({1,2,3}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : E = divisor(x*y^2*z^3)
│ │ │  
│ │ │ -o3 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o3 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : F = divisor(ideal(x*y^2*z^3))
│ │ │  
│ │ │ -o4 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o4 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o4 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : G = divisor({{1, ideal(x)}, {2, ideal(y)}, {3, ideal(z)}})
│ │ │  
│ │ │ -o5 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o5 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : H = divisor(x) + 2*divisor(y) + 3*divisor(z)
│ │ │  
│ │ │ -o6 = 3*Div(z) + 2*Div(y) + Div(x)
│ │ │ +o6 = Div(x) + 3*Div(z) + 2*Div(y)
│ │ │  
│ │ │  o6 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Next we construct the same divisor in two different ways.  We are working on the quadric cone, and we are working with a divisor of a ruling of the cone.  This divisor is not Cartier, but 2 times it is.
    
│ │ │ @@ -204,15 +204,15 @@
│ │ │  o14 : WeilDivisor on A
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : E = divisor(y2z)
│ │ │  
│ │ │ -o15 = 2*Div(yz2, y2z, y3, xyz, xy2, x2y) + Div(z3, yz2, y2z, xz2, xyz, x2z)
│ │ │ +o15 = Div(z3, yz2, y2z, xz2, xyz, x2z) + 2*Div(yz2, y2z, y3, xyz, xy2, x2y)
│ │ │  
│ │ │  o15 : WeilDivisor on A
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    We can construct a Q-divisor as well.  Here are two ways to do it (we work in $A^2$ this time).
    
│ │ │ @@ -223,24 +223,24 @@
│ │ │                
    i16 : R = ZZ/7[x,y];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : D = divisor({-1/2, 2/1}, {ideal(y^2-x^3), ideal(x)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o17 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ +o17 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │  
│ │ │  o17 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i18 : D = (-1/2)*divisor(y^2-x^3) + (2/1)*divisor(x)
│ │ │  
│ │ │ -o18 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ +o18 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │  
│ │ │  o18 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Or an R-divisor.  This time we work in the cone over $P^1 \times P^1$.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,35 +43,35 @@
│ │ │ │  call it. In our first example, we construct divisors on $A^3$ (which can also
│ │ │ │  be viewed as divisors on $P^2$ since the ideals are homogeneous). The following
│ │ │ │  creates the same Weil divisor with coefficients 1, 2 and 3 in five different
│ │ │ │  ways.
│ │ │ │  i1 : R = QQ[x,y,z];
│ │ │ │  i2 : D = divisor({1,2,3}, {ideal(x), ideal(y), ideal(z)})
│ │ │ │  
│ │ │ │ -o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ │ +o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : E = divisor(x*y^2*z^3)
│ │ │ │  
│ │ │ │ -o3 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ │ +o3 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ │ │  i4 : F = divisor(ideal(x*y^2*z^3))
│ │ │ │  
│ │ │ │ -o4 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ │ +o4 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ │  
│ │ │ │  o4 : WeilDivisor on R
│ │ │ │  i5 : G = divisor({{1, ideal(x)}, {2, ideal(y)}, {3, ideal(z)}})
│ │ │ │  
│ │ │ │ -o5 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ │ +o5 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  i6 : H = divisor(x) + 2*divisor(y) + 3*divisor(z)
│ │ │ │  
│ │ │ │ -o6 = 3*Div(z) + 2*Div(y) + Div(x)
│ │ │ │ +o6 = Div(x) + 3*Div(z) + 2*Div(y)
│ │ │ │  
│ │ │ │  o6 : WeilDivisor on R
│ │ │ │  Next we construct the same divisor in two different ways. We are working on the
│ │ │ │  quadric cone, and we are working with a divisor of a ruling of the cone. This
│ │ │ │  divisor is not Cartier, but 2 times it is.
│ │ │ │  i7 : R = QQ[x,y,z]/ideal(x^2-y*z);
│ │ │ │  i8 : D = divisor({2}, {ideal(x,y)})
│ │ │ │ @@ -95,28 +95,28 @@
│ │ │ │  i14 : D = divisor(x3)
│ │ │ │  
│ │ │ │  o14 = 3*Div(xz2, xyz, xy2, x2z, x2y, x3)
│ │ │ │  
│ │ │ │  o14 : WeilDivisor on A
│ │ │ │  i15 : E = divisor(y2z)
│ │ │ │  
│ │ │ │ -o15 = 2*Div(yz2, y2z, y3, xyz, xy2, x2y) + Div(z3, yz2, y2z, xz2, xyz, x2z)
│ │ │ │ +o15 = Div(z3, yz2, y2z, xz2, xyz, x2z) + 2*Div(yz2, y2z, y3, xyz, xy2, x2y)
│ │ │ │  
│ │ │ │  o15 : WeilDivisor on A
│ │ │ │  We can construct a Q-divisor as well. Here are two ways to do it (we work in
│ │ │ │  $A^2$ this time).
│ │ │ │  i16 : R = ZZ/7[x,y];
│ │ │ │  i17 : D = divisor({-1/2, 2/1}, {ideal(y^2-x^3), ideal(x)}, CoefficientType=>QQ)
│ │ │ │  
│ │ │ │ -o17 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ │ +o17 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │ │  
│ │ │ │  o17 : QWeilDivisor on R
│ │ │ │  i18 : D = (-1/2)*divisor(y^2-x^3) + (2/1)*divisor(x)
│ │ │ │  
│ │ │ │ -o18 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ │ +o18 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │ │  
│ │ │ │  o18 : QWeilDivisor on R
│ │ │ │  Or an R-divisor. This time we work in the cone over $P^1 \times P^1$.
│ │ │ │  i19 : R = ZZ/11[x,y,u,v]/ideal(x*y-u*v);
│ │ │ │  i20 : D = divisor({1.1, -3.14159}, {ideal(x,u), ideal(x, v)},
│ │ │ │  CoefficientType=>RR)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_dualize.html
│ │ │ @@ -163,43 +163,43 @@
│ │ │              
│ │ │                
    i11 : M = J*R^1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.135013s (cpu); 0.0782845s (thread); 0s (gc)
│ │ │ + -- used 0.15575s (cpu); 0.0816305s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.455344s (cpu); 0.455348s (thread); 0s (gc)
│ │ │ + -- used 0.548671s (cpu); 0.548681s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.57369s (cpu); 0.498594s (thread); 0s (gc)
    
│ │ │ + -- used 0.691125s (cpu); 0.608267s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00285362s (cpu); 0.00285446s (thread); 0s (gc)
    
│ │ │ + -- used 0.00346799s (cpu); 0.00347585s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00222665s (cpu); 0.0022275s (thread); 0s (gc)
│ │ │ + -- used 0.00285906s (cpu); 0.00286407s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    For monomial ideals in toric rings, frequently ModuleStrategy appears faster.
    
│ │ │ @@ -223,23 +223,23 @@
│ │ │  
│ │ │  o19 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.279007s (cpu); 0.150681s (thread); 0s (gc)
│ │ │ + -- used 0.321724s (cpu); 0.151468s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00604243s (cpu); 0.00604328s (thread); 0s (gc)
│ │ │ + -- used 0.00756043s (cpu); 0.00756775s (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 0.135013s (cpu); 0.0782845s (thread); 0s (gc)
│ │ │ │ + -- used 0.15575s (cpu); 0.0816305s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 : Ideal of R
│ │ │ │  i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.455344s (cpu); 0.455348s (thread); 0s (gc)
│ │ │ │ + -- used 0.548671s (cpu); 0.548681s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 : Ideal of R
│ │ │ │  i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ │ - -- used 0.57369s (cpu); 0.498594s (thread); 0s (gc)
│ │ │ │ + -- used 0.691125s (cpu); 0.608267s (thread); 0s (gc)
│ │ │ │  i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.00285362s (cpu); 0.00285446s (thread); 0s (gc)
│ │ │ │ + -- used 0.00346799s (cpu); 0.00347585s (thread); 0s (gc)
│ │ │ │  i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.00222665s (cpu); 0.0022275s (thread); 0s (gc)
│ │ │ │ + -- used 0.00285906s (cpu); 0.00286407s (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 0.279007s (cpu); 0.150681s (thread); 0s (gc)
│ │ │ │ + -- used 0.321724s (cpu); 0.151468s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 : Ideal of R
│ │ │ │  i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.00604243s (cpu); 0.00604328s (thread); 0s (gc)
│ │ │ │ + -- used 0.00756043s (cpu); 0.00756775s (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/_is__Cartier.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │                
    i4 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o5 = Div(x, z) + 2*Div(y, z)
│ │ │ +o5 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : isCartier( D )
│ │ │ @@ -181,15 +181,15 @@
│ │ │                
    i13 : R = QQ[x, y, z] / ideal(x * y - z^2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o14 = Div(x, z) + 2*Div(y, z)
│ │ │ +o14 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o14 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : isCartier(D, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  i3 : isCartier( D )
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  Neither is this divisor.
│ │ │ │  i4 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i5 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o5 = Div(x, z) + 2*Div(y, z)
│ │ │ │ +o5 = 2*Div(y, z) + Div(x, z)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  i6 : isCartier( D )
│ │ │ │  
│ │ │ │  o6 = false
│ │ │ │  Of course the next divisor is Cartier.
│ │ │ │  i7 : R = QQ[x, y, z];
│ │ │ │ @@ -55,15 +55,15 @@
│ │ │ │  o11 : WeilDivisor on R
│ │ │ │  i12 : isCartier(D, IsGraded => true)
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  i13 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │ │  i14 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o14 = Div(x, z) + 2*Div(y, z)
│ │ │ │ +o14 = 2*Div(y, z) + Div(x, z)
│ │ │ │  
│ │ │ │  o14 : WeilDivisor on R
│ │ │ │  i15 : isCartier(D, IsGraded => true)
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  The output value of this function is stored in the divisor's cache with the
│ │ │ │  value of the last IsGraded option. If you change the IsGraded option, the value
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Homogeneous_lp__Basic__Divisor_rp.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │                
    i1 : R = QQ[x, y, z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({1, 2, 3}, {ideal(x * y - z^2), ideal(y * z - x^2), ideal(x * z - y^2)})
│ │ │  
│ │ │ -o2 = 3*Div(-y^2+x*z) + Div(x*y-z^2) + 2*Div(-x^2+y*z)
│ │ │ +o2 = Div(x*y-z^2) + 2*Div(-x^2+y*z) + 3*Div(-y^2+x*z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : isHomogeneous( D )
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns true if the divisor is graded (homogeneous), otherwise it
│ │ │ │  returns false.
│ │ │ │  i1 : R = QQ[x, y, z];
│ │ │ │  i2 : D = divisor({1, 2, 3}, {ideal(x * y - z^2), ideal(y * z - x^2), ideal(x *
│ │ │ │  z - y^2)})
│ │ │ │  
│ │ │ │ -o2 = 3*Div(-y^2+x*z) + Div(x*y-z^2) + 2*Div(-x^2+y*z)
│ │ │ │ +o2 = Div(x*y-z^2) + 2*Div(-x^2+y*z) + 3*Div(-y^2+x*z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : isHomogeneous( D )
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : R = QQ[x, y, z];
│ │ │ │  i5 : D = divisor({1, 2}, {ideal(x * y - z^2), ideal(y^2 - z^3)})
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Linear__Equivalent.html
│ │ │ @@ -81,24 +81,24 @@
│ │ │                
    i1 : R = QQ[x, y, z]/ ideal(x * y - z^2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 8*Div(y, z) + 3*Div(x, z)
│ │ │ +o2 = 3*Div(x, z) + 8*Div(y, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │  
│ │ │ -o3 = 8*Div(y, z) + Div(x, z)
│ │ │ +o3 = Div(x, z) + 8*Div(y, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : isLinearEquivalent(D1, D2)
│ │ │ @@ -116,24 +116,24 @@
│ │ │                
    i5 : R = QQ[x, y, z]/ ideal(x * y - z^2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o6 = 3*Div(x, z) + 8*Div(y, z)
│ │ │ +o6 = 8*Div(y, z) + 3*Div(x, z)
│ │ │  
│ │ │  o6 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │  
│ │ │ -o7 = Div(x, z) + 8*Div(y, z)
│ │ │ +o7 = 8*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o7 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : isLinearEquivalent(D1, D2, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,36 +17,36 @@
│ │ │ │            o flag, a _B_o_o_l_e_a_n_ _v_a_l_u_e,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Given two Weil divisors, this method checks whether they are linearly
│ │ │ │  equivalent.
│ │ │ │  i1 : R = QQ[x, y, z]/ ideal(x * y - z^2);
│ │ │ │  i2 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o2 = 8*Div(y, z) + 3*Div(x, z)
│ │ │ │ +o2 = 3*Div(x, z) + 8*Div(y, z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │ │  
│ │ │ │ -o3 = 8*Div(y, z) + Div(x, z)
│ │ │ │ +o3 = Div(x, z) + 8*Div(y, z)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ │ │  i4 : isLinearEquivalent(D1, D2)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  If IsGraded is set to true (by default it is false), then it treats the
│ │ │ │  divisors as divisors on the $Proj$ of their ambient ring.
│ │ │ │  i5 : R = QQ[x, y, z]/ ideal(x * y - z^2);
│ │ │ │  i6 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o6 = 3*Div(x, z) + 8*Div(y, z)
│ │ │ │ +o6 = 8*Div(y, z) + 3*Div(x, z)
│ │ │ │  
│ │ │ │  o6 : WeilDivisor on R
│ │ │ │  i7 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │ │  
│ │ │ │ -o7 = Div(x, z) + 8*Div(y, z)
│ │ │ │ +o7 = 8*Div(y, z) + Div(x, z)
│ │ │ │  
│ │ │ │  o7 : WeilDivisor on R
│ │ │ │  i8 : isLinearEquivalent(D1, D2, IsGraded => true)
│ │ │ │  
│ │ │ │  o8 = false
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _O_O_ _R_W_e_i_l_D_i_v_i_s_o_r
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Q__Cartier.html
│ │ │ @@ -83,24 +83,24 @@
│ │ │                
    i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 2*Div(y, z) + Div(x, z)
│ │ │ +o2 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o3 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ +o3 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : isQCartier(10, D1)
│ │ │ @@ -164,24 +164,24 @@
│ │ │                
    i11 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o12 = 2*Div(y, z) + Div(x, z)
│ │ │ +o12 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o12 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o13 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ +o13 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │  
│ │ │  o13 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : isQCartier(10, D1, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,21 +21,21 @@
│ │ │ │  Check whether $m$ times a Weil or Q-divisor $D$ is Cartier for each $m$ from 1
│ │ │ │  to a fixed positive integer n1 (if the divisor is a QWeilDivisor, it can search
│ │ │ │  slightly higher than n1). If m * D1 is Cartier, it returns m. If it fails to
│ │ │ │  find an m, it returns 0.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i2 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o2 = 2*Div(y, z) + Div(x, z)
│ │ │ │ +o2 = Div(x, z) + 2*Div(y, z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType =>
│ │ │ │  QQ)
│ │ │ │  
│ │ │ │ -o3 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ │ +o3 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │ │  
│ │ │ │  o3 : QWeilDivisor on R
│ │ │ │  i4 : isQCartier(10, D1)
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  i5 : isQCartier(10, D2)
│ │ │ │  
│ │ │ │ @@ -59,21 +59,21 @@
│ │ │ │  
│ │ │ │  o10 = 0
│ │ │ │  If the option IsGraded is set to true (by default it is false), then it treats
│ │ │ │  the divisor as a divisor on the $Proj$ of their ambient ring.
│ │ │ │  i11 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i12 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o12 = 2*Div(y, z) + Div(x, z)
│ │ │ │ +o12 = Div(x, z) + 2*Div(y, z)
│ │ │ │  
│ │ │ │  o12 : WeilDivisor on R
│ │ │ │  i13 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType =>
│ │ │ │  QQ)
│ │ │ │  
│ │ │ │ -o13 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ │ +o13 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │ │  
│ │ │ │  o13 : QWeilDivisor on R
│ │ │ │  i14 : isQCartier(10, D1, IsGraded => true)
│ │ │ │  
│ │ │ │  o14 = 1
│ │ │ │  i15 : isQCartier(10, D2, IsGraded => true)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Q__Linear__Equivalent.html
│ │ │ @@ -155,24 +155,24 @@
│ │ │                
    i10 : R = QQ[x, y, z] / ideal(x * y - z^2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o11 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │ +o11 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │  
│ │ │  o11 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : E = divisor({3/2, -1/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o12 = 3/2*Div(y, z) + -1/4*Div(x, z)
│ │ │ +o12 = -1/4*Div(x, z) + 3/2*Div(y, z)
│ │ │  
│ │ │  o12 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : isQLinearEquivalent(10, D, E, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -52,21 +52,21 @@
│ │ │ │  o9 = true
│ │ │ │  If IsGraded=>true (the default is false), then it treats the divisors as if
│ │ │ │  they are divisors on the $Proj$ of their ambient ring.
│ │ │ │  i10 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │ │  i11 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType =>
│ │ │ │  QQ)
│ │ │ │  
│ │ │ │ -o11 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │ │ +o11 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ │  
│ │ │ │  o11 : QWeilDivisor on R
│ │ │ │  i12 : E = divisor({3/2, -1/4}, {ideal(y, z), ideal(x, z)}, CoefficientType =>
│ │ │ │  QQ)
│ │ │ │  
│ │ │ │ -o12 = 3/2*Div(y, z) + -1/4*Div(x, z)
│ │ │ │ +o12 = -1/4*Div(x, z) + 3/2*Div(y, z)
│ │ │ │  
│ │ │ │  o12 : QWeilDivisor on R
│ │ │ │  i13 : isQLinearEquivalent(10, D, E, IsGraded => true)
│ │ │ │  
│ │ │ │  o13 = true
│ │ │ │  i14 : isQLinearEquivalent(10, 3*D, E, IsGraded => true)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__S__N__C.html
│ │ │ @@ -80,15 +80,15 @@
│ │ │                
    i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = Div(x, z) + -2*Div(y, z)
│ │ │ +o2 = -2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : isSNC( D )
│ │ │ @@ -152,15 +152,15 @@
│ │ │                
    i10 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o11 = -2*Div(y, z) + Div(x, z)
│ │ │ +o11 = Div(x, z) + -2*Div(y, z)
│ │ │  
│ │ │  o11 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : isSNC( D, IsGraded => true )
│ │ │ @@ -198,15 +198,15 @@
│ │ │                
    i16 : R = QQ[x,y,z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o17 = Div(x) + Div(x+y) + Div(y)
│ │ │ +o17 = Div(x+y) + Div(y) + Div(x)
│ │ │  
│ │ │  o17 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i18 : isSNC( D, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns true if the divisor is simple normal crossings, this
│ │ │ │  includes checking that the ambient ring is regular.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i2 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o2 = Div(x, z) + -2*Div(y, z)
│ │ │ │ +o2 = -2*Div(y, z) + Div(x, z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : isSNC( D )
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : R = QQ[x, y];
│ │ │ │  i5 : D = divisor(x*y*(x+y))
│ │ │ │ @@ -45,15 +45,15 @@
│ │ │ │  o9 = true
│ │ │ │  If IsGraded is set to true (default false), then the divisor is treated as if
│ │ │ │  it is on the $Proj$ of the ambient ring. In particular, non-SNC behavior at the
│ │ │ │  origin is ignored.
│ │ │ │  i10 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i11 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o11 = -2*Div(y, z) + Div(x, z)
│ │ │ │ +o11 = Div(x, z) + -2*Div(y, z)
│ │ │ │  
│ │ │ │  o11 : WeilDivisor on R
│ │ │ │  i12 : isSNC( D, IsGraded => true )
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  i13 : R = QQ[x, y];
│ │ │ │  i14 : D = divisor(x*y*(x+y))
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │  o14 : WeilDivisor on R
│ │ │ │  i15 : isSNC( D, IsGraded => true )
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  i16 : R = QQ[x,y,z];
│ │ │ │  i17 : D = divisor(x*y*(x+y))
│ │ │ │  
│ │ │ │ -o17 = Div(x) + Div(x+y) + Div(y)
│ │ │ │ +o17 = Div(x+y) + Div(y) + Div(x)
│ │ │ │  
│ │ │ │  o17 : WeilDivisor on R
│ │ │ │  i18 : isSNC( D, IsGraded => true)
│ │ │ │  
│ │ │ │  o18 = false
│ │ │ │  The output value of this function is stored in the divisor's cache with the
│ │ │ │  value of the last IsGraded option. If you change the IsGraded option, the value
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_map__To__Projective__Space.html
│ │ │ @@ -112,15 +112,15 @@
│ │ │                
    i4 : R = ZZ/7[x,y,z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : D = divisor(x*y)
│ │ │  
│ │ │ -o5 = Div(y) + Div(x)
│ │ │ +o5 = Div(x) + Div(y)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : mapToProjectiveSpace(D, Variable=>"Z")
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │  
│ │ │ │  o3 : RingMap R <-- QQ[YY ..YY ]
│ │ │ │                          1    2
│ │ │ │  The user may also specify the variable name of the new projective space.
│ │ │ │  i4 : R = ZZ/7[x,y,z];
│ │ │ │  i5 : D = divisor(x*y)
│ │ │ │  
│ │ │ │ -o5 = Div(y) + Div(x)
│ │ │ │ +o5 = Div(x) + Div(y)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  i6 : mapToProjectiveSpace(D, Variable=>"Z")
│ │ │ │  
│ │ │ │               ZZ            2             2        2
│ │ │ │  o6 = map (R, --[Z ..Z ], {x , x*y, x*z, y , y*z, z })
│ │ │ │                7  1   6
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.html
│ │ │ @@ -94,24 +94,24 @@
│ │ │  o3 : RingMap T <-- R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : D = divisor(y*z)
│ │ │  
│ │ │ -o4 = 3*Div(w, z, y) + 3*Div(z, y, x)
│ │ │ +o4 = 3*Div(z, y, x) + 3*Div(w, z, y)
│ │ │  
│ │ │  o4 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : pullback(f, D, Strategy=>Primes)
│ │ │  
│ │ │ -o5 = 3*Div(b) + 3*Div(a)
│ │ │ +o5 = 3*Div(a) + 3*Div(b)
│ │ │  
│ │ │  o5 : WeilDivisor on T
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : pullback(f, D, Strategy=>Sheaves)
│ │ │ @@ -150,24 +150,24 @@
│ │ │  o10 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : D1 = pullback(f, D)
│ │ │  
│ │ │ -o11 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │ +o11 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │  
│ │ │  o11 : WeilDivisor on S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : f^* D
│ │ │  
│ │ │ -o12 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │ +o12 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │  
│ │ │  o12 : WeilDivisor on S
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    As illustrated by the previous example, the same functionality can also be accomplished by f^* (which creates a function which sends a divisor $D$ to $f^* D$).
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,20 +29,20 @@
│ │ │ │  i1 : R = QQ[x,y,z,w]/ideal(z^2-y*w,y*z-x*w,y^2-x*z);
│ │ │ │  i2 : T = QQ[a,b];
│ │ │ │  i3 : f = map(T, R, {a^3, a^2*b, a*b^2, b^3});
│ │ │ │  
│ │ │ │  o3 : RingMap T <-- R
│ │ │ │  i4 : D = divisor(y*z)
│ │ │ │  
│ │ │ │ -o4 = 3*Div(w, z, y) + 3*Div(z, y, x)
│ │ │ │ +o4 = 3*Div(z, y, x) + 3*Div(w, z, y)
│ │ │ │  
│ │ │ │  o4 : WeilDivisor on R
│ │ │ │  i5 : pullback(f, D, Strategy=>Primes)
│ │ │ │  
│ │ │ │ -o5 = 3*Div(b) + 3*Div(a)
│ │ │ │ +o5 = 3*Div(a) + 3*Div(b)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on T
│ │ │ │  i6 : pullback(f, D, Strategy=>Sheaves)
│ │ │ │  
│ │ │ │  o6 = 3*Div(b) + 3*Div(a)
│ │ │ │  
│ │ │ │  o6 : WeilDivisor on T
│ │ │ │ @@ -53,20 +53,20 @@
│ │ │ │  
│ │ │ │  o9 : RingMap S <-- R
│ │ │ │  i10 : D = divisor(x*y*(x+y));
│ │ │ │  
│ │ │ │  o10 : WeilDivisor on R
│ │ │ │  i11 : D1 = pullback(f, D)
│ │ │ │  
│ │ │ │ -o11 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │ │ +o11 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │ │  
│ │ │ │  o11 : WeilDivisor on S
│ │ │ │  i12 : f^* D
│ │ │ │  
│ │ │ │ -o12 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │ │ +o12 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │ │  
│ │ │ │  o12 : WeilDivisor on S
│ │ │ │  As illustrated by the previous example, the same functionality can also be
│ │ │ │  accomplished by f^* (which creates a function which sends a divisor $D$ to $f^*
│ │ │ │  D$).
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _P_r_i_m_e_s -- a value for the option Strategy for the pullback method
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexify.html
│ │ │ @@ -267,23 +267,23 @@
│ │ │  
│ │ │  o22 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : time reflexify(J);
│ │ │ - -- used 0.260716s (cpu); 0.201186s (thread); 0s (gc)
│ │ │ + -- used 0.305298s (cpu); 0.234436s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.527272s (cpu); 0.394308s (thread); 0s (gc)
    
│ │ │ + -- used 0.484222s (cpu); 0.39818s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Because of this, there are two strategies for computing a reflexification (at least if the module embeds as an ideal).
    
│ │ │          
    
│ │ │          
    
│ │ │ @@ -319,26 +319,26 @@
│ │ │              
│ │ │                
    i28 : M = J*R^1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : J1 = time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.273432s (cpu); 0.158699s (thread); 0s (gc)
│ │ │ + -- used 0.328963s (cpu); 0.156132s (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 6.95165s (cpu); 4.6997s (thread); 0s (gc)
│ │ │ + -- used 6.33337s (cpu); 4.98751s (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
    
│ │ │              
│ │ │            
│ │ │ @@ -348,21 +348,21 @@
│ │ │  
│ │ │  o31 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 5.67659s (cpu); 4.46311s (thread); 0s (gc)
    
│ │ │ + -- used 6.71641s (cpu); 5.39144s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.775911s (cpu); 0.448237s (thread); 0s (gc)
    
│ │ │ + -- used 0.663711s (cpu); 0.45656s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    However, sometimes ModuleStrategy is faster, especially for Monomial ideals.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -389,15 +389,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i37 : M = I^20*R^1;
    
│ │ │              
    
│ │ │                
    i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 1.05331s (cpu); 0.384322s (thread); 0s (gc)
│ │ │ + -- used 1.28401s (cpu); 0.46856s (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 ,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -406,15 +406,15 @@
│ │ │  
│ │ │  o38 : Ideal of R
    
│ │ │              
    
│ │ │                
    i39 : time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 0.238734s (cpu); 0.0653637s (thread); 0s (gc)
│ │ │ + -- used 0.259904s (cpu); 0.0660955s (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 ,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -423,21 +423,21 @@
│ │ │  
│ │ │  o39 : Ideal of R
    
│ │ │              
    
│ │ │                
    i40 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 0.276556s (cpu); 0.103744s (thread); 0s (gc)
    
│ │ │ + -- used 0.323261s (cpu); 0.106148s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00801402s (cpu); 0.00801143s (thread); 0s (gc)
    
│ │ │ + -- used 0.00832269s (cpu); 0.00833187s (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 0.260716s (cpu); 0.201186s (thread); 0s (gc)
│ │ │ │ + -- used 0.305298s (cpu); 0.234436s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 : Ideal of R
│ │ │ │  i24 : time reflexify(J*R^1);
│ │ │ │ - -- used 0.527272s (cpu); 0.394308s (thread); 0s (gc)
│ │ │ │ + -- used 0.484222s (cpu); 0.39818s (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 0.273432s (cpu); 0.158699s (thread); 0s (gc)
│ │ │ │ + -- used 0.328963s (cpu); 0.156132s (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 6.95165s (cpu); 4.6997s (thread); 0s (gc)
│ │ │ │ + -- used 6.33337s (cpu); 4.98751s (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 5.67659s (cpu); 4.46311s (thread); 0s (gc)
│ │ │ │ + -- used 6.71641s (cpu); 5.39144s (thread); 0s (gc)
│ │ │ │  i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ │ - -- used 0.775911s (cpu); 0.448237s (thread); 0s (gc)
│ │ │ │ + -- used 0.663711s (cpu); 0.45656s (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 1.05331s (cpu); 0.384322s (thread); 0s (gc)
│ │ │ │ + -- used 1.28401s (cpu); 0.46856s (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 0.238734s (cpu); 0.0653637s (thread); 0s (gc)
│ │ │ │ + -- used 0.259904s (cpu); 0.0660955s (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 0.276556s (cpu); 0.103744s (thread); 0s (gc)
│ │ │ │ + -- used 0.323261s (cpu); 0.106148s (thread); 0s (gc)
│ │ │ │  i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ │ - -- used 0.00801402s (cpu); 0.00801143s (thread); 0s (gc)
│ │ │ │ + -- used 0.00832269s (cpu); 0.00833187s (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
│ │ │ @@ -124,30 +124,30 @@
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time J20a = reflexivePower(20, I);
│ │ │ - -- used 0.0323817s (cpu); 0.0323825s (thread); 0s (gc)
│ │ │ + -- used 0.0477471s (cpu); 0.0477492s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.204644s (cpu); 0.153031s (thread); 0s (gc)
│ │ │ + -- used 0.234538s (cpu); 0.159666s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : J20a == J20b
│ │ │ @@ -171,23 +171,23 @@
│ │ │  
│ │ │  o12 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ - -- used 0.029315s (cpu); 0.0293196s (thread); 0s (gc)
│ │ │ + -- used 0.0376128s (cpu); 0.0376168s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.143657s (cpu); 0.0818326s (thread); 0s (gc)
│ │ │ + -- used 0.188293s (cpu); 0.101737s (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 0.0323817s (cpu); 0.0323825s (thread); 0s (gc)
│ │ │ │ + -- used 0.0477471s (cpu); 0.0477492s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : I20 = I^20;
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time J20b = reflexify(I20);
│ │ │ │ - -- used 0.204644s (cpu); 0.153031s (thread); 0s (gc)
│ │ │ │ + -- used 0.234538s (cpu); 0.159666s (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 0.029315s (cpu); 0.0293196s (thread); 0s (gc)
│ │ │ │ + -- used 0.0376128s (cpu); 0.0376168s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 : Ideal of R
│ │ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.143657s (cpu); 0.0818326s (thread); 0s (gc)
│ │ │ │ + -- used 0.188293s (cpu); 0.101737s (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/WeilDivisors/html/_ring_lp__Basic__Divisor_rp.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │                
    i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 2*Div(y, z) + Div(x, z)
│ │ │ +o2 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : ring( D )
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _r_i_n_g,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns the ambient ring of a divisor.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i2 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o2 = 2*Div(y, z) + Div(x, z)
│ │ │ │ +o2 = Div(x, z) + 2*Div(y, z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : ring( D )
│ │ │ │  
│ │ │ │  o3 = R
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_to__R__Weil__Divisor.html
│ │ │ @@ -79,42 +79,42 @@
│ │ │                
    i1 : R = ZZ/5[x,y];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({2, 0, -4}, {ideal(x), ideal(y), ideal(x-y)})
│ │ │  
│ │ │ -o2 = -4*Div(x-y) + 2*Div(x) + 0*Div(y)
│ │ │ +o2 = 2*Div(x) + 0*Div(y) + -4*Div(x-y)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : E = (1/2)*D
│ │ │  
│ │ │ -o3 = -2*Div(x-y) + Div(x)
│ │ │ +o3 = Div(x) + -2*Div(x-y)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : F = toRWeilDivisor(D)
│ │ │  
│ │ │ -o4 = -4*Div(x-y) + 2*Div(x)
│ │ │ +o4 = 2*Div(x) + -4*Div(x-y)
│ │ │  
│ │ │  o4 : RWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : G = toRWeilDivisor(E)
│ │ │  
│ │ │ -o5 = -2*Div(x-y) + Div(x)
│ │ │ +o5 = Div(x) + -2*Div(x-y)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : F == 2*G
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,30 +15,30 @@
│ │ │ │            o an instance of the type _R_W_e_i_l_D_i_v_i_s_o_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Turn a Weil divisor or a Q-divisor into a R-divisor (or do nothing to a R-
│ │ │ │  divisor).
│ │ │ │  i1 : R = ZZ/5[x,y];
│ │ │ │  i2 : D = divisor({2, 0, -4}, {ideal(x), ideal(y), ideal(x-y)})
│ │ │ │  
│ │ │ │ -o2 = -4*Div(x-y) + 2*Div(x) + 0*Div(y)
│ │ │ │ +o2 = 2*Div(x) + 0*Div(y) + -4*Div(x-y)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : E = (1/2)*D
│ │ │ │  
│ │ │ │ -o3 = -2*Div(x-y) + Div(x)
│ │ │ │ +o3 = Div(x) + -2*Div(x-y)
│ │ │ │  
│ │ │ │  o3 : QWeilDivisor on R
│ │ │ │  i4 : F = toRWeilDivisor(D)
│ │ │ │  
│ │ │ │ -o4 = -4*Div(x-y) + 2*Div(x)
│ │ │ │ +o4 = 2*Div(x) + -4*Div(x-y)
│ │ │ │  
│ │ │ │  o4 : RWeilDivisor on R
│ │ │ │  i5 : G = toRWeilDivisor(E)
│ │ │ │  
│ │ │ │ -o5 = -2*Div(x-y) + Div(x)
│ │ │ │ +o5 = Div(x) + -2*Div(x-y)
│ │ │ │  
│ │ │ │  o5 : RWeilDivisor on R
│ │ │ │  i6 : F == 2*G
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_o_W_e_i_l_D_i_v_i_s_o_r -- create a Weil divisor from a Q or R-divisor
│ │ ├── ./usr/share/doc/Macaulay2/WeylAlgebras/example-output/_factor__Weyl__Algebra.out
│ │ │ @@ -4,19 +4,19 @@
│ │ │  
│ │ │  o1 = R
│ │ │  
│ │ │  o1 : PolynomialRing, 1 differential variable(s)
│ │ │  
│ │ │  i2 : factorWA(x^5*dx^2+7*x^4*dx+8*x^3-x*dx^2+dx)
│ │ │  
│ │ │ -                                    2         3       2                  2  
│ │ │ -o2 = {(x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1), (x dx + 3x  - x*dx + 1)(dx)(x  +
│ │ │ +        3       2                                      3                  
│ │ │ +o2 = {(x dx + 3x  + x*dx - 1)(dx)(x - 1)(x + 1), (dx)(x dx + x*dx - 2)(x -
│ │ │       ------------------------------------------------------------------------
│ │ │ -               3                2         3       2                         
│ │ │ -     1), (dx)(x dx - x*dx + 2)(x  + 1), (x dx + 3x  + x*dx - 1)(dx)(x - 1)(x
│ │ │ +                  3       2                  2             3                2
│ │ │ +     1)(x + 1), (x dx + 3x  - x*dx + 1)(dx)(x  + 1), (dx)(x dx - x*dx + 2)(x 
│ │ │       ------------------------------------------------------------------------
│ │ │ -                 3
│ │ │ -     + 1), (dx)(x dx + x*dx - 2)(x - 1)(x + 1)}
│ │ │ +                                         2
│ │ │ +     + 1), (x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1)}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylAlgebras/html/_factor__Weyl__Algebra.html
│ │ │ @@ -84,22 +84,22 @@
│ │ │  o1 : PolynomialRing, 1 differential variable(s)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : factorWA(x^5*dx^2+7*x^4*dx+8*x^3-x*dx^2+dx)
│ │ │  
│ │ │ -                                    2         3       2                  2  
│ │ │ -o2 = {(x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1), (x dx + 3x  - x*dx + 1)(dx)(x  +
│ │ │ +        3       2                                      3                  
│ │ │ +o2 = {(x dx + 3x  + x*dx - 1)(dx)(x - 1)(x + 1), (dx)(x dx + x*dx - 2)(x -
│ │ │       ------------------------------------------------------------------------
│ │ │ -               3                2         3       2                         
│ │ │ -     1), (dx)(x dx - x*dx + 2)(x  + 1), (x dx + 3x  + x*dx - 1)(dx)(x - 1)(x
│ │ │ +                  3       2                  2             3                2
│ │ │ +     1)(x + 1), (x dx + 3x  - x*dx + 1)(dx)(x  + 1), (dx)(x dx - x*dx + 2)(x 
│ │ │       ------------------------------------------------------------------------
│ │ │ -                 3
│ │ │ -     + 1), (dx)(x dx + x*dx - 2)(x - 1)(x + 1)}
│ │ │ +                                         2
│ │ │ +     + 1), (x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1)}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    To reduce their number, two factorisations are considered equivalent if they can be related by (1)	switching commuting irreducible factors or (2) switching monomials and degree 0 factors; a normal order is chosen where commuting factors are sorted, and monomials are pushed to the right/left if they're differential/not.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,22 +20,22 @@
│ │ │ │  i1 : R = makeWA(QQ[x])
│ │ │ │  
│ │ │ │  o1 = R
│ │ │ │  
│ │ │ │  o1 : PolynomialRing, 1 differential variable(s)
│ │ │ │  i2 : factorWA(x^5*dx^2+7*x^4*dx+8*x^3-x*dx^2+dx)
│ │ │ │  
│ │ │ │ -                                    2         3       2                  2
│ │ │ │ -o2 = {(x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1), (x dx + 3x  - x*dx + 1)(dx)(x  +
│ │ │ │ +        3       2                                      3
│ │ │ │ +o2 = {(x dx + 3x  + x*dx - 1)(dx)(x - 1)(x + 1), (dx)(x dx + x*dx - 2)(x -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -               3                2         3       2
│ │ │ │ -     1), (dx)(x dx - x*dx + 2)(x  + 1), (x dx + 3x  + x*dx - 1)(dx)(x - 1)(x
│ │ │ │ +                  3       2                  2             3                2
│ │ │ │ +     1)(x + 1), (x dx + 3x  - x*dx + 1)(dx)(x  + 1), (dx)(x dx - x*dx + 2)(x
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -                 3
│ │ │ │ -     + 1), (dx)(x dx + x*dx - 2)(x - 1)(x + 1)}
│ │ │ │ +                                         2
│ │ │ │ +     + 1), (x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1)}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  To reduce their number, two factorisations are considered equivalent if they
│ │ │ │  can be related by (1) switching commuting irreducible factors or (2) switching
│ │ │ │  monomials and degree 0 factors; a normal order is chosen where commuting
│ │ │ │  factors are sorted, and monomials are pushed to the right/left if they're
│ │ │ │  differential/not.
│ │ ├── ./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 1.9378s (cpu); 1.16676s (thread); 0s (gc)
│ │ │ + -- used 2.65721s (cpu); 1.40532s (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 4.77653s (cpu); 2.98611s (thread); 0s (gc)
│ │ │ + -- used 7.83092s (cpu); 3.33045s (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 5.07329s (cpu); 2.97775s (thread); 0s (gc)
│ │ │ + -- used 8.275s (cpu); 3.65263s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : peek last ms
│ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/html/_map__Stratify.html
│ │ │ @@ -292,15 +292,15 @@
│ │ │  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 1.9378s (cpu); 1.16676s (thread); 0s (gc)
│ │ │ + -- used 2.65721s (cpu); 1.40532s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o23 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -318,15 +318,15 @@
│ │ │  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 4.77653s (cpu); 2.98611s (thread); 0s (gc)
│ │ │ + -- used 7.83092s (cpu); 3.33045s (thread); 0s (gc)
│ │ │  
│ │ │  o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o25 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -344,15 +344,15 @@
│ │ │  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 5.07329s (cpu); 2.97775s (thread); 0s (gc)
│ │ │ + -- used 8.275s (cpu); 3.65263s (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 1.9378s (cpu); 1.16676s (thread); 0s (gc)
│ │ │ │ + -- used 2.65721s (cpu); 1.40532s (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 4.77653s (cpu); 2.98611s (thread); 0s (gc)
│ │ │ │ + -- used 7.83092s (cpu); 3.33045s (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 5.07329s (cpu); 2.97775s (thread); 0s (gc)
│ │ │ │ + -- used 8.275s (cpu); 3.65263s (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/M2-16949-0/172
│ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-22504-0/172
│ │ │  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/M2-16949-0/172
│ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16949-0/174
│ │ │ +using temporary file /tmp/M2-22504-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-22504-0/174
│ │ │  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/M2-16949-0/174
│ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16949-0/176
│ │ │ +using temporary file /tmp/M2-22504-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-22504-0/176
│ │ │  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/M2-16949-0/176
│ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16949-0/178
│ │ │ +using temporary file /tmp/M2-22504-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-22504-0/178
│ │ │  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/M2-16949-0/178
│ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16949-0/180
│ │ │ +using temporary file /tmp/M2-22504-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-22504-0/180
│ │ │  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/M2-16949-0/180
│ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16949-0/182
│ │ │ +using temporary file /tmp/M2-22504-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-22504-0/182
│ │ │  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/M2-16949-0/182
│ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16949-0/184
│ │ │ +using temporary file /tmp/M2-22504-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-22504-0/184
│ │ │  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/M2-16949-0/184
│ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16949-0/186
│ │ │ +using temporary file /tmp/M2-22504-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-22504-0/186
│ │ │  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/M2-16949-0/186
│ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16949-0/188
│ │ │ +using temporary file /tmp/M2-22504-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-22504-0/188
│ │ │  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/M2-16949-0/188
│ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16949-0/190
│ │ │ +using temporary file /tmp/M2-22504-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-22504-0/190
│ │ │  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/M2-16949-0/190
│ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16949-0/192
│ │ │ +using temporary file /tmp/M2-22504-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-22504-0/192
│ │ │  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/M2-16949-0/192
│ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16949-0/194
│ │ │ +using temporary file /tmp/M2-22504-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-22504-0/194
│ │ │  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/M2-16949-0/194
│ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16949-0/196
│ │ │ +using temporary file /tmp/M2-22504-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-22504-0/196
│ │ │  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/M2-16949-0/196
│ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16949-0/198
│ │ │ +using temporary file /tmp/M2-22504-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-22504-0/198
│ │ │  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/M2-16949-0/198
│ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16949-0/200
│ │ │ +using temporary file /tmp/M2-22504-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-22504-0/200
│ │ │  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/M2-16949-0/200
│ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16949-0/202
│ │ │ +using temporary file /tmp/M2-22504-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-22504-0/202
│ │ │  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/M2-16949-0/202
│ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16949-0/204
│ │ │ +using temporary file /tmp/M2-22504-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-22504-0/204
│ │ │  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/M2-16949-0/204
│ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16949-0/206
│ │ │ +using temporary file /tmp/M2-22504-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-22504-0/206
│ │ │  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/M2-16949-0/206
│ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16949-0/208
│ │ │ +using temporary file /tmp/M2-22504-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-22504-0/208
│ │ │  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/M2-16949-0/208
│ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16949-0/210
│ │ │ +using temporary file /tmp/M2-22504-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-22504-0/210
│ │ │  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/M2-16949-0/210
│ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16949-0/212
│ │ │ +using temporary file /tmp/M2-22504-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-22504-0/212
│ │ │  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/M2-16949-0/212
│ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16949-0/214
│ │ │ +using temporary file /tmp/M2-22504-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-22504-0/214
│ │ │  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/M2-16949-0/214
│ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16949-0/216
│ │ │ +using temporary file /tmp/M2-22504-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-22504-0/216
│ │ │  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/M2-16949-0/216
│ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16949-0/218
│ │ │ +using temporary file /tmp/M2-22504-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-22504-0/218
│ │ │  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/M2-16949-0/218
│ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16949-0/220
│ │ │ +using temporary file /tmp/M2-22504-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-22504-0/220
│ │ │  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/M2-16949-0/220
│ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16949-0/222
│ │ │ +using temporary file /tmp/M2-22504-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-22504-0/222
│ │ │  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/M2-16949-0/222
│ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16949-0/224
│ │ │ +using temporary file /tmp/M2-22504-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-22504-0/224
│ │ │  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/M2-16949-0/224
│ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16949-0/226
│ │ │ +using temporary file /tmp/M2-22504-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-22504-0/226
│ │ │  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/M2-16949-0/226
│ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16949-0/228
│ │ │ +using temporary file /tmp/M2-22504-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-22504-0/228
│ │ │  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/M2-16949-0/228
│ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16949-0/230
│ │ │ +using temporary file /tmp/M2-22504-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-22504-0/230
│ │ │  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/M2-16949-0/230
│ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16949-0/232
│ │ │ +using temporary file /tmp/M2-22504-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-22504-0/232
│ │ │  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/M2-16949-0/232
│ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16949-0/234
│ │ │ +using temporary file /tmp/M2-22504-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-22504-0/234
│ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/234
│ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16949-0/236
│ │ │ +using temporary file /tmp/M2-22504-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-22504-0/236
│ │ │  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/M2-16949-0/236
│ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16949-0/238
│ │ │ +using temporary file /tmp/M2-22504-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-22504-0/238
│ │ │  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/M2-16949-0/238
│ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16949-0/240
│ │ │ +using temporary file /tmp/M2-22504-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-22504-0/240
│ │ │  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/M2-16949-0/240
│ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16949-0/242
│ │ │ +using temporary file /tmp/M2-22504-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-22504-0/242
│ │ │  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/M2-16949-0/242
│ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16949-0/244
│ │ │ +using temporary file /tmp/M2-22504-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-22504-0/244
│ │ │  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/M2-16949-0/244
│ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16949-0/246
│ │ │ +using temporary file /tmp/M2-22504-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-22504-0/246
│ │ │  This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/246
│ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16949-0/248
│ │ │ +using temporary file /tmp/M2-22504-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-22504-0/248
│ │ │  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/M2-16949-0/248
│ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16949-0/250
│ │ │ +using temporary file /tmp/M2-22504-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-22504-0/250
│ │ │  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/M2-16949-0/250
│ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16949-0/252
│ │ │ +using temporary file /tmp/M2-22504-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-22504-0/252
│ │ │  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/M2-16949-0/252
│ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16949-0/254
│ │ │ +using temporary file /tmp/M2-22504-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-22504-0/254
│ │ │  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/M2-16949-0/254
│ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16949-0/256
│ │ │ +using temporary file /tmp/M2-22504-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-22504-0/256
│ │ │  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/M2-16949-0/256
│ │ │ +using temporary file /tmp/M2-22504-0/256
│ │ │  
│ │ │  i6 : QQ[x,y];
│ │ │  
│ │ │  i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16949-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-22504-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-16949-0/258
│ │ │ +using temporary file /tmp/M2-22504-0/258
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/html/___Installation_spand_sp__Configuration_spof_spgfan__Interface.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │            
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    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/M2-16949-0/172
│ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-22504-0/172
│ │ │  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.
│ │ │ @@ -128,16 +128,16 @@
│ │ │  --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/M2-16949-0/172
│ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16949-0/174
│ │ │ +using temporary file /tmp/M2-22504-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-22504-0/174
│ │ │  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.
│ │ │ @@ -146,69 +146,69 @@
│ │ │   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/M2-16949-0/174
│ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16949-0/176
│ │ │ +using temporary file /tmp/M2-22504-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-22504-0/176
│ │ │  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/M2-16949-0/176
│ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16949-0/178
│ │ │ +using temporary file /tmp/M2-22504-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-22504-0/178
│ │ │  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/M2-16949-0/178
│ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16949-0/180
│ │ │ +using temporary file /tmp/M2-22504-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-22504-0/180
│ │ │  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/M2-16949-0/180
│ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16949-0/182
│ │ │ +using temporary file /tmp/M2-22504-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-22504-0/182
│ │ │  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/M2-16949-0/182
│ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16949-0/184
│ │ │ +using temporary file /tmp/M2-22504-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-22504-0/184
│ │ │  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/M2-16949-0/184
│ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16949-0/186
│ │ │ +using temporary file /tmp/M2-22504-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-22504-0/186
│ │ │  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/M2-16949-0/186
│ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16949-0/188
│ │ │ +using temporary file /tmp/M2-22504-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-22504-0/188
│ │ │  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}
│ │ │ @@ -216,30 +216,30 @@
│ │ │  -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/M2-16949-0/188
│ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16949-0/190
│ │ │ +using temporary file /tmp/M2-22504-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-22504-0/190
│ │ │  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/M2-16949-0/190
│ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16949-0/192
│ │ │ +using temporary file /tmp/M2-22504-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-22504-0/192
│ │ │  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.
│ │ │ @@ -254,57 +254,57 @@
│ │ │   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/M2-16949-0/192
│ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16949-0/194
│ │ │ +using temporary file /tmp/M2-22504-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-22504-0/194
│ │ │  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/M2-16949-0/194
│ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16949-0/196
│ │ │ +using temporary file /tmp/M2-22504-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-22504-0/196
│ │ │  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/M2-16949-0/196
│ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16949-0/198
│ │ │ +using temporary file /tmp/M2-22504-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-22504-0/198
│ │ │  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/M2-16949-0/198
│ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16949-0/200
│ │ │ +using temporary file /tmp/M2-22504-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-22504-0/200
│ │ │  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/M2-16949-0/200
│ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16949-0/202
│ │ │ +using temporary file /tmp/M2-22504-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-22504-0/202
│ │ │  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/M2-16949-0/202
│ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16949-0/204
│ │ │ +using temporary file /tmp/M2-22504-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-22504-0/204
│ │ │  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/M2-16949-0/204
│ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16949-0/206
│ │ │ +using temporary file /tmp/M2-22504-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-22504-0/206
│ │ │  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:
│ │ │ @@ -319,16 +319,16 @@
│ │ │   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/M2-16949-0/206
│ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16949-0/208
│ │ │ +using temporary file /tmp/M2-22504-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-22504-0/208
│ │ │  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:
│ │ │ @@ -339,44 +339,44 @@
│ │ │   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/M2-16949-0/208
│ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16949-0/210
│ │ │ +using temporary file /tmp/M2-22504-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-22504-0/210
│ │ │  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/M2-16949-0/210
│ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16949-0/212
│ │ │ +using temporary file /tmp/M2-22504-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-22504-0/212
│ │ │  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/M2-16949-0/212
│ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16949-0/214
│ │ │ +using temporary file /tmp/M2-22504-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-22504-0/214
│ │ │  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/M2-16949-0/214
│ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16949-0/216
│ │ │ +using temporary file /tmp/M2-22504-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-22504-0/216
│ │ │  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.
│ │ │ @@ -389,25 +389,25 @@
│ │ │  
│ │ │  --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/M2-16949-0/216
│ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16949-0/218
│ │ │ +using temporary file /tmp/M2-22504-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-22504-0/218
│ │ │  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/M2-16949-0/218
│ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16949-0/220
│ │ │ +using temporary file /tmp/M2-22504-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-22504-0/220
│ │ │  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:
│ │ │ @@ -416,70 +416,70 @@
│ │ │  --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/M2-16949-0/220
│ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16949-0/222
│ │ │ +using temporary file /tmp/M2-22504-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-22504-0/222
│ │ │  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/M2-16949-0/222
│ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16949-0/224
│ │ │ +using temporary file /tmp/M2-22504-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-22504-0/224
│ │ │  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/M2-16949-0/224
│ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16949-0/226
│ │ │ +using temporary file /tmp/M2-22504-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-22504-0/226
│ │ │  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/M2-16949-0/226
│ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16949-0/228
│ │ │ +using temporary file /tmp/M2-22504-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-22504-0/228
│ │ │  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/M2-16949-0/228
│ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16949-0/230
│ │ │ +using temporary file /tmp/M2-22504-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-22504-0/230
│ │ │  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/M2-16949-0/230
│ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16949-0/232
│ │ │ +using temporary file /tmp/M2-22504-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-22504-0/232
│ │ │  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/M2-16949-0/232
│ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16949-0/234
│ │ │ +using temporary file /tmp/M2-22504-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-22504-0/234
│ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/234
│ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16949-0/236
│ │ │ +using temporary file /tmp/M2-22504-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-22504-0/236
│ │ │  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/M2-16949-0/236
│ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16949-0/238
│ │ │ +using temporary file /tmp/M2-22504-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-22504-0/238
│ │ │  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/M2-16949-0/238
│ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16949-0/240
│ │ │ +using temporary file /tmp/M2-22504-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-22504-0/240
│ │ │  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.
│ │ │ @@ -491,16 +491,16 @@
│ │ │   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/M2-16949-0/240
│ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16949-0/242
│ │ │ +using temporary file /tmp/M2-22504-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-22504-0/242
│ │ │  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
│ │ │  
│ │ │ @@ -525,48 +525,48 @@
│ │ │  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/M2-16949-0/242
│ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16949-0/244
│ │ │ +using temporary file /tmp/M2-22504-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-22504-0/244
│ │ │  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/M2-16949-0/244
│ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16949-0/246
│ │ │ +using temporary file /tmp/M2-22504-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-22504-0/246
│ │ │  This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/246
│ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16949-0/248
│ │ │ +using temporary file /tmp/M2-22504-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-22504-0/248
│ │ │  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/M2-16949-0/248
│ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16949-0/250
│ │ │ +using temporary file /tmp/M2-22504-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-22504-0/250
│ │ │  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/M2-16949-0/250
│ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16949-0/252
│ │ │ +using temporary file /tmp/M2-22504-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-22504-0/252
│ │ │  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:
│ │ │ @@ -574,24 +574,24 @@
│ │ │  --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/M2-16949-0/252
│ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16949-0/254
│ │ │ +using temporary file /tmp/M2-22504-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-22504-0/254
│ │ │  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/M2-16949-0/254
│ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16949-0/256
│ │ │ +using temporary file /tmp/M2-22504-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-22504-0/256
│ │ │  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.
│ │ │ @@ -611,32 +611,32 @@
│ │ │  --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/M2-16949-0/256
    
│ │ │ +using temporary file /tmp/M2-22504-0/256
│ │ │              
    
│ │ │                
    i6 : QQ[x,y];
    
│ │ │              
    
│ │ │                
    i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16949-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-22504-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-16949-0/258
    
│ │ │ +using temporary file /tmp/M2-22504-0/258
│ │ │              
    
│ │ │          
    
│ │ │            
    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/M2-16949-0/172
│ │ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-22504-0/172
│ │ │ │  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/M2-16949-0/172
│ │ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16949-0/174
│ │ │ │ +using temporary file /tmp/M2-22504-0/172
│ │ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-22504-0/174
│ │ │ │  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/M2-16949-0/174
│ │ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16949-0/176
│ │ │ │ +using temporary file /tmp/M2-22504-0/174
│ │ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-22504-0/176
│ │ │ │  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/M2-16949-0/176
│ │ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16949-0/178
│ │ │ │ +using temporary file /tmp/M2-22504-0/176
│ │ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-22504-0/178
│ │ │ │  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/M2-16949-0/178
│ │ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16949-0/180
│ │ │ │ +using temporary file /tmp/M2-22504-0/178
│ │ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-22504-0/180
│ │ │ │  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/M2-16949-0/180
│ │ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16949-0/182
│ │ │ │ +using temporary file /tmp/M2-22504-0/180
│ │ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-22504-0/182
│ │ │ │  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/M2-16949-0/182
│ │ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16949-0/184
│ │ │ │ +using temporary file /tmp/M2-22504-0/182
│ │ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-22504-0/184
│ │ │ │  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/M2-16949-0/184
│ │ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16949-0/186
│ │ │ │ +using temporary file /tmp/M2-22504-0/184
│ │ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-22504-0/186
│ │ │ │  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/M2-16949-0/186
│ │ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16949-0/188
│ │ │ │ +using temporary file /tmp/M2-22504-0/186
│ │ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-22504-0/188
│ │ │ │  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/M2-16949-0/188
│ │ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16949-0/190
│ │ │ │ +using temporary file /tmp/M2-22504-0/188
│ │ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-22504-0/190
│ │ │ │  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/M2-16949-0/190
│ │ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16949-0/192
│ │ │ │ +using temporary file /tmp/M2-22504-0/190
│ │ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-22504-0/192
│ │ │ │  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/M2-16949-0/192
│ │ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16949-0/194
│ │ │ │ +using temporary file /tmp/M2-22504-0/192
│ │ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-22504-0/194
│ │ │ │  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/M2-16949-0/194
│ │ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16949-0/196
│ │ │ │ +using temporary file /tmp/M2-22504-0/194
│ │ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-22504-0/196
│ │ │ │  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/M2-16949-0/196
│ │ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16949-0/198
│ │ │ │ +using temporary file /tmp/M2-22504-0/196
│ │ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-22504-0/198
│ │ │ │  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/M2-16949-0/198
│ │ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16949-0/200
│ │ │ │ +using temporary file /tmp/M2-22504-0/198
│ │ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-22504-0/200
│ │ │ │  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/M2-16949-0/200
│ │ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16949-0/202
│ │ │ │ +using temporary file /tmp/M2-22504-0/200
│ │ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-22504-0/202
│ │ │ │  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/M2-16949-0/202
│ │ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16949-0/204
│ │ │ │ +using temporary file /tmp/M2-22504-0/202
│ │ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-22504-0/204
│ │ │ │  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/M2-16949-0/204
│ │ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16949-0/206
│ │ │ │ +using temporary file /tmp/M2-22504-0/204
│ │ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-22504-0/206
│ │ │ │  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/M2-16949-0/206
│ │ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16949-0/208
│ │ │ │ +using temporary file /tmp/M2-22504-0/206
│ │ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-22504-0/208
│ │ │ │  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/M2-16949-0/208
│ │ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16949-0/210
│ │ │ │ +using temporary file /tmp/M2-22504-0/208
│ │ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-22504-0/210
│ │ │ │  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/M2-16949-0/210
│ │ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16949-0/212
│ │ │ │ +using temporary file /tmp/M2-22504-0/210
│ │ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-22504-0/212
│ │ │ │  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/M2-16949-0/212
│ │ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16949-0/214
│ │ │ │ +using temporary file /tmp/M2-22504-0/212
│ │ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-22504-0/214
│ │ │ │  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/M2-16949-0/214
│ │ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16949-0/216
│ │ │ │ +using temporary file /tmp/M2-22504-0/214
│ │ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-22504-0/216
│ │ │ │  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/M2-16949-0/216
│ │ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16949-0/218
│ │ │ │ +using temporary file /tmp/M2-22504-0/216
│ │ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-22504-0/218
│ │ │ │  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/M2-16949-0/218
│ │ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16949-0/220
│ │ │ │ +using temporary file /tmp/M2-22504-0/218
│ │ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-22504-0/220
│ │ │ │  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/M2-16949-0/220
│ │ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16949-0/222
│ │ │ │ +using temporary file /tmp/M2-22504-0/220
│ │ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-22504-0/222
│ │ │ │  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/M2-16949-0/222
│ │ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16949-0/224
│ │ │ │ +using temporary file /tmp/M2-22504-0/222
│ │ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-22504-0/224
│ │ │ │  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/M2-16949-0/224
│ │ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16949-0/226
│ │ │ │ +using temporary file /tmp/M2-22504-0/224
│ │ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-22504-0/226
│ │ │ │  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/M2-16949-0/226
│ │ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16949-0/228
│ │ │ │ +using temporary file /tmp/M2-22504-0/226
│ │ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-22504-0/228
│ │ │ │  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/M2-16949-0/228
│ │ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16949-0/230
│ │ │ │ +using temporary file /tmp/M2-22504-0/228
│ │ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-22504-0/230
│ │ │ │  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/M2-16949-0/230
│ │ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16949-0/232
│ │ │ │ +using temporary file /tmp/M2-22504-0/230
│ │ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-22504-0/232
│ │ │ │  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/M2-16949-0/232
│ │ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16949-0/234
│ │ │ │ +using temporary file /tmp/M2-22504-0/232
│ │ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-22504-0/234
│ │ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16949-0/234
│ │ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16949-0/236
│ │ │ │ +using temporary file /tmp/M2-22504-0/234
│ │ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-22504-0/236
│ │ │ │  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/M2-16949-0/236
│ │ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16949-0/238
│ │ │ │ +using temporary file /tmp/M2-22504-0/236
│ │ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-22504-0/238
│ │ │ │  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/M2-16949-0/238
│ │ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16949-0/240
│ │ │ │ +using temporary file /tmp/M2-22504-0/238
│ │ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-22504-0/240
│ │ │ │  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/M2-16949-0/240
│ │ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16949-0/242
│ │ │ │ +using temporary file /tmp/M2-22504-0/240
│ │ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-22504-0/242
│ │ │ │  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/M2-16949-0/242
│ │ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16949-0/244
│ │ │ │ +using temporary file /tmp/M2-22504-0/242
│ │ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-22504-0/244
│ │ │ │  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/M2-16949-0/244
│ │ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16949-0/246
│ │ │ │ +using temporary file /tmp/M2-22504-0/244
│ │ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-22504-0/246
│ │ │ │  This program computes the multiplicity of a tropical cone given a marked
│ │ │ │  reduced Groebner basis for its initial ideal.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16949-0/246
│ │ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16949-0/248
│ │ │ │ +using temporary file /tmp/M2-22504-0/246
│ │ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-22504-0/248
│ │ │ │  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/M2-16949-0/248
│ │ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16949-0/250
│ │ │ │ +using temporary file /tmp/M2-22504-0/248
│ │ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-22504-0/250
│ │ │ │  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/M2-16949-0/250
│ │ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16949-0/252
│ │ │ │ +using temporary file /tmp/M2-22504-0/250
│ │ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-22504-0/252
│ │ │ │  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/M2-16949-0/252
│ │ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16949-0/254
│ │ │ │ +using temporary file /tmp/M2-22504-0/252
│ │ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-22504-0/254
│ │ │ │  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/M2-16949-0/254
│ │ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16949-0/256
│ │ │ │ +using temporary file /tmp/M2-22504-0/254
│ │ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-22504-0/256
│ │ │ │  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/M2-16949-0/256
│ │ │ │ +using temporary file /tmp/M2-22504-0/256
│ │ │ │  i6 : QQ[x,y];
│ │ │ │  i7 : gfan {x,y};
│ │ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16949-0/258
│ │ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-22504-0/258
│ │ │ │  Q[x1,x2]
│ │ │ │  {{
│ │ │ │  x2,
│ │ │ │  x1}
│ │ │ │  }
│ │ │ │ -using temporary file /tmp/M2-16949-0/258
│ │ │ │ +using temporary file /tmp/M2-22504-0/258
│ │ │ │  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-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/gfanInterface.m2:2630:0.
│ │ ├── ./usr/share/info/AInfinity.info.gz
│ │ │ ├── AInfinity.info
│ │ │ │ @@ -6133,16 +6133,16 @@
│ │ │ │  00017f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017f70: 2b0a 7c69 3320 3a20 656c 6170 7365 6454  +.|i3 : elapsedT
│ │ │ │  00017f80: 696d 6520 6275 726b 6552 6573 6f6c 7574  ime burkeResolut
│ │ │ │  00017f90: 696f 6e28 4d2c 2037 2c20 4368 6563 6b20  ion(M, 7, Check 
│ │ │ │  00017fa0: 3d3e 2066 616c 7365 2920 2020 2020 2020  => false)       
│ │ │ │ -00017fb0: 2020 2020 7c0a 7c20 2d2d 2031 2e37 3131      |.| -- 1.711
│ │ │ │ -00017fc0: 3338 7320 656c 6170 7365 6420 2020 2020  38s elapsed     
│ │ │ │ +00017fb0: 2020 2020 7c0a 7c20 2d2d 2031 2e35 3137      |.| -- 1.517
│ │ │ │ +00017fc0: 3273 2065 6c61 7073 6564 2020 2020 2020  2s elapsed      
│ │ │ │  00017fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ff0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00018000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018030: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ @@ -6176,16 +6176,16 @@
│ │ │ │  000181f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018210: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │  00018220: 656c 6170 7365 6454 696d 6520 6275 726b  elapsedTime burk
│ │ │ │  00018230: 6552 6573 6f6c 7574 696f 6e28 4d2c 2037  eResolution(M, 7
│ │ │ │  00018240: 2c20 4368 6563 6b20 3d3e 2074 7275 6529  , Check => true)
│ │ │ │  00018250: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00018260: 2d2d 2032 2e31 3536 3935 7320 656c 6170  -- 2.15695s elap
│ │ │ │ -00018270: 7365 6420 2020 2020 2020 2020 2020 2020  sed             
│ │ │ │ +00018260: 2d2d 2031 2e37 3834 3873 2065 6c61 7073  -- 1.7848s elaps
│ │ │ │ +00018270: 6564 2020 2020 2020 2020 2020 2020 2020  ed              
│ │ │ │  00018280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000182b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182e0: 2020 2020 7c0a 7c20 2020 2020 2031 2020      |.|      1
│ │ ├── ./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 3232 3837     |.| -- .02287
│ │ │ │ -00002ec0: 3931 7320 656c 6170 7365 6420 2020 2020  91s elapsed     
│ │ │ │ +00002eb0: 2020 207c 0a7c 202d 2d20 2e30 3330 3931     |.| -- .03091
│ │ │ │ +00002ec0: 3573 2065 6c61 7073 6564 2020 2020 2020  5s 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 3730 3236 3373 2065 6c61 7073 6564  .670263s elapsed
│ │ │ │ +00006420: 2e35 3833 3535 3973 2065 6c61 7073 6564  .583559s 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 2035 2e37 3737 3937 7320 656c  | -- 5.77797s el
│ │ │ │ +00006790: 7c20 2d2d 2035 2e35 3730 3333 7320 656c  | -- 5.57033s 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: 3133 3636 3338 7320 656c 6170 7365 6420  136638s elapsed 
│ │ │ │ +0000a360: 3134 3732 3431 7320 656c 6170 7365 6420  147241s 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 3633 3733 3173  |.| -- .0563731s
│ │ │ │ +0000a5e0: 7c0a 7c20 2d2d 202e 3036 3836 3233 3573  |.| -- .0686235s
│ │ │ │  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        |.|       
│ │ │ │ @@ -2700,15 +2700,15 @@
│ │ │ │  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 2e38 3334 3635 7320 656c 6170 7365   1.83465s elapse
│ │ │ │ +0000a920: 2031 2e36 3535 3038 7320 656c 6170 7365   1.65508s elapse
│ │ │ │  0000a930: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │  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
│ │ ├── ./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 3039 3235 7320 2863 7075 293b  0.440925s (cpu);
│ │ │ │ -00010f90: 2030 2e33 3637 3030 3473 2028 7468 7265   0.367004s (thre
│ │ │ │ +00010f80: 302e 3536 3035 3335 7320 2863 7075 293b  0.560535s (cpu);
│ │ │ │ +00010f90: 2030 2e34 3534 3731 3273 2028 7468 7265   0.454712s (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 3737 3333 3473 2028 6370  ed 0.477334s (cp
│ │ │ │ -000113a0: 7529 3b20 302e 3430 3035 3337 7320 2874  u); 0.400537s (t
│ │ │ │ +00011390: 6564 2030 2e35 3837 3032 3873 2028 6370  ed 0.587028s (cp
│ │ │ │ +000113a0: 7529 3b20 302e 3438 3438 3537 7320 2874  u); 0.484857s (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
│ │ │ │ @@ -200,71 +200,76 @@
│ │ │ │  00000c70: 2d2d 2d2d 2b0a 7c69 3120 3a20 7275 6e42  ----+.|i1 : runB
│ │ │ │  00000c80: 656e 6368 6d61 726b 7320 2272 6573 3339  enchmarks "res39
│ │ │ │  00000c90: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │  00000ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000cc0: 2020 2020 7c0a 7c2d 2d20 6265 6769 6e6e      |.|-- beginn
│ │ │ │  00000cd0: 696e 6720 636f 6d70 7574 6174 696f 6e20  ing computation 
│ │ │ │ -00000ce0: 5375 6e20 4465 6320 3134 2031 353a 3331  Sun Dec 14 15:31
│ │ │ │ -00000cf0: 3a34 3220 5554 4320 3230 3235 2020 2020  :42 UTC 2025    
│ │ │ │ +00000ce0: 5765 6420 4a61 6e20 2037 2031 323a 3236  Wed Jan  7 12:26
│ │ │ │ +00000cf0: 3a35 3020 5554 4320 3230 3236 2020 2020  :50 UTC 2026    
│ │ │ │  00000d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000d10: 2020 2020 7c0a 7c2d 2d20 4c69 6e75 7820      |.|-- Linux 
│ │ │ │  00000d20: 7362 7569 6c64 2036 2e31 322e 3537 2b64  sbuild 6.12.57+d
│ │ │ │ -00000d30: 6562 3133 2d61 6d64 3634 2023 3120 534d  eb13-amd64 #1 SM
│ │ │ │ -00000d40: 5020 5052 4545 4d50 545f 4459 4e41 4d49  P PREEMPT_DYNAMI
│ │ │ │ -00000d50: 4320 4465 6269 616e 2036 2e31 322e 3537  C Debian 6.12.57
│ │ │ │ -00000d60: 2d31 2020 7c0a 7c2d 2d20 414d 4420 4550  -1  |.|-- AMD EP
│ │ │ │ -00000d70: 5943 2037 3730 3250 2036 342d 436f 7265  YC 7702P 64-Core
│ │ │ │ -00000d80: 2050 726f 6365 7373 6f72 2020 4175 7468   Processor  Auth
│ │ │ │ -00000d90: 656e 7469 6341 4d44 2020 6370 7520 4d48  enticAMD  cpu MH
│ │ │ │ -00000da0: 7a20 3139 3936 2e32 3439 2020 2020 2020  z 1996.249      
│ │ │ │ +00000d30: 6562 3133 2d63 6c6f 7564 2d61 6d64 3634  eb13-cloud-amd64
│ │ │ │ +00000d40: 2023 3120 534d 5020 5052 4545 4d50 545f   #1 SMP PREEMPT_
│ │ │ │ +00000d50: 4459 4e41 4d49 4320 4465 6269 616e 2020  DYNAMIC Debian  
│ │ │ │ +00000d60: 2020 2020 7c0a 7c2d 2d20 496e 7465 6c20      |.|-- Intel 
│ │ │ │ +00000d70: 5865 6f6e 2050 726f 6365 7373 6f72 2028  Xeon Processor (
│ │ │ │ +00000d80: 536b 796c 616b 652c 2049 4252 5329 2020  Skylake, IBRS)  
│ │ │ │ +00000d90: 4765 6e75 696e 6549 6e74 656c 2020 6370  GenuineIntel  cp
│ │ │ │ +00000da0: 7520 4d48 7a20 3230 3939 2e39 3938 2020  u MHz 2099.998  
│ │ │ │  00000db0: 2020 2020 7c0a 7c2d 2d20 4d61 6361 756c      |.|-- Macaul
│ │ │ │  00000dc0: 6179 3220 312e 3235 2e31 312c 2063 6f6d  ay2 1.25.11, com
│ │ │ │  00000dd0: 7069 6c65 6420 7769 7468 2067 6363 2031  piled with gcc 1
│ │ │ │  00000de0: 352e 322e 3020 2020 2020 2020 2020 2020  5.2.0           
│ │ │ │  00000df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000e00: 2020 2020 7c0a 7c2d 2d20 7265 7333 393a      |.|-- res39:
│ │ │ │  00000e10: 2072 6573 206f 6620 6120 6765 6e65 7269   res of a generi
│ │ │ │  00000e20: 6320 3320 6279 2039 206d 6174 7269 7820  c 3 by 9 matrix 
│ │ │ │ -00000e30: 6f76 6572 205a 5a2f 3130 313a 202e 3135  over ZZ/101: .15
│ │ │ │ -00000e40: 3332 3135 2073 6563 6f6e 6473 2020 2020  3215 seconds    
│ │ │ │ +00000e30: 6f76 6572 205a 5a2f 3130 313a 202e 3137  over ZZ/101: .17
│ │ │ │ +00000e40: 3532 3234 2073 6563 6f6e 6473 2020 2020  5224 seconds    
│ │ │ │  00000e50: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │  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 7c0a 7c28 3230 3235 2d31 312d  ----|.|(2025-11-
│ │ │ │ -00000eb0: 3035 2920 7838 365f 3634 2047 4e55 2f4c  05) x86_64 GNU/L
│ │ │ │ -00000ec0: 696e 7578 2020 2020 2020 2020 2020 2020  inux            
│ │ │ │ +00000ea0: 2d2d 2d2d 7c0a 7c36 2e31 322e 3537 2d31  ----|.|6.12.57-1
│ │ │ │ +00000eb0: 2028 3230 3235 2d31 312d 3035 2920 7838   (2025-11-05) x8
│ │ │ │ +00000ec0: 365f 3634 2047 4e55 2f4c 696e 7578 2020  6_64 GNU/Linux  
│ │ │ │  00000ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00000ef0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00000f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00000f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00000f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00000f40: 2d2d 2d2d 2b0a 0a46 6f72 2074 6865 2070  ----+..For the p
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│ │ │ │ -00000f60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ -00000f70: 6520 6f62 6a65 6374 202a 6e6f 7465 2072  e object *note r
│ │ │ │ -00000f80: 756e 4265 6e63 686d 6172 6b73 3a20 7275  unBenchmarks: ru
│ │ │ │ -00000f90: 6e42 656e 6368 6d61 726b 732c 2069 7320  nBenchmarks, is 
│ │ │ │ -00000fa0: 6120 2a6e 6f74 6520 636f 6d6d 616e 643a  a *note command:
│ │ │ │ -00000fb0: 0a28 4d61 6361 756c 6179 3244 6f63 2943  .(Macaulay2Doc)C
│ │ │ │ -00000fc0: 6f6d 6d61 6e64 2c2e 0a0a 2d2d 2d2d 2d2d  ommand,...------
│ │ │ │ -00000fd0: 2d2d 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 2d0a 0a54 6865 2073  ---------..The s
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│ │ │ │ -00001030: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ -00001040: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ -00001050: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ -00001060: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ -00001070: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ -00001080: 2f42 656e 6368 6d61 726b 2e0a 6d32 3a32  /Benchmark..m2:2
│ │ │ │ -00001090: 3937 3a30 2e0a 1f0a 5461 6720 5461 626c  97:0....Tag Tabl
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│ │ │ │ +00000f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00000f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00000f40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00000f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00000f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00000f90: 2d2d 2d2d 2b0a 0a46 6f72 2074 6865 2070  ----+..For the p
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│ │ │ │ +00000fb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +00000fc0: 6520 6f62 6a65 6374 202a 6e6f 7465 2072  e object *note r
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│ │ │ │ +00001090: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +000010a0: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +000010b0: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ +000010c0: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ +000010d0: 2f42 656e 6368 6d61 726b 2e0a 6d32 3a32  /Benchmark..m2:2
│ │ │ │ +000010e0: 3937 3a30 2e0a 1f0a 5461 6720 5461 626c  97:0....Tag Tabl
│ │ │ │ +000010f0: 653a 0a4e 6f64 653a 2054 6f70 7f32 3334  e:.Node: Top.234
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│ │ │ │ +00001110: 6172 6b73 7f32 3033 350a 1f0a 456e 6420  arks.2035...End 
│ │ │ │ +00001120: 5461 6720 5461 626c 650a                 Tag Table.
│ │ ├── ./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 3238 3730     "/tmp/M2-2870
│ │ │ │ -00008d40: 362d 302f 3022 2c20 4f70 7469 6f6e 2074  6-0/0", Option t
│ │ │ │ +00008d30: 2020 2022 2f74 6d70 2f4d 322d 3430 3931     "/tmp/M2-4091
│ │ │ │ +00008d40: 332d 302f 3022 2c20 4f70 7469 6f6e 2074  3-0/0", Option t
│ │ │ │  00008d50: 6f20 6368 616e 6765 2064 6972 6563 746f  o change directo
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│ │ │ │  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 3837 3036 2d30 2f30 222c  p/M2-28706-0/0",
│ │ │ │ +00013710: 702f 4d32 2d34 3039 3133 2d30 2f30 222c  p/M2-40913-0/0",
│ │ │ │  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 3837      "/tmp/M2-287
│ │ │ │ -00015670: 3036 2d30 2f30 222c 204f 7074 696f 6e20  06-0/0", Option 
│ │ │ │ +00015660: 2020 2020 222f 746d 702f 4d32 2d34 3039      "/tmp/M2-409
│ │ │ │ +00015670: 3133 2d30 2f30 222c 204f 7074 696f 6e20  13-0/0", 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
│ │ │ │ @@ -12972,15 +12972,15 @@
│ │ │ │  00032ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00032ad0: 0a7c 6939 203a 2065 6c61 7073 6564 5469  .|i9 : elapsedTi
│ │ │ │  00032ae0: 6d65 2063 203d 2063 6861 7261 6374 6572  me c = character
│ │ │ │  00032af0: 2041 2020 2020 2020 2020 2020 2020 2020   A              
│ │ │ │  00032b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00032b20: 0a7c 202d 2d20 2e35 3539 3133 3573 2065  .| -- .559135s e
│ │ │ │ +00032b20: 0a7c 202d 2d20 2e34 3337 3039 3473 2065  .| -- .437094s e
│ │ │ │  00032b30: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │  00032b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00032b70: 0a7c 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                  
│ │ │ │ @@ -14183,15 +14183,15 @@
│ │ │ │  00037660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037680: 2b0a 7c69 3720 3a20 656c 6170 7365 6454  +.|i7 : elapsedT
│ │ │ │  00037690: 696d 6520 633d 6368 6172 6163 7465 7220  ime c=character 
│ │ │ │  000376a0: 4120 2020 2020 2020 2020 2020 2020 2020  A               
│ │ │ │  000376b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000376c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000376d0: 7c0a 7c20 2d2d 202e 3437 3439 3136 7320  |.| -- .474916s 
│ │ │ │ +000376d0: 7c0a 7c20 2d2d 202e 3439 3836 3533 7320  |.| -- .498653s 
│ │ │ │  000376e0: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │  000376f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037720: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00037730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -15614,16 +15614,16 @@
│ │ │ │  0003cfd0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  0003cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003cff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d010: 2d2b 0a7c 6932 3020 3a20 656c 6170 7365  -+.|i20 : elapse
│ │ │ │  0003d020: 6454 696d 6520 6131 203d 2063 6861 7261  dTime a1 = chara
│ │ │ │  0003d030: 6374 6572 2041 3120 2020 2020 2020 2020  cter A1         
│ │ │ │ -0003d040: 2020 2020 2020 7c0a 7c20 2d2d 202e 3835        |.| -- .85
│ │ │ │ -0003d050: 3237 3632 7320 656c 6170 7365 6420 2020  2762s elapsed   
│ │ │ │ +0003d040: 2020 2020 2020 7c0a 7c20 2d2d 202e 3833        |.| -- .83
│ │ │ │ +0003d050: 3933 3135 7320 656c 6170 7365 6420 2020  9315s elapsed   
│ │ │ │  0003d060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d070: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003d080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d0b0: 7c0a 7c6f 3230 203d 2043 6861 7261 6374  |.|o20 = Charact
│ │ │ │  0003d0c0: 6572 206f 7665 7220 5220 2020 2020 2020  er over R       
│ │ │ │ @@ -15654,16 +15654,16 @@
│ │ │ │  0003d250: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │  0003d260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  0003d290: 6932 3120 3a20 656c 6170 7365 6454 696d  i21 : elapsedTim
│ │ │ │  0003d2a0: 6520 6132 203d 2063 6861 7261 6374 6572  e a2 = character
│ │ │ │  0003d2b0: 2041 3220 2020 2020 2020 2020 2020 2020   A2             
│ │ │ │ -0003d2c0: 2020 7c0a 7c20 2d2d 2033 342e 3334 3473    |.| -- 34.344s
│ │ │ │ -0003d2d0: 2065 6c61 7073 6564 2020 2020 2020 2020   elapsed        
│ │ │ │ +0003d2c0: 2020 7c0a 7c20 2d2d 2032 372e 3132 3838    |.| -- 27.1288
│ │ │ │ +0003d2d0: 7320 656c 6170 7365 6420 2020 2020 2020  s elapsed       
│ │ │ │  0003d2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d2f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0003d300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d320: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │  0003d330: 3231 203d 2043 6861 7261 6374 6572 206f  21 = Character o
│ │ │ │  0003d340: 7665 7220 5220 2020 2020 2020 2020 2020  ver R           
│ │ │ │ @@ -16112,16 +16112,16 @@
│ │ │ │  0003eef0: 6f33 3120 3a20 4163 7469 6f6e 4f6e 4772  o31 : ActionOnGr
│ │ │ │  0003ef00: 6164 6564 4d6f 6475 6c65 2020 2020 2020  adedModule      
│ │ │ │  0003ef10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0003ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  0003ef40: 6933 3220 3a20 656c 6170 7365 6454 696d  i32 : elapsedTim
│ │ │ │  0003ef50: 6520 6220 3d20 6368 6172 6163 7465 7228  e b = character(
│ │ │ │ -0003ef60: 422c 3231 297c 0a7c 202d 2d20 3134 2e32  B,21)|.| -- 14.2
│ │ │ │ -0003ef70: 3631 3773 2065 6c61 7073 6564 2020 2020  617s elapsed    
│ │ │ │ +0003ef60: 422c 3231 297c 0a7c 202d 2d20 3132 2e35  B,21)|.| -- 12.5
│ │ │ │ +0003ef70: 3239 7320 656c 6170 7365 6420 2020 2020  29s elapsed     
│ │ │ │  0003ef80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0003ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003efb0: 2020 2020 207c 0a7c 6f33 3220 3d20 4368       |.|o32 = Ch
│ │ │ │  0003efc0: 6172 6163 7465 7220 6f76 6572 2052 2020  aracter over R  
│ │ │ │  0003efd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0003efe0: 2020 2020 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 2e33 3534 3231   -- used 0.35421
│ │ │ │ -000044e0: 3873 2028 6370 7529 3b20 302e 3238 3930  8s (cpu); 0.2890
│ │ │ │ -000044f0: 3037 7320 2874 6872 6561 6429 3b20 3073  07s (thread); 0s
│ │ │ │ -00004500: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +000044d0: 202d 2d20 7573 6564 2030 2e33 3339 3235   -- used 0.33925
│ │ │ │ +000044e0: 7320 2863 7075 293b 2030 2e32 3532 3138  s (cpu); 0.25218
│ │ │ │ +000044f0: 3173 2028 7468 7265 6164 293b 2030 7320  1s (thread); 0s 
│ │ │ │ +00004500: 2867 6329 2020 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
│ │ │ │ @@ -2973,23 +2973,23 @@
│ │ │ │  0000b9c0: 2020 2020 2020 2020 207c 0a7c 6f38 203d           |.|o8 =
│ │ │ │  0000b9d0: 2048 6173 6854 6162 6c65 7b30 203d 3e20   HashTable{0 => 
│ │ │ │  0000b9e0: 7b78 202c 2078 2079 2c20 7820 7920 2c20  {x , x y, x y , 
│ │ │ │  0000b9f0: 7820 7920 2c20 782a 7920 2c20 7820 7d20  x y , x*y , x } 
│ │ │ │  0000ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ba10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0000ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000ba30: 2020 3520 2020 2033 2033 2020 2035 2032    5    3 3   5 2
│ │ │ │ -0000ba40: 2020 2032 2034 2020 2035 2033 2020 2035     2 4   5 3   5
│ │ │ │ -0000ba50: 2034 2020 2035 2020 2020 3520 3220 2020   4   5    5 2   
│ │ │ │ -0000ba60: 3520 3320 2020 2020 207c 0a7c 2020 2020  5 3      |.|    
│ │ │ │ +0000ba30: 2020 3520 3220 2020 3220 3420 2020 3520    5 2   2 4   5 
│ │ │ │ +0000ba40: 3320 2020 3520 3420 2020 3520 2020 2035  3   5 4   5    5
│ │ │ │ +0000ba50: 2032 2020 2035 2033 2020 2035 2034 2020   2   5 3   5 4  
│ │ │ │ +0000ba60: 2034 2032 2020 2020 207c 0a7c 2020 2020   4 2     |.|    
│ │ │ │  0000ba70: 2020 2020 2020 2020 2020 2031 203d 3e20             1 => 
│ │ │ │ -0000ba80: 7b78 2079 2c20 7820 7920 2c20 7820 7920  {x y, x y , x y 
│ │ │ │ -0000ba90: 2c20 7820 7920 2c20 7820 7920 2c20 7820  , x y , x y , x 
│ │ │ │ -0000baa0: 7920 2c20 7820 792c 2078 2079 202c 2078  y , x y, x y , x
│ │ │ │ -0000bab0: 2079 202c 2020 2020 207c 0a7c 2020 2020   y ,     |.|    
│ │ │ │ +0000ba80: 7b78 2079 202c 2078 2079 202c 2078 2079  {x y , x y , x y
│ │ │ │ +0000ba90: 202c 2078 2079 202c 2078 2079 2c20 7820   , x y , x y, x 
│ │ │ │ +0000baa0: 7920 2c20 7820 7920 2c20 7820 7920 2c20  y , x y , x y , 
│ │ │ │ +0000bab0: 7820 7920 2c20 2020 207c 0a7c 2020 2020  x y ,    |.|    
│ │ │ │  0000bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bad0: 2020 3520 3220 2020 3520 3420 2020 3520    5 2   5 4   5 
│ │ │ │  0000bae0: 3320 2020 3520 3420 2020 3520 3220 2020  3   5 4   5 2   
│ │ │ │  0000baf0: 3520 3420 2020 3520 3320 2020 3520 3420  5 4   5 3   5 4 
│ │ │ │  0000bb00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0000bb10: 2020 2020 2020 2020 2020 2032 203d 3e20             2 => 
│ │ │ │  0000bb20: 7b78 2079 202c 2078 2079 202c 2078 2079  {x y , x y , x y
│ │ │ │ @@ -3007,25 +3007,25 @@
│ │ │ │  0000bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bbf0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │  0000bc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bc40: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020  ---------|.|    
│ │ │ │ -0000bc50: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ +0000bc50: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │  0000bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000bc90: 2020 2020 2020 2020 207c 0a7c 2035 2034           |.| 5 4
│ │ │ │ -0000bca0: 2020 2034 2032 2020 2034 2034 2020 2020     4 2   4 4    
│ │ │ │ +0000bc90: 2020 2020 2020 2020 207c 0a7c 2034 2034           |.| 4 4
│ │ │ │ +0000bca0: 2020 2035 2020 2020 3320 3320 2020 2020     5    3 3     
│ │ │ │  0000bcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bce0: 2020 2020 2020 2020 207c 0a7c 7820 7920           |.|x y 
│ │ │ │ -0000bcf0: 2c20 7820 7920 2c20 7820 7920 7d20 2020  , x y , x y }   
│ │ │ │ +0000bcf0: 2c20 7820 792c 2078 2079 207d 2020 2020  , x y, x y }    
│ │ │ │  0000bd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd30: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0000bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -6321,22 +6321,22 @@
│ │ │ │  00018b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018b10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00018b20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00018b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018b60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00018b70: 2020 2020 3520 3320 2020 2035 2034 2020      5 3    5 4  
│ │ │ │ -00018b80: 2033 2035 2020 2020 3420 3520 2020 3220   3 5    4 5   2 
│ │ │ │ -00018b90: 2020 2020 2020 2032 2020 2020 3420 3420         2    4 4 
│ │ │ │ +00018b70: 2020 2020 3420 3420 2020 2035 2033 2020      4 4    5 3  
│ │ │ │ +00018b80: 2020 3520 3420 2020 3320 3520 2020 2034    5 4   3 5    4
│ │ │ │ +00018b90: 2035 2020 2032 2020 2020 2020 2020 3220   5   2        2 
│ │ │ │  00018ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018bb0: 2020 2020 207c 0a7c 6f35 203d 207b 7820       |.|o5 = {x 
│ │ │ │ -00018bc0: 7920 7a2c 2078 2079 202c 2078 2079 207a  y z, x y , x y z
│ │ │ │ -00018bd0: 2c20 7820 7920 2c20 7820 792a 7a2c 2078  , x y , x y*z, x
│ │ │ │ -00018be0: 2a79 207a 2c20 7820 7920 7a7d 2020 2020  *y z, x y z}    
│ │ │ │ +00018bc0: 7920 7a2c 2078 2079 207a 2c20 7820 7920  y z, x y z, x y 
│ │ │ │ +00018bd0: 2c20 7820 7920 7a2c 2078 2079 202c 2078  , x y z, x y , x
│ │ │ │ +00018be0: 2079 2a7a 2c20 782a 7920 7a7d 2020 2020   y*z, x*y z}    
│ │ │ │  00018bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018c00: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00018c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018c40: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │  00018c50: 203a 204c 6973 7420 2020 2020 2020 2020   : List
│ │ ├── ./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 3239 3737 3232 7320 2863 7075 293b  0.297722s (cpu);
│ │ │ │ -00012da0: 2030 2e32 3339 3538 3473 2028 7468 7265   0.239584s (thre
│ │ │ │ +00012d90: 302e 3338 3039 3332 7320 2863 7075 293b  0.380932s (cpu);
│ │ │ │ +00012da0: 2030 2e32 3932 3637 3573 2028 7468 7265   0.292675s (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               
│ │ │ │ @@ -6579,32 +6579,32 @@
│ │ │ │  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 3638 3333  -- used 0.096833
│ │ │ │ -00019ba0: 3673 2028 6370 7529 3b20 302e 3039 3638  6s (cpu); 0.0968
│ │ │ │ -00019bb0: 3332 3773 2028 7468 7265 6164 293b 2030  327s (thread); 0
│ │ │ │ -00019bc0: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +00019b90: 2d2d 2075 7365 6420 302e 3131 3037 3135  -- used 0.110715
│ │ │ │ +00019ba0: 7320 2863 7075 293b 2030 2e31 3130 3731  s (cpu); 0.11071
│ │ │ │ +00019bb0: 3673 2028 7468 7265 6164 293b 2030 7320  6s (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 3232 3630 3032  -- used 0.226002
│ │ │ │ -00019c90: 7320 2863 7075 293b 2030 2e31 3536 3739  s (cpu); 0.15679
│ │ │ │ -00019ca0: 3673 2028 7468 7265 6164 293b 2030 7320  6s (thread); 0s 
│ │ │ │ +00019c80: 2d2d 2075 7365 6420 302e 3238 3134 3533  -- used 0.281453
│ │ │ │ +00019c90: 7320 2863 7075 293b 2030 2e31 3932 3337  s (cpu); 0.19237
│ │ │ │ +00019ca0: 3973 2028 7468 7265 6164 293b 2030 7320  9s (thread); 0s 
│ │ │ │  00019cb0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  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
│ │ ├── ./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 3436 3137 3473 2028  used 0.246174s (
│ │ │ │ -00004c60: 6370 7529 3b20 302e 3136 3436 3234 7320  cpu); 0.164624s 
│ │ │ │ +00004c50: 7573 6564 2030 2e32 3838 3734 3573 2028  used 0.288745s (
│ │ │ │ +00004c60: 6370 7529 3b20 302e 3138 3430 3831 7320  cpu); 0.184081s 
│ │ │ │  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                  
│ │ │ │ @@ -1300,16 +1300,16 @@
│ │ │ │  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 3734 3331 3273 2028  used 0.374312s (
│ │ │ │ -000051b0: 6370 7529 3b20 302e 3239 3531 3235 7320  cpu); 0.295125s 
│ │ │ │ +000051a0: 7573 6564 2030 2e34 3735 3733 3673 2028  used 0.475736s (
│ │ │ │ +000051b0: 6370 7529 3b20 302e 3337 3538 3633 7320  cpu); 0.375863s 
│ │ │ │  000051c0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │  000051d0: 2920 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                  
│ │ │ │ @@ -4341,18 +4341,18 @@
│ │ │ │  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 3632 3933 3032  -- used 0.629302
│ │ │ │ -00010fc0: 7320 2863 7075 293b 2030 2e33 3031 3631  s (cpu); 0.30161
│ │ │ │ -00010fd0: 3373 2028 7468 7265 6164 293b 2030 7320  3s (thread); 0s 
│ │ │ │ -00010fe0: 2867 6329 2020 2020 2020 2020 2020 207c  (gc)           |
│ │ │ │ +00010fb0: 2d2d 2075 7365 6420 302e 3933 3136 3273  -- used 0.93162s
│ │ │ │ +00010fc0: 2028 6370 7529 3b20 302e 3339 3031 3535   (cpu); 0.390155
│ │ │ │ +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   
│ │ │ │  00011040: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │  00011050: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ @@ -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 3133 3132 3773 2028 6370 7529  d 2.13127s (cpu)
│ │ │ │ -00011370: 3b20 312e 3832 3435 3573 2028 7468 7265  ; 1.82455s (thre
│ │ │ │ +00011360: 6420 322e 3436 3831 3773 2028 6370 7529  d 2.46817s (cpu)
│ │ │ │ +00011370: 3b20 322e 3139 3030 3773 2028 7468 7265  ; 2.19007s (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 3739 3130 3673 2028 6370 7529   0.279106s (cpu)
│ │ │ │ -000118f0: 3b20 302e 3139 3532 3732 7320 2874 6872  ; 0.195272s (thr
│ │ │ │ +000118e0: 2030 2e33 3231 3339 3173 2028 6370 7529   0.321391s (cpu)
│ │ │ │ +000118f0: 3b20 302e 3233 3336 3537 7320 2874 6872  ; 0.233657s (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 3832  .| -- used 0.082
│ │ │ │ -00011c90: 3531 3433 7320 2863 7075 293b 2030 2e30  5143s (cpu); 0.0
│ │ │ │ -00011ca0: 3832 3532 3039 7320 2874 6872 6561 6429  825209s (thread)
│ │ │ │ -00011cb0: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +00011c80: 0a7c 202d 2d20 7573 6564 2030 2e31 3032  .| -- used 0.102
│ │ │ │ +00011c90: 3938 3373 2028 6370 7529 3b20 302e 3130  983s (cpu); 0.10
│ │ │ │ +00011ca0: 3238 3632 7320 2874 6872 6561 6429 3b20  2862s (thread); 
│ │ │ │ +00011cb0: 3073 2028 6763 2920 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                  
│ │ │ │ @@ -5446,6793 +5446,6790 @@
│ │ │ │  00015450: 2072 6574 7572 6e65 6420 696e 2074 6865   returned in the
│ │ │ │  00015460: 2073 616d 6520 7269 6e67 2e20 5765 206d   same ring. We m
│ │ │ │  00015470: 6179 2061 6c73 6f20 7265 7475 726e 2061  ay also return a
│ │ │ │  00015480: 0a4d 7574 6162 6c65 4861 7368 5461 626c  .MutableHashTabl
│ │ │ │  00015490: 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  e...+-----------
│ │ │ │  000154a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000154b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000154c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131  ----------+.|i11
│ │ │ │ -000154d0: 203a 2052 3d4d 756c 7469 5072 6f6a 436f   : R=MultiProjCo
│ │ │ │ -000154e0: 6f72 6452 696e 6728 7b32 2c32 7d29 2020  ordRing({2,2})  
│ │ │ │ +000154c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120  ---------+.|i11 
│ │ │ │ +000154d0: 3a20 523d 4d75 6c74 6950 726f 6a43 6f6f  : R=MultiProjCoo
│ │ │ │ +000154e0: 7264 5269 6e67 287b 322c 327d 2920 2020  rdRing({2,2})   
│ │ │ │  000154f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015500: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00015500: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00015510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015530: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ -00015540: 203d 2052 2020 2020 2020 2020 2020 2020   = R            
│ │ │ │ +00015530: 2020 2020 2020 207c 0a7c 6f31 3120 3d20         |.|o11 = 
│ │ │ │ +00015540: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  00015550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015570: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00015560: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015570: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00015580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000155a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ -000155b0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ -000155c0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -000155d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000155e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000155a0: 2020 2020 207c 0a7c 6f31 3120 3a20 506f       |.|o11 : Po
│ │ │ │ +000155b0: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020  lynomialRing    
│ │ │ │ +000155c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000155d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000155e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000155f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015610: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132  ----------+.|i12
│ │ │ │ -00015620: 203a 2041 3d43 686f 7752 696e 6728 5229   : A=ChowRing(R)
│ │ │ │ +00015610: 2d2d 2d2b 0a7c 6931 3220 3a20 413d 4368  ---+.|i12 : A=Ch
│ │ │ │ +00015620: 6f77 5269 6e67 2852 2920 2020 2020 2020  owRing(R)       
│ │ │ │  00015630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015650: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00015640: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00015650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015680: 2020 2020 2020 2020 2020 7c0a 7c6f 3132            |.|o12
│ │ │ │ -00015690: 203d 2041 2020 2020 2020 2020 2020 2020   = A            
│ │ │ │ +00015680: 207c 0a7c 6f31 3220 3d20 4120 2020 2020   |.|o12 = A     
│ │ │ │ +00015690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000156a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000156b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000156c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000156b0: 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020                  
│ │ │ │ -000156f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3132            |.|o12
│ │ │ │ -00015700: 203a 2051 756f 7469 656e 7452 696e 6720   : QuotientRing 
│ │ │ │ +000156e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000156f0: 0a7c 6f31 3220 3a20 5175 6f74 6965 6e74  .|o12 : Quotient
│ │ │ │ +00015700: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │  00015710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015730: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00015720: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00015730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015760: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133  ----------+.|i13
│ │ │ │ -00015770: 203a 2072 3d67 656e 7320 5220 2020 2020   : r=gens R     
│ │ │ │ +00015750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00015760: 6931 3320 3a20 723d 6765 6e73 2052 2020  i13 : r=gens R  
│ │ │ │ +00015770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00015790: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000157a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000157b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3133            |.|o13
│ │ │ │ -000157e0: 203d 207b 7820 2c20 7820 2c20 7820 2c20   = {x , x , x , 
│ │ │ │ -000157f0: 7820 2c20 7820 2c20 7820 7d20 2020 2020  x , x , x }     
│ │ │ │ -00015800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015810: 2020 7c0a 7c20 2020 2020 2020 2030 2020    |.|        0  
│ │ │ │ -00015820: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -00015830: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ -00015840: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000157c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +000157d0: 3320 3d20 7b78 202c 2078 202c 2078 202c  3 = {x , x , x ,
│ │ │ │ +000157e0: 2078 202c 2078 202c 2078 207d 2020 2020   x , x , x }    
│ │ │ │ +000157f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015800: 2020 7c0a 7c20 2020 2020 2020 2030 2020    |.|        0  
│ │ │ │ +00015810: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ +00015820: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ +00015830: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015880: 2020 7c0a 7c6f 3133 203a 204c 6973 7420    |.|o13 : List 
│ │ │ │ +00015870: 7c0a 7c6f 3133 203a 204c 6973 7420 2020  |.|o13 : List   
│ │ │ │ +00015880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000158a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000158b0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000158a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000158b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000158c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000158d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000158e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000158f0: 2d2d 2b0a 7c69 3134 203a 204b 3d69 6465  --+.|i14 : K=ide
│ │ │ │ -00015900: 616c 2872 5f30 5e32 2a72 5f33 2d72 5f34  al(r_0^2*r_3-r_4
│ │ │ │ -00015910: 2a72 5f31 2a72 5f32 2c72 5f32 5e32 2a72  *r_1*r_2,r_2^2*r
│ │ │ │ -00015920: 5f35 2920 2020 2020 2020 7c0a 7c20 2020  _5)       |.|   
│ │ │ │ +000158d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000158e0: 7c69 3134 203a 204b 3d69 6465 616c 2872  |i14 : K=ideal(r
│ │ │ │ +000158f0: 5f30 5e32 2a72 5f33 2d72 5f34 2a72 5f31  _0^2*r_3-r_4*r_1
│ │ │ │ +00015900: 2a72 5f32 2c72 5f32 5e32 2a72 5f35 2920  *r_2,r_2^2*r_5) 
│ │ │ │ +00015910: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00015920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015960: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015970: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00015980: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00015990: 2020 2020 2020 2020 2020 7c0a 7c6f 3134            |.|o14
│ │ │ │ -000159a0: 203d 2069 6465 616c 2028 7820 7820 202d   = ideal (x x  -
│ │ │ │ -000159b0: 2078 2078 2078 202c 2078 2078 2029 2020   x x x , x x )  
│ │ │ │ -000159c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000159d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -000159e0: 2020 2030 2033 2020 2020 3120 3220 3420     0 3    1 2 4 
│ │ │ │ -000159f0: 2020 3220 3520 2020 2020 2020 2020 2020    2 5           
│ │ │ │ -00015a00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00015940: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00015950: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00015960: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00015970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015980: 2020 207c 0a7c 6f31 3420 3d20 6964 6561     |.|o14 = idea
│ │ │ │ +00015990: 6c20 2878 2078 2020 2d20 7820 7820 7820  l (x x  - x x x 
│ │ │ │ +000159a0: 2c20 7820 7820 2920 2020 2020 2020 2020  , x x )         
│ │ │ │ +000159b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000159c0: 2020 2020 2020 2020 2020 2030 2033 2020             0 3  
│ │ │ │ +000159d0: 2020 3120 3220 3420 2020 3220 3520 2020    1 2 4   2 5   
│ │ │ │ +000159e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000159f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00015a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a40: 2020 7c0a 7c6f 3134 203a 2049 6465 616c    |.|o14 : Ideal
│ │ │ │ -00015a50: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ -00015a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a70: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00015a20: 2020 2020 2020 2020 7c0a 7c6f 3134 203a          |.|o14 :
│ │ │ │ +00015a30: 2049 6465 616c 206f 6620 5220 2020 2020   Ideal of R     
│ │ │ │ +00015a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015a50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00015a60: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00015a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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 3938 3435 3736 7320   used 0.984576s 
│ │ │ │ -00015b00: 2863 7075 293b 2030 2e34 3633 3332 3673  (cpu); 0.463326s
│ │ │ │ -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   
│ │ │ │ -00015b80: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -00015b90: 2020 7c0a 7c6f 3135 203d 2037 6820 6820    |.|o15 = 7h h 
│ │ │ │ -00015ba0: 202b 2035 6820 6820 202b 2034 6820 6820   + 5h h  + 4h h 
│ │ │ │ -00015bb0: 202b 2068 2020 2b20 3368 2068 2020 2b20   + h  + 3h h  + 
│ │ │ │ -00015bc0: 6820 2020 2020 2020 2020 7c0a 7c20 2020  h         |.|   
│ │ │ │ -00015bd0: 2020 2020 2031 2032 2020 2020 2031 2032       1 2     1 2
│ │ │ │ -00015be0: 2020 2020 2031 2032 2020 2020 3120 2020       1 2    1   
│ │ │ │ -00015bf0: 2020 3120 3220 2020 2032 2020 2020 2020    1 2    2      
│ │ │ │ -00015c00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c30: 2020 2020 2020 2020 2020 7c0a 7c6f 3135            |.|o15
│ │ │ │ -00015c40: 203a 2041 2020 2020 2020 2020 2020 2020   : A            
│ │ │ │ -00015c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c70: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00015c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3136  ----------+.|i16
│ │ │ │ -00015cb0: 203a 2063 736d 4b48 6173 683d 2043 534d   : csmKHash= CSM
│ │ │ │ -00015cc0: 2841 2c4b 2c4f 7574 7075 743d 3e48 6173  (A,K,Output=>Has
│ │ │ │ -00015cd0: 6846 6f72 6d29 2020 2020 2020 2020 2020  hForm)          
│ │ │ │ -00015ce0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d10: 2020 2020 2020 2020 2020 7c0a 7c6f 3136            |.|o16
│ │ │ │ -00015d20: 203d 204d 7574 6162 6c65 4861 7368 5461   = MutableHashTa
│ │ │ │ -00015d30: 626c 657b 2e2e 2e34 2e2e 2e7d 2020 2020  ble{...4...}    
│ │ │ │ +00015a90: 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a 2074  ------+.|i15 : t
│ │ │ │ +00015aa0: 696d 6520 6373 6d4b 3d43 534d 2841 2c4b  ime csmK=CSM(A,K
│ │ │ │ +00015ab0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00015ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00015ad0: 202d 2d20 7573 6564 2031 2e34 3239 3235   -- used 1.42925
│ │ │ │ +00015ae0: 7320 2863 7075 293b 2030 2e35 3130 3432  s (cpu); 0.51042
│ │ │ │ +00015af0: 3773 2028 7468 7265 6164 293b 2030 7320  7s (thread); 0s 
│ │ │ │ +00015b00: 2867 6329 7c0a 7c20 2020 2020 2020 2020  (gc)|.|         
│ │ │ │ +00015b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015b30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00015b40: 2020 2020 2020 3220 3220 2020 2020 3220        2 2     2 
│ │ │ │ +00015b50: 2020 2020 2020 2020 3220 2020 2032 2020          2    2  
│ │ │ │ +00015b60: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00015b70: 2020 7c0a 7c6f 3135 203d 2037 6820 6820    |.|o15 = 7h h 
│ │ │ │ +00015b80: 202b 2035 6820 6820 202b 2034 6820 6820   + 5h h  + 4h h 
│ │ │ │ +00015b90: 202b 2068 2020 2b20 3368 2068 2020 2b20   + h  + 3h h  + 
│ │ │ │ +00015ba0: 6820 2020 2020 2020 207c 0a7c 2020 2020  h        |.|    
│ │ │ │ +00015bb0: 2020 2020 3120 3220 2020 2020 3120 3220      1 2     1 2 
│ │ │ │ +00015bc0: 2020 2020 3120 3220 2020 2031 2020 2020      1 2    1    
│ │ │ │ +00015bd0: 2031 2032 2020 2020 3220 2020 2020 2020   1 2    2       
│ │ │ │ +00015be0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00015bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c10: 2020 2020 2020 207c 0a7c 6f31 3520 3a20         |.|o15 : 
│ │ │ │ +00015c20: 4120 2020 2020 2020 2020 2020 2020 2020  A               
│ │ │ │ +00015c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015c50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00015c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015c80: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 6373  -----+.|i16 : cs
│ │ │ │ +00015c90: 6d4b 4861 7368 3d20 4353 4d28 412c 4b2c  mKHash= CSM(A,K,
│ │ │ │ +00015ca0: 4f75 7470 7574 3d3e 4861 7368 466f 726d  Output=>HashForm
│ │ │ │ +00015cb0: 2920 2020 2020 2020 2020 2020 7c0a 7c20  )           |.| 
│ │ │ │ +00015cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015cf0: 2020 207c 0a7c 6f31 3620 3d20 4d75 7461     |.|o16 = Muta
│ │ │ │ +00015d00: 626c 6548 6173 6854 6162 6c65 7b2e 2e2e  bleHashTable{...
│ │ │ │ +00015d10: 342e 2e2e 7d20 2020 2020 2020 2020 2020  4...}           
│ │ │ │ +00015d20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00015d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d80: 2020 2020 2020 2020 2020 7c0a 7c6f 3136            |.|o16
│ │ │ │ -00015d90: 203a 204d 7574 6162 6c65 4861 7368 5461   : MutableHashTa
│ │ │ │ -00015da0: 626c 6520 2020 2020 2020 2020 2020 2020  ble             
│ │ │ │ -00015db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015dc0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00015dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137  ----------+.|i17
│ │ │ │ -00015e00: 203a 2063 736d 4b3d 3d63 736d 4b48 6173   : csmK==csmKHas
│ │ │ │ -00015e10: 6823 2243 534d 2220 2020 2020 2020 2020  h#"CSM"         
│ │ │ │ +00015d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015d60: 207c 0a7c 6f31 3620 3a20 4d75 7461 626c   |.|o16 : Mutabl
│ │ │ │ +00015d70: 6548 6173 6854 6162 6c65 2020 2020 2020  eHashTable      
│ │ │ │ +00015d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015d90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00015da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00015dd0: 0a7c 6931 3720 3a20 6373 6d4b 3d3d 6373  .|i17 : csmK==cs
│ │ │ │ +00015de0: 6d4b 4861 7368 2322 4353 4d22 2020 2020  mKHash#"CSM"    
│ │ │ │ +00015df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015e00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00015e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e30: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015e30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00015e40: 6f31 3720 3d20 7472 7565 2020 2020 2020  o17 = true      
│ │ │ │  00015e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e60: 2020 2020 2020 2020 2020 7c0a 7c6f 3137            |.|o17
│ │ │ │ -00015e70: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ -00015e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ea0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00015eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138  ----------+.|i18
│ │ │ │ -00015ee0: 203a 2043 534d 2841 2c69 6465 616c 284b   : CSM(A,ideal(K
│ │ │ │ -00015ef0: 5f30 2929 3d3d 6373 6d4b 4861 7368 237b  _0))==csmKHash#{
│ │ │ │ -00015f00: 307d 2020 2020 2020 2020 2020 2020 2020  0}              
│ │ │ │ -00015f10: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015e70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00015e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00015eb0: 3820 3a20 4353 4d28 412c 6964 6561 6c28  8 : CSM(A,ideal(
│ │ │ │ +00015ec0: 4b5f 3029 293d 3d63 736d 4b48 6173 6823  K_0))==csmKHash#
│ │ │ │ +00015ed0: 7b30 7d20 2020 2020 2020 2020 2020 2020  {0}             
│ │ │ │ +00015ee0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00015ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015f10: 2020 2020 2020 2020 207c 0a7c 6f31 3820           |.|o18 
│ │ │ │ +00015f20: 3d20 7472 7565 2020 2020 2020 2020 2020  = true          
│ │ │ │  00015f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015f40: 2020 2020 2020 2020 2020 7c0a 7c6f 3138            |.|o18
│ │ │ │ -00015f50: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ -00015f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015f80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00015f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 7570  ----------+..Sup
│ │ │ │ -00015fc0: 706f 7365 2077 6520 6861 7665 2061 6c72  pose we have alr
│ │ │ │ -00015fd0: 6561 6479 2063 6f6d 7075 7465 6420 736f  eady computed so
│ │ │ │ -00015fe0: 6d65 206f 6620 4353 4d20 636c 6173 7365  me of CSM classe
│ │ │ │ -00015ff0: 7320 6f66 2068 7970 6572 7375 7266 6163  s of hypersurfac
│ │ │ │ -00016000: 6573 2069 6e76 6f6c 7665 640a 696e 2074  es involved.in t
│ │ │ │ -00016010: 6865 2069 6e63 6c75 7369 6f6e 2d65 7863  he inclusion-exc
│ │ │ │ -00016020: 6c75 7369 6f6e 2070 726f 6365 6475 7265  lusion procedure
│ │ │ │ -00016030: 2c20 7468 656e 2077 6520 6d61 7920 696e  , then we may in
│ │ │ │ -00016040: 7075 7420 7468 6573 6520 746f 2062 6520  put these to be 
│ │ │ │ -00016050: 7573 6564 2062 7920 7468 650a 4353 4d20  used by the.CSM 
│ │ │ │ -00016060: 6675 6e63 7469 6f6e 2e20 496e 2074 6865  function. In the
│ │ │ │ -00016070: 2065 7861 6d70 6c65 2062 656c 6f77 2077   example below w
│ │ │ │ -00016080: 6520 696e 7075 7420 7468 6520 4353 4d20  e input the CSM 
│ │ │ │ -00016090: 636c 6173 7320 6f66 2056 284b 5f30 2920  class of V(K_0) 
│ │ │ │ -000160a0: 2874 6861 7420 6973 206f 660a 7468 6520  (that is of.the 
│ │ │ │ -000160b0: 6879 7065 7273 7572 6661 6365 2064 6566  hypersurface def
│ │ │ │ -000160c0: 696e 6564 2062 7920 7468 6520 6669 7273  ined by the firs
│ │ │ │ -000160d0: 7420 706f 6c79 6e6f 6d69 616c 2067 656e  t polynomial gen
│ │ │ │ -000160e0: 6572 6174 696e 6720 4b29 2061 6e64 2074  erating K) and t
│ │ │ │ -000160f0: 6865 2043 534d 0a63 6c61 7373 206f 6620  he CSM.class of 
│ │ │ │ -00016100: 7468 6520 6879 7065 7273 7572 6661 6365  the hypersurface
│ │ │ │ -00016110: 2064 6566 696e 6564 2062 7920 7468 6520   defined by the 
│ │ │ │ -00016120: 7072 6f64 7563 7420 6f66 2074 6865 2067  product of the g
│ │ │ │ -00016130: 656e 6572 6174 6f72 7320 6f66 204b 2e0a  enerators of K..
│ │ │ │ -00016140: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00016150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00016180: 6931 3920 3a20 6d3d 6e65 7720 4d75 7461  i19 : m=new Muta
│ │ │ │ -00016190: 626c 6548 6173 6854 6162 6c65 3b20 2020  bleHashTable;   
│ │ │ │ -000161a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000161b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000161c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000161d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000161e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000161f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020  ---------+.|i20 
│ │ │ │ -00016200: 3a20 6d23 7b30 7d3d 6373 6d4b 4861 7368  : m#{0}=csmKHash
│ │ │ │ -00016210: 237b 307d 2020 2020 2020 2020 2020 2020  #{0}            
│ │ │ │ +00015f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015f50: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00015f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015f80: 2d2d 2d2d 2d2d 2d2b 0a0a 5375 7070 6f73  -------+..Suppos
│ │ │ │ +00015f90: 6520 7765 2068 6176 6520 616c 7265 6164  e we have alread
│ │ │ │ +00015fa0: 7920 636f 6d70 7574 6564 2073 6f6d 6520  y computed some 
│ │ │ │ +00015fb0: 6f66 2043 534d 2063 6c61 7373 6573 206f  of CSM classes o
│ │ │ │ +00015fc0: 6620 6879 7065 7273 7572 6661 6365 7320  f hypersurfaces 
│ │ │ │ +00015fd0: 696e 766f 6c76 6564 0a69 6e20 7468 6520  involved.in the 
│ │ │ │ +00015fe0: 696e 636c 7573 696f 6e2d 6578 636c 7573  inclusion-exclus
│ │ │ │ +00015ff0: 696f 6e20 7072 6f63 6564 7572 652c 2074  ion procedure, t
│ │ │ │ +00016000: 6865 6e20 7765 206d 6179 2069 6e70 7574  hen we may input
│ │ │ │ +00016010: 2074 6865 7365 2074 6f20 6265 2075 7365   these to be use
│ │ │ │ +00016020: 6420 6279 2074 6865 0a43 534d 2066 756e  d by the.CSM fun
│ │ │ │ +00016030: 6374 696f 6e2e 2049 6e20 7468 6520 6578  ction. In the ex
│ │ │ │ +00016040: 616d 706c 6520 6265 6c6f 7720 7765 2069  ample below we i
│ │ │ │ +00016050: 6e70 7574 2074 6865 2043 534d 2063 6c61  nput the CSM cla
│ │ │ │ +00016060: 7373 206f 6620 5628 4b5f 3029 2028 7468  ss of V(K_0) (th
│ │ │ │ +00016070: 6174 2069 7320 6f66 0a74 6865 2068 7970  at is of.the hyp
│ │ │ │ +00016080: 6572 7375 7266 6163 6520 6465 6669 6e65  ersurface define
│ │ │ │ +00016090: 6420 6279 2074 6865 2066 6972 7374 2070  d by the first p
│ │ │ │ +000160a0: 6f6c 796e 6f6d 6961 6c20 6765 6e65 7261  olynomial genera
│ │ │ │ +000160b0: 7469 6e67 204b 2920 616e 6420 7468 6520  ting K) and the 
│ │ │ │ +000160c0: 4353 4d0a 636c 6173 7320 6f66 2074 6865  CSM.class of the
│ │ │ │ +000160d0: 2068 7970 6572 7375 7266 6163 6520 6465   hypersurface de
│ │ │ │ +000160e0: 6669 6e65 6420 6279 2074 6865 2070 726f  fined by the pro
│ │ │ │ +000160f0: 6475 6374 206f 6620 7468 6520 6765 6e65  duct of the gene
│ │ │ │ +00016100: 7261 746f 7273 206f 6620 4b2e 0a0a 2b2d  rators of K...+-
│ │ │ │ +00016110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016140: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3139  ----------+.|i19
│ │ │ │ +00016150: 203a 206d 3d6e 6577 204d 7574 6162 6c65   : m=new Mutable
│ │ │ │ +00016160: 4861 7368 5461 626c 653b 2020 2020 2020  HashTable;      
│ │ │ │ +00016170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016180: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00016190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000161a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000161b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000161c0: 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a 206d  ------+.|i20 : m
│ │ │ │ +000161d0: 237b 307d 3d63 736d 4b48 6173 6823 7b30  #{0}=csmKHash#{0
│ │ │ │ +000161e0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +000161f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016200: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016230: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00016240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016270: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00016280: 3220 3220 2020 2020 3220 2020 2020 2020  2 2     2       
│ │ │ │ -00016290: 2020 3220 2020 2020 3220 2020 2020 2020    2     2       
│ │ │ │ -000162a0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -000162b0: 2020 207c 0a7c 6f32 3020 3d20 3868 2068     |.|o20 = 8h h
│ │ │ │ -000162c0: 2020 2b20 3768 2068 2020 2b20 3668 2068    + 7h h  + 6h h
│ │ │ │ -000162d0: 2020 2b20 3268 2020 2b20 3568 2068 2020    + 2h  + 5h h  
│ │ │ │ -000162e0: 2b20 3268 2020 2b20 3268 2020 2b20 6820  + 2h  + 2h  + h 
│ │ │ │ -000162f0: 207c 0a7c 2020 2020 2020 2020 3120 3220   |.|        1 2 
│ │ │ │ -00016300: 2020 2020 3120 3220 2020 2020 3120 3220      1 2     1 2 
│ │ │ │ -00016310: 2020 2020 3120 2020 2020 3120 3220 2020      1     1 2   
│ │ │ │ -00016320: 2020 3220 2020 2020 3120 2020 2032 207c    2     1    2 |
│ │ │ │ -00016330: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016240: 2020 7c0a 7c20 2020 2020 2020 2032 2032    |.|        2 2
│ │ │ │ +00016250: 2020 2020 2032 2020 2020 2020 2020 2032       2         2
│ │ │ │ +00016260: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00016270: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00016280: 7c0a 7c6f 3230 203d 2038 6820 6820 202b  |.|o20 = 8h h  +
│ │ │ │ +00016290: 2037 6820 6820 202b 2036 6820 6820 202b   7h h  + 6h h  +
│ │ │ │ +000162a0: 2032 6820 202b 2035 6820 6820 202b 2032   2h  + 5h h  + 2
│ │ │ │ +000162b0: 6820 202b 2032 6820 202b 2068 2020 7c0a  h  + 2h  + h  |.
│ │ │ │ +000162c0: 7c20 2020 2020 2020 2031 2032 2020 2020  |        1 2    
│ │ │ │ +000162d0: 2031 2032 2020 2020 2031 2032 2020 2020   1 2     1 2    
│ │ │ │ +000162e0: 2031 2020 2020 2031 2032 2020 2020 2032   1     1 2     2
│ │ │ │ +000162f0: 2020 2020 2031 2020 2020 3220 7c0a 7c20       1    2 |.| 
│ │ │ │ +00016300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016330: 2020 2020 2020 2020 2020 7c0a 7c6f 3230            |.|o20
│ │ │ │ +00016340: 203a 2041 2020 2020 2020 2020 2020 2020   : A            
│ │ │ │  00016350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016360: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00016370: 6f32 3020 3a20 4120 2020 2020 2020 2020  o20 : A         
│ │ │ │ -00016380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000163a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000163b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000163c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000163d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000163e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120  ---------+.|i21 
│ │ │ │ -000163f0: 3a20 6d23 7b30 2c31 7d3d 6373 6d4b 4861  : m#{0,1}=csmKHa
│ │ │ │ -00016400: 7368 237b 302c 317d 2020 2020 2020 2020  sh#{0,1}        
│ │ │ │ +00016360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016370: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00016380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000163a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000163b0: 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a 206d  ------+.|i21 : m
│ │ │ │ +000163c0: 237b 302c 317d 3d63 736d 4b48 6173 6823  #{0,1}=csmKHash#
│ │ │ │ +000163d0: 7b30 2c31 7d20 2020 2020 2020 2020 2020  {0,1}           
│ │ │ │ +000163e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000163f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016420: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -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                  
│ │ │ │ -00016460: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00016470: 3220 3220 2020 2020 3220 2020 2020 2020  2 2     2       
│ │ │ │ -00016480: 2020 3220 2020 2020 3220 2020 2020 2020    2     2       
│ │ │ │ -00016490: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -000164a0: 2020 207c 0a7c 6f32 3120 3d20 3968 2068     |.|o21 = 9h h
│ │ │ │ -000164b0: 2020 2b20 3968 2068 2020 2b20 3968 2068    + 9h h  + 9h h
│ │ │ │ -000164c0: 2020 2b20 3368 2020 2b20 3768 2068 2020    + 3h  + 7h h  
│ │ │ │ -000164d0: 2b20 3368 2020 2b20 3368 2020 2b20 3268  + 3h  + 3h  + 2h
│ │ │ │ -000164e0: 207c 0a7c 2020 2020 2020 2020 3120 3220   |.|        1 2 
│ │ │ │ -000164f0: 2020 2020 3120 3220 2020 2020 3120 3220      1 2     1 2 
│ │ │ │ -00016500: 2020 2020 3120 2020 2020 3120 3220 2020      1     1 2   
│ │ │ │ -00016510: 2020 3220 2020 2020 3120 2020 2020 327c    2     1     2|
│ │ │ │ -00016520: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016430: 2020 7c0a 7c20 2020 2020 2020 2032 2032    |.|        2 2
│ │ │ │ +00016440: 2020 2020 2032 2020 2020 2020 2020 2032       2         2
│ │ │ │ +00016450: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00016460: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00016470: 7c0a 7c6f 3231 203d 2039 6820 6820 202b  |.|o21 = 9h h  +
│ │ │ │ +00016480: 2039 6820 6820 202b 2039 6820 6820 202b   9h h  + 9h h  +
│ │ │ │ +00016490: 2033 6820 202b 2037 6820 6820 202b 2033   3h  + 7h h  + 3
│ │ │ │ +000164a0: 6820 202b 2033 6820 202b 2032 6820 7c0a  h  + 3h  + 2h |.
│ │ │ │ +000164b0: 7c20 2020 2020 2020 2031 2032 2020 2020  |        1 2    
│ │ │ │ +000164c0: 2031 2032 2020 2020 2031 2032 2020 2020   1 2     1 2    
│ │ │ │ +000164d0: 2031 2020 2020 2031 2032 2020 2020 2032   1     1 2     2
│ │ │ │ +000164e0: 2020 2020 2031 2020 2020 2032 7c0a 7c20       1     2|.| 
│ │ │ │ +000164f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016520: 2020 2020 2020 2020 2020 7c0a 7c6f 3231            |.|o21
│ │ │ │ +00016530: 203a 2041 2020 2020 2020 2020 2020 2020   : A            
│ │ │ │  00016540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016550: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00016560: 6f32 3120 3a20 4120 2020 2020 2020 2020  o21 : A         
│ │ │ │ -00016570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016590: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000165a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 2e31 3131 3430 3373 2028 6370  ed 0.111403s (cp
│ │ │ │ -00016630: 7529 3b20 302e 3035 3830 3932 3573 2028  u); 0.0580925s (
│ │ │ │ -00016640: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ -00016650: 2020 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         
│ │ │ │ -000166b0: 3220 2020 2032 2020 2020 2020 2020 2020  2    2          
│ │ │ │ -000166c0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -000166d0: 207c 0a7c 6f32 3220 3d20 3768 2068 2020   |.|o22 = 7h h  
│ │ │ │ -000166e0: 2b20 3568 2068 2020 2b20 3468 2068 2020  + 5h h  + 4h h  
│ │ │ │ -000166f0: 2b20 6820 202b 2033 6820 6820 202b 2068  + h  + 3h h  + h
│ │ │ │ -00016700: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016710: 0a7c 2020 2020 2020 2020 3120 3220 2020  .|        1 2   
│ │ │ │ -00016720: 2020 3120 3220 2020 2020 3120 3220 2020    1 2     1 2   
│ │ │ │ -00016730: 2031 2020 2020 2031 2032 2020 2020 3220   1     1 2    2 
│ │ │ │ -00016740: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00016750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016560: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00016570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000165a0: 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a 2074  ------+.|i22 : t
│ │ │ │ +000165b0: 696d 6520 4353 4d28 412c 4b2c 6d29 2020  ime CSM(A,K,m)  
│ │ │ │ +000165c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000165d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000165e0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +000165f0: 302e 3130 3537 3737 7320 2863 7075 293b  0.105777s (cpu);
│ │ │ │ +00016600: 2030 2e30 3834 3136 3439 7320 2874 6872   0.0841649s (thr
│ │ │ │ +00016610: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ +00016620: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00016630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016660: 7c0a 7c20 2020 2020 2020 2032 2032 2020  |.|        2 2  
│ │ │ │ +00016670: 2020 2032 2020 2020 2020 2020 2032 2020     2         2  
│ │ │ │ +00016680: 2020 3220 2020 2020 2020 2020 2020 2032    2            2
│ │ │ │ +00016690: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000166a0: 7c6f 3232 203d 2037 6820 6820 202b 2035  |o22 = 7h h  + 5
│ │ │ │ +000166b0: 6820 6820 202b 2034 6820 6820 202b 2068  h h  + 4h h  + h
│ │ │ │ +000166c0: 2020 2b20 3368 2068 2020 2b20 6820 2020    + 3h h  + h   
│ │ │ │ +000166d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000166e0: 2020 2020 2020 2031 2032 2020 2020 2031         1 2     1
│ │ │ │ +000166f0: 2032 2020 2020 2031 2032 2020 2020 3120   2     1 2    1 
│ │ │ │ +00016700: 2020 2020 3120 3220 2020 2032 2020 2020      1 2    2    
│ │ │ │ +00016710: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00016720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016750: 2020 2020 2020 2020 7c0a 7c6f 3232 203a          |.|o22 :
│ │ │ │ +00016760: 2041 2020 2020 2020 2020 2020 2020 2020   A              
│ │ │ │  00016770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016780: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00016790: 3220 3a20 4120 2020 2020 2020 2020 2020  2 : A           
│ │ │ │ -000167a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000167b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000167c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -000167d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000167e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000167f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016800: 2d2d 2d2d 2d2d 2d2b 0a0a 496e 2074 6865  -------+..In the
│ │ │ │ -00016810: 2063 6173 6520 7768 6572 6520 7468 6520   case where the 
│ │ │ │ -00016820: 616d 6269 656e 7420 7370 6163 6520 6973  ambient space is
│ │ │ │ -00016830: 2061 2074 6f72 6963 2076 6172 6965 7479   a toric variety
│ │ │ │ -00016840: 2077 6869 6368 2069 7320 6e6f 7420 6120   which is not a 
│ │ │ │ -00016850: 7072 6f64 7563 740a 6f66 2070 726f 6a65  product.of proje
│ │ │ │ -00016860: 6374 6976 6520 7370 6163 6573 2077 6520  ctive spaces we 
│ │ │ │ -00016870: 6d75 7374 206c 6f61 6420 7468 6520 4e6f  must load the No
│ │ │ │ -00016880: 726d 616c 546f 7269 6356 6172 6965 7469  rmalToricVarieti
│ │ │ │ -00016890: 6573 2070 6163 6b61 6765 2061 6e64 206d  es package and m
│ │ │ │ -000168a0: 7573 740a 616c 736f 2069 6e70 7574 2074  ust.also input t
│ │ │ │ -000168b0: 6865 2074 6f72 6963 2076 6172 6965 7479  he toric variety
│ │ │ │ -000168c0: 2e20 4966 2074 6865 2074 6f72 6963 2076  . If the toric v
│ │ │ │ -000168d0: 6172 6965 7479 2069 7320 6120 7072 6f64  ariety is a prod
│ │ │ │ -000168e0: 7563 7420 6f66 2070 726f 6a65 6374 6976  uct of projectiv
│ │ │ │ -000168f0: 650a 7370 6163 6520 6974 2069 7320 7265  e.space it is re
│ │ │ │ -00016900: 636f 6d6d 656e 6420 746f 2075 7365 2074  commend to use t
│ │ │ │ -00016910: 6865 2066 6f72 6d20 6162 6f76 6520 7261  he form above ra
│ │ │ │ -00016920: 7468 6572 2074 6861 6e20 696e 7075 7474  ther than inputt
│ │ │ │ -00016930: 696e 6720 7468 6520 746f 7269 630a 7661  ing the toric.va
│ │ │ │ -00016940: 7269 6574 7920 666f 7220 6566 6669 6369  riety for effici
│ │ │ │ -00016950: 656e 6379 2072 6561 736f 6e73 2e0a 0a2b  ency reasons...+
│ │ │ │ +00016780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016790: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000167a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000167b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000167c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000167d0: 2d2d 2d2d 2b0a 0a49 6e20 7468 6520 6361  ----+..In the ca
│ │ │ │ +000167e0: 7365 2077 6865 7265 2074 6865 2061 6d62  se where the amb
│ │ │ │ +000167f0: 6965 6e74 2073 7061 6365 2069 7320 6120  ient space is a 
│ │ │ │ +00016800: 746f 7269 6320 7661 7269 6574 7920 7768  toric variety wh
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│ │ │ │ +00016820: 6475 6374 0a6f 6620 7072 6f6a 6563 7469  duct.of projecti
│ │ │ │ +00016830: 7665 2073 7061 6365 7320 7765 206d 7573  ve spaces we mus
│ │ │ │ +00016840: 7420 6c6f 6164 2074 6865 204e 6f72 6d61  t load the Norma
│ │ │ │ +00016850: 6c54 6f72 6963 5661 7269 6574 6965 7320  lToricVarieties 
│ │ │ │ +00016860: 7061 636b 6167 6520 616e 6420 6d75 7374  package and must
│ │ │ │ +00016870: 0a61 6c73 6f20 696e 7075 7420 7468 6520  .also input the 
│ │ │ │ +00016880: 746f 7269 6320 7661 7269 6574 792e 2049  toric variety. I
│ │ │ │ +00016890: 6620 7468 6520 746f 7269 6320 7661 7269  f the toric vari
│ │ │ │ +000168a0: 6574 7920 6973 2061 2070 726f 6475 6374  ety is a product
│ │ │ │ +000168b0: 206f 6620 7072 6f6a 6563 7469 7665 0a73   of projective.s
│ │ │ │ +000168c0: 7061 6365 2069 7420 6973 2072 6563 6f6d  pace it is recom
│ │ │ │ +000168d0: 6d65 6e64 2074 6f20 7573 6520 7468 6520  mend to use the 
│ │ │ │ +000168e0: 666f 726d 2061 626f 7665 2072 6174 6865  form above rathe
│ │ │ │ +000168f0: 7220 7468 616e 2069 6e70 7574 7469 6e67  r than inputting
│ │ │ │ +00016900: 2074 6865 2074 6f72 6963 0a76 6172 6965   the toric.varie
│ │ │ │ +00016910: 7479 2066 6f72 2065 6666 6963 6965 6e63  ty for efficienc
│ │ │ │ +00016920: 7920 7265 6173 6f6e 732e 0a0a 2b2d 2d2d  y reasons...+---
│ │ │ │ +00016930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000169a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a  --------+.|i23 :
│ │ │ │ -000169b0: 206e 6565 6473 5061 636b 6167 6520 224e   needsPackage "N
│ │ │ │ -000169c0: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ -000169d0: 6965 7322 2020 2020 2020 2020 2020 2020  ies"            
│ │ │ │ +00016970: 2d2d 2d2d 2d2b 0a7c 6932 3320 3a20 6e65  -----+.|i23 : ne
│ │ │ │ +00016980: 6564 7350 6163 6b61 6765 2022 4e6f 726d  edsPackage "Norm
│ │ │ │ +00016990: 616c 546f 7269 6356 6172 6965 7469 6573  alToricVarieties
│ │ │ │ +000169a0: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ +000169b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000169c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000169d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000169e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000169f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00016a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00016a40: 7c6f 3233 203d 204e 6f72 6d61 6c54 6f72  |o23 = NormalTor
│ │ │ │ -00016a50: 6963 5661 7269 6574 6965 7320 2020 2020  icVarieties     
│ │ │ │ +000169f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016a00: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00016a10: 3320 3d20 4e6f 726d 616c 546f 7269 6356  3 = NormalToricV
│ │ │ │ +00016a20: 6172 6965 7469 6573 2020 2020 2020 2020  arieties        
│ │ │ │ +00016a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016a50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00016a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00016a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016aa0: 207c 0a7c 6f32 3320 3a20 5061 636b 6167   |.|o23 : Packag
│ │ │ │ +00016ab0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │  00016ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ad0: 2020 2020 7c0a 7c6f 3233 203a 2050 6163      |.|o23 : Pac
│ │ │ │ -00016ae0: 6b61 6765 2020 2020 2020 2020 2020 2020  kage            
│ │ │ │ -00016af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016b10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016b20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00016b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3234  ----------+.|i24
│ │ │ │ -00016b70: 203a 2052 686f 203d 207b 7b31 2c30 2c30   : Rho = {{1,0,0
│ │ │ │ -00016b80: 7d2c 7b30 2c31 2c30 7d2c 7b30 2c30 2c31  },{0,1,0},{0,0,1
│ │ │ │ -00016b90: 7d2c 7b2d 312c 2d31 2c30 7d2c 7b30 2c30  },{-1,-1,0},{0,0
│ │ │ │ -00016ba0: 2c2d 317d 7d20 2020 2020 2020 2020 2020  ,-1}}           
│ │ │ │ -00016bb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00016bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c00: 7c0a 7c6f 3234 203d 207b 7b31 2c20 302c  |.|o24 = {{1, 0,
│ │ │ │ -00016c10: 2030 7d2c 207b 302c 2031 2c20 307d 2c20   0}, {0, 1, 0}, 
│ │ │ │ -00016c20: 7b30 2c20 302c 2031 7d2c 207b 2d31 2c20  {0, 0, 1}, {-1, 
│ │ │ │ -00016c30: 2d31 2c20 307d 2c20 7b30 2c20 302c 202d  -1, 0}, {0, 0, -
│ │ │ │ -00016c40: 317d 7d20 2020 2020 2020 207c 0a7c 2020  1}}        |.|  
│ │ │ │ +00016ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ae0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00016af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b30: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3420 3a20  -------+.|i24 : 
│ │ │ │ +00016b40: 5268 6f20 3d20 7b7b 312c 302c 307d 2c7b  Rho = {{1,0,0},{
│ │ │ │ +00016b50: 302c 312c 307d 2c7b 302c 302c 317d 2c7b  0,1,0},{0,0,1},{
│ │ │ │ +00016b60: 2d31 2c2d 312c 307d 2c7b 302c 302c 2d31  -1,-1,0},{0,0,-1
│ │ │ │ +00016b70: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +00016b80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00016b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016bc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00016bd0: 6f32 3420 3d20 7b7b 312c 2030 2c20 307d  o24 = {{1, 0, 0}
│ │ │ │ +00016be0: 2c20 7b30 2c20 312c 2030 7d2c 207b 302c  , {0, 1, 0}, {0,
│ │ │ │ +00016bf0: 2030 2c20 317d 2c20 7b2d 312c 202d 312c   0, 1}, {-1, -1,
│ │ │ │ +00016c00: 2030 7d2c 207b 302c 2030 2c20 2d31 7d7d   0}, {0, 0, -1}}
│ │ │ │ +00016c10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00016c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c60: 2020 207c 0a7c 6f32 3420 3a20 4c69 7374     |.|o24 : List
│ │ │ │  00016c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c90: 2020 2020 2020 7c0a 7c6f 3234 203a 204c        |.|o24 : L
│ │ │ │ -00016ca0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ -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: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00016cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00016d30: 3235 203a 2053 6967 6d61 203d 207b 7b30  25 : Sigma = {{0
│ │ │ │ -00016d40: 2c31 2c32 7d2c 7b31 2c32 2c33 7d2c 7b30  ,1,2},{1,2,3},{0
│ │ │ │ -00016d50: 2c32 2c33 7d2c 7b30 2c31 2c34 7d2c 7b31  ,2,3},{0,1,4},{1
│ │ │ │ -00016d60: 2c33 2c34 7d2c 7b30 2c33 2c34 7d7d 2020  ,3,4},{0,3,4}}  
│ │ │ │ -00016d70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00016d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016dc0: 2020 7c0a 7c6f 3235 203d 207b 7b30 2c20    |.|o25 = {{0, 
│ │ │ │ -00016dd0: 312c 2032 7d2c 207b 312c 2032 2c20 337d  1, 2}, {1, 2, 3}
│ │ │ │ -00016de0: 2c20 7b30 2c20 322c 2033 7d2c 207b 302c  , {0, 2, 3}, {0,
│ │ │ │ -00016df0: 2031 2c20 347d 2c20 7b31 2c20 332c 2034   1, 4}, {1, 3, 4
│ │ │ │ -00016e00: 7d2c 207b 302c 2033 2c20 347d 7d7c 0a7c  }, {0, 3, 4}}|.|
│ │ │ │ +00016c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ca0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016cb0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00016cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016cf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3520  ---------+.|i25 
│ │ │ │ +00016d00: 3a20 5369 676d 6120 3d20 7b7b 302c 312c  : Sigma = {{0,1,
│ │ │ │ +00016d10: 327d 2c7b 312c 322c 337d 2c7b 302c 322c  2},{1,2,3},{0,2,
│ │ │ │ +00016d20: 337d 2c7b 302c 312c 347d 2c7b 312c 332c  3},{0,1,4},{1,3,
│ │ │ │ +00016d30: 347d 2c7b 302c 332c 347d 7d20 2020 2020  4},{0,3,4}}     
│ │ │ │ +00016d40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00016d90: 0a7c 6f32 3520 3d20 7b7b 302c 2031 2c20  .|o25 = {{0, 1, 
│ │ │ │ +00016da0: 327d 2c20 7b31 2c20 322c 2033 7d2c 207b  2}, {1, 2, 3}, {
│ │ │ │ +00016db0: 302c 2032 2c20 337d 2c20 7b30 2c20 312c  0, 2, 3}, {0, 1,
│ │ │ │ +00016dc0: 2034 7d2c 207b 312c 2033 2c20 347d 2c20   4}, {1, 3, 4}, 
│ │ │ │ +00016dd0: 7b30 2c20 332c 2034 7d7d 7c0a 7c20 2020  {0, 3, 4}}|.|   
│ │ │ │ +00016de0: 2020 2020 2020 2020 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 2020                  
│ │ │ │ +00016e20: 2020 2020 207c 0a7c 6f32 3520 3a20 4c69       |.|o25 : Li
│ │ │ │ +00016e30: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │  00016e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e50: 2020 2020 2020 2020 7c0a 7c6f 3235 203a          |.|o25 :
│ │ │ │ -00016e60: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │ -00016e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ea0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00016eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00016ef0: 7c69 3236 203a 2058 203d 206e 6f72 6d61  |i26 : X = norma
│ │ │ │ -00016f00: 6c54 6f72 6963 5661 7269 6574 7928 5268  lToricVariety(Rh
│ │ │ │ -00016f10: 6f2c 5369 676d 612c 436f 6566 6669 6369  o,Sigma,Coeffici
│ │ │ │ -00016f20: 656e 7452 696e 6720 3d3e 5a5a 2f33 3237  entRing =>ZZ/327
│ │ │ │ -00016f30: 3439 2920 2020 2020 207c 0a7c 2020 2020  49)      |.|    
│ │ │ │ +00016e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016e70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00016e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00016ec0: 3620 3a20 5820 3d20 6e6f 726d 616c 546f  6 : X = normalTo
│ │ │ │ +00016ed0: 7269 6356 6172 6965 7479 2852 686f 2c53  ricVariety(Rho,S
│ │ │ │ +00016ee0: 6967 6d61 2c43 6f65 6666 6963 6965 6e74  igma,Coefficient
│ │ │ │ +00016ef0: 5269 6e67 203d 3e5a 5a2f 3332 3734 3929  Ring =>ZZ/32749)
│ │ │ │ +00016f00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f50: 207c 0a7c 6f32 3620 3d20 5820 2020 2020   |.|o26 = X     
│ │ │ │  00016f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f80: 2020 2020 7c0a 7c6f 3236 203d 2058 2020      |.|o26 = X  
│ │ │ │ -00016f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  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  .|              
│ │ │ │ -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 7c0a 7c6f 3236            |.|o26
│ │ │ │ -00017020: 203a 204e 6f72 6d61 6c54 6f72 6963 5661   : NormalToricVa
│ │ │ │ -00017030: 7269 6574 7920 2020 2020 2020 2020 2020  riety           
│ │ │ │ -00017040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017060: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00017070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000170a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000170b0: 2b0a 7c69 3237 203a 2063 736d 583d 4353  +.|i27 : csmX=CS
│ │ │ │ -000170c0: 4d20 5820 2020 2020 2020 2020 2020 2020  M X             
│ │ │ │ +00016fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016fe0: 2020 2020 2020 207c 0a7c 6f32 3620 3a20         |.|o26 : 
│ │ │ │ +00016ff0: 4e6f 726d 616c 546f 7269 6356 6172 6965  NormalToricVarie
│ │ │ │ +00017000: 7479 2020 2020 2020 2020 2020 2020 2020  ty              
│ │ │ │ +00017010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017030: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00017040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00017080: 6932 3720 3a20 6373 6d58 3d43 534d 2058  i27 : csmX=CSM X
│ │ │ │ +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 2020                  
│ │ │ │ +000170c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  000170d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000170e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000170f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000170f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017110: 2020 207c 0a7c 2020 2020 2020 2020 3220     |.|        2 
│ │ │ │ +00017120: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00017130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017140: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00017150: 2032 2020 2020 2020 2032 2020 2020 2020   2       2      
│ │ │ │ -00017160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017190: 207c 0a7c 6f32 3720 3d20 3678 2078 2020   |.|o27 = 6x x  
│ │ │ │ -000171a0: 2b20 3378 2020 2b20 3678 2078 2020 2b20  + 3x  + 6x x  + 
│ │ │ │ -000171b0: 3378 2020 2b20 3278 2020 2b20 3120 2020  3x  + 2x  + 1   
│ │ │ │ -000171c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000171d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000171e0: 2020 2020 2020 2033 2034 2020 2020 2033         3 4     3
│ │ │ │ -000171f0: 2020 2020 2033 2034 2020 2020 2033 2020       3 4     3  
│ │ │ │ -00017200: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +00017140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017150: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017160: 7c6f 3237 203d 2036 7820 7820 202b 2033  |o27 = 6x x  + 3
│ │ │ │ +00017170: 7820 202b 2036 7820 7820 202b 2033 7820  x  + 6x x  + 3x 
│ │ │ │ +00017180: 202b 2032 7820 202b 2031 2020 2020 2020   + 2x  + 1      
│ │ │ │ +00017190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000171a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000171b0: 2020 2020 3320 3420 2020 2020 3320 2020      3 4     3   
│ │ │ │ +000171c0: 2020 3320 3420 2020 2020 3320 2020 2020    3 4     3     
│ │ │ │ +000171d0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +000171e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000171f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00017200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017220: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00017230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017270: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00017280: 2020 2020 2020 2020 2020 205a 5a5b 7820             ZZ[x 
│ │ │ │ -00017290: 2e2e 7820 5d20 2020 2020 2020 2020 2020  ..x ]           
│ │ │ │ -000172a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017230: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00017240: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00017250: 2020 2020 2020 2020 5a5a 5b78 202e 2e78          ZZ[x ..x
│ │ │ │ +00017260: 205d 2020 2020 2020 2020 2020 2020 2020   ]              
│ │ │ │ +00017270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017280: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00017290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000172a0: 2020 2020 2020 2030 2020 2034 2020 2020         0   4    
│ │ │ │ +000172b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000172c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172d0: 2020 2020 2020 2020 2020 3020 2020 3420            0   4 
│ │ │ │ -000172e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017300: 2020 2020 2020 2020 7c0a 7c6f 3237 203a          |.|o27 :
│ │ │ │ -00017310: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   ---------------
│ │ │ │ -00017320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017330: 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020 2020  ----------      
│ │ │ │ -00017340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017350: 2020 207c 0a7c 2020 2020 2020 2878 2078     |.|      (x x
│ │ │ │ -00017360: 202c 2078 2078 2078 202c 2078 2020 2d20   , x x x , x  - 
│ │ │ │ -00017370: 7820 2c20 7820 202d 2078 202c 2078 2020  x , x  - x , x  
│ │ │ │ -00017380: 2d20 7820 2920 2020 2020 2020 2020 2020  - x )           
│ │ │ │ -00017390: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000173a0: 7c20 2020 2020 2020 2032 2034 2020 2030  |        2 4   0
│ │ │ │ -000173b0: 2031 2033 2020 2030 2020 2020 3320 2020   1 3   0    3   
│ │ │ │ -000173c0: 3120 2020 2033 2020 2032 2020 2020 3420  1    3   2    4 
│ │ │ │ -000173d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000173e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000172d0: 2020 2020 207c 0a7c 6f32 3720 3a20 2d2d       |.|o27 : --
│ │ │ │ +000172e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000172f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017300: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020  -------         
│ │ │ │ +00017310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017320: 7c0a 7c20 2020 2020 2028 7820 7820 2c20  |.|      (x x , 
│ │ │ │ +00017330: 7820 7820 7820 2c20 7820 202d 2078 202c  x x x , x  - x ,
│ │ │ │ +00017340: 2078 2020 2d20 7820 2c20 7820 202d 2078   x  - x , x  - x
│ │ │ │ +00017350: 2029 2020 2020 2020 2020 2020 2020 2020   )              
│ │ │ │ +00017360: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00017370: 2020 2020 2020 3220 3420 2020 3020 3120        2 4   0 1 
│ │ │ │ +00017380: 3320 2020 3020 2020 2033 2020 2031 2020  3   0    3   1  
│ │ │ │ +00017390: 2020 3320 2020 3220 2020 2034 2020 2020    3   2    4    
│ │ │ │ +000173a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000173b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000173c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000173d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000173e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000173f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017430: 2d2d 2d2d 2b0a 7c69 3238 203a 2043 683d  ----+.|i28 : Ch=
│ │ │ │ -00017440: 7269 6e67 2063 736d 5820 2020 2020 2020  ring csmX       
│ │ │ │ +00017400: 2d2b 0a7c 6932 3820 3a20 4368 3d72 696e  -+.|i28 : Ch=rin
│ │ │ │ +00017410: 6720 6373 6d58 2020 2020 2020 2020 2020  g csmX          
│ │ │ │ +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 7c0a 7c20              |.| 
│ │ │ │  00017450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017480: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00017490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000174a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017490: 2020 2020 2020 207c 0a7c 6f32 3820 3d20         |.|o28 = 
│ │ │ │ +000174a0: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │  000174b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000174c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3238            |.|o28
│ │ │ │ -000174d0: 203d 2043 6820 2020 2020 2020 2020 2020   = Ch           
│ │ │ │ -000174e0: 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 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000174f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017510: 2020 2020 207c 0a7c 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 2020                  
│ │ │ │ +00017510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017520: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017530: 6f32 3820 3a20 5175 6f74 6965 6e74 5269  o28 : QuotientRi
│ │ │ │ +00017540: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  00017550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017560: 7c0a 7c6f 3238 203a 2051 756f 7469 656e  |.|o28 : Quotien
│ │ │ │ -00017570: 7452 696e 6720 2020 2020 2020 2020 2020  tRing           
│ │ │ │ -00017580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000175a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00017560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017570: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00017580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +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 2b0a 7c69 3239 203a 2043  ------+.|i29 : C
│ │ │ │ -00017600: 6865 636b 546f 7269 6356 6172 6965 7479  heckToricVariety
│ │ │ │ -00017610: 5661 6c69 6428 5829 2020 2020 2020 2020  Valid(X)        
│ │ │ │ +000175c0: 2d2d 2d2b 0a7c 6932 3920 3a20 4368 6563  ---+.|i29 : Chec
│ │ │ │ +000175d0: 6b54 6f72 6963 5661 7269 6574 7956 616c  kToricVarietyVal
│ │ │ │ +000175e0: 6964 2858 2920 2020 2020 2020 2020 2020  id(X)           
│ │ │ │ +000175f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017600: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017610: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00017620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017640: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00017650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017650: 2020 2020 2020 2020 207c 0a7c 6f32 3920           |.|o29 
│ │ │ │ +00017660: 3d20 7472 7565 2020 2020 2020 2020 2020  = true          
│ │ │ │  00017670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017680: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00017690: 3239 203d 2074 7275 6520 2020 2020 2020  29 = 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 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -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 2b0a 7c69 3330 203a 2052 3d72 696e  --+.|i30 : R=rin
│ │ │ │ -00017730: 6728 5829 2020 2020 2020 2020 2020 2020  g(X)            
│ │ │ │ +00017680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000176a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000176b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000176c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000176d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000176e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000176f0: 0a7c 6933 3020 3a20 523d 7269 6e67 2858  .|i30 : R=ring(X
│ │ │ │ +00017700: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00017710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017730: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00017740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017760: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017780: 2020 2020 207c 0a7c 6f33 3020 3d20 5220       |.|o30 = R 
│ │ │ │  00017790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000177a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000177b0: 2020 2020 2020 2020 7c0a 7c6f 3330 203d          |.|o30 =
│ │ │ │ -000177c0: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ -000177d0: 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: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000177e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000177f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017800: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00017810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00017850: 7c6f 3330 203a 2050 6f6c 796e 6f6d 6961  |o30 : Polynomia
│ │ │ │ -00017860: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │ -00017870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017890: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00017800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017810: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +00017820: 3020 3a20 506f 6c79 6e6f 6d69 616c 5269  0 : PolynomialRi
│ │ │ │ +00017830: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +00017840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017860: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00017870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000178a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000178b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000178c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000178d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000178e0: 2d2d 2d2d 2b0a 7c69 3331 203a 2049 3d69  ----+.|i31 : I=i
│ │ │ │ -000178f0: 6465 616c 2852 5f30 5e34 2a52 5f31 2c52  deal(R_0^4*R_1,R
│ │ │ │ -00017900: 5f30 2a52 5f33 2a52 5f34 2a52 5f32 2d52  _0*R_3*R_4*R_2-R
│ │ │ │ -00017910: 5f32 5e32 2a52 5f30 5e32 2920 2020 2020  _2^2*R_0^2)     
│ │ │ │ -00017920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017930: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00017940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017970: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00017980: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ -00017990: 2020 2032 2032 2020 2020 2020 2020 2020     2 2          
│ │ │ │ -000179a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000179b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000179c0: 2020 2020 207c 0a7c 6f33 3120 3d20 6964       |.|o31 = id
│ │ │ │ -000179d0: 6561 6c20 2878 2078 202c 202d 2078 2078  eal (x x , - x x
│ │ │ │ -000179e0: 2020 2b20 7820 7820 7820 7820 2920 2020    + x x x x )   
│ │ │ │ -000179f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a10: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00017a20: 2030 2031 2020 2020 2030 2032 2020 2020   0 1     0 2    
│ │ │ │ -00017a30: 3020 3220 3320 3420 2020 2020 2020 2020  0 2 3 4         
│ │ │ │ +000178b0: 2d2b 0a7c 6933 3120 3a20 493d 6964 6561  -+.|i31 : I=idea
│ │ │ │ +000178c0: 6c28 525f 305e 342a 525f 312c 525f 302a  l(R_0^4*R_1,R_0*
│ │ │ │ +000178d0: 525f 332a 525f 342a 525f 322d 525f 325e  R_3*R_4*R_2-R_2^
│ │ │ │ +000178e0: 322a 525f 305e 3229 2020 2020 2020 2020  2*R_0^2)        
│ │ │ │ +000178f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017940: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00017950: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +00017960: 3220 3220 2020 2020 2020 2020 2020 2020  2 2             
│ │ │ │ +00017970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017990: 2020 7c0a 7c6f 3331 203d 2069 6465 616c    |.|o31 = ideal
│ │ │ │ +000179a0: 2028 7820 7820 2c20 2d20 7820 7820 202b   (x x , - x x  +
│ │ │ │ +000179b0: 2078 2078 2078 2078 2029 2020 2020 2020   x x x x )      
│ │ │ │ +000179c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000179d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000179e0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +000179f0: 3120 2020 2020 3020 3220 2020 2030 2032  1     0 2    0 2
│ │ │ │ +00017a00: 2033 2034 2020 2020 2020 2020 2020 2020   3 4            
│ │ │ │ +00017a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017a20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00017a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00017a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017a70: 2020 207c 0a7c 6f33 3120 3a20 4964 6561     |.|o31 : Idea
│ │ │ │ +00017a80: 6c20 6f66 2052 2020 2020 2020 2020 2020  l of R          
│ │ │ │  00017a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017aa0: 2020 2020 2020 7c0a 7c6f 3331 203a 2049        |.|o31 : I
│ │ │ │ -00017ab0: 6465 616c 206f 6620 5220 2020 2020 2020  deal of R       
│ │ │ │ -00017ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017af0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00017b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00017b40: 3332 203a 2043 534d 2858 2c49 2920 2020  32 : CSM(X,I)   
│ │ │ │ -00017b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017ab0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017ac0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00017ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017b00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 3220  ---------+.|i32 
│ │ │ │ +00017b10: 3a20 4353 4d28 582c 4929 2020 2020 2020  : CSM(X,I)      
│ │ │ │ +00017b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00017b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017b80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00017b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00017ba0: 0a7c 2020 2020 2020 2020 3220 2020 2020  .|        2     
│ │ │ │ +00017bb0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  00017bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017bd0: 2020 7c0a 7c20 2020 2020 2020 2032 2020    |.|        2  
│ │ │ │ -00017be0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -00017bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00017c20: 6f33 3220 3d20 3578 2078 2020 2b20 3378  o32 = 5x x  + 3x
│ │ │ │ -00017c30: 2020 2b20 3478 2078 2020 2b20 7820 2020    + 4x x  + x   
│ │ │ │ -00017c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00017c70: 2020 2033 2034 2020 2020 2033 2020 2020     3 4     3    
│ │ │ │ -00017c80: 2033 2034 2020 2020 3320 2020 2020 2020   3 4    3       
│ │ │ │ +00017bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017be0: 2020 2020 2020 2020 2020 7c0a 7c6f 3332            |.|o32
│ │ │ │ +00017bf0: 203d 2035 7820 7820 202b 2033 7820 202b   = 5x x  + 3x  +
│ │ │ │ +00017c00: 2034 7820 7820 202b 2078 2020 2020 2020   4x x  + x      
│ │ │ │ +00017c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017c30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00017c40: 3320 3420 2020 2020 3320 2020 2020 3320  3 4     3     3 
│ │ │ │ +00017c50: 3420 2020 2033 2020 2020 2020 2020 2020  4    3          
│ │ │ │ +00017c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017c80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00017c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017cb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00017cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017cc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00017cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017cf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00017d00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00017d10: 2020 2020 2020 205a 5a5b 7820 2e2e 7820         ZZ[x ..x 
│ │ │ │ -00017d20: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ -00017d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00017ce0: 2020 2020 5a5a 5b78 202e 2e78 205d 2020      ZZ[x ..x ]  
│ │ │ │ +00017cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017d10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017d30: 2020 2030 2020 2034 2020 2020 2020 2020     0   4        
│ │ │ │ +00017d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d60: 2020 2020 2020 3020 2020 3420 2020 2020        0   4     
│ │ │ │ -00017d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d90: 2020 2020 7c0a 7c6f 3332 203a 202d 2d2d      |.|o32 : ---
│ │ │ │ -00017da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017dc0: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020  ------          
│ │ │ │ -00017dd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017de0: 0a7c 2020 2020 2020 2878 2078 202c 2078  .|      (x x , x
│ │ │ │ -00017df0: 2078 2078 202c 2078 2020 2d20 7820 2c20   x x , x  - x , 
│ │ │ │ -00017e00: 7820 202d 2078 202c 2078 2020 2d20 7820  x  - x , x  - x 
│ │ │ │ -00017e10: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00017e20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00017e30: 2020 2020 2032 2034 2020 2030 2031 2033       2 4   0 1 3
│ │ │ │ -00017e40: 2020 2030 2020 2020 3320 2020 3120 2020     0    3   1   
│ │ │ │ -00017e50: 2033 2020 2032 2020 2020 3420 2020 2020   3   2    4     
│ │ │ │ -00017e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017e70: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00017e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017ec0: 2b0a 7c69 3333 203a 2043 534d 2843 682c  +.|i33 : CSM(Ch,
│ │ │ │ -00017ed0: 582c 4929 2020 2020 2020 2020 2020 2020  X,I)            
│ │ │ │ +00017d60: 207c 0a7c 6f33 3220 3a20 2d2d 2d2d 2d2d   |.|o32 : ------
│ │ │ │ +00017d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017d90: 2d2d 2d20 2020 2020 2020 2020 2020 2020  ---             
│ │ │ │ +00017da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017db0: 2020 2020 2028 7820 7820 2c20 7820 7820       (x x , x x 
│ │ │ │ +00017dc0: 7820 2c20 7820 202d 2078 202c 2078 2020  x , x  - x , x  
│ │ │ │ +00017dd0: 2d20 7820 2c20 7820 202d 2078 2029 2020  - x , x  - x )  
│ │ │ │ +00017de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017df0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00017e00: 2020 3220 3420 2020 3020 3120 3320 2020    2 4   0 1 3   
│ │ │ │ +00017e10: 3020 2020 2033 2020 2031 2020 2020 3320  0    3   1    3 
│ │ │ │ +00017e20: 2020 3220 2020 2034 2020 2020 2020 2020    2    4        
│ │ │ │ +00017e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017e40: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +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 2d2b 0a7c  -------------+.|
│ │ │ │ +00017e90: 6933 3320 3a20 4353 4d28 4368 2c58 2c49  i33 : CSM(Ch,X,I
│ │ │ │ +00017ea0: 2920 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: 2020 2020 2020 2020 7c0a 7c20 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 207c 0a7c 2020             |.|  
│ │ │ │ +00017f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +00017f20: 2020 207c 0a7c 2020 2020 2020 2020 3220     |.|        2 
│ │ │ │ +00017f30: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00017f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00017f60: 2032 2020 2020 2020 2032 2020 2020 2020   2       2      
│ │ │ │ -00017f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f80: 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 7c0a                |.
│ │ │ │ +00017f70: 7c6f 3333 203d 2038 7820 7820 202b 2033  |o33 = 8x x  + 3
│ │ │ │ +00017f80: 7820 202b 2035 7820 7820 202b 2078 2020  x  + 5x x  + x  
│ │ │ │  00017f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017fa0: 207c 0a7c 6f33 3320 3d20 3878 2078 2020   |.|o33 = 8x x  
│ │ │ │ -00017fb0: 2b20 3378 2020 2b20 3578 2078 2020 2b20  + 3x  + 5x x  + 
│ │ │ │ -00017fc0: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ -00017fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017fe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00017ff0: 2020 2020 2020 2033 2034 2020 2020 2033         3 4     3
│ │ │ │ -00018000: 2020 2020 2033 2034 2020 2020 3320 2020       3 4    3   
│ │ │ │ +00017fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017fb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00017fc0: 2020 2020 3320 3420 2020 2020 3320 2020      3 4     3   
│ │ │ │ +00017fd0: 2020 3320 3420 2020 2033 2020 2020 2020    3 4    3      
│ │ │ │ +00017fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018000: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00018010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018030: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00018040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018050: 0a7c 6f33 3320 3a20 4368 2020 2020 2020  .|o33 : Ch      
│ │ │ │  00018060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018080: 2020 7c0a 7c6f 3333 203a 2043 6820 2020    |.|o33 : Ch   
│ │ │ │ -00018090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180c0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00018080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018090: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000180a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000180b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000180c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000180d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000180e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000180f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018110: 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 6869 7320  --------+..This 
│ │ │ │ -00018120: 6675 6e63 7469 6f6e 206d 6179 2061 6c73  function may als
│ │ │ │ -00018130: 6f20 636f 6d70 7574 6520 7468 6520 4353  o compute the CS
│ │ │ │ -00018140: 4d20 636c 6173 7320 6f66 2061 206e 6f72  M class of a nor
│ │ │ │ -00018150: 6d61 6c20 746f 7269 6320 7661 7269 6574  mal toric variet
│ │ │ │ -00018160: 7920 6465 6669 6e65 640a 6279 2061 2066  y defined.by a f
│ │ │ │ -00018170: 616e 2e20 496e 2074 6869 7320 6361 7365  an. In this case
│ │ │ │ -00018180: 2061 2063 6f6d 6269 6e61 746f 7269 616c   a combinatorial
│ │ │ │ -00018190: 206d 6574 686f 6420 6973 2075 7365 642e   method is used.
│ │ │ │ -000181a0: 2054 6869 7320 6d65 7468 6f64 2069 7320   This method is 
│ │ │ │ -000181b0: 6163 6365 7373 6564 0a77 6974 6820 7468  accessed.with th
│ │ │ │ -000181c0: 6520 7573 7561 6c20 4353 4d20 636f 6d6d  e usual CSM comm
│ │ │ │ -000181d0: 616e 6420 7769 7468 2065 6974 6865 7220  and with either 
│ │ │ │ -000181e0: 6f6e 6c79 2061 2074 6f72 6963 2076 6172  only a toric var
│ │ │ │ -000181f0: 6965 7479 206f 7220 6120 746f 7269 6320  iety or a toric 
│ │ │ │ -00018200: 7661 7269 6574 790a 616e 6420 6120 4368  variety.and a Ch
│ │ │ │ -00018210: 6f77 2072 696e 6720 6173 2069 6e70 7574  ow ring as input
│ │ │ │ -00018220: 2e20 496e 2074 6869 7320 6361 7365 2077  . In this case w
│ │ │ │ -00018230: 6520 6f6e 6c79 2072 6571 7569 7265 2074  e only require t
│ │ │ │ -00018240: 6861 7420 7468 6520 696e 7075 7420 746f  hat the input to
│ │ │ │ -00018250: 7269 630a 7661 7269 6574 7920 6973 2063  ric.variety is c
│ │ │ │ -00018260: 6f6d 706c 6574 6520 616e 6420 7369 6d70  omplete and simp
│ │ │ │ -00018270: 6c69 6369 616c 2028 696e 2070 6172 7469  licial (in parti
│ │ │ │ -00018280: 6375 6c61 7220 7765 2064 6f20 6e6f 7420  cular we do not 
│ │ │ │ -00018290: 6e65 6564 2069 7420 746f 2062 650a 736d  need it to be.sm
│ │ │ │ -000182a0: 6f6f 7468 292e 0a0a 2b2d 2d2d 2d2d 2d2d  ooth)...+-------
│ │ │ │ -000182b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000182c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000182d0: 2d2d 2d2b 0a7c 6933 3420 3a20 6e65 6564  ---+.|i34 : need
│ │ │ │ -000182e0: 7350 6163 6b61 6765 2022 4e6f 726d 616c  sPackage "Normal
│ │ │ │ -000182f0: 546f 7269 6356 6172 6965 7469 6573 2220  ToricVarieties" 
│ │ │ │ -00018300: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00018310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018320: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00018330: 6f33 3420 3d20 4e6f 726d 616c 546f 7269  o34 = NormalTori
│ │ │ │ -00018340: 6356 6172 6965 7469 6573 2020 2020 2020  cVarieties      
│ │ │ │ -00018350: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00018360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000180e0: 2d2d 2d2d 2d2b 0a0a 5468 6973 2066 756e  -----+..This fun
│ │ │ │ +000180f0: 6374 696f 6e20 6d61 7920 616c 736f 2063  ction may also c
│ │ │ │ +00018100: 6f6d 7075 7465 2074 6865 2043 534d 2063  ompute the CSM c
│ │ │ │ +00018110: 6c61 7373 206f 6620 6120 6e6f 726d 616c  lass of a normal
│ │ │ │ +00018120: 2074 6f72 6963 2076 6172 6965 7479 2064   toric variety d
│ │ │ │ +00018130: 6566 696e 6564 0a62 7920 6120 6661 6e2e  efined.by a fan.
│ │ │ │ +00018140: 2049 6e20 7468 6973 2063 6173 6520 6120   In this case a 
│ │ │ │ +00018150: 636f 6d62 696e 6174 6f72 6961 6c20 6d65  combinatorial me
│ │ │ │ +00018160: 7468 6f64 2069 7320 7573 6564 2e20 5468  thod is used. Th
│ │ │ │ +00018170: 6973 206d 6574 686f 6420 6973 2061 6363  is method is acc
│ │ │ │ +00018180: 6573 7365 640a 7769 7468 2074 6865 2075  essed.with the u
│ │ │ │ +00018190: 7375 616c 2043 534d 2063 6f6d 6d61 6e64  sual CSM command
│ │ │ │ +000181a0: 2077 6974 6820 6569 7468 6572 206f 6e6c   with either onl
│ │ │ │ +000181b0: 7920 6120 746f 7269 6320 7661 7269 6574  y a toric variet
│ │ │ │ +000181c0: 7920 6f72 2061 2074 6f72 6963 2076 6172  y or a toric var
│ │ │ │ +000181d0: 6965 7479 0a61 6e64 2061 2043 686f 7720  iety.and a Chow 
│ │ │ │ +000181e0: 7269 6e67 2061 7320 696e 7075 742e 2049  ring as input. I
│ │ │ │ +000181f0: 6e20 7468 6973 2063 6173 6520 7765 206f  n this case we o
│ │ │ │ +00018200: 6e6c 7920 7265 7175 6972 6520 7468 6174  nly require that
│ │ │ │ +00018210: 2074 6865 2069 6e70 7574 2074 6f72 6963   the input toric
│ │ │ │ +00018220: 0a76 6172 6965 7479 2069 7320 636f 6d70  .variety is comp
│ │ │ │ +00018230: 6c65 7465 2061 6e64 2073 696d 706c 6963  lete and simplic
│ │ │ │ +00018240: 6961 6c20 2869 6e20 7061 7274 6963 756c  ial (in particul
│ │ │ │ +00018250: 6172 2077 6520 646f 206e 6f74 206e 6565  ar we do not nee
│ │ │ │ +00018260: 6420 6974 2074 6f20 6265 0a73 6d6f 6f74  d it to be.smoot
│ │ │ │ +00018270: 6829 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  h)...+----------
│ │ │ │ +00018280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000182a0: 2b0a 7c69 3334 203a 206e 6565 6473 5061  +.|i34 : needsPa
│ │ │ │ +000182b0: 636b 6167 6520 224e 6f72 6d61 6c54 6f72  ckage "NormalTor
│ │ │ │ +000182c0: 6963 5661 7269 6574 6965 7322 207c 0a7c  icVarieties" |.|
│ │ │ │ +000182d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000182e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000182f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3334            |.|o34
│ │ │ │ +00018300: 203d 204e 6f72 6d61 6c54 6f72 6963 5661   = NormalToricVa
│ │ │ │ +00018310: 7269 6574 6965 7320 2020 2020 2020 2020  rieties         
│ │ │ │ +00018320: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00018330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018350: 2020 2020 7c0a 7c6f 3334 203a 2050 6163      |.|o34 : Pac
│ │ │ │ +00018360: 6b61 6765 2020 2020 2020 2020 2020 2020  kage            
│ │ │ │  00018370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018380: 2020 2020 2020 207c 0a7c 6f33 3420 3a20         |.|o34 : 
│ │ │ │ -00018390: 5061 636b 6167 6520 2020 2020 2020 2020  Package         
│ │ │ │ -000183a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000183b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000183c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000183d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000183e0: 2d2b 0a7c 6933 3520 3a20 5520 3d20 6869  -+.|i35 : U = hi
│ │ │ │ -000183f0: 727a 6562 7275 6368 5375 7266 6163 6520  rzebruchSurface 
│ │ │ │ -00018400: 3720 2020 2020 2020 2020 2020 2020 7c0a  7             |.
│ │ │ │ -00018410: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00018380: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00018390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000183a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000183b0: 7c69 3335 203a 2055 203d 2068 6972 7a65  |i35 : U = hirze
│ │ │ │ +000183c0: 6272 7563 6853 7572 6661 6365 2037 2020  bruchSurface 7  
│ │ │ │ +000183d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000183e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000183f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018400: 2020 2020 2020 2020 7c0a 7c6f 3335 203d          |.|o35 =
│ │ │ │ +00018410: 2055 2020 2020 2020 2020 2020 2020 2020   U              
│ │ │ │  00018420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018430: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ -00018440: 3520 3d20 5520 2020 2020 2020 2020 2020  5 = U           
│ │ │ │ +00018430: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00018440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018460: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00018470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018490: 2020 2020 207c 0a7c 6f33 3520 3a20 4e6f       |.|o35 : No
│ │ │ │ -000184a0: 726d 616c 546f 7269 6356 6172 6965 7479  rmalToricVariety
│ │ │ │ -000184b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000184c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -000184d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000184e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000184f0: 0a7c 6933 3620 3a20 4368 3d54 6f72 6963  .|i36 : Ch=Toric
│ │ │ │ -00018500: 4368 6f77 5269 6e67 2855 2920 2020 2020  ChowRing(U)     
│ │ │ │ -00018510: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00018520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018460: 2020 7c0a 7c6f 3335 203a 204e 6f72 6d61    |.|o35 : Norma
│ │ │ │ +00018470: 6c54 6f72 6963 5661 7269 6574 7920 2020  lToricVariety   
│ │ │ │ +00018480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018490: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000184a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000184b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000184c0: 3336 203a 2043 683d 546f 7269 6343 686f  36 : Ch=ToricCho
│ │ │ │ +000184d0: 7752 696e 6728 5529 2020 2020 2020 2020  wRing(U)        
│ │ │ │ +000184e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000184f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018510: 2020 2020 2020 7c0a 7c6f 3336 203d 2043        |.|o36 = C
│ │ │ │ +00018520: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │  00018530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018540: 2020 2020 2020 2020 207c 0a7c 6f33 3620           |.|o36 
│ │ │ │ -00018550: 3d20 4368 2020 2020 2020 2020 2020 2020  = Ch            
│ │ │ │ +00018540: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00018550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018570: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00018580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000185a0: 2020 207c 0a7c 6f33 3620 3a20 5175 6f74     |.|o36 : Quot
│ │ │ │ -000185b0: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ -000185c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000185d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -000185e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000185f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00018600: 6933 3720 3a20 4353 4d20 5520 2020 2020  i37 : CSM U     
│ │ │ │ +00018570: 7c0a 7c6f 3336 203a 2051 756f 7469 656e  |.|o36 : Quotien
│ │ │ │ +00018580: 7452 696e 6720 2020 2020 2020 2020 2020  tRing           
│ │ │ │ +00018590: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000185a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000185b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000185c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3337  ----------+.|i37
│ │ │ │ +000185d0: 203a 2043 534d 2055 2020 2020 2020 2020   : CSM U        
│ │ │ │ +000185e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000185f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00018600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018620: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00018630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018620: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00018630: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  00018640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018650: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00018660: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00018670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018680: 2020 2020 7c0a 7c6f 3337 203d 202d 2033      |.|o37 = - 3
│ │ │ │ -00018690: 7820 7820 202b 2078 2020 2d20 3578 2020  x x  + x  - 5x  
│ │ │ │ -000186a0: 2b20 3278 2020 2b20 3120 2020 2020 2020  + 2x  + 1       
│ │ │ │ -000186b0: 207c 0a7c 2020 2020 2020 2020 2020 3220   |.|          2 
│ │ │ │ -000186c0: 3320 2020 2033 2020 2020 2032 2020 2020  3    3     2    
│ │ │ │ -000186d0: 2033 2020 2020 2020 2020 2020 2020 7c0a   3            |.
│ │ │ │ -000186e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000186f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018700: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00018710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018720: 205a 5a5b 7820 2e2e 7820 5d20 2020 2020   ZZ[x ..x ]     
│ │ │ │ -00018730: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00018740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018750: 2020 3020 2020 3320 2020 2020 2020 2020    0   3         
│ │ │ │ -00018760: 2020 2020 207c 0a7c 6f33 3720 3a20 2d2d       |.|o37 : --
│ │ │ │ -00018770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018790: 2d2d 7c0a 7c20 2020 2020 2028 7820 7820  --|.|      (x x 
│ │ │ │ -000187a0: 2c20 7820 7820 2c20 7820 202d 2078 202c  , x x , x  - x ,
│ │ │ │ -000187b0: 2078 2020 2b20 3778 2020 2d20 7820 297c   x  + 7x  - x )|
│ │ │ │ -000187c0: 0a7c 2020 2020 2020 2020 3020 3220 2020  .|        0 2   
│ │ │ │ -000187d0: 3120 3320 2020 3020 2020 2032 2020 2031  1 3   0    2   1
│ │ │ │ -000187e0: 2020 2020 2032 2020 2020 3320 7c0a 2b2d       2    3 |.+-
│ │ │ │ -000187f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00018820: 3a20 6373 6d31 3d43 534d 2843 682c 5529  : csm1=CSM(Ch,U)
│ │ │ │ +00018650: 207c 0a7c 6f33 3720 3d20 2d20 3378 2078   |.|o37 = - 3x x
│ │ │ │ +00018660: 2020 2b20 7820 202d 2035 7820 202b 2032    + x  - 5x  + 2
│ │ │ │ +00018670: 7820 202b 2031 2020 2020 2020 2020 7c0a  x  + 1        |.
│ │ │ │ +00018680: 7c20 2020 2020 2020 2020 2032 2033 2020  |          2 3  
│ │ │ │ +00018690: 2020 3320 2020 2020 3220 2020 2020 3320    3     2     3 
│ │ │ │ +000186a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000186b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000186c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000186d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000186e0: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │ +000186f0: 5b78 202e 2e78 205d 2020 2020 2020 2020  [x ..x ]        
│ │ │ │ +00018700: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00018710: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00018720: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +00018730: 2020 7c0a 7c6f 3337 203a 202d 2d2d 2d2d    |.|o37 : -----
│ │ │ │ +00018740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
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│ │ │ │ +00018780: 202b 2037 7820 202d 2078 2029 7c0a 7c20   + 7x  - x )|.| 
│ │ │ │ +00018790: 2020 2020 2020 2030 2032 2020 2031 2033         0 2   1 3
│ │ │ │ +000187a0: 2020 2030 2020 2020 3220 2020 3120 2020     0    2   1   
│ │ │ │ +000187b0: 2020 3220 2020 2033 207c 0a2b 2d2d 2d2d    2    3 |.+----
│ │ │ │ +000187c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000187d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000187e0: 2d2d 2d2d 2d2d 2b0a 7c69 3338 203a 2063  ------+.|i38 : c
│ │ │ │ +000187f0: 736d 313d 4353 4d28 4368 2c55 2920 2020  sm1=CSM(Ch,U)   
│ │ │ │ +00018800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018810: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00018820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018840: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00018850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018870: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00018880: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -00018890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000188a0: 7c0a 7c6f 3338 203d 202d 2033 7820 7820  |.|o38 = - 3x x 
│ │ │ │ -000188b0: 202b 2078 2020 2d20 3578 2020 2b20 3278   + x  - 5x  + 2x
│ │ │ │ -000188c0: 2020 2b20 3120 2020 2020 2020 207c 0a7c    + 1        |.|
│ │ │ │ -000188d0: 2020 2020 2020 2020 2020 3220 3320 2020            2 3   
│ │ │ │ -000188e0: 2033 2020 2020 2032 2020 2020 2033 2020   3     2     3  
│ │ │ │ -000188f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00018840: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00018850: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00018860: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00018870: 6f33 3820 3d20 2d20 3378 2078 2020 2b20  o38 = - 3x x  + 
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│ │ │ │ +00018890: 2031 2020 2020 2020 2020 7c0a 7c20 2020   1        |.|   
│ │ │ │ +000188a0: 2020 2020 2020 2032 2033 2020 2020 3320         2 3    3 
│ │ │ │ +000188b0: 2020 2020 3220 2020 2020 3320 2020 2020      2     3     
│ │ │ │ +000188c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000188d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000188e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000188f0: 2020 2020 7c0a 7c6f 3338 203a 2043 6820      |.|o38 : Ch 
│ │ │ │  00018900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018920: 2020 2020 2020 207c 0a7c 6f33 3820 3a20         |.|o38 : 
│ │ │ │ -00018930: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │ -00018940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018950: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
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│ │ │ │ -00018970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00018b40: 6e6f 6e7a 6572 6f20 7072 6f62 6162 696c  nonzero probabil
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│ │ │ │ -000192b0: 2020 2020 2a20 4d65 7468 6f64 2028 6d69      * Method (mi
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│ │ │ │ -000192f0: 2020 2049 6e63 6c75 7369 6f6e 4578 636c     InclusionExcl
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│ │ │ │ -00019310: 4578 636c 7573 696f 6e2c 2061 7070 6c69  Exclusion, appli
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│ │ │ │ -000193c0: 2020 2070 6572 666f 726d 616e 6365 2069     performance i
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│ │ │ │ -000195e0: 2076 616c 7565 2043 686f 7752 696e 6745   value ChowRingE
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│ │ │ │ -00019620: 2064 6566 6175 6c74 206f 7574 7075 7420   default output 
│ │ │ │ -00019630: 6973 2061 6e20 696e 7465 6765 720a 2020  is an integer.  
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│ │ │ │ -00019650: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ -00019660: 7565 2043 686f 7752 696e 6745 6c65 6d65  ue ChowRingEleme
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│ │ │ │ -00019690: 2020 6f75 7470 7574 2074 6f20 7265 7475    output to retu
│ │ │ │ -000196a0: 726e 2c20 4861 7368 466f 726d 2072 6574  rn, HashForm ret
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│ │ │ │ -00019710: 2020 2020 2020 205c 7b30 5c7d 2c5c 7b31         \{0\},\{1
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│ │ │ │ -00019790: 7375 6273 6574 7320 6f66 2074 6865 2067  subsets of the g
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│ │ │ │ -00019800: 7375 7266 6163 6520 6769 7665 6e20 6279  surface given by
│ │ │ │ -00019810: 2074 6865 2070 726f 6475 6374 206f 6620   the product of 
│ │ │ │ -00019820: 616c 6c20 706f 6c79 6e6f 6d69 616c 7320  all polynomials 
│ │ │ │ -00019830: 696e 2074 6865 0a20 2020 2020 2020 2063  in the.        c
│ │ │ │ -00019840: 6f72 7265 7370 6f6e 6469 6e67 2073 6574  orresponding set
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│ │ │ │ -00019930: 496e 740a 2020 2020 2020 2020 4d65 7468  Int.        Meth
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│ │ │ │ +00018ab0: 7468 6174 2074 6865 2061 6c67 6f72 6974  that the algorit
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│ │ │ │ +00018b10: 7a65 726f 2070 726f 6261 6269 6c69 7479  zero probability
│ │ │ │ +00018b20: 2e20 5265 6164 206d 6f72 6520 756e 6465  . Read more unde
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│ │ │ │ +00018b50: 2070 726f 6261 6269 6c69 7374 6963 2061   probabilistic a
│ │ │ │ +00018b60: 6c67 6f72 6974 686d 2c2e 0a0a 0a0a 5761  lgorithm,.....Wa
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│ │ │ │ +00018b80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00018b90: 0a20 202a 2022 4353 4d28 4964 6561 6c29  .  * "CSM(Ideal)
│ │ │ │ +00018ba0: 220a 2020 2a20 2243 534d 2849 6465 616c  ".  * "CSM(Ideal
│ │ │ │ +00018bb0: 2c53 796d 626f 6c29 220a 2020 2a20 2243  ,Symbol)".  * "C
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│ │ │ │ +00018bd0: 4964 6561 6c29 220a 2020 2a20 2243 534d  Ideal)".  * "CSM
│ │ │ │ +00018be0: 2851 756f 7469 656e 7452 696e 672c 4964  (QuotientRing,Id
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│ │ │ │ +00018c10: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ +00018c20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
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│ │ │ │ +00018c40: 4353 4d3a 2043 534d 2c20 6973 2061 202a  CSM: CSM, is a *
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│ │ │ │ +00018cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
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│ │ │ │ +00018d70: 3232 313a 302e 0a1f 0a46 696c 653a 2043  221:0....File: C
│ │ │ │ +00018d80: 6861 7261 6374 6572 6973 7469 6343 6c61  haracteristicCla
│ │ │ │ +00018d90: 7373 6573 2e69 6e66 6f2c 204e 6f64 653a  sses.info, Node:
│ │ │ │ +00018da0: 2045 756c 6572 2c20 4e65 7874 3a20 4575   Euler, Next: Eu
│ │ │ │ +00018db0: 6c65 7241 6666 696e 652c 2050 7265 763a  lerAffine, Prev:
│ │ │ │ +00018dc0: 2043 534d 2c20 5570 3a20 546f 700a 0a45   CSM, Up: Top..E
│ │ │ │ +00018dd0: 756c 6572 202d 2d20 5468 6520 4575 6c65  uler -- The Eule
│ │ │ │ +00018de0: 7220 4368 6172 6163 7465 7269 7374 6963  r Characteristic
│ │ │ │ +00018df0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +00018e00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00018e10: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a  **..  * Usage: .
│ │ │ │ +00018e20: 2020 2020 2020 2020 4575 6c65 7220 490a          Euler I.
│ │ │ │ +00018e30: 2020 2020 2020 2020 4575 6c65 7228 582c          Euler(X,
│ │ │ │ +00018e40: 4a29 0a20 2020 2020 2020 2045 756c 6572  J).        Euler
│ │ │ │ +00018e50: 2063 736d 0a20 202a 2049 6e70 7574 733a   csm.  * Inputs:
│ │ │ │ +00018e60: 0a20 2020 2020 202a 2049 2c20 616e 202a  .      * I, an *
│ │ │ │ +00018e70: 6e6f 7465 2069 6465 616c 3a20 284d 6163  note ideal: (Mac
│ │ │ │ +00018e80: 6175 6c61 7932 446f 6329 4964 6561 6c2c  aulay2Doc)Ideal,
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│ │ │ │ +00018ea0: 6e65 6f75 7320 6964 6561 6c20 696e 2061  neous ideal in a
│ │ │ │ +00018eb0: 0a20 2020 2020 2020 2067 7261 6465 6420  .        graded 
│ │ │ │ +00018ec0: 706f 6c79 6e6f 6d69 616c 2072 696e 6720  polynomial ring 
│ │ │ │ +00018ed0: 6f76 6572 2061 2066 6965 6c64 2064 6566  over a field def
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│ │ │ │ +000195a0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
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│ │ │ │ +00019610: 202a 204f 7574 7075 7420 3d3e 202e 2e2e   * Output => ...
│ │ │ │ +00019620: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +00019630: 4368 6f77 5269 6e67 456c 656d 656e 742c  ChowRingElement,
│ │ │ │ +00019640: 2048 6173 6846 6f72 6d2c 2074 6865 2074   HashForm, the t
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│ │ │ │ +000196a0: 7468 650a 2020 2020 2020 2020 6b65 7920  the.        key 
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│ │ │ │ +000198e0: 7768 656e 2075 7369 6e67 2074 6865 2044  when using the D
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│ │ │ │  00019bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019bd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00019bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019bf0: 2020 7c0a 7c6f 3220 3d20 5220 2020 2020    |.|o2 = R     
│ │ │ │  00019c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c20: 2020 2020 207c 0a7c 6f32 203d 2052 2020       |.|o2 = R  
│ │ │ │ +00019c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019c40: 2020 7c0a 7c20 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +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                  
│ │ │ │ +00019c90: 2020 7c0a 7c6f 3220 3a20 506f 6c79 6e6f    |.|o2 : Polyno
│ │ │ │ +00019ca0: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │  00019cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019cc0: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c       |.|o2 : Pol
│ │ │ │ -00019cd0: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ -00019ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019d10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00019cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019ce0: 2020 7c0a 2b2d 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 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 2d2b 0a7c 6933 203a 2049 3d69  -----+.|i3 : I=i
│ │ │ │ -00019d70: 6465 616c 2872 616e 646f 6d28 312c 5229  deal(random(1,R)
│ │ │ │ -00019d80: 2c72 616e 646f 6d28 322c 5229 2920 2020  ,random(2,R))   
│ │ │ │ +00019d30: 2d2d 2b0a 7c69 3320 3a20 493d 6964 6561  --+.|i3 : I=idea
│ │ │ │ +00019d40: 6c28 7261 6e64 6f6d 2831 2c52 292c 7261  l(random(1,R),ra
│ │ │ │ +00019d50: 6e64 6f6d 2832 2c52 2929 2020 2020 2020  ndom(2,R))      
│ │ │ │ +00019d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019d80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00019d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019db0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00019db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019dd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00019de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00019e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e40: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -00019e50: 2020 2020 207c 0a7c 6f33 203d 2069 6465       |.|o3 = ide
│ │ │ │ -00019e60: 616c 2028 3130 3778 2020 2b20 3433 3736  al (107x  + 4376
│ │ │ │ -00019e70: 7820 202d 2036 3331 3678 2020 2b20 3331  x  - 6316x  + 31
│ │ │ │ -00019e80: 3837 7820 202b 2033 3738 3378 202c 202d  87x  + 3783x , -
│ │ │ │ -00019e90: 2036 3035 3378 2020 2b20 3835 3730 7820   6053x  + 8570x 
│ │ │ │ -00019ea0: 7820 202b 207c 0a7c 2020 2020 2020 2020  x  + |.|        
│ │ │ │ -00019eb0: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ -00019ec0: 2031 2020 2020 2020 2020 3220 2020 2020   1        2     
│ │ │ │ -00019ed0: 2020 2033 2020 2020 2020 2020 3420 2020     3        4   
│ │ │ │ -00019ee0: 2020 2020 2020 3020 2020 2020 2020 2030        0        0
│ │ │ │ -00019ef0: 2031 2020 207c 0a7c 2020 2020 202d 2d2d   1   |.|     ---
│ │ │ │ +00019e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019e10: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00019e20: 2020 7c0a 7c6f 3320 3d20 6964 6561 6c20    |.|o3 = ideal 
│ │ │ │ +00019e30: 2831 3037 7820 202b 2034 3337 3678 2020  (107x  + 4376x  
│ │ │ │ +00019e40: 2d20 3633 3136 7820 202b 2033 3138 3778  - 6316x  + 3187x
│ │ │ │ +00019e50: 2020 2b20 3337 3833 7820 2c20 2d20 3630    + 3783x , - 60
│ │ │ │ +00019e60: 3533 7820 202b 2038 3537 3078 2078 2020  53x  + 8570x x  
│ │ │ │ +00019e70: 2b20 7c0a 7c20 2020 2020 2020 2020 2020  + |.|           
│ │ │ │ +00019e80: 2020 2020 2030 2020 2020 2020 2020 3120       0        1 
│ │ │ │ +00019e90: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00019ea0: 3320 2020 2020 2020 2034 2020 2020 2020  3        4      
│ │ │ │ +00019eb0: 2020 2030 2020 2020 2020 2020 3020 3120     0        0 1 
│ │ │ │ +00019ec0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d    |.|     ------
│ │ │ │ +00019ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019f40: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -00019f50: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00019f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f70: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00019f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f90: 2020 2020 207c 0a7c 2020 2020 2031 3033       |.|     103
│ │ │ │ -00019fa0: 3539 7820 202d 2031 3630 3930 7820 7820  59x  - 16090x x 
│ │ │ │ -00019fb0: 202d 2038 3231 3078 2078 2020 2b20 3530   - 8210x x  + 50
│ │ │ │ -00019fc0: 3731 7820 202b 2038 3434 3478 2078 2020  71x  + 8444x x  
│ │ │ │ -00019fd0: 2d20 3839 3937 7820 7820 202d 2036 3934  - 8997x x  - 694
│ │ │ │ -00019fe0: 3978 2078 207c 0a7c 2020 2020 2020 2020  9x x |.|        
│ │ │ │ -00019ff0: 2020 2031 2020 2020 2020 2020 2030 2032     1         0 2
│ │ │ │ -0001a000: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ -0001a010: 2020 2032 2020 2020 2020 2020 3020 3320     2        0 3 
│ │ │ │ -0001a020: 2020 2020 2020 2031 2033 2020 2020 2020         1 3      
│ │ │ │ -0001a030: 2020 3220 337c 0a7c 2020 2020 202d 2d2d    2 3|.|     ---
│ │ │ │ +00019f10: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │ +00019f20: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00019f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019f40: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00019f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019f60: 2020 7c0a 7c20 2020 2020 3130 3335 3978    |.|     10359x
│ │ │ │ +00019f70: 2020 2d20 3136 3039 3078 2078 2020 2d20    - 16090x x  - 
│ │ │ │ +00019f80: 3832 3130 7820 7820 202b 2035 3037 3178  8210x x  + 5071x
│ │ │ │ +00019f90: 2020 2b20 3834 3434 7820 7820 202d 2038    + 8444x x  - 8
│ │ │ │ +00019fa0: 3939 3778 2078 2020 2d20 3639 3439 7820  997x x  - 6949x 
│ │ │ │ +00019fb0: 7820 7c0a 7c20 2020 2020 2020 2020 2020  x |.|           
│ │ │ │ +00019fc0: 3120 2020 2020 2020 2020 3020 3220 2020  1         0 2   
│ │ │ │ +00019fd0: 2020 2020 2031 2032 2020 2020 2020 2020       1 2        
│ │ │ │ +00019fe0: 3220 2020 2020 2020 2030 2033 2020 2020  2        0 3    
│ │ │ │ +00019ff0: 2020 2020 3120 3320 2020 2020 2020 2032      1 3        2
│ │ │ │ +0001a000: 2033 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d   3|.|     ------
│ │ │ │ +0001a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a080: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -0001a090: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0001a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a0c0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0001a0d0: 2020 2020 207c 0a7c 2020 2020 202d 2031       |.|     - 1
│ │ │ │ -0001a0e0: 3432 3534 7820 202d 2031 3132 3236 7820  4254x  - 11226x 
│ │ │ │ -0001a0f0: 7820 202b 2032 3635 3378 2078 2020 2b20  x  + 2653x x  + 
│ │ │ │ -0001a100: 3132 3336 3578 2078 2020 2d20 3130 3232  12365x x  - 1022
│ │ │ │ -0001a110: 3678 2078 2020 2d20 3132 3639 3678 2029  6x x  - 12696x )
│ │ │ │ -0001a120: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001a130: 2020 2020 2033 2020 2020 2020 2020 2030       3         0
│ │ │ │ -0001a140: 2034 2020 2020 2020 2020 3120 3420 2020   4        1 4   
│ │ │ │ -0001a150: 2020 2020 2020 3220 3420 2020 2020 2020        2 4       
│ │ │ │ -0001a160: 2020 3320 3420 2020 2020 2020 2020 3420    3 4         4 
│ │ │ │ -0001a170: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a050: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │ +0001a060: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0001a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a090: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0001a0a0: 2020 7c0a 7c20 2020 2020 2d20 3134 3235    |.|     - 1425
│ │ │ │ +0001a0b0: 3478 2020 2d20 3131 3232 3678 2078 2020  4x  - 11226x x  
│ │ │ │ +0001a0c0: 2b20 3236 3533 7820 7820 202b 2031 3233  + 2653x x  + 123
│ │ │ │ +0001a0d0: 3635 7820 7820 202d 2031 3032 3236 7820  65x x  - 10226x 
│ │ │ │ +0001a0e0: 7820 202d 2031 3236 3936 7820 2920 2020  x  - 12696x )   
│ │ │ │ +0001a0f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001a100: 2020 3320 2020 2020 2020 2020 3020 3420    3         0 4 
│ │ │ │ +0001a110: 2020 2020 2020 2031 2034 2020 2020 2020         1 4      
│ │ │ │ +0001a120: 2020 2032 2034 2020 2020 2020 2020 2033     2 4         3
│ │ │ │ +0001a130: 2034 2020 2020 2020 2020 2034 2020 2020   4         4    
│ │ │ │ +0001a140: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a190: 2020 7c0a 7c6f 3320 3a20 4964 6561 6c20    |.|o3 : Ideal 
│ │ │ │ +0001a1a0: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │  0001a1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a1c0: 2020 2020 207c 0a7c 6f33 203a 2049 6465       |.|o3 : Ide
│ │ │ │ -0001a1d0: 616c 206f 6620 5220 2020 2020 2020 2020  al of R         
│ │ │ │ -0001a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a210: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a1e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3537 3639 3539 7320 2863 7075   0.0576959s (cpu
│ │ │ │ -0001a2d0: 293b 2030 2e30 3335 3439 3238 7320 2874  ); 0.0354928s (t
│ │ │ │ -0001a2e0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +0001a230: 2d2d 2b0a 7c69 3420 3a20 7469 6d65 2045  --+.|i4 : time E
│ │ │ │ +0001a240: 756c 6572 2849 2c49 6e70 7574 4973 536d  uler(I,InputIsSm
│ │ │ │ +0001a250: 6f6f 7468 3d3e 7472 7565 2920 2020 2020  ooth=>true)     
│ │ │ │ +0001a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a280: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ +0001a290: 3130 3432 3031 7320 2863 7075 293b 2030  104201s (cpu); 0
│ │ │ │ +0001a2a0: 2e30 3531 3433 3337 7320 2874 6872 6561  .0514337s (threa
│ │ │ │ +0001a2b0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +0001a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a2d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001a2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a300: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a320: 2020 7c0a 7c6f 3420 3d20 3420 2020 2020    |.|o4 = 4     
│ │ │ │  0001a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a350: 2020 2020 207c 0a7c 6f34 203d 2034 2020       |.|o4 = 4  
│ │ │ │ +0001a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a3a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a370: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3534 3131 3473 2028 6370 7529   0.254114s (cpu)
│ │ │ │ -0001a460: 3b20 302e 3134 3638 3333 7320 2874 6872  ; 0.146833s (thr
│ │ │ │ -0001a470: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ +0001a3c0: 2d2d 2b0a 7c69 3520 3a20 7469 6d65 2045  --+.|i5 : time E
│ │ │ │ +0001a3d0: 756c 6572 2049 2020 2020 2020 2020 2020  uler I          
│ │ │ │ +0001a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a410: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ +0001a420: 3335 3935 3939 7320 2863 7075 293b 2030  359599s (cpu); 0
│ │ │ │ +0001a430: 2e32 3031 3238 3173 2028 7468 7265 6164  .201281s (thread
│ │ │ │ +0001a440: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +0001a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a460: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a490: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a4b0: 2020 7c0a 7c6f 3520 3d20 3420 2020 2020    |.|o5 = 4     
│ │ │ │  0001a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a4e0: 2020 2020 207c 0a7c 6f35 203d 2034 2020       |.|o5 = 4  
│ │ │ │ +0001a4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a530: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a500: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a580: 2d2d 2d2d 2d2b 0a7c 6936 203a 2045 756c  -----+.|i6 : Eul
│ │ │ │ -0001a590: 6572 4948 6173 683d 4575 6c65 7228 492c  erIHash=Euler(I,
│ │ │ │ -0001a5a0: 4f75 7470 7574 3d3e 4861 7368 466f 726d  Output=>HashForm
│ │ │ │ -0001a5b0: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ -0001a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a5d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a550: 2d2d 2b0a 7c69 3620 3a20 4575 6c65 7249  --+.|i6 : EulerI
│ │ │ │ +0001a560: 4861 7368 3d45 756c 6572 2849 2c4f 7574  Hash=Euler(I,Out
│ │ │ │ +0001a570: 7075 743d 3e48 6173 6846 6f72 6d29 3b20  put=>HashForm); 
│ │ │ │ +0001a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a5a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a620: 2d2d 2d2d 2d2b 0a7c 6937 203a 2041 3d72  -----+.|i7 : A=r
│ │ │ │ -0001a630: 696e 6720 4575 6c65 7249 4861 7368 2322  ing EulerIHash#"
│ │ │ │ -0001a640: 4353 4d22 2020 2020 2020 2020 2020 2020  CSM"            
│ │ │ │ +0001a5f0: 2d2d 2b0a 7c69 3720 3a20 413d 7269 6e67  --+.|i7 : A=ring
│ │ │ │ +0001a600: 2045 756c 6572 4948 6173 6823 2243 534d   EulerIHash#"CSM
│ │ │ │ +0001a610: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ +0001a620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a640: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a670: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a690: 2020 7c0a 7c6f 3720 3d20 4120 2020 2020    |.|o7 = A     
│ │ │ │  0001a6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a6c0: 2020 2020 207c 0a7c 6f37 203d 2041 2020       |.|o7 = A  
│ │ │ │ +0001a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a6e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a710: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a730: 2020 7c0a 7c6f 3720 3a20 5175 6f74 6965    |.|o7 : Quotie
│ │ │ │ +0001a740: 6e74 5269 6e67 2020 2020 2020 2020 2020  ntRing          
│ │ │ │  0001a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a760: 2020 2020 207c 0a7c 6f37 203a 2051 756f       |.|o7 : Quo
│ │ │ │ -0001a770: 7469 656e 7452 696e 6720 2020 2020 2020  tientRing       
│ │ │ │ -0001a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a7b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a780: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a800: 2d2d 2d2d 2d2b 0a7c 6938 203a 2045 756c  -----+.|i8 : Eul
│ │ │ │ -0001a810: 6572 4948 6173 6823 7b30 2c31 7d3d 3d43  erIHash#{0,1}==C
│ │ │ │ -0001a820: 534d 2841 2c69 6465 616c 2849 5f30 2a49  SM(A,ideal(I_0*I
│ │ │ │ -0001a830: 5f31 2929 2020 2020 2020 2020 2020 2020  _1))            
│ │ │ │ +0001a7d0: 2d2d 2b0a 7c69 3820 3a20 4575 6c65 7249  --+.|i8 : EulerI
│ │ │ │ +0001a7e0: 4861 7368 237b 302c 317d 3d3d 4353 4d28  Hash#{0,1}==CSM(
│ │ │ │ +0001a7f0: 412c 6964 6561 6c28 495f 302a 495f 3129  A,ideal(I_0*I_1)
│ │ │ │ +0001a800: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0001a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a820: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a850: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a870: 2020 7c0a 7c6f 3820 3d20 7472 7565 2020    |.|o8 = true  
│ │ │ │  0001a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8a0: 2020 2020 207c 0a7c 6f38 203d 2074 7275       |.|o8 = tru
│ │ │ │ -0001a8b0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ -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 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a8c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a940: 2d2d 2d2d 2d2b 0a7c 6939 203a 204a 3d49  -----+.|i9 : J=I
│ │ │ │ -0001a950: 2b69 6465 616c 2878 5f30 2a78 5f32 2d78  +ideal(x_0*x_2-x
│ │ │ │ -0001a960: 5f33 2a78 5f30 2920 2020 2020 2020 2020  _3*x_0)         
│ │ │ │ +0001a910: 2d2d 2b0a 7c69 3920 3a20 4a3d 492b 6964  --+.|i9 : J=I+id
│ │ │ │ +0001a920: 6561 6c28 785f 302a 785f 322d 785f 332a  eal(x_0*x_2-x_3*
│ │ │ │ +0001a930: 785f 3029 2020 2020 2020 2020 2020 2020  x_0)            
│ │ │ │ +0001a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a960: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a990: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa20: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -0001aa30: 2020 2020 207c 0a7c 6f39 203d 2069 6465       |.|o9 = ide
│ │ │ │ -0001aa40: 616c 2028 3130 3778 2020 2b20 3433 3736  al (107x  + 4376
│ │ │ │ -0001aa50: 7820 202d 2036 3331 3678 2020 2b20 3331  x  - 6316x  + 31
│ │ │ │ -0001aa60: 3837 7820 202b 2033 3738 3378 202c 202d  87x  + 3783x , -
│ │ │ │ -0001aa70: 2036 3035 3378 2020 2b20 3835 3730 7820   6053x  + 8570x 
│ │ │ │ -0001aa80: 7820 202b 207c 0a7c 2020 2020 2020 2020  x  + |.|        
│ │ │ │ -0001aa90: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ -0001aaa0: 2031 2020 2020 2020 2020 3220 2020 2020   1        2     
│ │ │ │ -0001aab0: 2020 2033 2020 2020 2020 2020 3420 2020     3        4   
│ │ │ │ -0001aac0: 2020 2020 2020 3020 2020 2020 2020 2030        0        0
│ │ │ │ -0001aad0: 2031 2020 207c 0a7c 2020 2020 202d 2d2d   1   |.|     ---
│ │ │ │ +0001a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9f0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0001aa00: 2020 7c0a 7c6f 3920 3d20 6964 6561 6c20    |.|o9 = ideal 
│ │ │ │ +0001aa10: 2831 3037 7820 202b 2034 3337 3678 2020  (107x  + 4376x  
│ │ │ │ +0001aa20: 2d20 3633 3136 7820 202b 2033 3138 3778  - 6316x  + 3187x
│ │ │ │ +0001aa30: 2020 2b20 3337 3833 7820 2c20 2d20 3630    + 3783x , - 60
│ │ │ │ +0001aa40: 3533 7820 202b 2038 3537 3078 2078 2020  53x  + 8570x x  
│ │ │ │ +0001aa50: 2b20 7c0a 7c20 2020 2020 2020 2020 2020  + |.|           
│ │ │ │ +0001aa60: 2020 2020 2030 2020 2020 2020 2020 3120       0        1 
│ │ │ │ +0001aa70: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0001aa80: 3320 2020 2020 2020 2034 2020 2020 2020  3        4      
│ │ │ │ +0001aa90: 2020 2030 2020 2020 2020 2020 3020 3120     0        0 1 
│ │ │ │ +0001aaa0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d    |.|     ------
│ │ │ │ +0001aab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001aac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001aad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001aae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001aaf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ab00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ab10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ab20: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -0001ab30: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -0001ab40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab50: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -0001ab60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab70: 2020 2020 207c 0a7c 2020 2020 2031 3033       |.|     103
│ │ │ │ -0001ab80: 3539 7820 202d 2031 3630 3930 7820 7820  59x  - 16090x x 
│ │ │ │ -0001ab90: 202d 2038 3231 3078 2078 2020 2b20 3530   - 8210x x  + 50
│ │ │ │ -0001aba0: 3731 7820 202b 2038 3434 3478 2078 2020  71x  + 8444x x  
│ │ │ │ -0001abb0: 2d20 3839 3937 7820 7820 202d 2036 3934  - 8997x x  - 694
│ │ │ │ -0001abc0: 3978 2078 207c 0a7c 2020 2020 2020 2020  9x x |.|        
│ │ │ │ -0001abd0: 2020 2031 2020 2020 2020 2020 2030 2032     1         0 2
│ │ │ │ -0001abe0: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ -0001abf0: 2020 2032 2020 2020 2020 2020 3020 3320     2        0 3 
│ │ │ │ -0001ac00: 2020 2020 2020 2031 2033 2020 2020 2020         1 3      
│ │ │ │ -0001ac10: 2020 3220 337c 0a7c 2020 2020 202d 2d2d    2 3|.|     ---
│ │ │ │ +0001aaf0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │ +0001ab00: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0001ab10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ab20: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0001ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ab40: 2020 7c0a 7c20 2020 2020 3130 3335 3978    |.|     10359x
│ │ │ │ +0001ab50: 2020 2d20 3136 3039 3078 2078 2020 2d20    - 16090x x  - 
│ │ │ │ +0001ab60: 3832 3130 7820 7820 202b 2035 3037 3178  8210x x  + 5071x
│ │ │ │ +0001ab70: 2020 2b20 3834 3434 7820 7820 202d 2038    + 8444x x  - 8
│ │ │ │ +0001ab80: 3939 3778 2078 2020 2d20 3639 3439 7820  997x x  - 6949x 
│ │ │ │ +0001ab90: 7820 7c0a 7c20 2020 2020 2020 2020 2020  x |.|           
│ │ │ │ +0001aba0: 3120 2020 2020 2020 2020 3020 3220 2020  1         0 2   
│ │ │ │ +0001abb0: 2020 2020 2031 2032 2020 2020 2020 2020       1 2        
│ │ │ │ +0001abc0: 3220 2020 2020 2020 2030 2033 2020 2020  2        0 3    
│ │ │ │ +0001abd0: 2020 2020 3120 3320 2020 2020 2020 2032      1 3        2
│ │ │ │ +0001abe0: 2033 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d   3|.|     ------
│ │ │ │ +0001abf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ac00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ac10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ac20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ac30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ac40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ac50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ac60: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -0001ac70: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0001ac80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aca0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0001acb0: 2020 2020 207c 0a7c 2020 2020 202d 2031       |.|     - 1
│ │ │ │ -0001acc0: 3432 3534 7820 202d 2031 3132 3236 7820  4254x  - 11226x 
│ │ │ │ -0001acd0: 7820 202b 2032 3635 3378 2078 2020 2b20  x  + 2653x x  + 
│ │ │ │ -0001ace0: 3132 3336 3578 2078 2020 2d20 3130 3232  12365x x  - 1022
│ │ │ │ -0001acf0: 3678 2078 2020 2d20 3132 3639 3678 202c  6x x  - 12696x ,
│ │ │ │ -0001ad00: 2078 2078 207c 0a7c 2020 2020 2020 2020   x x |.|        
│ │ │ │ -0001ad10: 2020 2020 2033 2020 2020 2020 2020 2030       3         0
│ │ │ │ -0001ad20: 2034 2020 2020 2020 2020 3120 3420 2020   4        1 4   
│ │ │ │ -0001ad30: 2020 2020 2020 3220 3420 2020 2020 2020        2 4       
│ │ │ │ -0001ad40: 2020 3320 3420 2020 2020 2020 2020 3420    3 4         4 
│ │ │ │ -0001ad50: 2020 3020 327c 0a7c 2020 2020 202d 2d2d    0 2|.|     ---
│ │ │ │ +0001ac30: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │ +0001ac40: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0001ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ac60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ac70: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0001ac80: 2020 7c0a 7c20 2020 2020 2d20 3134 3235    |.|     - 1425
│ │ │ │ +0001ac90: 3478 2020 2d20 3131 3232 3678 2078 2020  4x  - 11226x x  
│ │ │ │ +0001aca0: 2b20 3236 3533 7820 7820 202b 2031 3233  + 2653x x  + 123
│ │ │ │ +0001acb0: 3635 7820 7820 202d 2031 3032 3236 7820  65x x  - 10226x 
│ │ │ │ +0001acc0: 7820 202d 2031 3236 3936 7820 2c20 7820  x  - 12696x , x 
│ │ │ │ +0001acd0: 7820 7c0a 7c20 2020 2020 2020 2020 2020  x |.|           
│ │ │ │ +0001ace0: 2020 3320 2020 2020 2020 2020 3020 3420    3         0 4 
│ │ │ │ +0001acf0: 2020 2020 2020 2031 2034 2020 2020 2020         1 4      
│ │ │ │ +0001ad00: 2020 2032 2034 2020 2020 2020 2020 2033     2 4         3
│ │ │ │ +0001ad10: 2034 2020 2020 2020 2020 2034 2020 2030   4         4   0
│ │ │ │ +0001ad20: 2032 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d   2|.|     ------
│ │ │ │ +0001ad30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ad40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ad50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ad60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ad80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ad90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ada0: 2d2d 2d2d 2d7c 0a7c 2020 2020 202d 2078  -----|.|     - x
│ │ │ │ -0001adb0: 2078 2029 2020 2020 2020 2020 2020 2020   x )            
│ │ │ │ -0001adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ad70: 2d2d 7c0a 7c20 2020 2020 2d20 7820 7820  --|.|     - x x 
│ │ │ │ +0001ad80: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0001ad90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001adc0: 2020 7c0a 7c20 2020 2020 2020 2030 2033    |.|        0 3
│ │ │ │  0001add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001adf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001ae00: 3020 3320 2020 2020 2020 2020 2020 2020  0 3             
│ │ │ │ -0001ae10: 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 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ae60: 2020 7c0a 7c6f 3920 3a20 4964 6561 6c20    |.|o9 : Ideal 
│ │ │ │ +0001ae70: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │  0001ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae90: 2020 2020 207c 0a7c 6f39 203a 2049 6465       |.|o9 : Ide
│ │ │ │ -0001aea0: 616c 206f 6620 5220 2020 2020 2020 2020  al of R         
│ │ │ │ -0001aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aee0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aeb0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001aec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001aed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001aef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001af00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001af10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001af20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001af30: 2d2d 2d2d 2d2b 0a0a 4e6f 7465 2074 6861  -----+..Note tha
│ │ │ │ -0001af40: 7420 7468 6520 6964 6561 6c20 4a20 6162  t the ideal J ab
│ │ │ │ -0001af50: 6f76 6520 6973 2061 2063 6f6d 706c 6574  ove is a complet
│ │ │ │ -0001af60: 6520 696e 7465 7273 6563 7469 6f6e 2c20  e intersection, 
│ │ │ │ -0001af70: 7468 7573 2077 6520 6d61 7920 6368 616e  thus we may chan
│ │ │ │ -0001af80: 6765 2074 6865 0a6d 6574 686f 6420 6f70  ge the.method op
│ │ │ │ -0001af90: 7469 6f6e 2077 6869 6368 206d 6179 2073  tion which may s
│ │ │ │ -0001afa0: 7065 6564 2063 6f6d 7075 7461 7469 6f6e  peed computation
│ │ │ │ -0001afb0: 2069 6e20 736f 6d65 2063 6173 6573 2e20   in some cases. 
│ │ │ │ -0001afc0: 5765 206d 6179 2061 6c73 6f20 6e6f 7465  We may also note
│ │ │ │ -0001afd0: 2074 6861 740a 7468 6520 6964 6561 6c20   that.the ideal 
│ │ │ │ -0001afe0: 6765 6e65 7261 7465 6420 6279 2074 6865  generated by the
│ │ │ │ -0001aff0: 2066 6972 7374 2032 2067 656e 6572 6174   first 2 generat
│ │ │ │ -0001b000: 6f72 7320 6f66 2049 2064 6566 696e 6573  ors of I defines
│ │ │ │ -0001b010: 2061 2073 6d6f 6f74 6820 7363 6865 6d65   a smooth scheme
│ │ │ │ -0001b020: 2061 6e64 0a69 6e70 7574 2074 6869 7320   and.input this 
│ │ │ │ -0001b030: 696e 666f 726d 6174 696f 6e20 696e 746f  information into
│ │ │ │ -0001b040: 2074 6865 206d 6574 686f 642e 2054 6869   the method. Thi
│ │ │ │ -0001b050: 7320 6d61 7920 616c 736f 2069 6d70 726f  s may also impro
│ │ │ │ -0001b060: 7665 2063 6f6d 7075 7461 7469 6f6e 0a73  ve computation.s
│ │ │ │ -0001b070: 7065 6564 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  peed...+--------
│ │ │ │ -0001b080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 2e31 3832 3936   -- used 0.18296
│ │ │ │ -0001b110: 3773 2028 6370 7529 3b20 302e 3036 3930  7s (cpu); 0.0690
│ │ │ │ -0001b120: 3139 3973 2028 7468 7265 6164 293b 2030  199s (thread); 0
│ │ │ │ -0001b130: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ -0001b140: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001af00: 2d2d 2b0a 0a4e 6f74 6520 7468 6174 2074  --+..Note that t
│ │ │ │ +0001af10: 6865 2069 6465 616c 204a 2061 626f 7665  he ideal J above
│ │ │ │ +0001af20: 2069 7320 6120 636f 6d70 6c65 7465 2069   is a complete i
│ │ │ │ +0001af30: 6e74 6572 7365 6374 696f 6e2c 2074 6875  ntersection, thu
│ │ │ │ +0001af40: 7320 7765 206d 6179 2063 6861 6e67 6520  s we may change 
│ │ │ │ +0001af50: 7468 650a 6d65 7468 6f64 206f 7074 696f  the.method optio
│ │ │ │ +0001af60: 6e20 7768 6963 6820 6d61 7920 7370 6565  n which may spee
│ │ │ │ +0001af70: 6420 636f 6d70 7574 6174 696f 6e20 696e  d computation in
│ │ │ │ +0001af80: 2073 6f6d 6520 6361 7365 732e 2057 6520   some cases. We 
│ │ │ │ +0001af90: 6d61 7920 616c 736f 206e 6f74 6520 7468  may also note th
│ │ │ │ +0001afa0: 6174 0a74 6865 2069 6465 616c 2067 656e  at.the ideal gen
│ │ │ │ +0001afb0: 6572 6174 6564 2062 7920 7468 6520 6669  erated by the fi
│ │ │ │ +0001afc0: 7273 7420 3220 6765 6e65 7261 746f 7273  rst 2 generators
│ │ │ │ +0001afd0: 206f 6620 4920 6465 6669 6e65 7320 6120   of I defines a 
│ │ │ │ +0001afe0: 736d 6f6f 7468 2073 6368 656d 6520 616e  smooth scheme an
│ │ │ │ +0001aff0: 640a 696e 7075 7420 7468 6973 2069 6e66  d.input this inf
│ │ │ │ +0001b000: 6f72 6d61 7469 6f6e 2069 6e74 6f20 7468  ormation into th
│ │ │ │ +0001b010: 6520 6d65 7468 6f64 2e20 5468 6973 206d  e method. This m
│ │ │ │ +0001b020: 6179 2061 6c73 6f20 696d 7072 6f76 6520  ay also improve 
│ │ │ │ +0001b030: 636f 6d70 7574 6174 696f 6e0a 7370 6565  computation.spee
│ │ │ │ +0001b040: 642e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  d...+-----------
│ │ │ │ +0001b050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b080: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 2074  ------+.|i10 : t
│ │ │ │ +0001b090: 696d 6520 4575 6c65 7228 4a2c 4d65 7468  ime Euler(J,Meth
│ │ │ │ +0001b0a0: 6f64 3d3e 4469 7265 6374 436f 6d70 6c65  od=>DirectComple
│ │ │ │ +0001b0b0: 7465 496e 7429 2020 2020 2020 2020 2020  teInt)          
│ │ │ │ +0001b0c0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ +0001b0d0: 2075 7365 6420 302e 3139 3734 3339 7320   used 0.197439s 
│ │ │ │ +0001b0e0: 2863 7075 293b 2030 2e31 3033 3033 3373  (cpu); 0.103033s
│ │ │ │ +0001b0f0: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ +0001b100: 6329 2020 2020 2020 2020 2020 2020 7c0a  c)            |.
│ │ │ │ +0001b110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b150: 2020 7c0a 7c6f 3130 203d 2032 2020 2020    |.|o10 = 2    
│ │ │ │  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                  
│ │ │ │ -0001b1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b1c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0001b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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: 3930 3537 3573 2028 6370 7529 3b20 302e  90575s (cpu); 0.
│ │ │ │ -0001b270: 3038 3435 3639 3373 2028 7468 7265 6164  0845693s (thread
│ │ │ │ -0001b280: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ -0001b290: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b190: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 2b0a 7c69 3131  ----------+.|i11
│ │ │ │ +0001b1e0: 203a 2074 696d 6520 4575 6c65 7228 4a2c   : time Euler(J,
│ │ │ │ +0001b1f0: 4d65 7468 6f64 3d3e 4469 7265 6374 436f  Method=>DirectCo
│ │ │ │ +0001b200: 6d70 6c65 7465 496e 742c 496e 6473 4f66  mpleteInt,IndsOf
│ │ │ │ +0001b210: 536d 6f6f 7468 3d3e 7b30 2c31 7d29 7c0a  Smooth=>{0,1})|.
│ │ │ │ +0001b220: 7c20 2d2d 2075 7365 6420 302e 3238 3335  | -- used 0.2835
│ │ │ │ +0001b230: 3436 7320 2863 7075 293b 2030 2e31 3138  46s (cpu); 0.118
│ │ │ │ +0001b240: 3033 3573 2028 7468 7265 6164 293b 2030  035s (thread); 0
│ │ │ │ +0001b250: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +0001b260: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b2a0: 2020 2020 2020 7c0a 7c6f 3131 203d 2032        |.|o11 = 2
│ │ │ │  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                  
│ │ │ │ -0001b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b310: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -0001b320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b360: 2d2b 0a0a 4e6f 7720 636f 6e73 6964 6572  -+..Now consider
│ │ │ │ -0001b370: 2061 6e20 6578 616d 706c 6520 696e 205c   an example in \
│ │ │ │ -0001b380: 5050 5e32 205c 7469 6d65 7320 5c50 505e  PP^2 \times \PP^
│ │ │ │ -0001b390: 322e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  2...+-----------
│ │ │ │ -0001b3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b3c0: 2d2d 2d2b 0a7c 6931 3220 3a20 523d 4d75  ---+.|i12 : R=Mu
│ │ │ │ -0001b3d0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67  ltiProjCoordRing
│ │ │ │ -0001b3e0: 287b 322c 327d 2920 2020 2020 2020 2020  ({2,2})         
│ │ │ │ -0001b3f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b2e0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001b2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0001b330: 0a4e 6f77 2063 6f6e 7369 6465 7220 616e  .Now consider an
│ │ │ │ +0001b340: 2065 7861 6d70 6c65 2069 6e20 5c50 505e   example in \PP^
│ │ │ │ +0001b350: 3220 5c74 696d 6573 205c 5050 5e32 2e0a  2 \times \PP^2..
│ │ │ │ +0001b360: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0001b370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b390: 2b0a 7c69 3132 203a 2052 3d4d 756c 7469  +.|i12 : R=Multi
│ │ │ │ +0001b3a0: 5072 6f6a 436f 6f72 6452 696e 6728 7b32  ProjCoordRing({2
│ │ │ │ +0001b3b0: 2c32 7d29 2020 2020 2020 2020 2020 2020  ,2})            
│ │ │ │ +0001b3c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b3f0: 2020 7c0a 7c6f 3132 203d 2052 2020 2020    |.|o12 = R    
│ │ │ │  0001b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b420: 2020 2020 207c 0a7c 6f31 3220 3d20 5220       |.|o12 = R 
│ │ │ │ +0001b420: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0001b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b450: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b450: 2020 2020 7c0a 7c6f 3132 203a 2050 6f6c      |.|o12 : Pol
│ │ │ │ +0001b460: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │  0001b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b480: 2020 2020 2020 207c 0a7c 6f31 3220 3a20         |.|o12 : 
│ │ │ │ -0001b490: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ -0001b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b4b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0001b4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b4e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320  ---------+.|i13 
│ │ │ │ -0001b4f0: 3a20 723d 6765 6e73 2052 2020 2020 2020  : r=gens R      
│ │ │ │ +0001b480: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001b490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b4b0: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2072  ------+.|i13 : r
│ │ │ │ +0001b4c0: 3d67 656e 7320 5220 2020 2020 2020 2020  =gens R         
│ │ │ │ +0001b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b4e0: 2020 2020 2020 207c 0a7c 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 7c20 2020            |.|   
│ │ │ │ -0001b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b540: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0001b550: 3320 3d20 7b78 202c 2078 202c 2078 202c  3 = {x , x , x ,
│ │ │ │ -0001b560: 2078 202c 2078 202c 2078 207d 2020 2020   x , x , x }    
│ │ │ │ -0001b570: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001b580: 2020 2020 2020 2030 2020 2031 2020 2032         0   1   2
│ │ │ │ -0001b590: 2020 2033 2020 2034 2020 2035 2020 2020     3   4   5    
│ │ │ │ -0001b5a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b510: 2020 2020 2020 2020 7c0a 7c6f 3133 203d          |.|o13 =
│ │ │ │ +0001b520: 207b 7820 2c20 7820 2c20 7820 2c20 7820   {x , x , x , x 
│ │ │ │ +0001b530: 2c20 7820 2c20 7820 7d20 2020 2020 2020  , x , x }       
│ │ │ │ +0001b540: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001b550: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ +0001b560: 3320 2020 3420 2020 3520 2020 2020 2020  3   4   5       
│ │ │ │ +0001b570: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b5a0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001b5b0: 3320 3a20 4c69 7374 2020 2020 2020 2020  3 : List        
│ │ │ │  0001b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b5d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001b5e0: 7c6f 3133 203a 204c 6973 7420 2020 2020  |o13 : List     
│ │ │ │ -0001b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001b610: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0001b620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b640: 2b0a 7c69 3134 203a 204b 3d69 6465 616c  +.|i14 : K=ideal
│ │ │ │ -0001b650: 2872 5f30 5e32 2a72 5f33 2d72 5f34 2a72  (r_0^2*r_3-r_4*r
│ │ │ │ -0001b660: 5f31 2a72 5f32 2c72 5f32 5e32 2a72 5f35  _1*r_2,r_2^2*r_5
│ │ │ │ -0001b670: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ -0001b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b5d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +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 2d2b 0a7c  -------------+.|
│ │ │ │ +0001b610: 6931 3420 3a20 4b3d 6964 6561 6c28 725f  i14 : K=ideal(r_
│ │ │ │ +0001b620: 305e 322a 725f 332d 725f 342a 725f 312a  0^2*r_3-r_4*r_1*
│ │ │ │ +0001b630: 725f 322c 725f 325e 322a 725f 3529 7c0a  r_2,r_2^2*r_5)|.
│ │ │ │ +0001b640: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001b650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b660: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001b670: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001b680: 3220 2020 2020 2020 2020 2020 2020 2032  2              2
│ │ │ │  0001b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b6a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0001b6b0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -0001b6c0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0001b6d0: 2020 207c 0a7c 6f31 3420 3d20 6964 6561     |.|o14 = idea
│ │ │ │ -0001b6e0: 6c20 2878 2078 2020 2d20 7820 7820 7820  l (x x  - x x x 
│ │ │ │ -0001b6f0: 2c20 7820 7820 2920 2020 2020 2020 2020  , x x )         
│ │ │ │ -0001b700: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001b710: 2020 2020 2030 2033 2020 2020 3120 3220       0 3    1 2 
│ │ │ │ -0001b720: 3420 2020 3220 3520 2020 2020 2020 2020  4   2 5         
│ │ │ │ -0001b730: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001b740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b6a0: 7c0a 7c6f 3134 203d 2069 6465 616c 2028  |.|o14 = ideal (
│ │ │ │ +0001b6b0: 7820 7820 202d 2078 2078 2078 202c 2078  x x  - x x x , x
│ │ │ │ +0001b6c0: 2078 2029 2020 2020 2020 2020 2020 2020   x )            
│ │ │ │ +0001b6d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001b6e0: 2020 3020 3320 2020 2031 2032 2034 2020    0 3    1 2 4  
│ │ │ │ +0001b6f0: 2032 2035 2020 2020 2020 2020 2020 2020   2 5            
│ │ │ │ +0001b700: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b730: 2020 207c 0a7c 6f31 3420 3a20 4964 6561     |.|o14 : Idea
│ │ │ │ +0001b740: 6c20 6f66 2052 2020 2020 2020 2020 2020  l of R          
│ │ │ │  0001b750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b760: 2020 2020 2020 7c0a 7c6f 3134 203a 2049        |.|o14 : I
│ │ │ │ -0001b770: 6465 616c 206f 6620 5220 2020 2020 2020  deal of R       
│ │ │ │ -0001b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b790: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -0001b7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b7c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a  --------+.|i15 :
│ │ │ │ -0001b7d0: 2045 756c 6572 4b3d 4575 6c65 7228 4b29   EulerK=Euler(K)
│ │ │ │ +0001b760: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b790: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 4575  -----+.|i15 : Eu
│ │ │ │ +0001b7a0: 6c65 724b 3d45 756c 6572 284b 2920 2020  lerK=Euler(K)   
│ │ │ │ +0001b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b7c0: 2020 2020 2020 7c0a 7c20 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 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001b800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b7f0: 2020 2020 2020 207c 0a7c 6f31 3520 3d20         |.|o15 = 
│ │ │ │ +0001b800: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │  0001b810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b820: 2020 2020 2020 2020 2020 7c0a 7c6f 3135            |.|o15
│ │ │ │ -0001b830: 203d 2037 2020 2020 2020 2020 2020 2020   = 7            
│ │ │ │ -0001b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b850: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0001b860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001b890: 3136 203a 2063 736d 4b3d 2043 534d 284b  16 : csmK= CSM(K
│ │ │ │ -0001b8a0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0001b8b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b8e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001b8f0: 7c20 2020 2020 2020 2032 2032 2020 2020  |        2 2    
│ │ │ │ -0001b900: 2032 2020 2020 2020 2020 2032 2020 2020   2         2    
│ │ │ │ -0001b910: 3220 2020 2020 2020 2020 2020 2032 207c  2            2 |
│ │ │ │ -0001b920: 0a7c 6f31 3620 3d20 3768 2068 2020 2b20  .|o16 = 7h h  + 
│ │ │ │ -0001b930: 3568 2068 2020 2b20 3468 2068 2020 2b20  5h h  + 4h h  + 
│ │ │ │ -0001b940: 6820 202b 2033 6820 6820 202b 2068 2020  h  + 3h h  + h  
│ │ │ │ -0001b950: 7c0a 7c20 2020 2020 2020 2031 2032 2020  |.|        1 2  
│ │ │ │ -0001b960: 2020 2031 2032 2020 2020 2031 2032 2020     1 2     1 2  
│ │ │ │ -0001b970: 2020 3120 2020 2020 3120 3220 2020 2032    1     1 2    2
│ │ │ │ -0001b980: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b820: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001b830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b850: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620  ---------+.|i16 
│ │ │ │ +0001b860: 3a20 6373 6d4b 3d20 4353 4d28 4b29 2020  : csmK= CSM(K)  
│ │ │ │ +0001b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b880: 2020 2020 2020 2020 2020 7c0a 7c20 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 207c 0a7c 2020             |.|  
│ │ │ │ +0001b8c0: 2020 2020 2020 3220 3220 2020 2020 3220        2 2     2 
│ │ │ │ +0001b8d0: 2020 2020 2020 2020 3220 2020 2032 2020          2    2  
│ │ │ │ +0001b8e0: 2020 2020 2020 2020 2020 3220 7c0a 7c6f            2 |.|o
│ │ │ │ +0001b8f0: 3136 203d 2037 6820 6820 202b 2035 6820  16 = 7h h  + 5h 
│ │ │ │ +0001b900: 6820 202b 2034 6820 6820 202b 2068 2020  h  + 4h h  + h  
│ │ │ │ +0001b910: 2b20 3368 2068 2020 2b20 6820 207c 0a7c  + 3h h  + h  |.|
│ │ │ │ +0001b920: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ +0001b930: 3120 3220 2020 2020 3120 3220 2020 2031  1 2     1 2    1
│ │ │ │ +0001b940: 2020 2020 2031 2032 2020 2020 3220 7c0a       1 2    2 |.
│ │ │ │ +0001b950: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001b960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b970: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001b980: 0a7c 2020 2020 2020 5a5a 5b68 202e 2e68  .|      ZZ[h ..h
│ │ │ │ +0001b990: 205d 2020 2020 2020 2020 2020 2020 2020   ]              
│ │ │ │  0001b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9b0: 2020 7c0a 7c20 2020 2020 205a 5a5b 6820    |.|      ZZ[h 
│ │ │ │ -0001b9c0: 2e2e 6820 5d20 2020 2020 2020 2020 2020  ..h ]           
│ │ │ │ +0001b9b0: 7c0a 7c20 2020 2020 2020 2020 2031 2020  |.|          1  
│ │ │ │ +0001b9c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  0001b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001b9f0: 3120 2020 3220 2020 2020 2020 2020 2020  1   2           
│ │ │ │ +0001b9e0: 207c 0a7c 6f31 3620 3a20 2d2d 2d2d 2d2d   |.|o16 : ------
│ │ │ │ +0001b9f0: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │  0001ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba10: 2020 2020 7c0a 7c6f 3136 203a 202d 2d2d      |.|o16 : ---
│ │ │ │ -0001ba20: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020  -------         
│ │ │ │ +0001ba10: 2020 7c0a 7c20 2020 2020 2020 2020 3320    |.|         3 
│ │ │ │ +0001ba20: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │  0001ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001ba50: 2033 2020 2033 2020 2020 2020 2020 2020   3   3          
│ │ │ │ +0001ba40: 2020 207c 0a7c 2020 2020 2020 2028 6820     |.|       (h 
│ │ │ │ +0001ba50: 2c20 6820 2920 2020 2020 2020 2020 2020  , h )           
│ │ │ │  0001ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001ba80: 2868 202c 2068 2029 2020 2020 2020 2020  (h , h )        
│ │ │ │ +0001ba70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001ba80: 3120 2020 3220 2020 2020 2020 2020 2020  1   2           
│ │ │ │  0001ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001baa0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0001bab0: 2020 2031 2020 2032 2020 2020 2020 2020     1   2        
│ │ │ │ -0001bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bad0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0001bae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001baf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bb00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720  ---------+.|i17 
│ │ │ │ -0001bb10: 3a20 4575 6c65 724b 3d3d 4575 6c65 7228  : EulerK==Euler(
│ │ │ │ -0001bb20: 6373 6d4b 2920 2020 2020 2020 2020 2020  csmK)           
│ │ │ │ -0001bb30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001baa0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001bab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bad0: 2d2d 2d2d 2d2d 2b0a 7c69 3137 203a 2045  ------+.|i17 : E
│ │ │ │ +0001bae0: 756c 6572 4b3d 3d45 756c 6572 2863 736d  ulerK==Euler(csm
│ │ │ │ +0001baf0: 4b29 2020 2020 2020 2020 2020 2020 2020  K)              
│ │ │ │ +0001bb00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bb30: 2020 2020 2020 2020 7c0a 7c6f 3137 203d          |.|o17 =
│ │ │ │ +0001bb40: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │  0001bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bb60: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0001bb70: 3720 3d20 7472 7565 2020 2020 2020 2020  7 = true        
│ │ │ │ -0001bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bb90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0001bba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -0001bbd0: 496e 2074 6865 2063 6173 6520 7768 6572  In the case wher
│ │ │ │ -0001bbe0: 6520 7468 6520 616d 6269 656e 7420 7370  e the ambient sp
│ │ │ │ -0001bbf0: 6163 6520 6973 2061 2074 6f72 6963 2076  ace is a toric v
│ │ │ │ -0001bc00: 6172 6965 7479 2077 6869 6368 2069 7320  ariety which is 
│ │ │ │ -0001bc10: 6e6f 7420 6120 7072 6f64 7563 740a 6f66  not a product.of
│ │ │ │ -0001bc20: 2070 726f 6a65 6374 6976 6520 7370 6163   projective spac
│ │ │ │ -0001bc30: 6573 2077 6520 6d75 7374 206c 6f61 6420  es we must load 
│ │ │ │ -0001bc40: 7468 6520 4e6f 726d 616c 546f 7269 6356  the NormalToricV
│ │ │ │ -0001bc50: 6172 6965 7469 6573 2070 6163 6b61 6765  arieties package
│ │ │ │ -0001bc60: 2061 6e64 206d 7573 740a 616c 736f 2069   and must.also i
│ │ │ │ -0001bc70: 6e70 7574 2074 6865 2074 6f72 6963 2076  nput the toric v
│ │ │ │ -0001bc80: 6172 6965 7479 2e20 4966 2074 6865 2074  ariety. If the t
│ │ │ │ -0001bc90: 6f72 6963 2076 6172 6965 7479 2069 7320  oric variety is 
│ │ │ │ -0001bca0: 6120 7072 6f64 7563 7420 6f66 2070 726f  a product of pro
│ │ │ │ -0001bcb0: 6a65 6374 6976 650a 7370 6163 6520 6974  jective.space it
│ │ │ │ -0001bcc0: 2069 7320 7265 636f 6d6d 656e 6465 6420   is recommended 
│ │ │ │ -0001bcd0: 746f 2075 7365 2074 6865 2066 6f72 6d20  to use the form 
│ │ │ │ -0001bce0: 6162 6f76 6520 7261 7468 6572 2074 6861  above rather tha
│ │ │ │ -0001bcf0: 6e20 696e 7075 7474 696e 6720 7468 6520  n inputting the 
│ │ │ │ -0001bd00: 746f 7269 630a 7661 7269 6574 7920 666f  toric.variety fo
│ │ │ │ -0001bd10: 7220 6566 6669 6369 656e 6379 2072 6561  r efficiency rea
│ │ │ │ -0001bd20: 736f 6e73 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  sons...+--------
│ │ │ │ -0001bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd70: 2b0a 7c69 3138 203a 206e 6565 6473 5061  +.|i18 : needsPa
│ │ │ │ -0001bd80: 636b 6167 6520 224e 6f72 6d61 6c54 6f72  ckage "NormalTor
│ │ │ │ -0001bd90: 6963 5661 7269 6574 6965 7322 2020 2020  icVarieties"    
│ │ │ │ +0001bb60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6e20  ----------+..In 
│ │ │ │ +0001bba0: 7468 6520 6361 7365 2077 6865 7265 2074  the case where t
│ │ │ │ +0001bbb0: 6865 2061 6d62 6965 6e74 2073 7061 6365  he ambient space
│ │ │ │ +0001bbc0: 2069 7320 6120 746f 7269 6320 7661 7269   is a toric vari
│ │ │ │ +0001bbd0: 6574 7920 7768 6963 6820 6973 206e 6f74  ety which is not
│ │ │ │ +0001bbe0: 2061 2070 726f 6475 6374 0a6f 6620 7072   a product.of pr
│ │ │ │ +0001bbf0: 6f6a 6563 7469 7665 2073 7061 6365 7320  ojective spaces 
│ │ │ │ +0001bc00: 7765 206d 7573 7420 6c6f 6164 2074 6865  we must load the
│ │ │ │ +0001bc10: 204e 6f72 6d61 6c54 6f72 6963 5661 7269   NormalToricVari
│ │ │ │ +0001bc20: 6574 6965 7320 7061 636b 6167 6520 616e  eties package an
│ │ │ │ +0001bc30: 6420 6d75 7374 0a61 6c73 6f20 696e 7075  d must.also inpu
│ │ │ │ +0001bc40: 7420 7468 6520 746f 7269 6320 7661 7269  t the toric vari
│ │ │ │ +0001bc50: 6574 792e 2049 6620 7468 6520 746f 7269  ety. If the tori
│ │ │ │ +0001bc60: 6320 7661 7269 6574 7920 6973 2061 2070  c variety is a p
│ │ │ │ +0001bc70: 726f 6475 6374 206f 6620 7072 6f6a 6563  roduct of projec
│ │ │ │ +0001bc80: 7469 7665 0a73 7061 6365 2069 7420 6973  tive.space it is
│ │ │ │ +0001bc90: 2072 6563 6f6d 6d65 6e64 6564 2074 6f20   recommended to 
│ │ │ │ +0001bca0: 7573 6520 7468 6520 666f 726d 2061 626f  use the form abo
│ │ │ │ +0001bcb0: 7665 2072 6174 6865 7220 7468 616e 2069  ve rather than i
│ │ │ │ +0001bcc0: 6e70 7574 7469 6e67 2074 6865 2074 6f72  nputting the tor
│ │ │ │ +0001bcd0: 6963 0a76 6172 6965 7479 2066 6f72 2065  ic.variety for e
│ │ │ │ +0001bce0: 6666 6963 6965 6e63 7920 7265 6173 6f6e  fficiency reason
│ │ │ │ +0001bcf0: 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  s...+-----------
│ │ │ │ +0001bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001bd40: 6931 3820 3a20 6e65 6564 7350 6163 6b61  i18 : needsPacka
│ │ │ │ +0001bd50: 6765 2022 4e6f 726d 616c 546f 7269 6356  ge "NormalToricV
│ │ │ │ +0001bd60: 6172 6965 7469 6573 2220 2020 2020 2020  arieties"       
│ │ │ │ +0001bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bd80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001bd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bdb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bdd0: 2020 207c 0a7c 6f31 3820 3d20 4e6f 726d     |.|o18 = Norm
│ │ │ │ +0001bde0: 616c 546f 7269 6356 6172 6965 7469 6573  alToricVarieties
│ │ │ │  0001bdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be00: 2020 2020 2020 7c0a 7c6f 3138 203d 204e        |.|o18 = N
│ │ │ │ -0001be10: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ -0001be20: 6965 7320 2020 2020 2020 2020 2020 2020  ies             
│ │ │ │ +0001be00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001be10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001be20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0001be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001be60: 2020 2020 2020 2020 207c 0a7c 6f31 3820           |.|o18 
│ │ │ │ +0001be70: 3a20 5061 636b 6167 6520 2020 2020 2020  : Package       
│ │ │ │  0001be80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001bea0: 3138 203a 2050 6163 6b61 6765 2020 2020  18 : Package    
│ │ │ │ -0001beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bee0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -0001bef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bf00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bf10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bf20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bf30: 2d2d 2b0a 7c69 3139 203a 2052 686f 203d  --+.|i19 : Rho =
│ │ │ │ -0001bf40: 207b 7b31 2c30 2c30 7d2c 7b30 2c31 2c30   {{1,0,0},{0,1,0
│ │ │ │ -0001bf50: 7d2c 7b30 2c30 2c31 7d2c 7b2d 312c 2d31  },{0,0,1},{-1,-1
│ │ │ │ -0001bf60: 2c30 7d2c 7b30 2c30 2c2d 317d 7d20 2020  ,0},{0,0,-1}}   
│ │ │ │ -0001bf70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001beb0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001bec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001bf00: 0a7c 6931 3920 3a20 5268 6f20 3d20 7b7b  .|i19 : Rho = {{
│ │ │ │ +0001bf10: 312c 302c 307d 2c7b 302c 312c 307d 2c7b  1,0,0},{0,1,0},{
│ │ │ │ +0001bf20: 302c 302c 317d 2c7b 2d31 2c2d 312c 307d  0,0,1},{-1,-1,0}
│ │ │ │ +0001bf30: 2c7b 302c 302c 2d31 7d7d 2020 2020 2020  ,{0,0,-1}}      
│ │ │ │ +0001bf40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bfc0: 2020 2020 2020 2020 7c0a 7c6f 3139 203d          |.|o19 =
│ │ │ │ -0001bfd0: 207b 7b31 2c20 302c 2030 7d2c 207b 302c   {{1, 0, 0}, {0,
│ │ │ │ -0001bfe0: 2031 2c20 307d 2c20 7b30 2c20 302c 2031   1, 0}, {0, 0, 1
│ │ │ │ -0001bff0: 7d2c 207b 2d31 2c20 2d31 2c20 307d 2c20  }, {-1, -1, 0}, 
│ │ │ │ -0001c000: 7b30 2c20 302c 202d 317d 7d20 2020 2020  {0, 0, -1}}     
│ │ │ │ -0001c010: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bf90: 2020 2020 207c 0a7c 6f31 3920 3d20 7b7b       |.|o19 = {{
│ │ │ │ +0001bfa0: 312c 2030 2c20 307d 2c20 7b30 2c20 312c  1, 0, 0}, {0, 1,
│ │ │ │ +0001bfb0: 2030 7d2c 207b 302c 2030 2c20 317d 2c20   0}, {0, 0, 1}, 
│ │ │ │ +0001bfc0: 7b2d 312c 202d 312c 2030 7d2c 207b 302c  {-1, -1, 0}, {0,
│ │ │ │ +0001bfd0: 2030 2c20 2d31 7d7d 2020 2020 2020 2020   0, -1}}        
│ │ │ │ +0001bfe0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001bff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c020: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001c030: 3920 3a20 4c69 7374 2020 2020 2020 2020  9 : List        
│ │ │ │  0001c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c050: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001c060: 7c6f 3139 203a 204c 6973 7420 2020 2020  |o19 : List     
│ │ │ │ -0001c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c0a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001c050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c070: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001c080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c0f0: 2d2d 2d2d 2b0a 7c69 3230 203a 2053 6967  ----+.|i20 : Sig
│ │ │ │ -0001c100: 6d61 203d 207b 7b30 2c31 2c32 7d2c 7b31  ma = {{0,1,2},{1
│ │ │ │ -0001c110: 2c32 2c33 7d2c 7b30 2c32 2c33 7d2c 7b30  ,2,3},{0,2,3},{0
│ │ │ │ -0001c120: 2c31 2c34 7d2c 7b31 2c33 2c34 7d2c 7b30  ,1,4},{1,3,4},{0
│ │ │ │ -0001c130: 2c33 2c34 7d7d 2020 2020 2020 2020 207c  ,3,4}}         |
│ │ │ │ -0001c140: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c180: 2020 2020 2020 2020 2020 7c0a 7c6f 3230            |.|o20
│ │ │ │ -0001c190: 203d 207b 7b30 2c20 312c 2032 7d2c 207b   = {{0, 1, 2}, {
│ │ │ │ -0001c1a0: 312c 2032 2c20 337d 2c20 7b30 2c20 322c  1, 2, 3}, {0, 2,
│ │ │ │ -0001c1b0: 2033 7d2c 207b 302c 2031 2c20 347d 2c20   3}, {0, 1, 4}, 
│ │ │ │ -0001c1c0: 7b31 2c20 332c 2034 7d2c 207b 302c 2033  {1, 3, 4}, {0, 3
│ │ │ │ -0001c1d0: 2c20 347d 7d7c 0a7c 2020 2020 2020 2020  , 4}}|.|        
│ │ │ │ -0001c1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c0c0: 2d2b 0a7c 6932 3020 3a20 5369 676d 6120  -+.|i20 : Sigma 
│ │ │ │ +0001c0d0: 3d20 7b7b 302c 312c 327d 2c7b 312c 322c  = {{0,1,2},{1,2,
│ │ │ │ +0001c0e0: 337d 2c7b 302c 322c 337d 2c7b 302c 312c  3},{0,2,3},{0,1,
│ │ │ │ +0001c0f0: 347d 2c7b 312c 332c 347d 2c7b 302c 332c  4},{1,3,4},{0,3,
│ │ │ │ +0001c100: 347d 7d20 2020 2020 2020 2020 7c0a 7c20  4}}         |.| 
│ │ │ │ +0001c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c150: 2020 2020 2020 207c 0a7c 6f32 3020 3d20         |.|o20 = 
│ │ │ │ +0001c160: 7b7b 302c 2031 2c20 327d 2c20 7b31 2c20  {{0, 1, 2}, {1, 
│ │ │ │ +0001c170: 322c 2033 7d2c 207b 302c 2032 2c20 337d  2, 3}, {0, 2, 3}
│ │ │ │ +0001c180: 2c20 7b30 2c20 312c 2034 7d2c 207b 312c  , {0, 1, 4}, {1,
│ │ │ │ +0001c190: 2033 2c20 347d 2c20 7b30 2c20 332c 2034   3, 4}, {0, 3, 4
│ │ │ │ +0001c1a0: 7d7d 7c0a 7c20 2020 2020 2020 2020 2020  }}|.|           
│ │ │ │ +0001c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c1e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001c1f0: 6f32 3020 3a20 4c69 7374 2020 2020 2020  o20 : List      
│ │ │ │  0001c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c220: 7c0a 7c6f 3230 203a 204c 6973 7420 2020  |.|o20 : List   
│ │ │ │ -0001c230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c260: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001c220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c230: 2020 2020 2020 2020 7c0a 2b2d 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 2d2d 2d2d  ----------------
│ │ │ │  0001c270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c2b0: 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a 2058  ------+.|i21 : X
│ │ │ │ -0001c2c0: 203d 206e 6f72 6d61 6c54 6f72 6963 5661   = normalToricVa
│ │ │ │ -0001c2d0: 7269 6574 7928 5268 6f2c 5369 676d 612c  riety(Rho,Sigma,
│ │ │ │ -0001c2e0: 436f 6566 6669 6369 656e 7452 696e 6720  CoefficientRing 
│ │ │ │ -0001c2f0: 3d3e 5a5a 2f33 3237 3439 2920 2020 2020  =>ZZ/32749)     
│ │ │ │ -0001c300: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c280: 2d2d 2d2b 0a7c 6932 3120 3a20 5820 3d20  ---+.|i21 : X = 
│ │ │ │ +0001c290: 6e6f 726d 616c 546f 7269 6356 6172 6965  normalToricVarie
│ │ │ │ +0001c2a0: 7479 2852 686f 2c53 6967 6d61 2c43 6f65  ty(Rho,Sigma,Coe
│ │ │ │ +0001c2b0: 6666 6963 6965 6e74 5269 6e67 203d 3e5a  fficientRing =>Z
│ │ │ │ +0001c2c0: 5a2f 3332 3734 3929 2020 2020 2020 7c0a  Z/32749)      |.
│ │ │ │ +0001c2d0: 7c20 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c310: 2020 2020 2020 2020 207c 0a7c 6f32 3120           |.|o21 
│ │ │ │ +0001c320: 3d20 5820 2020 2020 2020 2020 2020 2020  = X             
│ │ │ │  0001c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c340: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001c350: 3231 203d 2058 2020 2020 2020 2020 2020  21 = X          
│ │ │ │ -0001c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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      |.|         
│ │ │ │  0001c370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c390: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0001c3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c3a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001c3b0: 0a7c 6f32 3120 3a20 4e6f 726d 616c 546f  .|o21 : NormalTo
│ │ │ │ +0001c3c0: 7269 6356 6172 6965 7479 2020 2020 2020  ricVariety      
│ │ │ │  0001c3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c3e0: 2020 7c0a 7c6f 3231 203a 204e 6f72 6d61    |.|o21 : Norma
│ │ │ │ -0001c3f0: 6c54 6f72 6963 5661 7269 6574 7920 2020  lToricVariety   
│ │ │ │ -0001c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c420: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001c3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c3f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001c400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c470: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a  --------+.|i22 :
│ │ │ │ -0001c480: 2043 6865 636b 546f 7269 6356 6172 6965   CheckToricVarie
│ │ │ │ -0001c490: 7479 5661 6c69 6428 5829 2020 2020 2020  tyValid(X)      
│ │ │ │ +0001c440: 2d2d 2d2d 2d2b 0a7c 6932 3220 3a20 4368  -----+.|i22 : Ch
│ │ │ │ +0001c450: 6563 6b54 6f72 6963 5661 7269 6574 7956  eckToricVarietyV
│ │ │ │ +0001c460: 616c 6964 2858 2920 2020 2020 2020 2020  alid(X)         
│ │ │ │ +0001c470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c490: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c4c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001c4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c4d0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0001c4e0: 3220 3d20 7472 7565 2020 2020 2020 2020  2 = true        
│ │ │ │  0001c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c500: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001c510: 7c6f 3232 203d 2074 7275 6520 2020 2020  |o22 = true     
│ │ │ │ -0001c520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c550: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c520: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001c530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c5a0: 2d2d 2d2d 2b0a 7c69 3233 203a 2052 3d72  ----+.|i23 : R=r
│ │ │ │ -0001c5b0: 696e 6728 5829 2020 2020 2020 2020 2020  ing(X)          
│ │ │ │ +0001c570: 2d2b 0a7c 6932 3320 3a20 523d 7269 6e67  -+.|i23 : R=ring
│ │ │ │ +0001c580: 2858 2920 2020 2020 2020 2020 2020 2020  (X)             
│ │ │ │ +0001c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c5e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001c5f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0001c600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c600: 2020 2020 2020 207c 0a7c 6f32 3320 3d20         |.|o23 = 
│ │ │ │ +0001c610: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  0001c620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c630: 2020 2020 2020 2020 2020 7c0a 7c6f 3233            |.|o23
│ │ │ │ -0001c640: 203d 2052 2020 2020 2020 2020 2020 2020   = R            
│ │ │ │ -0001c650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c650: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001c660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c680: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001c690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c690: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001c6a0: 6f32 3320 3a20 506f 6c79 6e6f 6d69 616c  o23 : Polynomial
│ │ │ │ +0001c6b0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │  0001c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c6d0: 7c0a 7c6f 3233 203a 2050 6f6c 796e 6f6d  |.|o23 : Polynom
│ │ │ │ -0001c6e0: 6961 6c52 696e 6720 2020 2020 2020 2020  ialRing         
│ │ │ │ -0001c6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c710: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c6e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001c6f0: 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c760: 2d2d 2d2d 2d2d 2b0a 7c69 3234 203a 2049  ------+.|i24 : I
│ │ │ │ -0001c770: 3d69 6465 616c 2852 5f30 5e34 2a52 5f31  =ideal(R_0^4*R_1
│ │ │ │ -0001c780: 2c52 5f30 2a52 5f33 2a52 5f34 2a52 5f32  ,R_0*R_3*R_4*R_2
│ │ │ │ -0001c790: 2d52 5f32 5e32 2a52 5f30 5e32 2920 2020  -R_2^2*R_0^2)   
│ │ │ │ +0001c730: 2d2d 2d2b 0a7c 6932 3420 3a20 493d 6964  ---+.|i24 : I=id
│ │ │ │ +0001c740: 6561 6c28 525f 305e 342a 525f 312c 525f  eal(R_0^4*R_1,R_
│ │ │ │ +0001c750: 302a 525f 332a 525f 342a 525f 322d 525f  0*R_3*R_4*R_2-R_
│ │ │ │ +0001c760: 325e 322a 525f 305e 3229 2020 2020 2020  2^2*R_0^2)      
│ │ │ │ +0001c770: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001c780: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001c7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001c800: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ -0001c810: 2020 2020 2032 2032 2020 2020 2020 2020       2 2        
│ │ │ │ -0001c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c840: 2020 2020 2020 207c 0a7c 6f32 3420 3d20         |.|o24 = 
│ │ │ │ -0001c850: 6964 6561 6c20 2878 2078 202c 202d 2078  ideal (x x , - x
│ │ │ │ -0001c860: 2078 2020 2b20 7820 7820 7820 7820 2920   x  + x x x x ) 
│ │ │ │ -0001c870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c890: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0001c8a0: 2020 2030 2031 2020 2020 2030 2032 2020     0 1     0 2  
│ │ │ │ -0001c8b0: 2020 3020 3220 3320 3420 2020 2020 2020    0 2 3 4       
│ │ │ │ +0001c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c7c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001c7d0: 2020 2020 2020 2020 2020 3420 2020 2020            4     
│ │ │ │ +0001c7e0: 2020 3220 3220 2020 2020 2020 2020 2020    2 2           
│ │ │ │ +0001c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c810: 2020 2020 7c0a 7c6f 3234 203d 2069 6465      |.|o24 = ide
│ │ │ │ +0001c820: 616c 2028 7820 7820 2c20 2d20 7820 7820  al (x x , - x x 
│ │ │ │ +0001c830: 202b 2078 2078 2078 2078 2029 2020 2020   + x x x x )    
│ │ │ │ +0001c840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c850: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001c860: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001c870: 3020 3120 2020 2020 3020 3220 2020 2030  0 1     0 2    0
│ │ │ │ +0001c880: 2032 2033 2034 2020 2020 2020 2020 2020   2 3 4          
│ │ │ │ +0001c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c8a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c8d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001c8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c8f0: 2020 2020 207c 0a7c 6f32 3420 3a20 4964       |.|o24 : Id
│ │ │ │ +0001c900: 6561 6c20 6f66 2052 2020 2020 2020 2020  eal of R        
│ │ │ │  0001c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c920: 2020 2020 2020 2020 7c0a 7c6f 3234 203a          |.|o24 :
│ │ │ │ -0001c930: 2049 6465 616c 206f 6620 5220 2020 2020   Ideal of R     
│ │ │ │ -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 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0001c980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001c9c0: 7c69 3235 203a 2063 736d 493d 4353 4d28  |i25 : csmI=CSM(
│ │ │ │ -0001c9d0: 582c 4929 2020 2020 2020 2020 2020 2020  X,I)            
│ │ │ │ +0001c920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c940: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0001c950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0001c990: 3520 3a20 6373 6d49 3d43 534d 2858 2c49  5 : csmI=CSM(X,I
│ │ │ │ +0001c9a0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0001c9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c9d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001ca00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ca10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ca20: 207c 0a7c 2020 2020 2020 2020 3220 2020   |.|        2   
│ │ │ │ +0001ca30: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  0001ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca50: 2020 2020 7c0a 7c20 2020 2020 2020 2032      |.|        2
│ │ │ │ -0001ca60: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0001ca70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001caa0: 0a7c 6f32 3520 3d20 3578 2078 2020 2b20  .|o25 = 5x x  + 
│ │ │ │ -0001cab0: 3378 2020 2b20 3478 2078 2020 2b20 7820  3x  + 4x x  + x 
│ │ │ │ -0001cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001caf0: 2020 2020 2033 2034 2020 2020 2033 2020       3 4     3  
│ │ │ │ -0001cb00: 2020 2033 2034 2020 2020 3320 2020 2020     3 4    3     
│ │ │ │ +0001ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ca60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0001ca70: 3235 203d 2035 7820 7820 202b 2033 7820  25 = 5x x  + 3x 
│ │ │ │ +0001ca80: 202b 2034 7820 7820 202b 2078 2020 2020   + 4x x  + x    
│ │ │ │ +0001ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cab0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001cac0: 2020 3320 3420 2020 2020 3320 2020 2020    3 4     3     
│ │ │ │ +0001cad0: 3320 3420 2020 2033 2020 2020 2020 2020  3 4    3        
│ │ │ │ +0001cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001caf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cb00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cb30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cb40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cb60: 2020 2020 2020 5a5a 5b78 202e 2e78 205d        ZZ[x ..x ]
│ │ │ │  0001cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cb80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0001cb90: 2020 2020 2020 2020 205a 5a5b 7820 2e2e           ZZ[x ..
│ │ │ │ -0001cba0: 7820 5d20 2020 2020 2020 2020 2020 2020  x ]             
│ │ │ │ -0001cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cbc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cb90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001cba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cbb0: 2020 2020 2030 2020 2034 2020 2020 2020       0   4      
│ │ │ │ +0001cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cbe0: 2020 2020 2020 2020 3020 2020 3420 2020          0   4   
│ │ │ │ -0001cbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc10: 2020 2020 2020 7c0a 7c6f 3235 203a 202d        |.|o25 : -
│ │ │ │ -0001cc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cc40: 2d2d 2d2d 2d2d 2d2d 2020 2020 2020 2020  --------        
│ │ │ │ -0001cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc60: 207c 0a7c 2020 2020 2020 2878 2078 202c   |.|      (x x ,
│ │ │ │ -0001cc70: 2078 2078 2078 202c 2078 2020 2d20 7820   x x x , x  - x 
│ │ │ │ -0001cc80: 2c20 7820 202d 2078 202c 2078 2020 2d20  , x  - x , x  - 
│ │ │ │ -0001cc90: 7820 2920 2020 2020 2020 2020 2020 2020  x )             
│ │ │ │ -0001cca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001ccb0: 2020 2020 2020 2032 2034 2020 2030 2031         2 4   0 1
│ │ │ │ -0001ccc0: 2033 2020 2030 2020 2020 3320 2020 3120   3   0    3   1 
│ │ │ │ -0001ccd0: 2020 2033 2020 2032 2020 2020 3420 2020     3   2    4   
│ │ │ │ -0001cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ccf0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -0001cd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cd40: 2d2d 2b0a 7c69 3236 203a 2045 756c 6572  --+.|i26 : Euler
│ │ │ │ -0001cd50: 493d 4575 6c65 7228 582c 4929 2020 2020  I=Euler(X,I)    
│ │ │ │ +0001cbe0: 2020 207c 0a7c 6f32 3520 3a20 2d2d 2d2d     |.|o25 : ----
│ │ │ │ +0001cbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cc10: 2d2d 2d2d 2d20 2020 2020 2020 2020 2020  -----           
│ │ │ │ +0001cc20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001cc30: 7c20 2020 2020 2028 7820 7820 2c20 7820  |      (x x , x 
│ │ │ │ +0001cc40: 7820 7820 2c20 7820 202d 2078 202c 2078  x x , x  - x , x
│ │ │ │ +0001cc50: 2020 2d20 7820 2c20 7820 202d 2078 2029    - x , x  - x )
│ │ │ │ +0001cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cc70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001cc80: 2020 2020 3220 3420 2020 3020 3120 3320      2 4   0 1 3 
│ │ │ │ +0001cc90: 2020 3020 2020 2033 2020 2031 2020 2020    0    3   1    
│ │ │ │ +0001cca0: 3320 2020 3220 2020 2034 2020 2020 2020  3   2    4      
│ │ │ │ +0001ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ccc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001ccd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ccf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001cd10: 0a7c 6932 3620 3a20 4575 6c65 7249 3d45  .|i26 : EulerI=E
│ │ │ │ +0001cd20: 756c 6572 2858 2c49 2920 2020 2020 2020  uler(X,I)       
│ │ │ │ +0001cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cd50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0001cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cda0: 2020 2020 207c 0a7c 6f32 3620 3d20 3520       |.|o26 = 5 
│ │ │ │  0001cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cdd0: 2020 2020 2020 2020 7c0a 7c6f 3236 203d          |.|o26 =
│ │ │ │ -0001cde0: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ -0001cdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0001ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001ce70: 7c69 3237 203a 2045 756c 6572 2863 736d  |i27 : Euler(csm
│ │ │ │ -0001ce80: 4929 3d3d 4575 6c65 7249 2020 2020 2020  I)==EulerI      
│ │ │ │ +0001cdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cdf0: 7c0a 2b2d 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 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0001ce40: 3720 3a20 4575 6c65 7228 6373 6d49 293d  7 : Euler(csmI)=
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│ │ │ │ +0001ce60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ce70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ce80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001ce90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ceb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001ceb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ced0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ced0: 207c 0a7c 6f32 3720 3d20 7472 7565 2020   |.|o27 = true  
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│ │ │ │  0001cef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf00: 2020 2020 7c0a 7c6f 3237 203d 2074 7275      |.|o27 = tru
│ │ │ │ -0001cf10: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ -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 207c                 |
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│ │ │ │ -0001cf70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cf80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001d160: 2070 726f 6261 6269 6c69 7479 2e20 5265   probability. Re
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│ │ │ │ +0001cf30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cf40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cf50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0001cfd0: 2043 6f6d 704d 6574 686f 643a 2043 6f6d   CompMethod: Com
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│ │ │ │ +0001d180: 6d2c 2e0a 0a57 6179 7320 746f 2075 7365  m,...Ways to use
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│ │ │ │ +0001d1a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +0001d1b0: 2245 756c 6572 2849 6465 616c 2922 0a20  "Euler(Ideal)". 
│ │ │ │ +0001d1c0: 202a 2022 4575 6c65 7228 5269 6e67 456c   * "Euler(RingEl
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│ │ │ │ +0001d1f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +0001d200: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
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│ │ │ │ +0001d290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001d3e0: 4575 6c65 7241 6666 696e 6520 2d2d 2054  EulerAffine -- T
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│ │ │ │ -0001d420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ -0001d460: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -0001d470: 2045 756c 6572 4166 6669 6e65 2049 0a20   EulerAffine I. 
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│ │ │ │ -0001d490: 202a 2049 2c20 616e 202a 6e6f 7465 2069   * I, an *note i
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│ │ │ │ -0001d4d0: 6d69 616c 2072 696e 670a 2020 2020 2020  mial ring.      
│ │ │ │ -0001d4e0: 2020 6f76 6572 2061 2066 6965 6c64 2064    over a field d
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│ │ │ │ -0001d510: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ -0001d520: 6120 2a6e 6f74 6520 7269 6e67 2065 6c65  a *note ring ele
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│ │ │ │ -0001d5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d610: 2d2d 2b0a 7c69 3120 3a20 6b6b 3d5a 5a2f  --+.|i1 : kk=ZZ/
│ │ │ │ -0001d620: 3332 3734 393b 2020 2020 2020 2020 2020  32749;          
│ │ │ │ -0001d630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d640: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0001d650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d680: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 523d  ------+.|i2 : R=
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│ │ │ │ +0001d350: 6c65 3a20 4368 6172 6163 7465 7269 7374  le: Characterist
│ │ │ │ +0001d360: 6963 436c 6173 7365 732e 696e 666f 2c20  icClasses.info, 
│ │ │ │ +0001d370: 4e6f 6465 3a20 4575 6c65 7241 6666 696e  Node: EulerAffin
│ │ │ │ +0001d380: 652c 204e 6578 743a 2049 6e64 734f 6653  e, Next: IndsOfS
│ │ │ │ +0001d390: 6d6f 6f74 682c 2050 7265 763a 2045 756c  mooth, Prev: Eul
│ │ │ │ +0001d3a0: 6572 2c20 5570 3a20 546f 700a 0a45 756c  er, Up: Top..Eul
│ │ │ │ +0001d3b0: 6572 4166 6669 6e65 202d 2d20 5468 6520  erAffine -- The 
│ │ │ │ +0001d3c0: 4575 6c65 7220 4368 6172 6163 7465 7269  Euler Characteri
│ │ │ │ +0001d3d0: 7374 6963 206f 6620 616e 2061 6666 696e  stic of an affin
│ │ │ │ +0001d3e0: 6520 7661 7269 6574 792e 0a2a 2a2a 2a2a  e variety..*****
│ │ │ │ +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 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ +0001d430: 6167 653a 200a 2020 2020 2020 2020 4575  age: .        Eu
│ │ │ │ +0001d440: 6c65 7241 6666 696e 6520 490a 2020 2a20  lerAffine I.  * 
│ │ │ │ +0001d450: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +0001d460: 492c 2061 6e20 2a6e 6f74 6520 6964 6561  I, an *note idea
│ │ │ │ +0001d470: 6c3a 2028 4d61 6361 756c 6179 3244 6f63  l: (Macaulay2Doc
│ │ │ │ +0001d480: 2949 6465 616c 2c2c 2061 6e20 6964 6561  )Ideal,, an idea
│ │ │ │ +0001d490: 6c20 696e 2061 2070 6f6c 796e 6f6d 6961  l in a polynomia
│ │ │ │ +0001d4a0: 6c20 7269 6e67 0a20 2020 2020 2020 206f  l ring.        o
│ │ │ │ +0001d4b0: 7665 7220 6120 6669 656c 6420 6465 6669  ver a field defi
│ │ │ │ +0001d4c0: 6e69 6e67 2061 6e20 6166 6669 6e65 2076  ning an affine v
│ │ │ │ +0001d4d0: 6172 6965 7479 2e0a 2020 2a20 4f75 7470  ariety..  * Outp
│ │ │ │ +0001d4e0: 7574 733a 0a20 2020 2020 202a 2061 202a  uts:.      * a *
│ │ │ │ +0001d4f0: 6e6f 7465 2072 696e 6720 656c 656d 656e  note ring elemen
│ │ │ │ +0001d500: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ +0001d510: 2952 696e 6745 6c65 6d65 6e74 2c2c 2074  )RingElement,, t
│ │ │ │ +0001d520: 6865 2045 756c 6572 0a20 2020 2020 2020  he Euler.       
│ │ │ │ +0001d530: 2063 6861 7261 6374 6572 6973 7469 630a   characteristic.
│ │ │ │ +0001d540: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +0001d550: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 2063  ========..This c
│ │ │ │ +0001d560: 6f6d 6d61 6e64 2063 6f6d 7075 7465 7320  ommand computes 
│ │ │ │ +0001d570: 7468 6520 4575 6c65 7220 6368 6172 6163  the Euler charac
│ │ │ │ +0001d580: 7465 7269 7374 6963 206f 6620 6120 636f  teristic of a co
│ │ │ │ +0001d590: 6d70 6c65 7820 6166 6669 6e65 2076 6172  mplex affine var
│ │ │ │ +0001d5a0: 6965 7479 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  iety...+--------
│ │ │ │ +0001d5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001d5e0: 0a7c 6931 203a 206b 6b3d 5a5a 2f33 3237  .|i1 : kk=ZZ/327
│ │ │ │ +0001d5f0: 3439 3b20 2020 2020 2020 2020 2020 2020  49;             
│ │ │ │ +0001d600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d610: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001d620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d650: 2d2d 2d2b 0a7c 6932 203a 2052 3d6b 6b5b  ---+.|i2 : R=kk[
│ │ │ │ +0001d660: 785f 312e 2e78 5f33 5d20 2020 2020 2020  x_1..x_3]       
│ │ │ │ +0001d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d680: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001d690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d6c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001d6c0: 2020 2020 2020 207c 0a7c 6f32 203d 2052         |.|o2 = R
│ │ │ │  0001d6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d6f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -0001d700: 3d20 5220 2020 2020 2020 2020 2020 2020  = R             
│ │ │ │ +0001d6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d700: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001d710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d730: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001d740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d760: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001d770: 7c6f 3220 3a20 506f 6c79 6e6f 6d69 616c  |o2 : Polynomial
│ │ │ │ -0001d780: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ -0001d790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d7a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0001d7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d7e0: 2d2d 2b0a 7c69 3320 3a20 493d 6964 6561  --+.|i3 : I=idea
│ │ │ │ -0001d7f0: 6c28 785f 315e 322b 785f 325e 322b 785f  l(x_1^2+x_2^2+x_
│ │ │ │ -0001d800: 335e 322d 3129 2020 2020 2020 2020 2020  3^2-1)          
│ │ │ │ -0001d810: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001d820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d730: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0001d740: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ +0001d750: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +0001d760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d770: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001d780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001d7b0: 0a7c 6933 203a 2049 3d69 6465 616c 2878  .|i3 : I=ideal(x
│ │ │ │ +0001d7c0: 5f31 5e32 2b78 5f32 5e32 2b78 5f33 5e32  _1^2+x_2^2+x_3^2
│ │ │ │ +0001d7d0: 2d31 2920 2020 2020 2020 2020 2020 2020  -1)             
│ │ │ │ +0001d7e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001d7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d820: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001d830: 2020 3220 2020 2032 2020 2020 3220 2020    2    2    2   
│ │ │ │  0001d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d850: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001d860: 2020 2020 2032 2020 2020 3220 2020 2032       2    2    2
│ │ │ │ -0001d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d850: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001d860: 6f33 203d 2069 6465 616c 2878 2020 2b20  o3 = ideal(x  + 
│ │ │ │ +0001d870: 7820 202b 2078 2020 2d20 3129 2020 2020  x  + x  - 1)    
│ │ │ │  0001d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d890: 7c0a 7c6f 3320 3d20 6964 6561 6c28 7820  |.|o3 = ideal(x 
│ │ │ │ -0001d8a0: 202b 2078 2020 2b20 7820 202d 2031 2920   + x  + x  - 1) 
│ │ │ │ -0001d8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d8c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001d8d0: 2020 2020 2020 2020 2031 2020 2020 3220           1    2 
│ │ │ │ -0001d8e0: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +0001d890: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001d8a0: 2020 2020 2020 3120 2020 2032 2020 2020        1    2    
│ │ │ │ +0001d8b0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0001d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d8d0: 207c 0a7c 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d900: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +0001d910: 203a 2049 6465 616c 206f 6620 5220 2020   : Ideal of R   
│ │ │ │  0001d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d930: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001d940: 7c6f 3320 3a20 4964 6561 6c20 6f66 2052  |o3 : Ideal of R
│ │ │ │ -0001d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d970: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -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 3038 3734 3133  -- used 0.087413
│ │ │ │ -0001da00: 3173 2028 6370 7529 3b20 302e 3034 3932  1s (cpu); 0.0492
│ │ │ │ -0001da10: 3135 3973 2028 7468 7265 6164 293b 2030  159s (thread); 0
│ │ │ │ -0001da20: 7320 2867 6329 7c0a 7c20 2020 2020 2020  s (gc)|.|       
│ │ │ │ -0001da30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d940: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001d950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001d980: 0a7c 6934 203a 2074 696d 6520 4575 6c65  .|i4 : time Eule
│ │ │ │ +0001d990: 7241 6666 696e 6520 4920 2020 2020 2020  rAffine I       
│ │ │ │ +0001d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d9b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ +0001d9c0: 7573 6564 2030 2e30 3835 3136 3331 7320  used 0.0851631s 
│ │ │ │ +0001d9d0: 2863 7075 293b 2030 2e30 3638 3836 3233  (cpu); 0.0688623
│ │ │ │ +0001d9e0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +0001d9f0: 6763 297c 0a7c 2020 2020 2020 2020 2020  gc)|.|          
│ │ │ │ +0001da00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001da20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001da30: 6f34 203d 2032 2020 2020 2020 2020 2020  o4 = 2          
│ │ │ │  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                  
│ │ │ │ -0001da90: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0001daa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dad0: 2d2d 2d2d 2b0a 0a4f 6273 6572 7665 2074  ----+..Observe t
│ │ │ │ -0001dae0: 6861 7420 7468 6520 616c 676f 7269 7468  hat the algorith
│ │ │ │ -0001daf0: 6d20 6973 2061 2070 726f 6261 6269 6c69  m is a probabili
│ │ │ │ -0001db00: 7374 6963 2061 6c67 6f72 6974 686d 2061  stic algorithm a
│ │ │ │ -0001db10: 6e64 206d 6179 2067 6976 6520 6120 7772  nd may give a wr
│ │ │ │ -0001db20: 6f6e 670a 616e 7377 6572 2077 6974 6820  ong.answer with 
│ │ │ │ -0001db30: 6120 736d 616c 6c20 6275 7420 6e6f 6e7a  a small but nonz
│ │ │ │ -0001db40: 6572 6f20 7072 6f62 6162 696c 6974 792e  ero probability.
│ │ │ │ -0001db50: 2052 6561 6420 6d6f 7265 2075 6e64 6572   Read more under
│ │ │ │ -0001db60: 202a 6e6f 7465 0a70 726f 6261 6269 6c69   *note.probabili
│ │ │ │ -0001db70: 7374 6963 2061 6c67 6f72 6974 686d 3a20  stic algorithm: 
│ │ │ │ -0001db80: 7072 6f62 6162 696c 6973 7469 6320 616c  probabilistic al
│ │ │ │ -0001db90: 676f 7269 7468 6d2c 2e0a 0a57 6179 7320  gorithm,...Ways 
│ │ │ │ -0001dba0: 746f 2075 7365 2045 756c 6572 4166 6669  to use EulerAffi
│ │ │ │ -0001dbb0: 6e65 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ne:.============
│ │ │ │ -0001dbc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ -0001dbd0: 2a20 2245 756c 6572 4166 6669 6e65 2849  * "EulerAffine(I
│ │ │ │ -0001dbe0: 6465 616c 2922 0a0a 466f 7220 7468 6520  deal)"..For the 
│ │ │ │ -0001dbf0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -0001dc00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ -0001dc10: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ -0001dc20: 4575 6c65 7241 6666 696e 653a 2045 756c  EulerAffine: Eul
│ │ │ │ -0001dc30: 6572 4166 6669 6e65 2c20 6973 2061 202a  erAffine, is a *
│ │ │ │ -0001dc40: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ -0001dc50: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932  tion:.(Macaulay2
│ │ │ │ -0001dc60: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ -0001dc70: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  on,...----------
│ │ │ │ +0001da60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0001da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001daa0: 2d2b 0a0a 4f62 7365 7276 6520 7468 6174  -+..Observe that
│ │ │ │ +0001dab0: 2074 6865 2061 6c67 6f72 6974 686d 2069   the algorithm i
│ │ │ │ +0001dac0: 7320 6120 7072 6f62 6162 696c 6973 7469  s a probabilisti
│ │ │ │ +0001dad0: 6320 616c 676f 7269 7468 6d20 616e 6420  c algorithm and 
│ │ │ │ +0001dae0: 6d61 7920 6769 7665 2061 2077 726f 6e67  may give a wrong
│ │ │ │ +0001daf0: 0a61 6e73 7765 7220 7769 7468 2061 2073  .answer with a s
│ │ │ │ +0001db00: 6d61 6c6c 2062 7574 206e 6f6e 7a65 726f  mall but nonzero
│ │ │ │ +0001db10: 2070 726f 6261 6269 6c69 7479 2e20 5265   probability. Re
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│ │ │ │ +0001db30: 6f74 650a 7072 6f62 6162 696c 6973 7469  ote.probabilisti
│ │ │ │ +0001db40: 6320 616c 676f 7269 7468 6d3a 2070 726f  c algorithm: pro
│ │ │ │ +0001db50: 6261 6269 6c69 7374 6963 2061 6c67 6f72  babilistic algor
│ │ │ │ +0001db60: 6974 686d 2c2e 0a0a 5761 7973 2074 6f20  ithm,...Ways to 
│ │ │ │ +0001db70: 7573 6520 4575 6c65 7241 6666 696e 653a  use EulerAffine:
│ │ │ │ +0001db80: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +0001db90: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ +0001dba0: 4575 6c65 7241 6666 696e 6528 4964 6561  EulerAffine(Idea
│ │ │ │ +0001dbb0: 6c29 220a 0a46 6f72 2074 6865 2070 726f  l)"..For the pro
│ │ │ │ +0001dbc0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ +0001dbd0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ +0001dbe0: 6f62 6a65 6374 202a 6e6f 7465 2045 756c  object *note Eul
│ │ │ │ +0001dbf0: 6572 4166 6669 6e65 3a20 4575 6c65 7241  erAffine: EulerA
│ │ │ │ +0001dc00: 6666 696e 652c 2069 7320 6120 2a6e 6f74  ffine, is a *not
│ │ │ │ +0001dc10: 6520 6d65 7468 6f64 2066 756e 6374 696f  e method functio
│ │ │ │ +0001dc20: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63  n:.(Macaulay2Doc
│ │ │ │ +0001dc30: 294d 6574 686f 6446 756e 6374 696f 6e2c  )MethodFunction,
│ │ │ │ +0001dc40: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │ +0001dc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001dc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001dc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0001dea0: 6769 7669 6e67 2074 6865 206d 6574 686f  giving the metho
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│ │ │ │ +0001def0: 7768 656e 2074 616b 656e 2074 6f67 6574  when taken toget
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│ │ │ │ +0001df60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001df70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001dfa0: 2d2d 2b0a 7c69 3120 3a20 5220 3d20 4d75  --+.|i1 : R = Mu
│ │ │ │ +0001dfb0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67  ltiProjCoordRing
│ │ │ │ +0001dfc0: 287b 322c 327d 2920 2020 2020 2020 2020  ({2,2})         
│ │ │ │ +0001dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001dfe0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e010: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  0001e020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e030: 7c0a 7c6f 3120 3d20 5220 2020 2020 2020  |.|o1 = R       
│ │ │ │  0001e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e060: 2020 207c 0a7c 6f31 203d 2052 2020 2020     |.|o1 = R    
│ │ │ │ -0001e070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e070: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0001e080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e0a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e0b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001e0c0: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c  |o1 : Polynomial
│ │ │ │ +0001e0d0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
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│ │ │ │ -0001e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e130: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0001e140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0001e180: 0a7c 6932 203a 2049 3d69 6465 616c 2852  .|i2 : I=ideal(R
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│ │ │ │ -0001e1a0: 2a52 5f33 2c72 616e 646f 6d28 7b30 2c31  *R_3,random({0,1
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│ │ │ │ -0001e1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e100: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001e110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0001e150: 3220 3a20 493d 6964 6561 6c28 525f 302a  2 : I=ideal(R_0*
│ │ │ │ +0001e160: 525f 312a 525f 332d 525f 305e 322a 525f  R_1*R_3-R_0^2*R_
│ │ │ │ +0001e170: 332c 7261 6e64 6f6d 287b 302c 317d 2c52  3,random({0,1},R
│ │ │ │ +0001e180: 292c 7261 6e64 6f6d 287b 312c 327d 2c52  ),random({1,2},R
│ │ │ │ +0001e190: 2929 3b7c 0a7c 2020 2020 2020 2020 2020  ));|.|          
│ │ │ │ +0001e1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e1d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +0001e1e0: 3a20 4964 6561 6c20 6f66 2052 2020 2020  : Ideal of R    
│ │ │ │  0001e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e200: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001e210: 6f32 203a 2049 6465 616c 206f 6620 5220  o2 : Ideal of R 
│ │ │ │ -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 2020 2020                  
│ │ │ │ -0001e250: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0001e260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 322e    |.| -- used 2.
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│ │ │ │ -0001e300: 3132 3030 3373 2028 7468 7265 6164 293b  12003s (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                  
│ │ │ │ +0001e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e220: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0001e230: 2d2d 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 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +0001e270: 7469 6d65 2043 534d 2849 2c4d 6574 686f  time CSM(I,Metho
│ │ │ │ +0001e280: 643d 3e44 6972 6563 7443 6f6d 706c 6574  d=>DirectComplet
│ │ │ │ +0001e290: 496e 7429 2020 2020 2020 2020 2020 2020  Int)            
│ │ │ │ +0001e2a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001e2b0: 0a7c 202d 2d20 7573 6564 2036 2e35 3433  .| -- used 6.543
│ │ │ │ +0001e2c0: 3837 7320 2863 7075 293b 2031 2e35 3734  87s (cpu); 1.574
│ │ │ │ +0001e2d0: 3333 7320 2874 6872 6561 6429 3b20 3073  33s (thread); 0s
│ │ │ │ +0001e2e0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +0001e2f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e330: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001e340: 2020 2020 2020 2032 2032 2020 2020 2032         2 2     2
│ │ │ │ +0001e350: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  0001e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e370: 7c0a 7c20 2020 2020 2020 3220 3220 2020  |.|       2 2   
│ │ │ │ -0001e380: 2020 3220 2020 2020 2020 2020 3220 2020    2         2   
│ │ │ │ -0001e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e380: 2020 2020 7c0a 7c6f 3320 3d20 3268 2068      |.|o3 = 2h h
│ │ │ │ +0001e390: 2020 2b20 3268 2068 2020 2b20 3568 2068    + 2h h  + 5h h
│ │ │ │  0001e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e3b0: 2020 2020 2020 207c 0a7c 6f33 203d 2032         |.|o3 = 2
│ │ │ │ -0001e3c0: 6820 6820 202b 2032 6820 6820 202b 2035  h h  + 2h h  + 5
│ │ │ │ -0001e3d0: 6820 6820 2020 2020 2020 2020 2020 2020  h h             
│ │ │ │ -0001e3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e3f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001e400: 7c20 2020 2020 2020 3120 3220 2020 2020  |       1 2     
│ │ │ │ -0001e410: 3120 3220 2020 2020 3120 3220 2020 2020  1 2     1 2     
│ │ │ │ +0001e3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e3c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001e3d0: 2020 2020 2031 2032 2020 2020 2031 2032       1 2     1 2
│ │ │ │ +0001e3e0: 2020 2020 2031 2032 2020 2020 2020 2020       1 2        
│ │ │ │ +0001e3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001e410: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
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│ │ │ │  0001e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e440: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ +0001e460: 205a 5a5b 6820 2e2e 6820 5d20 2020 2020   ZZ[h ..h ]     
│ │ │ │  0001e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e480: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001e490: 2020 2020 5a5a 5b68 202e 2e68 205d 2020      ZZ[h ..h ]  
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│ │ │ │ +0001e4a0: 7c0a 7c20 2020 2020 2020 2020 3120 2020  |.|         1   
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│ │ │ │  0001e4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001e4e0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
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│ │ │ │ +0001e4e0: 2020 2020 2020 207c 0a7c 6f33 203a 202d         |.|o3 : -
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│ │ │ │  0001e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e510: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
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│ │ │ │ +0001e520: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001e530: 7c20 2020 2020 2020 2033 2020 2033 2020  |        3   3  
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│ │ │ │ -0001e570: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -0001e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e570: 2020 2020 207c 0a7c 2020 2020 2020 2868       |.|      (h
│ │ │ │ +0001e580: 202c 2068 2029 2020 2020 2020 2020 2020   , h )          
│ │ │ │  0001e590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e5a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001e5b0: 2028 6820 2c20 6820 2920 2020 2020 2020   (h , h )       
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│ │ │ │ +0001e5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001e5c0: 2020 2020 2020 2031 2020 2032 2020 2020         1   2    
│ │ │ │  0001e5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -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,
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│ │ │ │ -0001e6a0: 6d70 6c65 7449 6e74 2c49 6e64 734f 6653  mpletInt,IndsOfS
│ │ │ │ -0001e6b0: 6d6f 6f74 683d 3e7b 312c 327d 2920 2020  mooth=>{1,2})   
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│ │ │ │ -0001e6d0: 322e 3733 3530 3573 2028 6370 7529 3b20  2.73505s (cpu); 
│ │ │ │ -0001e6e0: 312e 3234 3237 7320 2874 6872 6561 6429  1.2427s (thread)
│ │ │ │ -0001e6f0: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ -0001e700: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ +0001e610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0001e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ +0001e650: 3a20 7469 6d65 2043 534d 2849 2c4d 6574  : time CSM(I,Met
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│ │ │ │ +0001e690: 207c 0a7c 202d 2d20 7573 6564 2037 2e30   |.| -- used 7.0
│ │ │ │ +0001e6a0: 3935 3539 7320 2863 7075 293b 2031 2e35  9559s (cpu); 1.5
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│ │ │ │ +0001e6c0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ +0001e6d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001e710: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001e720: 0a7c 2020 2020 2020 2032 2032 2020 2020  .|       2 2    
│ │ │ │ +0001e730: 2032 2020 2020 2020 2020 2032 2020 2020   2         2    
│ │ │ │  0001e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e750: 2020 7c0a 7c20 2020 2020 2020 3220 3220    |.|       2 2 
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│ │ │ │ -0001e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001e7b0: 2035 6820 6820 2020 2020 2020 2020 2020   5h h           
│ │ │ │ -0001e7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e760: 2020 2020 2020 7c0a 7c6f 3420 3d20 3268        |.|o4 = 2h
│ │ │ │ +0001e770: 2068 2020 2b20 3268 2068 2020 2b20 3568   h  + 2h h  + 5h
│ │ │ │ +0001e780: 2068 2020 2020 2020 2020 2020 2020 2020   h              
│ │ │ │ +0001e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e7a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001e7b0: 2020 2020 2020 2031 2032 2020 2020 2031         1 2     1
│ │ │ │ +0001e7c0: 2032 2020 2020 2031 2032 2020 2020 2020   2     1 2      
│ │ │ │  0001e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e7e0: 7c0a 7c20 2020 2020 2020 3120 3220 2020  |.|       1 2   
│ │ │ │ -0001e7f0: 2020 3120 3220 2020 2020 3120 3220 2020    1 2     1 2   
│ │ │ │ +0001e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e7f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │  0001e810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e820: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
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│ │ │ │ +0001e840: 2020 205a 5a5b 6820 2e2e 6820 5d20 2020     ZZ[h ..h ]   
│ │ │ │  0001e850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e860: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001e870: 7c20 2020 2020 5a5a 5b68 202e 2e68 205d  |     ZZ[h ..h ]
│ │ │ │ -0001e880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e880: 2020 7c0a 7c20 2020 2020 2020 2020 3120    |.|         1 
│ │ │ │ +0001e890: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0001e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e8b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001e8c0: 2031 2020 2032 2020 2020 2020 2020 2020   1   2          
│ │ │ │ -0001e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e8c0: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
│ │ │ │ +0001e8d0: 202d 2d2d 2d2d 2d2d 2d2d 2d20 2020 2020   ----------     
│ │ │ │  0001e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e8f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001e900: 3420 3a20 2d2d 2d2d 2d2d 2d2d 2d2d 2020  4 : ----------  
│ │ │ │ -0001e910: 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: 7c0a 7c20 2020 2020 2020 2033 2020 2033  |.|        3   3
│ │ │ │  0001e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e940: 2020 207c 0a7c 2020 2020 2020 2020 3320     |.|        3 
│ │ │ │ -0001e950: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -0001e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e950: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001e960: 2868 202c 2068 2029 2020 2020 2020 2020  (h , h )        
│ │ │ │  0001e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e980: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001e990: 2020 2028 6820 2c20 6820 2920 2020 2020     (h , h )     
│ │ │ │ -0001e9a0: 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 7c0a                |.
│ │ │ │ +0001e9a0: 7c20 2020 2020 2020 2031 2020 2032 2020  |        1   2  
│ │ │ │  0001e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e9d0: 207c 0a7c 2020 2020 2020 2020 3120 2020   |.|        1   
│ │ │ │ -0001e9e0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0001e9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ea00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ea10: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001ea40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ea50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0001ea60: 0a0a 4675 6e63 7469 6f6e 7320 7769 7468  ..Functions with
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│ │ │ │ -0001eaa0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001eab0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001eac0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
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│ │ │ │ -0001eb00: 4353 4d2c 202d 2d20 5468 650a 2020 2020  CSM, -- The.    
│ │ │ │ -0001eb10: 4368 6572 6e2d 5363 6877 6172 747a 2d4d  Chern-Schwartz-M
│ │ │ │ -0001eb20: 6163 5068 6572 736f 6e20 636c 6173 730a  acPherson class.
│ │ │ │ -0001eb30: 2020 2a20 4575 6c65 7228 2e2e 2e2c 496e    * Euler(...,In
│ │ │ │ -0001eb40: 6473 4f66 536d 6f6f 7468 3d3e 2e2e 2e29  dsOfSmooth=>...)
│ │ │ │ -0001eb50: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
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│ │ │ │ -0001ebd0: 0a28 4d61 6361 756c 6179 3244 6f63 2953  .(Macaulay2Doc)S
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│ │ │ │ +0001e9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e9e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0001ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46  ------------+..F
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│ │ │ │ +0001ea70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -0001ed30: 0a0a 496e 7075 7449 7353 6d6f 6f74 680a  ..InputIsSmooth.
│ │ │ │ -0001ed40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44  *************..D
│ │ │ │ -0001ed50: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -0001ed60: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f70 7469  ======..The opti
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│ │ │ │ -0001ed80: 2069 7320 6f6e 6c79 2075 7365 6420 6279   is only used by
│ │ │ │ -0001ed90: 2074 6865 2063 6f6d 6d61 6e64 7320 2a6e   the commands *n
│ │ │ │ -0001eda0: 6f74 6520 4353 4d3a 2043 534d 2c2c 2061  ote CSM: CSM,, a
│ │ │ │ -0001edb0: 6e64 0a2a 6e6f 7465 2045 756c 6572 3a20  nd.*note Euler: 
│ │ │ │ -0001edc0: 4575 6c65 722c 2e20 4966 2074 6865 2069  Euler,. If the i
│ │ │ │ -0001edd0: 6e70 7574 2069 6465 616c 2069 7320 6b6e  nput ideal is kn
│ │ │ │ -0001ede0: 6f77 6e20 746f 2064 6566 696e 6520 6120  own to define a 
│ │ │ │ -0001edf0: 736d 6f6f 7468 2073 7562 7363 6865 6d65  smooth subscheme
│ │ │ │ -0001ee00: 0a73 6574 7469 6e67 2074 6869 7320 6f70  .setting this op
│ │ │ │ -0001ee10: 7469 6f6e 2074 6f20 7472 7565 2077 696c  tion to true wil
│ │ │ │ -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: 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: 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                  
│ │ │ │ +0001ec00: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
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│ │ │ │ +0001ec30: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ +0001ec40: 682f 6d61 6361 756c 6179 322d 312e 3235  h/macaulay2-1.25
│ │ │ │ +0001ec50: 2e31 312b 6473 2f4d 322f 4d61 6361 756c  .11+ds/M2/Macaul
│ │ │ │ +0001ec60: 6179 322f 7061 636b 6167 6573 2f0a 4368  ay2/packages/.Ch
│ │ │ │ +0001ec70: 6172 6163 7465 7269 7374 6963 436c 6173  aracteristicClas
│ │ │ │ +0001ec80: 7365 732e 6d32 3a32 3438 323a 302e 0a1f  ses.m2:2482:0...
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│ │ │ │ +0001ecc0: 536d 6f6f 7468 2c20 4e65 7874 3a20 6973  Smooth, Next: is
│ │ │ │ +0001ecd0: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573  MultiHomogeneous
│ │ │ │ +0001ece0: 2c20 5072 6576 3a20 496e 6473 4f66 536d  , Prev: IndsOfSm
│ │ │ │ +0001ecf0: 6f6f 7468 2c20 5570 3a20 546f 700a 0a49  ooth, Up: Top..I
│ │ │ │ +0001ed00: 6e70 7574 4973 536d 6f6f 7468 0a2a 2a2a  nputIsSmooth.***
│ │ │ │ +0001ed10: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363  **********..Desc
│ │ │ │ +0001ed20: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +0001ed30: 3d3d 3d0a 0a54 6865 206f 7074 696f 6e20  ===..The option 
│ │ │ │ +0001ed40: 496e 7075 7449 7353 6d6f 6f74 6820 6973  InputIsSmooth is
│ │ │ │ +0001ed50: 206f 6e6c 7920 7573 6564 2062 7920 7468   only used by th
│ │ │ │ +0001ed60: 6520 636f 6d6d 616e 6473 202a 6e6f 7465  e commands *note
│ │ │ │ +0001ed70: 2043 534d 3a20 4353 4d2c 2c20 616e 640a   CSM: CSM,, and.
│ │ │ │ +0001ed80: 2a6e 6f74 6520 4575 6c65 723a 2045 756c  *note Euler: Eul
│ │ │ │ +0001ed90: 6572 2c2e 2049 6620 7468 6520 696e 7075  er,. If the inpu
│ │ │ │ +0001eda0: 7420 6964 6561 6c20 6973 206b 6e6f 776e  t ideal is known
│ │ │ │ +0001edb0: 2074 6f20 6465 6669 6e65 2061 2073 6d6f   to define a smo
│ │ │ │ +0001edc0: 6f74 6820 7375 6273 6368 656d 650a 7365  oth subscheme.se
│ │ │ │ +0001edd0: 7474 696e 6720 7468 6973 206f 7074 696f  tting this optio
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│ │ │ │ +0001edf0: 7065 6564 2075 7020 636f 6d70 7574 6174  peed up computat
│ │ │ │ +0001ee00: 696f 6e73 2028 6974 2069 7320 7365 7420  ions (it is set 
│ │ │ │ +0001ee10: 746f 2066 616c 7365 2062 790a 6465 6661  to false by.defa
│ │ │ │ +0001ee20: 756c 7429 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  ult)...+--------
│ │ │ │ +0001ee30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ee40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ee50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001ee60: 0a7c 6931 203a 2052 203d 205a 5a2f 3332  .|i1 : R = ZZ/32
│ │ │ │ +0001ee70: 3734 395b 785f 302e 2e78 5f34 5d3b 2020  749[x_0..x_4];  
│ │ │ │ +0001ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ee90: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001eed0: 2d2d 2d2b 0a7c 6932 203a 2049 3d69 6465  ---+.|i2 : I=ide
│ │ │ │ +0001eee0: 616c 2872 616e 646f 6d28 322c 5229 2c72  al(random(2,R),r
│ │ │ │ +0001eef0: 616e 646f 6d28 322c 5229 2c72 616e 646f  andom(2,R),rando
│ │ │ │ +0001ef00: 6d28 312c 5229 293b 2020 2020 207c 0a7c  m(1,R));     |.|
│ │ │ │ +0001ef10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef40: 2020 2020 2020 207c 0a7c 6f32 203a 2049         |.|o2 : I
│ │ │ │ +0001ef50: 6465 616c 206f 6620 5220 2020 2020 2020  deal of R       
│ │ │ │  0001ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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 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: 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 2020 2020 2020 2020                  
│ │ │ │ -0001f020: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -0001f030: 7365 6420 302e 3835 3232 3433 7320 2863  sed 0.852243s (c
│ │ │ │ -0001f040: 7075 293b 2030 2e34 3338 3333 3273 2028  pu); 0.438332s (
│ │ │ │ -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                  
│ │ │ │ +0001ef70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef80: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0001ef90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001efa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001efb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ +0001efc0: 203a 2074 696d 6520 4353 4d20 4920 2020   : time CSM I   
│ │ │ │ +0001efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001eff0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +0001f000: 2031 2e30 3939 3138 7320 2863 7075 293b   1.09918s (cpu);
│ │ │ │ +0001f010: 2030 2e35 3334 3834 3573 2028 7468 7265   0.534845s (thre
│ │ │ │ +0001f020: 6164 293b 2030 7320 2867 6329 2020 207c  ad); 0s (gc)   |
│ │ │ │ +0001f030: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f060: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001f070: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │  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         
│ │ │ │ +0001f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f0a0: 2020 207c 0a7c 6f33 203d 2034 6820 2020     |.|o3 = 4h   
│ │ │ │  0001f0b0: 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                  
│ │ │ │ +0001f0d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f0e0: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │  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     
│ │ │ │ +0001f110: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0001f120: 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                  
│ │ │ │ +0001f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f150: 207c 0a7c 2020 2020 205a 5a5b 6820 5d20   |.|     ZZ[h ] 
│ │ │ │  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   ]              
│ │ │ │ +0001f180: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001f190: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │  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                  
│ │ │ │ +0001f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f1c0: 2020 2020 207c 0a7c 6f33 203a 202d 2d2d       |.|o3 : ---
│ │ │ │ +0001f1d0: 2d2d 2d20 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  ------          
│ │ │ │ +0001f1f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f200: 0a7c 2020 2020 2020 2020 3520 2020 2020  .|        5     
│ │ │ │  0001f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f230: 2020 7c0a 7c20 2020 2020 2020 2035 2020    |.|        5  
│ │ │ │ -0001f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f230: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001f240: 2020 2068 2020 2020 2020 2020 2020 2020     h            
│ │ │ │  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         
│ │ │ │ +0001f260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f270: 2020 207c 0a7c 2020 2020 2020 2020 3120     |.|        1 
│ │ │ │  0001f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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 3039 3131 3232 3873 2028 6370 7529  0.0911228s (cpu)
│ │ │ │ -0001f370: 3b20 302e 3033 3136 3439 3573 2028 7468  ; 0.0316495s (th
│ │ │ │ -0001f380: 7265 6164 293b 2030 7320 2867 6329 7c0a  read); 0s (gc)|.
│ │ │ │ -0001f390: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001f2a0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001f2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f2e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2074  -------+.|i4 : t
│ │ │ │ +0001f2f0: 696d 6520 4353 4d28 492c 496e 7075 7449  ime CSM(I,InputI
│ │ │ │ +0001f300: 7353 6d6f 6f74 683d 3e74 7275 6529 2020  sSmooth=>true)  
│ │ │ │ +0001f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f320: 207c 0a7c 202d 2d20 7573 6564 2030 2e30   |.| -- used 0.0
│ │ │ │ +0001f330: 3836 3435 3837 7320 2863 7075 293b 2030  864587s (cpu); 0
│ │ │ │ +0001f340: 2e30 3437 3838 3635 7320 2874 6872 6561  .0478865s (threa
│ │ │ │ +0001f350: 6429 3b20 3073 2028 6763 297c 0a7c 2020  d); 0s (gc)|.|  
│ │ │ │ +0001f360: 2020 2020 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 207c 0a7c 2020 2020 2020 2033       |.|       3
│ │ │ │  0001f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f3c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001f3d0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +0001f3c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f3d0: 0a7c 6f34 203d 2034 6820 2020 2020 2020  .|o4 = 4h       
│ │ │ │  0001f3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f400: 2020 7c0a 7c6f 3420 3d20 3468 2020 2020    |.|o4 = 4h    
│ │ │ │ -0001f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f400: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001f410: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │  0001f420: 2020 2020 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         
│ │ │ │ +0001f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f440: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0001f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f470: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f470: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f480: 2020 2020 205a 5a5b 6820 5d20 2020 2020       ZZ[h ]     
│ │ │ │  0001f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f4b0: 7c0a 7c20 2020 2020 5a5a 5b68 205d 2020  |.|     ZZ[h ]  
│ │ │ │ -0001f4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f4b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001f4c0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │  0001f4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f4e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001f4f0: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +0001f4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f4f0: 207c 0a7c 6f34 203a 202d 2d2d 2d2d 2d20   |.|o4 : ------ 
│ │ │ │  0001f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f510: 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  --              
│ │ │ │ +0001f520: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001f530: 2020 2020 2020 3520 2020 2020 2020 2020        5         
│ │ │ │  0001f540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f550: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f560: 7c20 2020 2020 2020 2035 2020 2020 2020  |        5      
│ │ │ │ +0001f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f560: 2020 2020 207c 0a7c 2020 2020 2020 2068       |.|       h
│ │ │ │  0001f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f580: 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             
│ │ │ │ +0001f590: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f5a0: 0a7c 2020 2020 2020 2020 3120 2020 2020  .|        1     
│ │ │ │  0001f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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
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│ │ │ │ -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  case...+--------
│ │ │ │ -0001f6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0001f6e0: 0a7c 6935 203a 2074 696d 6520 4368 6572  .|i5 : time Cher
│ │ │ │ -0001f6f0: 6e20 4920 2020 2020 2020 2020 2020 2020  n I             
│ │ │ │ -0001f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f710: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -0001f720: 7573 6564 2030 2e30 3735 3131 3431 7320  used 0.0751141s 
│ │ │ │ -0001f730: 2863 7075 293b 2030 2e30 3330 3936 3934  (cpu); 0.0309694
│ │ │ │ -0001f740: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -0001f750: 6763 297c 0a7c 2020 2020 2020 2020 2020  gc)|.|          
│ │ │ │ -0001f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f5d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001f5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f610: 2d2d 2d2b 0a0a 4e6f 7465 2074 6861 7420  ---+..Note that 
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│ │ │ │ +0001f630: 616c 656e 746c 792c 2075 7365 2074 6865  alently, use the
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│ │ │ │ +0001f660: 7465 6164 0a69 6e20 7468 6973 2063 6173  tead.in this cas
│ │ │ │ +0001f670: 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  e...+-----------
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│ │ │ │ +0001f690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0001f6b0: 3520 3a20 7469 6d65 2043 6865 726e 2049  5 : time Chern I
│ │ │ │ +0001f6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f6e0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ +0001f6f0: 6420 302e 3036 3231 3432 3573 2028 6370  d 0.0621425s (cp
│ │ │ │ +0001f700: 7529 3b20 302e 3034 3138 3637 3273 2028  u); 0.0418672s (
│ │ │ │ +0001f710: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ +0001f720: 7c0a 7c20 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            |.|   
│ │ │ │ +0001f760: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │  0001f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f780: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001f790: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +0001f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f790: 2020 2020 7c0a 7c6f 3520 3d20 3468 2020      |.|o5 = 4h  
│ │ │ │  0001f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f7c0: 2020 2020 2020 207c 0a7c 6f35 203d 2034         |.|o5 = 4
│ │ │ │ -0001f7d0: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +0001f7c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001f7d0: 7c20 2020 2020 2020 3120 2020 2020 2020  |       1       
│ │ │ │  0001f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f800: 207c 0a7c 2020 2020 2020 2031 2020 2020   |.|       1    
│ │ │ │ +0001f800: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0001f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f830: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f840: 2020 7c0a 7c20 2020 2020 5a5a 5b68 205d    |.|     ZZ[h ]
│ │ │ │  0001f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f870: 2020 2020 207c 0a7c 2020 2020 205a 5a5b       |.|     ZZ[
│ │ │ │ -0001f880: 6820 5d20 2020 2020 2020 2020 2020 2020  h ]             
│ │ │ │ +0001f870: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001f880: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │  0001f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f8a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001f8b0: 0a7c 2020 2020 2020 2020 2031 2020 2020  .|         1    
│ │ │ │ -0001f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f8b0: 2020 2020 2020 7c0a 7c6f 3520 3a20 2d2d        |.|o5 : --
│ │ │ │ +0001f8c0: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │  0001f8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f8e0: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ -0001f8f0: 202d 2d2d 2d2d 2d20 2020 2020 2020 2020   ------         
│ │ │ │ +0001f8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f8f0: 7c0a 7c20 2020 2020 2020 2035 2020 2020  |.|        5    
│ │ │ │  0001f900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f920: 2020 207c 0a7c 2020 2020 2020 2020 3520     |.|        5 
│ │ │ │ -0001f930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f920: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f930: 2020 2020 6820 2020 2020 2020 2020 2020      h           
│ │ │ │  0001f940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f950: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001f960: 2020 2020 2020 2068 2020 2020 2020 2020         h        
│ │ │ │ +0001f950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f960: 2020 2020 7c0a 7c20 2020 2020 2020 2031      |.|        1
│ │ │ │  0001f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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
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│ │ │ │ -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
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│ │ │ │ -0001fac0: 6865 726e 2d53 6368 7761 7274 7a2d 4d61  hern-Schwartz-Ma
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│ │ │ │ -0001fae0: 202a 2045 756c 6572 282e 2e2e 2c49 6e70   * Euler(...,Inp
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│ │ │ │ -0001fb00: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
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│ │ │ │ -0001fb50: 6520 496e 7075 7449 7353 6d6f 6f74 683a  e InputIsSmooth:
│ │ │ │ -0001fb60: 2049 6e70 7574 4973 536d 6f6f 7468 2c20   InputIsSmooth, 
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│ │ │ │ -0001fb80: 6c3a 0a28 4d61 6361 756c 6179 3244 6f63  l:.(Macaulay2Doc
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│ │ │ │ +0001f9a0: 2b2d 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: 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 756e 6374  --------+..Funct
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│ │ │ │ +0001fa00: 6420 496e 7075 7449 7353 6d6f 6f74 683a  d InputIsSmooth:
│ │ │ │ +0001fa10: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +0001fa20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001fa30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001fa40: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2243 534d  ======..  * "CSM
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│ │ │ │ +0001fa70: 202a 6e6f 7465 2043 534d 3a20 4353 4d2c   *note CSM: CSM,
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│ │ │ │ +0001fa90: 6e2d 5363 6877 6172 747a 2d4d 6163 5068  n-Schwartz-MacPh
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│ │ │ │ +0001fab0: 4575 6c65 7228 2e2e 2e2c 496e 7075 7449  Euler(...,InputI
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│ │ │ │ +0001fb00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
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│ │ │ │ +0001fb20: 6e70 7574 4973 536d 6f6f 7468 3a20 496e  nputIsSmooth: In
│ │ │ │ +0001fb30: 7075 7449 7353 6d6f 6f74 682c 2069 7320  putIsSmooth, is 
│ │ │ │ +0001fb40: 6120 2a6e 6f74 6520 7379 6d62 6f6c 3a0a  a *note symbol:.
│ │ │ │ +0001fb50: 284d 6163 6175 6c61 7932 446f 6329 5379  (Macaulay2Doc)Sy
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│ │ │ │ +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 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 
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│ │ │ │ -0001fcc0: 5072 6576 3a20 496e 7075 7449 7353 6d6f  Prev: InputIsSmo
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│ │ │ │ +000203f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020400: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00020410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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                  
│ │ │ │ +00020430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020440: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00020450: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │  00020460: 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)  
│ │ │ │ +00020470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020480: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +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 2d2d 2d2d  ----------------
│ │ │ │ +000204c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000204d0: 0a7c 6933 203a 2049 3d69 6465 616c 2878  .|i3 : I=ideal(x
│ │ │ │ +000204e0: 5f30 5e32 2a78 5f33 2d78 5f31 2a78 5f30  _0^2*x_3-x_1*x_0
│ │ │ │ +000204f0: 2a78 5f34 2c78 5f36 5e33 2920 2020 2020  *x_4,x_6^3)     
│ │ │ │ +00020500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020510: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00020520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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 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 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 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      
│ │ │ │ +00020540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020550: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00020560: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00020570: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +00020580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020590: 2020 2020 207c 0a7c 6f33 203d 2069 6465       |.|o3 = ide
│ │ │ │ +000205a0: 616c 2028 7820 7820 202d 2078 2078 2078  al (x x  - x x x
│ │ │ │ +000205b0: 202c 2078 2029 2020 2020 2020 2020 2020   , x )          
│ │ │ │ +000205c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000205d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000205e0: 2020 2020 2020 2030 2033 2020 2020 3020         0 3    0 
│ │ │ │ +000205f0: 3120 3420 2020 3620 2020 2020 2020 2020  1 4   6         
│ │ │ │ +00020600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020610: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00020620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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                  
│ │ │ │ +00020640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020650: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +00020660: 203a 2049 6465 616c 206f 6620 5220 2020   : Ideal of R   
│ │ │ │  00020670: 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  
│ │ │ │ +00020680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020690: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +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 2d2d 2d2d  ----------------
│ │ │ │ +000206d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000206e0: 0a7c 6934 203a 2069 734d 756c 7469 486f  .|i4 : isMultiHo
│ │ │ │ +000206f0: 6d6f 6765 6e65 6f75 7320 4920 2020 2020  mogeneous I     
│ │ │ │ +00020700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020720: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00020730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020750: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00020760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020760: 2020 207c 0a7c 6f34 203d 2074 7275 6520     |.|o4 = true 
│ │ │ │  00020770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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                  
│ │ │ │ +00020790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000207a0: 2020 2020 207c 0a2b 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000207e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2069  -------+.|i5 : i
│ │ │ │ +000207f0: 734d 756c 7469 486f 6d6f 6765 6e65 6f75  sMultiHomogeneou
│ │ │ │ +00020800: 7320 6964 6561 6c28 785f 302a 785f 332d  s ideal(x_0*x_3-
│ │ │ │ +00020810: 785f 312a 785f 302a 785f 342c 785f 365e  x_1*x_0*x_4,x_6^
│ │ │ │ +00020820: 3329 2020 2020 2020 207c 0a7c 496e 7075  3)       |.|Inpu
│ │ │ │ +00020830: 7420 7465 726d 2062 656c 6f77 2069 7320  t term below is 
│ │ │ │ +00020840: 6e6f 7420 686f 6d6f 6765 6e65 6f75 7320  not homogeneous 
│ │ │ │ +00020850: 7769 7468 2072 6573 7065 6374 2074 6f20  with respect to 
│ │ │ │ +00020860: 7468 6520 6772 6164 696e 677c 0a7c 2d20  the grading|.|- 
│ │ │ │ +00020870: 7820 7820 7820 202b 2078 2078 2020 2020  x x x  + x x    
│ │ │ │ +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 207c 0a7c               |.|
│ │ │ │ +000208b0: 2020 2030 2031 2034 2020 2020 3020 3320     0 1 4    0 3 
│ │ │ │  000208c0: 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              
│ │ │ │ +000208e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000208f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00020900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +00020920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020930: 207c 0a7c 6f35 203d 2066 616c 7365 2020   |.|o5 = false  
│ │ │ │  00020940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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,...-------
│ │ │ │ +00020960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020970: 2020 207c 0a2b 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000209b0: 2d2d 2d2d 2d2b 0a0a 4e6f 7465 2074 6861  -----+..Note tha
│ │ │ │ +000209c0: 7420 666f 7220 616e 2069 6465 616c 2074  t for an ideal t
│ │ │ │ +000209d0: 6f20 6265 206d 756c 7469 2d68 6f6d 6f67  o be multi-homog
│ │ │ │ +000209e0: 656e 656f 7573 2074 6865 2064 6567 7265  eneous the degre
│ │ │ │ +000209f0: 6520 7665 6374 6f72 206f 6620 616c 6c0a  e vector of all.
│ │ │ │ +00020a00: 6d6f 6e6f 6d69 616c 7320 696e 2061 2067  monomials in a g
│ │ │ │ +00020a10: 6976 656e 2067 656e 6572 6174 6f72 206d  iven generator m
│ │ │ │ +00020a20: 7573 7420 6265 2074 6865 2073 616d 652e  ust be the same.
│ │ │ │ +00020a30: 0a0a 5761 7973 2074 6f20 7573 6520 6973  ..Ways to use is
│ │ │ │ +00020a40: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573  MultiHomogeneous
│ │ │ │ +00020a50: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +00020a60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00020a70: 3d0a 0a20 202a 2022 6973 4d75 6c74 6948  =..  * "isMultiH
│ │ │ │ +00020a80: 6f6d 6f67 656e 656f 7573 2849 6465 616c  omogeneous(Ideal
│ │ │ │ +00020a90: 2922 0a20 202a 2022 6973 4d75 6c74 6948  )".  * "isMultiH
│ │ │ │ +00020aa0: 6f6d 6f67 656e 656f 7573 2852 696e 6745  omogeneous(RingE
│ │ │ │ +00020ab0: 6c65 6d65 6e74 2922 0a0a 466f 7220 7468  lement)"..For th
│ │ │ │ +00020ac0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ +00020ad0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00020ae0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ +00020af0: 6520 6973 4d75 6c74 6948 6f6d 6f67 656e  e isMultiHomogen
│ │ │ │ +00020b00: 656f 7573 3a20 6973 4d75 6c74 6948 6f6d  eous: isMultiHom
│ │ │ │ +00020b10: 6f67 656e 656f 7573 2c20 6973 2061 202a  ogeneous, is a *
│ │ │ │ +00020b20: 6e6f 7465 206d 6574 686f 640a 6675 6e63  note method.func
│ │ │ │ +00020b30: 7469 6f6e 3a20 284d 6163 6175 6c61 7932  tion: (Macaulay2
│ │ │ │ +00020b40: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ +00020b50: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  on,...----------
│ │ │ │ +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 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
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│ │ │ │ -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
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│ │ │ │ -00020d90: 6563 7469 7665 4465 6772 6565 2e20 5468  ectiveDegree. Th
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│ │ │ │ -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
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│ │ │ │ -00020f70: 6520 436f 6d70 4d65 7468 6f64 3a20 436f  e CompMethod: Co
│ │ │ │ -00020f80: 6d70 4d65 7468 6f64 2c20 506e 5265 7369  mpMethod, PnResi
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│ │ │ │ -00020fa0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
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│ │ │ │ -00020fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -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]       
│ │ │ │ +00020ba0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
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│ │ │ │ +00020c60: 204e 6578 743a 204d 756c 7469 5072 6f6a   Next: MultiProj
│ │ │ │ +00020c70: 436f 6f72 6452 696e 672c 2050 7265 763a  CoordRing, Prev:
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│ │ │ │ +00020c90: 6f75 732c 2055 703a 2054 6f70 0a0a 4d65  ous, Up: Top..Me
│ │ │ │ +00020ca0: 7468 6f64 0a2a 2a2a 2a2a 2a0a 0a44 6573  thod.******..Des
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│ │ │ │ +00020cd0: 204d 6574 686f 6420 6973 206f 6e6c 7920   Method is only 
│ │ │ │ +00020ce0: 7573 6564 2062 7920 7468 6520 636f 6d6d  used by the comm
│ │ │ │ +00020cf0: 616e 6473 202a 6e6f 7465 2043 534d 3a20  ands *note CSM: 
│ │ │ │ +00020d00: 4353 4d2c 2061 6e64 202a 6e6f 7465 2045  CSM, and *note E
│ │ │ │ +00020d10: 756c 6572 3a0a 4575 6c65 722c 2061 6e64  uler:.Euler, and
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│ │ │ │ +00020d40: 436f 6d70 4d65 7468 6f64 3a0a 436f 6d70  CompMethod:.Comp
│ │ │ │ +00020d50: 4d65 7468 6f64 2c3d 3e50 726f 6a65 6374  Method,=>Project
│ │ │ │ +00020d60: 6976 6544 6567 7265 652e 2054 6865 204d  iveDegree. The M
│ │ │ │ +00020d70: 6574 686f 6420 496e 636c 7573 696f 6e45  ethod InclusionE
│ │ │ │ +00020d80: 7863 6c75 7369 6f6e 2077 696c 6c20 616c  xclusion will al
│ │ │ │ +00020d90: 7761 7973 2062 650a 7573 6564 2077 6974  ways be.used wit
│ │ │ │ +00020da0: 6820 2a6e 6f74 6520 436f 6d70 4d65 7468  h *note CompMeth
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│ │ │ │ +00020dc0: 506e 5265 7369 6475 616c 206f 7220 6265  PnResidual or be
│ │ │ │ +00020dd0: 7274 696e 692e 2057 6865 6e20 7468 6520  rtini. When the 
│ │ │ │ +00020de0: 696e 7075 740a 6964 6561 6c20 6973 2061  input.ideal is a
│ │ │ │ +00020df0: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
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│ │ │ │ +00020e10: 706f 7465 6e74 6961 6c6c 792c 2073 7065  potentially, spe
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│ │ │ │ +00020e30: 6174 696f 6e0a 6279 2073 6574 7469 6e67  ation.by setting
│ │ │ │ +00020e40: 204d 6574 686f 643d 3e20 4469 7265 6374   Method=> Direct
│ │ │ │ +00020e50: 436f 6d70 6c65 7465 496e 742e 2054 6865  CompleteInt. The
│ │ │ │ +00020e60: 206f 7074 696f 6e20 4d65 7468 6f64 2069   option Method i
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│ │ │ │ +00020e80: 6865 0a63 6f6d 6d61 6e64 7320 2a6e 6f74  he.commands *not
│ │ │ │ +00020e90: 6520 4353 4d3a 2043 534d 2c20 616e 6420  e CSM: CSM, and 
│ │ │ │ +00020ea0: 2a6e 6f74 6520 4575 6c65 723a 2045 756c  *note Euler: Eul
│ │ │ │ +00020eb0: 6572 2c20 616e 6420 6f6e 6c79 2069 6e20  er, and only in 
│ │ │ │ +00020ec0: 636f 6d62 696e 6174 696f 6e20 7769 7468  combination with
│ │ │ │ +00020ed0: 0a2a 6e6f 7465 2043 6f6d 704d 6574 686f  .*note CompMetho
│ │ │ │ +00020ee0: 643a 2043 6f6d 704d 6574 686f 642c 3d3e  d: CompMethod,=>
│ │ │ │ +00020ef0: 5072 6f6a 6563 7469 7665 4465 6772 6565  ProjectiveDegree
│ │ │ │ +00020f00: 2e20 5468 6520 4d65 7468 6f64 2049 6e63  . The Method Inc
│ │ │ │ +00020f10: 6c75 7369 6f6e 4578 636c 7573 696f 6e0a  lusionExclusion.
│ │ │ │ +00020f20: 7769 6c6c 2061 6c77 6179 7320 6265 2075  will always be u
│ │ │ │ +00020f30: 7365 6420 7769 7468 202a 6e6f 7465 2043  sed with *note C
│ │ │ │ +00020f40: 6f6d 704d 6574 686f 643a 2043 6f6d 704d  ompMethod: CompM
│ │ │ │ +00020f50: 6574 686f 642c 2050 6e52 6573 6964 7561  ethod, PnResidua
│ │ │ │ +00020f60: 6c20 6f72 2062 6572 7469 6e69 2e0a 0a2b  l or bertini...+
│ │ │ │ +00020f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020fa0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +00020fb0: 5220 3d20 5a5a 2f33 3237 3439 5b78 5f30  R = ZZ/32749[x_0
│ │ │ │ +00020fc0: 2e2e 785f 365d 2020 2020 2020 2020 2020  ..x_6]          
│ │ │ │ +00020fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020fe0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00020ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +00021010: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021020: 7c6f 3120 3d20 5220 2020 2020 2020 2020  |o1 = R         
│ │ │ │  00021030: 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      
│ │ │ │ +00021050: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00021060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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                  
│ │ │ │ +00021080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021090: 2020 2020 7c0a 7c6f 3120 3a20 506f 6c79      |.|o1 : Poly
│ │ │ │ +000210a0: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │  000210b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -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(
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│ │ │ │ -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                  
│ │ │ │ +000210c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000210d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000210e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000210f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021100: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +00021110: 3a20 493d 6964 6561 6c28 7261 6e64 6f6d  : I=ideal(random
│ │ │ │ +00021120: 2832 2c52 292c 7261 6e64 6f6d 2831 2c52  (2,R),random(1,R
│ │ │ │ +00021130: 292c 525f 302a 525f 312a 525f 362d 525f  ),R_0*R_1*R_6-R_
│ │ │ │ +00021140: 305e 3329 3b7c 0a7c 2020 2020 2020 2020  0^3);|.|        
│ │ │ │ +00021150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021180: 7c0a 7c6f 3220 3a20 4964 6561 6c20 6f66  |.|o2 : Ideal of
│ │ │ │ +00021190: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │  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: 312e 3731 3030 3273 2028 6370 7529 3b20  1.71002s (cpu); 
│ │ │ │ -00021280: 302e 3938 3437 3132 7320 2874 6872 6561  0.984712s (threa
│ │ │ │ -00021290: 6429 3b20 3073 2028 6763 2920 2020 207c  d); 0s (gc)    |
│ │ │ │ -000212a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000212b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000211b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000211c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000211d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000211e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000211f0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 7469  ------+.|i3 : ti
│ │ │ │ +00021200: 6d65 2043 534d 2049 2020 2020 2020 2020  me CSM I        
│ │ │ │ +00021210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021230: 207c 0a7c 202d 2d20 7573 6564 2034 2e30   |.| -- used 4.0
│ │ │ │ +00021240: 3833 3237 7320 2863 7075 293b 2031 2e33  8327s (cpu); 1.3
│ │ │ │ +00021250: 3738 3932 7320 2874 6872 6561 6429 3b20  7892s (thread); 
│ │ │ │ +00021260: 3073 2028 6763 2920 2020 2020 7c0a 7c20  0s (gc)     |.| 
│ │ │ │ +00021270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000212a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000212b0: 2020 3520 2020 2020 2034 2020 2020 2033    5      4     3
│ │ │ │  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             
│ │ │ │ +000212d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00021330: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00021320: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
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│ │ │ │ -000213d0: 5b68 205d 2020 2020 2020 2020 2020 2020  [h ]            
│ │ │ │ +000213c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000213d0: 7c20 2020 2020 2020 2020 3120 2020 2020  |         1     
│ │ │ │  000213e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000213f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00021430: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00021450: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000215c0: 3233 3273 2028 7468 7265 6164 293b 2030  232s (thread); 0
│ │ │ │ -000215d0: 7320 2867 6329 2020 207c 0a7c 2020 2020  s (gc)   |.|    
│ │ │ │ -000215e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000214e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00021550: 7443 6f6d 706c 6574 6549 6e74 2920 2020  tCompleteInt)   
│ │ │ │ +00021560: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +00021570: 2d20 7573 6564 2030 2e38 3633 3435 3673  - used 0.863456s
│ │ │ │ +00021580: 2028 6370 7529 3b20 302e 3332 3733 3633   (cpu); 0.327363
│ │ │ │ +00021590: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
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│ │ │ │ +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 2020                  
│ │ │ │ +000215e0: 207c 0a7c 2020 2020 2020 2020 3520 2020   |.|        5   
│ │ │ │ +000215f0: 2020 2034 2020 2020 2033 2020 2020 2020     4     3      
│ │ │ │  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 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021650: 0a7c 6f34 203d 2031 3268 2020 2b20 3130  .|o4 = 12h  + 10
│ │ │ │ -00021660: 6820 202b 2036 6820 2020 2020 2020 2020  h  + 6h         
│ │ │ │ +00021610: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00021620: 3420 3d20 3132 6820 202b 2031 3068 2020  4 = 12h  + 10h  
│ │ │ │ +00021630: 2b20 3668 2020 2020 2020 2020 2020 2020  + 6h            
│ │ │ │ +00021640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021650: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00021660: 2020 3120 2020 2020 2031 2020 2020 2031    1      1     1
│ │ │ │  00021670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021680: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00021690: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
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│ │ │ │ +00021680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021690: 2020 7c0a 7c20 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000216d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000216c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000216d0: 2020 2020 205a 5a5b 6820 5d20 2020 2020       ZZ[h ]     
│ │ │ │  000216e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021700: 7c0a 7c20 2020 2020 5a5a 5b68 205d 2020  |.|     ZZ[h ]  
│ │ │ │ -00021710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021700: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00021710: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │  00021720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021730: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00021740: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ -00021750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021740: 2020 207c 0a7c 6f34 203a 202d 2d2d 2d2d     |.|o4 : -----
│ │ │ │ +00021750: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │  00021760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021770: 2020 2020 2020 7c0a 7c6f 3420 3a20 2d2d        |.|o4 : --
│ │ │ │ -00021780: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │ +00021770: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021780: 7c20 2020 2020 2020 2037 2020 2020 2020  |        7      
│ │ │ │  00021790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000217a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217b0: 207c 0a7c 2020 2020 2020 2020 3720 2020   |.|        7   
│ │ │ │ -000217c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000217b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000217c0: 2020 2068 2020 2020 2020 2020 2020 2020     h            
│ │ │ │  000217d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000217f0: 2020 2020 2020 6820 2020 2020 2020 2020        h         
│ │ │ │ +000217e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000217f0: 2020 2020 7c0a 7c20 2020 2020 2020 2031      |.|        1
│ │ │ │  00021800: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00021830: 2020 3120 2020 2020 2020 2020 2020 2020    1             
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│ │ │ │ -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  -------------+..
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -00021ad0: 2a6e 6f74 6520 7379 6d62 6f6c 3a20 284d  *note symbol: (M
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│ │ │ │ +00021830: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00021840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00021950: 496e 6473 4f66 536d 6f6f 7468 2c2e 0a0a  IndsOfSmooth,...
│ │ │ │ +00021960: 4675 6e63 7469 6f6e 7320 7769 7468 206f  Functions with o
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│ │ │ │ +000219a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -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  ----------------
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│ │ │ │ +00022180: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ +00022190: 3a20 533d 4d75 6c74 6950 726f 6a43 6f6f  : S=MultiProjCoo
│ │ │ │ +000221a0: 7264 5269 6e67 2851 512c 7379 6d62 6f6c  rdRing(QQ,symbol
│ │ │ │ +000221b0: 207a 2c7b 312c 332c 337d 2920 2020 2020   z,{1,3,3})     
│ │ │ │ +000221c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000221d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000221e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000221f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022200: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +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                  
│ │ │ │ +00022220: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +00022230: 3d20 5320 2020 2020 2020 2020 2020 2020  = S             
│ │ │ │  00022240: 2020 2020 2020 2020 2020 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                  
│ │ │ │ +00022250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022270: 2020 2020 2020 2020 2020 7c0a 7c20 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 207c 0a7c               |.|
│ │ │ │ +000222a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +000222c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +000222d0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │  000222e0: 2020 2020 2020 2020 2020 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               |.+
│ │ │ │ +000222f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022310: 2020 2020 2020 2020 2020 7c0a 2b2d 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 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                  
│ │ │ │ +00022360: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +00022370: 3a20 6465 6772 6565 7320 5320 2020 2020  : degrees S     
│ │ │ │ +00022380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000223a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000223b0: 2020 2020 2020 2020 2020 7c0a 7c20 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 207c 0a7c               |.|
│ │ │ │ +000223e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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}}    
│ │ │ │ +00022400: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +00022410: 3d20 7b7b 312c 2030 2c20 307d 2c20 7b31  = {{1, 0, 0}, {1
│ │ │ │ +00022420: 2c20 302c 2030 7d2c 207b 302c 2031 2c20  , 0, 0}, {0, 1, 
│ │ │ │ +00022430: 307d 2c20 7b30 2c20 312c 2030 7d2c 207b  0}, {0, 1, 0}, {
│ │ │ │ +00022440: 302c 2031 2c20 307d 2c20 7b30 2c20 312c  0, 1, 0}, {0, 1,
│ │ │ │ +00022450: 2030 7d2c 207b 302c 2020 7c0a 7c20 2020   0}, {0,  |.|   
│ │ │ │ +00022460: 2020 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 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000224a0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020  ----------|.|   
│ │ │ │ +000224b0: 2020 302c 2031 7d2c 207b 302c 2030 2c20    0, 1}, {0, 0, 
│ │ │ │ +000224c0: 317d 2c20 7b30 2c20 302c 2031 7d2c 207b  1}, {0, 0, 1}, {
│ │ │ │ +000224d0: 302c 2030 2c20 317d 7d20 2020 2020 2020  0, 0, 1}}       
│ │ │ │ +000224e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000224f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022520: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +00022540: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +00022550: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │  00022560: 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               |.+
│ │ │ │ +00022570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022590: 2020 2020 2020 2020 2020 7c0a 2b2d 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 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} 
│ │ │ │ +000225e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ +000225f0: 3a20 523d 4d75 6c74 6950 726f 6a43 6f6f  : R=MultiProjCoo
│ │ │ │ +00022600: 7264 5269 6e67 207b 322c 337d 2020 2020  rdRing {2,3}    
│ │ │ │ +00022610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022630: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  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 207c 0a7c               |.|
│ │ │ │ +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 2020 2020                  
│ │ │ │ -00022690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022680: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ +00022690: 3d20 5220 2020 2020 2020 2020 2020 2020  = R             
│ │ │ │  000226a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000226b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000226c0: 6f33 203d 2052 2020 2020 2020 2020 2020  o3 = R          
│ │ │ │ -000226d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000226b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000226c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000226d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000226e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000226f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022700: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +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                  
│ │ │ │ +00022720: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ +00022730: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │  00022740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022750: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022760: 6f33 203a 2050 6f6c 796e 6f6d 6961 6c52  o3 : PolynomialR
│ │ │ │ -00022770: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -00022780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000227a0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00022750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022770: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00022780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000227a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000227b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000227c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000227d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000227e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000227f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00022800: 6934 203a 2063 6f65 6666 6963 6965 6e74  i4 : coefficient
│ │ │ │ -00022810: 5269 6e67 2052 2020 2020 2020 2020 2020  Ring R          
│ │ │ │ +000227c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ +000227d0: 3a20 636f 6566 6669 6369 656e 7452 696e  : coefficientRin
│ │ │ │ +000227e0: 6720 5220 2020 2020 2020 2020 2020 2020  g R             
│ │ │ │ +000227f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022810: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022840: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022860: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022870: 2020 2020 5a5a 2020 2020 2020 2020 2020      ZZ          
│ │ │ │  00022880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022890: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000228a0: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │ -000228b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000228c0: 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 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ +000228c0: 3d20 2d2d 2d2d 2d20 2020 2020 2020 2020  = -----         
│ │ │ │  000228d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000228e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000228f0: 6f34 203d 202d 2d2d 2d2d 2020 2020 2020  o4 = -----      
│ │ │ │ -00022900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000228e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000228f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022900: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022910: 2020 3332 3734 3920 2020 2020 2020 2020    32749         
│ │ │ │  00022920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022930: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022940: 2020 2020 2033 3237 3439 2020 2020 2020       32749      
│ │ │ │ -00022950: 2020 2020 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 7c0a 7c20 2020            |.|   
│ │ │ │  00022960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000229a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ +000229b0: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │  000229c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000229e0: 6f34 203a 2051 756f 7469 656e 7452 696e  o4 : QuotientRin
│ │ │ │ -000229f0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -00022a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a20: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000229d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000229e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000229f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00022a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022a10: 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d  ----------------
│ │ │ │ -00022a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00022a80: 6935 203a 2064 6573 6372 6962 6520 5220  i5 : describe R 
│ │ │ │ -00022a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ +00022a50: 3a20 6465 7363 7269 6265 2052 2020 2020  : describe R    
│ │ │ │ +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            |.|   
│ │ │ │  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 207c 0a7c               |.|
│ │ │ │ +00022ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022af0: 2020 2020 5a5a 2020 2020 2020 2020 2020      ZZ          
│ │ │ │  00022b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022b20: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │ -00022b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022b70: 6f35 203d 202d 2d2d 2d2d 5b78 202e 2e78  o5 = -----[x ..x
│ │ │ │ -00022b80: 202c 2044 6567 7265 6573 203d 3e20 7b33   , Degrees => {3
│ │ │ │ -00022b90: 3a7b 317d 2c20 343a 7b30 7d7d 2c20 4865  :{1}, 4:{0}}, He
│ │ │ │ -00022ba0: 6674 203d 3e20 7b32 3a31 7d5d 2020 2020  ft => {2:1}]    
│ │ │ │ -00022bb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022bc0: 2020 2020 2033 3237 3439 2020 3020 2020       32749  0   
│ │ │ │ -00022bd0: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ -00022be0: 207b 307d 2020 2020 7b31 7d20 2020 2020   {0}    {1}     
│ │ │ │ -00022bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c00: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00022b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022b30: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ +00022b40: 3d20 2d2d 2d2d 2d5b 7820 2e2e 7820 2c20  = -----[x ..x , 
│ │ │ │ +00022b50: 4465 6772 6565 7320 3d3e 207b 333a 7b31  Degrees => {3:{1
│ │ │ │ +00022b60: 7d2c 2034 3a7b 307d 7d2c 2048 6566 7420  }, 4:{0}}, Heft 
│ │ │ │ +00022b70: 3d3e 207b 323a 317d 5d20 2020 2020 2020  => {2:1}]       
│ │ │ │ +00022b80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022b90: 2020 3332 3734 3920 2030 2020 2036 2020    32749  0   6  
│ │ │ │ +00022ba0: 2020 2020 2020 2020 2020 2020 2020 7b30                {0
│ │ │ │ +00022bb0: 7d20 2020 207b 317d 2020 2020 2020 2020  }    {1}        
│ │ │ │ +00022bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022bd0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00022be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00022c60: 6936 203a 2041 3d43 686f 7752 696e 6720  i6 : A=ChowRing 
│ │ │ │ -00022c70: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +00022c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ +00022c30: 3a20 413d 4368 6f77 5269 6e67 2052 2020  : A=ChowRing R  
│ │ │ │ +00022c40: 2020 2020 2020 2020 2020 2020 2020 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 7c0a 7c20 2020            |.|   
│ │ │ │  00022c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ca0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022cc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ +00022cd0: 3d20 4120 2020 2020 2020 2020 2020 2020  = A             
│ │ │ │  00022ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022cf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022d00: 6f36 203d 2041 2020 2020 2020 2020 2020  o6 = A          
│ │ │ │ -00022d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022d10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022d60: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ +00022d70: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │  00022d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022da0: 6f36 203a 2051 756f 7469 656e 7452 696e  o6 : QuotientRin
│ │ │ │ -00022db0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -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 207c 0a2b               |.+
│ │ │ │ +00022d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022db0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00022dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00022e40: 6937 203a 2064 6573 6372 6962 6520 4120  i7 : describe A 
│ │ │ │ -00022e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ +00022e10: 3a20 6465 7363 7269 6265 2041 2020 2020  : describe A    
│ │ │ │ +00022e20: 2020 2020 2020 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 7c0a 7c20 2020            |.|   
│ │ │ │  00022e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ea0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022eb0: 2020 5a5a 5b68 202e 2e68 205d 2020 2020    ZZ[h ..h ]    
│ │ │ │  00022ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ed0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022ee0: 2020 2020 205a 5a5b 6820 2e2e 6820 5d20       ZZ[h ..h ] 
│ │ │ │ -00022ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ef0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022f00: 2020 2020 2020 3120 2020 3220 2020 2020        1   2     
│ │ │ │  00022f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022f30: 2020 2020 2020 2020 2031 2020 2032 2020           1   2  
│ │ │ │ -00022f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022f40: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ +00022f50: 3d20 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020  = ----------    
│ │ │ │  00022f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022f80: 6f37 203d 202d 2d2d 2d2d 2d2d 2d2d 2d20  o7 = ---------- 
│ │ │ │ -00022f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022f90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022fa0: 2020 2020 2033 2020 2034 2020 2020 2020       3   4      
│ │ │ │  00022fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022fc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022fd0: 2020 2020 2020 2020 3320 2020 3420 2020          3   4   
│ │ │ │ -00022fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ff0: 2020 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                  
│ │ │ │ +00022fe0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022ff0: 2020 2028 6820 2c20 6820 2920 2020 2020     (h , h )     
│ │ │ │  00023000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023010: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023020: 2020 2020 2020 2868 202c 2068 2029 2020        (h , h )  
│ │ │ │ -00023030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023030: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00023040: 2020 2020 2031 2020 2032 2020 2020 2020       1   2      
│ │ │ │  00023050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023060: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023070: 2020 2020 2020 2020 3120 2020 3220 2020          1   2   
│ │ │ │ -00023080: 2020 2020 2020 2020 2020 2020 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               |.+
│ │ │ │ +00023060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023080: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00023090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000230a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000230b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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))             
│ │ │ │ +000230d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ +000230e0: 3a20 5365 6772 6528 412c 6964 6561 6c20  : Segre(A,ideal 
│ │ │ │ +000230f0: 7261 6e64 6f6d 287b 312c 317d 2c52 2929  random({1,1},R))
│ │ │ │ +00023100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023120: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00023130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023150: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 |.|
│ │ │ │ +00023170: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00023180: 2020 2020 2032 2033 2020 2020 2032 2032       2 3     2 2
│ │ │ │ +00023190: 2020 2020 2020 2033 2020 2020 2032 2020         3     2  
│ │ │ │ +000231a0: 2020 2020 2020 2032 2020 2020 3320 2020         2    3   
│ │ │ │ +000231b0: 2032 2020 2020 2020 2020 2020 2020 3220   2            2 
│ │ │ │ +000231c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ +000231d0: 3d20 3130 6820 6820 202d 2036 6820 6820  = 10h h  - 6h h 
│ │ │ │ +000231e0: 202d 2034 6820 6820 202b 2033 6820 6820   - 4h h  + 3h h 
│ │ │ │ +000231f0: 202b 2033 6820 6820 202b 2068 2020 2d20   + 3h h  + h  - 
│ │ │ │ +00023200: 6820 202d 2032 6820 6820 202d 2068 2020  h  - 2h h  - h  
│ │ │ │ +00023210: 2b20 6820 202b 2068 2020 7c0a 7c20 2020  + h  + h  |.|   
│ │ │ │ +00023220: 2020 2020 2031 2032 2020 2020 2031 2032       1 2     1 2
│ │ │ │ +00023230: 2020 2020 2031 2032 2020 2020 2031 2032       1 2     1 2
│ │ │ │ +00023240: 2020 2020 2031 2032 2020 2020 3220 2020       1 2    2   
│ │ │ │ +00023250: 2031 2020 2020 2031 2032 2020 2020 3220   1     1 2    2 
│ │ │ │ +00023260: 2020 2031 2020 2020 3220 7c0a 7c20 2020     1    2 |.|   
│ │ │ │ +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 2020 2020 2020 2020 2020                  
│ │ │ │ -000232b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000232c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000232b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ +000232c0: 3a20 4120 2020 2020 2020 2020 2020 2020  : A             
│ │ │ │  000232d0: 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               |.+
│ │ │ │ +000232e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000232f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023300: 2020 2020 2020 2020 2020 7c0a 2b2d 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 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,.
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│ │ │ │  00023b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00023b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023b80: 2020 207c 0a7c 6f32 203d 2041 2020 2020     |.|o2 = A    
│ │ │ │  00023b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023bb0: 2020 2020 2020 7c0a 7c6f 3220 3d20 4120        |.|o2 = A 
│ │ │ │ +00023bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023bd0: 2020 207c 0a7c 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 7c0a 7c20 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 2020 2020                  
│ │ │ │ +00023c20: 2020 207c 0a7c 6f32 203a 2051 756f 7469     |.|o2 : Quoti
│ │ │ │ +00023c30: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │  00023c40: 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        |.+-------
│ │ │ │ +00023c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023c70: 2020 207c 0a2b 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: 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);           
│ │ │ │ +00023cc0: 2d2d 2d2b 0a7c 6933 203a 2049 3d69 6465  ---+.|i3 : I=ide
│ │ │ │ +00023cd0: 616c 2872 616e 646f 6d28 322c 5229 2c52  al(random(2,R),R
│ │ │ │ +00023ce0: 5f30 2a52 5f31 2a52 5f36 2d52 5f30 5e33  _0*R_1*R_6-R_0^3
│ │ │ │ +00023cf0: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ +00023d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023d10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00023d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023d40: 2020 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │ -00023d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023d60: 2020 207c 0a7c 6f33 203a 2049 6465 616c     |.|o3 : Ideal
│ │ │ │ +00023d70: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │  00023d80: 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        |.+-------
│ │ │ │ +00023d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023db0: 2020 207c 0a2b 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: 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)     
│ │ │ │ +00023e00: 2d2d 2d2b 0a7c 6934 203a 2063 736d 3d43  ---+.|i4 : csm=C
│ │ │ │ +00023e10: 534d 2841 2c49 2c4f 7574 7075 743d 3e48  SM(A,I,Output=>H
│ │ │ │ +00023e20: 6173 6846 6f72 6d29 2020 2020 2020 2020  ashForm)        
│ │ │ │ +00023e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023e50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00023e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00023e80: 2020 2020 2020 2020 2020 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 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...}         
│ │ │ │ +00023ea0: 2020 207c 0a7c 6f34 203d 204d 7574 6162     |.|o4 = Mutab
│ │ │ │ +00023eb0: 6c65 4861 7368 5461 626c 657b 2e2e 2e34  leHashTable{...4
│ │ │ │ +00023ec0: 2e2e 2e7d 2020 2020 2020 2020 2020 2020  ...}            
│ │ │ │ +00023ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023ef0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00023f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00023f20: 2020 2020 2020 2020 2020 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                  
│ │ │ │ +00023f40: 2020 207c 0a7c 6f34 203a 204d 7574 6162     |.|o4 : Mutab
│ │ │ │ +00023f50: 6c65 4861 7368 5461 626c 6520 2020 2020  leHashTable     
│ │ │ │  00023f60: 2020 2020 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        |.+-------
│ │ │ │ +00023f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023f90: 2020 207c 0a2b 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: 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
│ │ │ │ -00024020: 656b 2063 736d 2020 2020 2020 2020 2020  ek csm          
│ │ │ │ -00024030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023fe0: 2d2d 2d2b 0a7c 6935 203a 2070 6565 6b20  ---+.|i5 : peek 
│ │ │ │ +00023ff0: 6373 6d20 2020 2020 2020 2020 2020 2020  csm             
│ │ │ │ +00024000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024030: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024060: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024080: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000240a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000240b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000240c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000240d0: 2020 2020 2020 2020 2020 2036 2020 2020             6    
│ │ │ │ -000240e0: 2020 3520 2020 2020 2034 2020 2020 2020    5      4      
│ │ │ │ -000240f0: 3320 2020 2020 2032 2020 2020 2020 2020  3      2        
│ │ │ │ -00024100: 2020 2020 2020 7c0a 7c6f 3520 3d20 4d75        |.|o5 = Mu
│ │ │ │ -00024110: 7461 626c 6548 6173 6854 6162 6c65 7b7b  tableHashTable{{
│ │ │ │ -00024120: 302c 2031 7d20 3d3e 2032 6820 202b 2032  0, 1} => 2h  + 2
│ │ │ │ -00024130: 3368 2020 2b20 3332 6820 202b 2033 3368  3h  + 32h  + 33h
│ │ │ │ -00024140: 2020 2b20 3138 6820 202b 2035 6820 7d20    + 18h  + 5h } 
│ │ │ │ -00024150: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024170: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -00024180: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ -00024190: 3120 2020 2020 2031 2020 2020 2031 2020  1      1     1  
│ │ │ │ -000241a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000241b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000241c0: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ -000241d0: 3520 2020 2020 2034 2020 2020 2020 3320  5      4      3 
│ │ │ │ -000241e0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -000241f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024200: 2020 2020 2020 2020 2020 2020 2020 2043                 C
│ │ │ │ -00024210: 534d 203d 3e20 3130 6820 202b 2031 3268  SM => 10h  + 12h
│ │ │ │ -00024220: 2020 2b20 3232 6820 202b 2031 3668 2020    + 22h  + 16h  
│ │ │ │ -00024230: 2b20 3668 2020 2020 2020 2020 2020 2020  + 6h            
│ │ │ │ -00024240: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024260: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ -00024270: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ -00024280: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ -00024290: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000242a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000242b0: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ -000242c0: 2020 2020 2020 3420 2020 2020 2033 2020        4      3  
│ │ │ │ -000242d0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -000242e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000242f0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00024300: 307d 203d 3e20 3668 2020 2b20 3138 6820  0} => 6h  + 18h 
│ │ │ │ -00024310: 202b 2032 3668 2020 2b20 3232 6820 202b   + 26h  + 22h  +
│ │ │ │ -00024320: 2031 3068 2020 2b20 3268 2020 2020 2020   10h  + 2h      
│ │ │ │ -00024330: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024350: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ -00024360: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00024370: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ -00024380: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000243a0: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ -000243b0: 2020 2020 2020 3420 2020 2020 2033 2020        4      3  
│ │ │ │ -000243c0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -000243d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000243e0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -000243f0: 317d 203d 3e20 3668 2020 2b20 3137 6820  1} => 6h  + 17h 
│ │ │ │ -00024400: 202b 2032 3868 2020 2b20 3237 6820 202b   + 28h  + 27h  +
│ │ │ │ -00024410: 2031 3468 2020 2b20 3368 2020 2020 2020   14h  + 3h      
│ │ │ │ -00024420: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024440: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ -00024450: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00024460: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ -00024470: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000240a0: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ +000240b0: 2020 2020 2020 3420 2020 2020 2033 2020        4      3  
│ │ │ │ +000240c0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +000240d0: 2020 207c 0a7c 6f35 203d 204d 7574 6162     |.|o5 = Mutab
│ │ │ │ +000240e0: 6c65 4861 7368 5461 626c 657b 7b30 2c20  leHashTable{{0, 
│ │ │ │ +000240f0: 317d 203d 3e20 3268 2020 2b20 3233 6820  1} => 2h  + 23h 
│ │ │ │ +00024100: 202b 2033 3268 2020 2b20 3333 6820 202b   + 32h  + 33h  +
│ │ │ │ +00024110: 2031 3868 2020 2b20 3568 207d 2020 2020   18h  + 5h }    
│ │ │ │ +00024120: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024140: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ +00024150: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00024160: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ +00024170: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024190: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ +000241a0: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ +000241b0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000241c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000241d0: 2020 2020 2020 2020 2020 2020 4353 4d20              CSM 
│ │ │ │ +000241e0: 3d3e 2031 3068 2020 2b20 3132 6820 202b  => 10h  + 12h  +
│ │ │ │ +000241f0: 2032 3268 2020 2b20 3136 6820 202b 2036   22h  + 16h  + 6
│ │ │ │ +00024200: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +00024210: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024230: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00024240: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00024250: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +00024260: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024280: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ +00024290: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ +000242a0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000242b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000242c0: 2020 2020 2020 2020 2020 2020 7b30 7d20              {0} 
│ │ │ │ +000242d0: 3d3e 2036 6820 202b 2031 3868 2020 2b20  => 6h  + 18h  + 
│ │ │ │ +000242e0: 3236 6820 202b 2032 3268 2020 2b20 3130  26h  + 22h  + 10
│ │ │ │ +000242f0: 6820 202b 2032 6820 2020 2020 2020 2020  h  + 2h         
│ │ │ │ +00024300: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024320: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00024330: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00024340: 2031 2020 2020 2031 2020 2020 2020 2020   1     1        
│ │ │ │ +00024350: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024370: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ +00024380: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ +00024390: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000243a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000243b0: 2020 2020 2020 2020 2020 2020 7b31 7d20              {1} 
│ │ │ │ +000243c0: 3d3e 2036 6820 202b 2031 3768 2020 2b20  => 6h  + 17h  + 
│ │ │ │ +000243d0: 3238 6820 202b 2032 3768 2020 2b20 3134  28h  + 27h  + 14
│ │ │ │ +000243e0: 6820 202b 2033 6820 2020 2020 2020 2020  h  + 3h         
│ │ │ │ +000243f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024410: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00024420: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00024430: 2031 2020 2020 2031 2020 2020 2020 2020   1     1        
│ │ │ │ +00024440: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00024450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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 2d2d 2d2d  ----------------
│ │ │ │ -000244c0: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 4353  ------+.|i6 : CS
│ │ │ │ -000244d0: 4d28 412c 6964 6561 6c20 495f 3029 3d3d  M(A,ideal I_0)==
│ │ │ │ -000244e0: 6373 6d23 7b30 7d20 2020 2020 2020 2020  csm#{0}         
│ │ │ │ +00024490: 2d2d 2d2b 0a7c 6936 203a 2043 534d 2841  ---+.|i6 : CSM(A
│ │ │ │ +000244a0: 2c69 6465 616c 2049 5f30 293d 3d63 736d  ,ideal I_0)==csm
│ │ │ │ +000244b0: 237b 307d 2020 2020 2020 2020 2020 2020  #{0}            
│ │ │ │ +000244c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000244d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000244e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000244f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024510: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024530: 2020 207c 0a7c 6f36 203d 2074 7275 6520     |.|o6 = true 
│ │ │ │  00024540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024560: 2020 2020 2020 7c0a 7c6f 3620 3d20 7472        |.|o6 = tr
│ │ │ │ -00024570: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ -00024580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000245a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000245b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00024560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024580: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00024590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000245a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000245b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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 2b0a 7c69 3720 3a20 4353  ------+.|i7 : CS
│ │ │ │ -00024610: 4d28 412c 6964 6561 6c28 495f 302a 495f  M(A,ideal(I_0*I_
│ │ │ │ -00024620: 3129 293d 3d63 736d 237b 302c 317d 2020  1))==csm#{0,1}  
│ │ │ │ +000245d0: 2d2d 2d2b 0a7c 6937 203a 2043 534d 2841  ---+.|i7 : CSM(A
│ │ │ │ +000245e0: 2c69 6465 616c 2849 5f30 2a49 5f31 2929  ,ideal(I_0*I_1))
│ │ │ │ +000245f0: 3d3d 6373 6d23 7b30 2c31 7d20 2020 2020  ==csm#{0,1}     
│ │ │ │ +00024600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024620: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024650: 2020 2020 2020 7c0a 7c20 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                  
│ │ │ │ +00024670: 2020 207c 0a7c 6f37 203d 2074 7275 6520     |.|o7 = true 
│ │ │ │  00024680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246a0: 2020 2020 2020 7c0a 7c6f 3720 3d20 7472        |.|o7 = tr
│ │ │ │ -000246b0: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ -000246c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246f0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000246a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000246b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000246c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000246d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000246e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000246f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024740: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 633d  ------+.|i8 : c=
│ │ │ │ -00024750: 4368 6572 6e28 2049 2c20 4f75 7470 7574  Chern( I, Output
│ │ │ │ -00024760: 3d3e 4861 7368 466f 726d 2920 2020 2020  =>HashForm)     
│ │ │ │ +00024710: 2d2d 2d2b 0a7c 6938 203a 2063 3d43 6865  ---+.|i8 : c=Che
│ │ │ │ +00024720: 726e 2820 492c 204f 7574 7075 743d 3e48  rn( I, Output=>H
│ │ │ │ +00024730: 6173 6846 6f72 6d29 2020 2020 2020 2020  ashForm)        
│ │ │ │ +00024740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024760: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024790: 2020 2020 2020 7c0a 7c20 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                  
│ │ │ │ -000247c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247e0: 2020 2020 2020 7c0a 7c6f 3820 3d20 4d75        |.|o8 = Mu
│ │ │ │ -000247f0: 7461 626c 6548 6173 6854 6162 6c65 7b2e  tableHashTable{.
│ │ │ │ -00024800: 2e2e 362e 2e2e 7d20 2020 2020 2020 2020  ..6...}         
│ │ │ │ +000247b0: 2020 207c 0a7c 6f38 203d 204d 7574 6162     |.|o8 = Mutab
│ │ │ │ +000247c0: 6c65 4861 7368 5461 626c 657b 2e2e 2e36  leHashTable{...6
│ │ │ │ +000247d0: 2e2e 2e7d 2020 2020 2020 2020 2020 2020  ...}            
│ │ │ │ +000247e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000247f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024800: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024830: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +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 2020 2020 2020 2020 2020                  
│ │ │ │ +00024850: 2020 207c 0a7c 6f38 203a 204d 7574 6162     |.|o8 : Mutab
│ │ │ │ +00024860: 6c65 4861 7368 5461 626c 6520 2020 2020  leHashTable     
│ │ │ │  00024870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024880: 2020 2020 2020 7c0a 7c6f 3820 3a20 4d75        |.|o8 : Mu
│ │ │ │ -00024890: 7461 626c 6548 6173 6854 6162 6c65 2020  tableHashTable  
│ │ │ │ -000248a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000248b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000248c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000248d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00024880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000248a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000248b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000248c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d  ----------------
│ │ │ │ -00024910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024920: 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 7065  ------+.|i9 : pe
│ │ │ │ -00024930: 656b 2063 2020 2020 2020 2020 2020 2020  ek c            
│ │ │ │ -00024940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000248f0: 2d2d 2d2b 0a7c 6939 203a 2070 6565 6b20  ---+.|i9 : peek 
│ │ │ │ +00024900: 6320 2020 2020 2020 2020 2020 2020 2020  c               
│ │ │ │ +00024910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024940: 2020 207c 0a7c 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 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024990: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000249a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000249b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000249d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249f0: 2020 2020 2032 2020 2020 2020 3320 2020       2      3   
│ │ │ │ -00024a00: 2020 2034 2020 2020 2020 2035 2020 2020     4       5    
│ │ │ │ -00024a10: 2020 2036 2020 7c0a 7c6f 3920 3d20 4d75     6  |.|o9 = Mu
│ │ │ │ -00024a20: 7461 626c 6548 6173 6854 6162 6c65 7b53  tableHashTable{S
│ │ │ │ -00024a30: 6567 7265 4c69 7374 203d 3e20 7b30 2c20  egreList => {0, 
│ │ │ │ -00024a40: 302c 2036 6820 2c20 2d33 3068 202c 2031  0, 6h , -30h , 1
│ │ │ │ -00024a50: 3134 6820 2c20 2d33 3930 6820 2c20 3132  14h , -390h , 12
│ │ │ │ -00024a60: 3636 6820 7d7d 7c0a 7c20 2020 2020 2020  66h }}|.|       
│ │ │ │ -00024a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a90: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00024aa0: 2020 2031 2020 2020 2020 2031 2020 2020     1       1    
│ │ │ │ -00024ab0: 2020 2031 2020 7c0a 7c20 2020 2020 2020     1  |.|       
│ │ │ │ -00024ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ae0: 2020 2032 2020 2020 3320 2020 2034 2020     2    3    4  
│ │ │ │ -00024af0: 2020 3520 2020 2036 2020 2020 2020 2020    5    6        
│ │ │ │ -00024b00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024b10: 2020 2020 2020 2020 2020 2020 2020 2047                 G
│ │ │ │ -00024b20: 6c69 7374 203d 3e20 7b31 2c20 3368 202c  list => {1, 3h ,
│ │ │ │ -00024b30: 2033 6820 2c20 3368 202c 2033 6820 2c20   3h , 3h , 3h , 
│ │ │ │ -00024b40: 3368 202c 2033 6820 7d20 2020 2020 2020  3h , 3h }       
│ │ │ │ -00024b50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024b70: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00024b80: 2020 2031 2020 2020 3120 2020 2031 2020     1    1    1  
│ │ │ │ -00024b90: 2020 3120 2020 2031 2020 2020 2020 2020    1    1        
│ │ │ │ -00024ba0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024bc0: 2020 2020 2020 2020 2020 2020 2036 2020               6  
│ │ │ │ -00024bd0: 2020 2020 2035 2020 2020 2020 2034 2020       5       4  
│ │ │ │ -00024be0: 2020 2020 3320 2020 2020 3220 2020 2020      3     2     
│ │ │ │ -00024bf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024c00: 2020 2020 2020 2020 2020 2020 2020 2053                 S
│ │ │ │ -00024c10: 6567 7265 203d 3e20 3132 3636 6820 202d  egre => 1266h  -
│ │ │ │ -00024c20: 2033 3930 6820 202b 2031 3134 6820 202d   390h  + 114h  -
│ │ │ │ -00024c30: 2033 3068 2020 2b20 3668 2020 2020 2020   30h  + 6h      
│ │ │ │ -00024c40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c60: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -00024c70: 2020 2020 2031 2020 2020 2020 2031 2020       1       1  
│ │ │ │ -00024c80: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ -00024c90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024cb0: 2020 2020 2020 2020 2020 2036 2020 2020             6    
│ │ │ │ -00024cc0: 2020 3520 2020 2020 2034 2020 2020 2020    5      4      
│ │ │ │ -00024cd0: 3320 2020 2020 3220 2020 2020 2020 2020  3     2         
│ │ │ │ -00024ce0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024cf0: 2020 2020 2020 2020 2020 2020 2020 2043                 C
│ │ │ │ -00024d00: 6865 726e 203d 3e20 3930 6820 202d 2031  hern => 90h  - 1
│ │ │ │ -00024d10: 3268 2020 2b20 3330 6820 202b 2031 3268  2h  + 30h  + 12h
│ │ │ │ -00024d20: 2020 2b20 3668 2020 2020 2020 2020 2020    + 6h          
│ │ │ │ -00024d30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d50: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -00024d60: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ -00024d70: 3120 2020 2020 3120 2020 2020 2020 2020  1     1         
│ │ │ │ -00024d80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024da0: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ -00024db0: 2020 2020 2020 3420 2020 2020 2033 2020        4      3  
│ │ │ │ -00024dc0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00024dd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024de0: 2020 2020 2020 2020 2020 2020 2020 2043                 C
│ │ │ │ -00024df0: 4620 3d3e 2039 3068 2020 2d20 3132 6820  F => 90h  - 12h 
│ │ │ │ -00024e00: 202b 2033 3068 2020 2b20 3132 6820 202b   + 30h  + 12h  +
│ │ │ │ -00024e10: 2036 6820 2020 2020 2020 2020 2020 2020   6h             
│ │ │ │ -00024e20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e40: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ -00024e50: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00024e60: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ -00024e70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000249c0: 2020 3220 2020 2020 2033 2020 2020 2020    2      3      
│ │ │ │ +000249d0: 3420 2020 2020 2020 3520 2020 2020 2020  4       5       
│ │ │ │ +000249e0: 3620 207c 0a7c 6f39 203d 204d 7574 6162  6  |.|o9 = Mutab
│ │ │ │ +000249f0: 6c65 4861 7368 5461 626c 657b 5365 6772  leHashTable{Segr
│ │ │ │ +00024a00: 654c 6973 7420 3d3e 207b 302c 2030 2c20  eList => {0, 0, 
│ │ │ │ +00024a10: 3668 202c 202d 3330 6820 2c20 3131 3468  6h , -30h , 114h
│ │ │ │ +00024a20: 202c 202d 3339 3068 202c 2031 3236 3668   , -390h , 1266h
│ │ │ │ +00024a30: 207d 7d7c 0a7c 2020 2020 2020 2020 2020   }}|.|          
│ │ │ │ +00024a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024a60: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ +00024a70: 3120 2020 2020 2020 3120 2020 2020 2020  1       1       
│ │ │ │ +00024a80: 3120 207c 0a7c 2020 2020 2020 2020 2020  1  |.|          
│ │ │ │ +00024a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024ab0: 3220 2020 2033 2020 2020 3420 2020 2035  2    3    4    5
│ │ │ │ +00024ac0: 2020 2020 3620 2020 2020 2020 2020 2020      6           
│ │ │ │ +00024ad0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024ae0: 2020 2020 2020 2020 2020 2020 476c 6973              Glis
│ │ │ │ +00024af0: 7420 3d3e 207b 312c 2033 6820 2c20 3368  t => {1, 3h , 3h
│ │ │ │ +00024b00: 202c 2033 6820 2c20 3368 202c 2033 6820   , 3h , 3h , 3h 
│ │ │ │ +00024b10: 2c20 3368 207d 2020 2020 2020 2020 2020  , 3h }          
│ │ │ │ +00024b20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024b40: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +00024b50: 3120 2020 2031 2020 2020 3120 2020 2031  1    1    1    1
│ │ │ │ +00024b60: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ +00024b70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024b90: 2020 2020 2020 2020 2020 3620 2020 2020            6     
│ │ │ │ +00024ba0: 2020 3520 2020 2020 2020 3420 2020 2020    5       4     
│ │ │ │ +00024bb0: 2033 2020 2020 2032 2020 2020 2020 2020   3     2        
│ │ │ │ +00024bc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024bd0: 2020 2020 2020 2020 2020 2020 5365 6772              Segr
│ │ │ │ +00024be0: 6520 3d3e 2031 3236 3668 2020 2d20 3339  e => 1266h  - 39
│ │ │ │ +00024bf0: 3068 2020 2b20 3131 3468 2020 2d20 3330  0h  + 114h  - 30
│ │ │ │ +00024c00: 6820 202b 2036 6820 2020 2020 2020 2020  h  + 6h         
│ │ │ │ +00024c10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024c30: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ +00024c40: 2020 3120 2020 2020 2020 3120 2020 2020    1       1     
│ │ │ │ +00024c50: 2031 2020 2020 2031 2020 2020 2020 2020   1     1        
│ │ │ │ +00024c60: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024c80: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ +00024c90: 2020 2020 2020 3420 2020 2020 2033 2020        4      3  
│ │ │ │ +00024ca0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00024cb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024cc0: 2020 2020 2020 2020 2020 2020 4368 6572              Cher
│ │ │ │ +00024cd0: 6e20 3d3e 2039 3068 2020 2d20 3132 6820  n => 90h  - 12h 
│ │ │ │ +00024ce0: 202b 2033 3068 2020 2b20 3132 6820 202b   + 30h  + 12h  +
│ │ │ │ +00024cf0: 2036 6820 2020 2020 2020 2020 2020 2020   6h             
│ │ │ │ +00024d00: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024d20: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ +00024d30: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00024d40: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +00024d50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024d70: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ +00024d80: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ +00024d90: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00024da0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024db0: 2020 2020 2020 2020 2020 2020 4346 203d              CF =
│ │ │ │ +00024dc0: 3e20 3930 6820 202d 2031 3268 2020 2b20  > 90h  - 12h  + 
│ │ │ │ +00024dd0: 3330 6820 202b 2031 3268 2020 2b20 3668  30h  + 12h  + 6h
│ │ │ │ +00024de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024df0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024e10: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00024e20: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00024e30: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00024e40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024e60: 2020 2036 2020 2020 2035 2020 2020 2034     6     5     4
│ │ │ │ +00024e70: 2020 2020 2033 2020 2020 2032 2020 2020       3     2    
│ │ │ │  00024e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e90: 2020 2020 2020 3620 2020 2020 3520 2020        6     5   
│ │ │ │ -00024ea0: 2020 3420 2020 2020 3320 2020 2020 3220    4     3     2 
│ │ │ │ -00024eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ec0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024ed0: 2020 2020 2020 2020 2020 2020 2020 2047                 G
│ │ │ │ -00024ee0: 203d 3e20 3368 2020 2b20 3368 2020 2b20   => 3h  + 3h  + 
│ │ │ │ -00024ef0: 3368 2020 2b20 3368 2020 2b20 3368 2020  3h  + 3h  + 3h  
│ │ │ │ -00024f00: 2b20 3368 2020 2b20 3120 2020 2020 2020  + 3h  + 1       
│ │ │ │ -00024f10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024f30: 2020 2020 2020 3120 2020 2020 3120 2020        1     1   
│ │ │ │ -00024f40: 2020 3120 2020 2020 3120 2020 2020 3120    1     1     1 
│ │ │ │ -00024f50: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ -00024f60: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00024e90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024ea0: 2020 2020 2020 2020 2020 2020 4720 3d3e              G =>
│ │ │ │ +00024eb0: 2033 6820 202b 2033 6820 202b 2033 6820   3h  + 3h  + 3h 
│ │ │ │ +00024ec0: 202b 2033 6820 202b 2033 6820 202b 2033   + 3h  + 3h  + 3
│ │ │ │ +00024ed0: 6820 202b 2031 2020 2020 2020 2020 2020  h  + 1          
│ │ │ │ +00024ee0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024f00: 2020 2031 2020 2020 2031 2020 2020 2031     1     1     1
│ │ │ │ +00024f10: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ +00024f20: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +00024f30: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00024f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024f70: 2d2d 2d2d 2d2d 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 2d2d 2b0a 7c69 3130 203a 2073  ------+.|i10 : s
│ │ │ │ -00024fc0: 6567 3d53 6567 7265 2820 492c 204f 7574  eg=Segre( I, Out
│ │ │ │ -00024fd0: 7075 743d 3e48 6173 6846 6f72 6d29 2020  put=>HashForm)  
│ │ │ │ +00024f80: 2d2d 2d2b 0a7c 6931 3020 3a20 7365 673d  ---+.|i10 : seg=
│ │ │ │ +00024f90: 5365 6772 6528 2049 2c20 4f75 7470 7574  Segre( I, Output
│ │ │ │ +00024fa0: 3d3e 4861 7368 466f 726d 2920 2020 2020  =>HashForm)     
│ │ │ │ +00024fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024fd0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025000: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00025000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025050: 2020 2020 2020 7c0a 7c6f 3130 203d 204d        |.|o10 = M
│ │ │ │ -00025060: 7574 6162 6c65 4861 7368 5461 626c 657b  utableHashTable{
│ │ │ │ -00025070: 2e2e 2e34 2e2e 2e7d 2020 2020 2020 2020  ...4...}        
│ │ │ │ +00025020: 2020 207c 0a7c 6f31 3020 3d20 4d75 7461     |.|o10 = Muta
│ │ │ │ +00025030: 626c 6548 6173 6854 6162 6c65 7b2e 2e2e  bleHashTable{...
│ │ │ │ +00025040: 342e 2e2e 7d20 2020 2020 2020 2020 2020  4...}           
│ │ │ │ +00025050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025070: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000250a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000250b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000250c0: 2020 207c 0a7c 6f31 3020 3a20 4d75 7461     |.|o10 : Muta
│ │ │ │ +000250d0: 626c 6548 6173 6854 6162 6c65 2020 2020  bleHashTable    
│ │ │ │  000250e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250f0: 2020 2020 2020 7c0a 7c6f 3130 203a 204d        |.|o10 : M
│ │ │ │ -00025100: 7574 6162 6c65 4861 7368 5461 626c 6520  utableHashTable 
│ │ │ │ -00025110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025140: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000250f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025110: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00025120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d  ----------------
│ │ │ │ -00025170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025190: 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a 2070  ------+.|i11 : p
│ │ │ │ -000251a0: 6565 6b20 7365 6720 2020 2020 2020 2020  eek seg         
│ │ │ │ -000251b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025160: 2d2d 2d2b 0a7c 6931 3120 3a20 7065 656b  ---+.|i11 : peek
│ │ │ │ +00025170: 2073 6567 2020 2020 2020 2020 2020 2020   seg            
│ │ │ │ +00025180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000251a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000251b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000251c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000251d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000251e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000251e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000251f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025200: 2020 207c 0a7c 2020 2020 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 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025260: 2020 2020 2020 3220 2020 2020 2033 2020        2      3  
│ │ │ │ -00025270: 2020 2020 3420 2020 2020 2020 3520 2020      4       5   
│ │ │ │ -00025280: 2020 2020 2020 7c0a 7c6f 3131 203d 204d        |.|o11 = M
│ │ │ │ -00025290: 7574 6162 6c65 4861 7368 5461 626c 657b  utableHashTable{
│ │ │ │ -000252a0: 5365 6772 654c 6973 7420 3d3e 207b 302c  SegreList => {0,
│ │ │ │ -000252b0: 2030 2c20 3668 202c 202d 3330 6820 2c20   0, 6h , -30h , 
│ │ │ │ -000252c0: 3131 3468 202c 202d 3339 3068 202c 2020  114h , -390h ,  
│ │ │ │ -000252d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000252e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000252f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025300: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00025310: 2020 2020 3120 2020 2020 2020 3120 2020      1       1   
│ │ │ │ -00025320: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025350: 2020 2020 3220 2020 2033 2020 2020 3420      2    3    4 
│ │ │ │ -00025360: 2020 2035 2020 2020 3620 2020 2020 2020     5    6       
│ │ │ │ -00025370: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025390: 476c 6973 7420 3d3e 207b 312c 2033 6820  Glist => {1, 3h 
│ │ │ │ -000253a0: 2c20 3368 202c 2033 6820 2c20 3368 202c  , 3h , 3h , 3h ,
│ │ │ │ -000253b0: 2033 6820 2c20 3368 207d 2020 2020 2020   3h , 3h }      
│ │ │ │ -000253c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000253d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000253e0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -000253f0: 2020 2020 3120 2020 2031 2020 2020 3120      1    1    1 
│ │ │ │ -00025400: 2020 2031 2020 2020 3120 2020 2020 2020     1    1       
│ │ │ │ -00025410: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025430: 2020 2020 2020 2020 2020 2020 2020 3620                6 
│ │ │ │ -00025440: 2020 2020 2020 3520 2020 2020 2020 3420        5       4 
│ │ │ │ -00025450: 2020 2020 2033 2020 2020 2032 2020 2020       3     2    
│ │ │ │ -00025460: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025480: 5365 6772 6520 3d3e 2031 3236 3668 2020  Segre => 1266h  
│ │ │ │ -00025490: 2d20 3339 3068 2020 2b20 3131 3468 2020  - 390h  + 114h  
│ │ │ │ -000254a0: 2d20 3330 6820 202b 2036 6820 2020 2020  - 30h  + 6h     
│ │ │ │ -000254b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000254c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000254d0: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -000254e0: 2020 2020 2020 3120 2020 2020 2020 3120        1       1 
│ │ │ │ -000254f0: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ -00025500: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00025230: 2020 2032 2020 2020 2020 3320 2020 2020     2      3     
│ │ │ │ +00025240: 2034 2020 2020 2020 2035 2020 2020 2020   4       5      
│ │ │ │ +00025250: 2020 207c 0a7c 6f31 3120 3d20 4d75 7461     |.|o11 = Muta
│ │ │ │ +00025260: 626c 6548 6173 6854 6162 6c65 7b53 6567  bleHashTable{Seg
│ │ │ │ +00025270: 7265 4c69 7374 203d 3e20 7b30 2c20 302c  reList => {0, 0,
│ │ │ │ +00025280: 2036 6820 2c20 2d33 3068 202c 2031 3134   6h , -30h , 114
│ │ │ │ +00025290: 6820 2c20 2d33 3930 6820 2c20 2020 2020  h , -390h ,     
│ │ │ │ +000252a0: 2020 207c 0a7c 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 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +000252e0: 2031 2020 2020 2020 2031 2020 2020 2020   1       1      
│ │ │ │ +000252f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025320: 2032 2020 2020 3320 2020 2034 2020 2020   2    3    4    
│ │ │ │ +00025330: 3520 2020 2036 2020 2020 2020 2020 2020  5    6          
│ │ │ │ +00025340: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025350: 2020 2020 2020 2020 2020 2020 2047 6c69               Gli
│ │ │ │ +00025360: 7374 203d 3e20 7b31 2c20 3368 202c 2033  st => {1, 3h , 3
│ │ │ │ +00025370: 6820 2c20 3368 202c 2033 6820 2c20 3368  h , 3h , 3h , 3h
│ │ │ │ +00025380: 202c 2033 6820 7d20 2020 2020 2020 2020   , 3h }         
│ │ │ │ +00025390: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000253a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000253b0: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +000253c0: 2031 2020 2020 3120 2020 2031 2020 2020   1    1    1    
│ │ │ │ +000253d0: 3120 2020 2031 2020 2020 2020 2020 2020  1    1          
│ │ │ │ +000253e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000253f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025400: 2020 2020 2020 2020 2020 2036 2020 2020             6    
│ │ │ │ +00025410: 2020 2035 2020 2020 2020 2034 2020 2020     5       4    
│ │ │ │ +00025420: 2020 3320 2020 2020 3220 2020 2020 2020    3     2       
│ │ │ │ +00025430: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025440: 2020 2020 2020 2020 2020 2020 2053 6567               Seg
│ │ │ │ +00025450: 7265 203d 3e20 3132 3636 6820 202d 2033  re => 1266h  - 3
│ │ │ │ +00025460: 3930 6820 202b 2031 3134 6820 202d 2033  90h  + 114h  - 3
│ │ │ │ +00025470: 3068 2020 2b20 3668 2020 2020 2020 2020  0h  + 6h        
│ │ │ │ +00025480: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000254a0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +000254b0: 2020 2031 2020 2020 2020 2031 2020 2020     1       1    
│ │ │ │ +000254c0: 2020 3120 2020 2020 3120 2020 2020 2020    1     1       
│ │ │ │ +000254d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000254e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000254f0: 2020 2020 3620 2020 2020 3520 2020 2020      6     5     
│ │ │ │ +00025500: 3420 2020 2020 3320 2020 2020 3220 2020  4     3     2   
│ │ │ │  00025510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025520: 2020 2020 2020 2036 2020 2020 2035 2020         6     5  
│ │ │ │ -00025530: 2020 2034 2020 2020 2033 2020 2020 2032     4     3     2
│ │ │ │ -00025540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025550: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025570: 4720 3d3e 2033 6820 202b 2033 6820 202b  G => 3h  + 3h  +
│ │ │ │ -00025580: 2033 6820 202b 2033 6820 202b 2033 6820   3h  + 3h  + 3h 
│ │ │ │ -00025590: 202b 2033 6820 202b 2031 2020 2020 2020   + 3h  + 1      
│ │ │ │ -000255a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000255b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000255c0: 2020 2020 2020 2031 2020 2020 2031 2020         1     1  
│ │ │ │ -000255d0: 2020 2031 2020 2020 2031 2020 2020 2031     1     1     1
│ │ │ │ -000255e0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ -000255f0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00025520: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025530: 2020 2020 2020 2020 2020 2020 2047 203d               G =
│ │ │ │ +00025540: 3e20 3368 2020 2b20 3368 2020 2b20 3368  > 3h  + 3h  + 3h
│ │ │ │ +00025550: 2020 2b20 3368 2020 2b20 3368 2020 2b20    + 3h  + 3h  + 
│ │ │ │ +00025560: 3368 2020 2b20 3120 2020 2020 2020 2020  3h  + 1         
│ │ │ │ +00025570: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025590: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ +000255a0: 3120 2020 2020 3120 2020 2020 3120 2020  1     1     1   
│ │ │ │ +000255b0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +000255c0: 2020 207c 0a7c 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 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025640: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 3620  ------|.|     6 
│ │ │ │ +00025610: 2d2d 2d7c 0a7c 2020 2020 2036 2020 2020  ---|.|     6    
│ │ │ │ +00025620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025660: 2020 207c 0a7c 3132 3636 6820 7d7d 2020     |.|1266h }}  
│ │ │ │  00025670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025690: 2020 2020 2020 7c0a 7c31 3236 3668 207d        |.|1266h }
│ │ │ │ -000256a0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ -000256b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000256a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000256b0: 2020 207c 0a7c 2020 2020 2031 2020 2020     |.|     1    
│ │ │ │  000256c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000256d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000256e0: 2020 2020 2020 7c0a 7c20 2020 2020 3120        |.|     1 
│ │ │ │ +000256e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000256f0: 2020 2020 2020 2020 2020 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 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00025700: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00025710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025780: 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a 2065  ------+.|i12 : e
│ │ │ │ -00025790: 753d 4575 6c65 7228 2049 2c20 4f75 7470  u=Euler( I, Outp
│ │ │ │ -000257a0: 7574 3d3e 4861 7368 466f 726d 2920 2020  ut=>HashForm)   
│ │ │ │ +00025750: 2d2d 2d2b 0a7c 6931 3220 3a20 6575 3d45  ---+.|i12 : eu=E
│ │ │ │ +00025760: 756c 6572 2820 492c 204f 7574 7075 743d  uler( I, Output=
│ │ │ │ +00025770: 3e48 6173 6846 6f72 6d29 2020 2020 2020  >HashForm)      
│ │ │ │ +00025780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000257a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000257b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000257c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000257d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000257d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000257e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000257f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025820: 2020 2020 2020 7c0a 7c6f 3132 203d 204d        |.|o12 = M
│ │ │ │ -00025830: 7574 6162 6c65 4861 7368 5461 626c 657b  utableHashTable{
│ │ │ │ -00025840: 2e2e 2e35 2e2e 2e7d 2020 2020 2020 2020  ...5...}        
│ │ │ │ +000257f0: 2020 207c 0a7c 6f31 3220 3d20 4d75 7461     |.|o12 = Muta
│ │ │ │ +00025800: 626c 6548 6173 6854 6162 6c65 7b2e 2e2e  bleHashTable{...
│ │ │ │ +00025810: 352e 2e2e 7d20 2020 2020 2020 2020 2020  5...}           
│ │ │ │ +00025820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025840: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025870: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00025870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000258a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025890: 2020 207c 0a7c 6f31 3220 3a20 4d75 7461     |.|o12 : Muta
│ │ │ │ +000258a0: 626c 6548 6173 6854 6162 6c65 2020 2020  bleHashTable    
│ │ │ │  000258b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000258c0: 2020 2020 2020 7c0a 7c6f 3132 203a 204d        |.|o12 : M
│ │ │ │ -000258d0: 7574 6162 6c65 4861 7368 5461 626c 6520  utableHashTable 
│ │ │ │ -000258e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000258f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025910: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000258c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000258d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000258e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000258f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025920: 2d2d 2d2d 2d2d 2d2d 2d2d 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  ----------------
│ │ │ │ -00025960: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2070  ------+.|i13 : p
│ │ │ │ -00025970: 6565 6b20 6575 2020 2020 2020 2020 2020  eek eu          
│ │ │ │ -00025980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025930: 2d2d 2d2b 0a7c 6931 3320 3a20 7065 656b  ---+.|i13 : peek
│ │ │ │ +00025940: 2065 7520 2020 2020 2020 2020 2020 2020   eu             
│ │ │ │ +00025950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025980: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000259a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000259b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000259b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000259c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000259d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000259e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000259f0: 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     
│ │ │ │ +000259d0: 2020 207c 0a7c 6f31 3320 3d20 4d75 7461     |.|o13 = Muta
│ │ │ │ +000259e0: 626c 6548 6173 6854 6162 6c65 7b45 756c  bleHashTable{Eul
│ │ │ │ +000259f0: 6572 203d 3e20 3130 2020 2020 2020 2020  er => 10        
│ │ │ │ +00025a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025a10: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ +00025a20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  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        |.+-------
│ │ │ │ +00025a40: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ +00025a50: 3520 2020 2020 2034 2020 2020 2020 3320  5      4      3 
│ │ │ │ +00025a60: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00025a70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025a80: 2020 2020 2020 2020 2020 2020 207b 302c               {0,
│ │ │ │ +00025a90: 2031 7d20 3d3e 2032 6820 202b 2032 3368   1} => 2h  + 23h
│ │ │ │ +00025aa0: 2020 2b20 3332 6820 202b 2033 3368 2020    + 32h  + 33h  
│ │ │ │ +00025ab0: 2b20 3138 6820 202b 2035 6820 2020 2020  + 18h  + 5h     
│ │ │ │ +00025ac0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025ae0: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ +00025af0: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ +00025b00: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ +00025b10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025b30: 2020 2020 2020 2036 2020 2020 2020 3520         6      5 
│ │ │ │ +00025b40: 2020 2020 2034 2020 2020 2020 3320 2020       4      3   
│ │ │ │ +00025b50: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00025b60: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025b70: 2020 2020 2020 2020 2020 2020 2043 534d               CSM
│ │ │ │ +00025b80: 203d 3e20 3130 6820 202b 2031 3268 2020   => 10h  + 12h  
│ │ │ │ +00025b90: 2b20 3232 6820 202b 2031 3668 2020 2b20  + 22h  + 16h  + 
│ │ │ │ +00025ba0: 3668 2020 2020 2020 2020 2020 2020 2020  6h              
│ │ │ │ +00025bb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025bd0: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ +00025be0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00025bf0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +00025c00: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025c20: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ +00025c30: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ +00025c40: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00025c50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025c60: 2020 2020 2020 2020 2020 2020 207b 307d               {0}
│ │ │ │ +00025c70: 203d 3e20 3668 2020 2b20 3138 6820 202b   => 6h  + 18h  +
│ │ │ │ +00025c80: 2032 3668 2020 2b20 3232 6820 202b 2031   26h  + 22h  + 1
│ │ │ │ +00025c90: 3068 2020 2b20 3268 2020 2020 2020 2020  0h  + 2h        
│ │ │ │ +00025ca0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025cc0: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00025cd0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00025ce0: 2020 3120 2020 2020 3120 2020 2020 2020    1     1       
│ │ │ │ +00025cf0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025d10: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ +00025d20: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ +00025d30: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00025d40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025d50: 2020 2020 2020 2020 2020 2020 207b 317d               {1}
│ │ │ │ +00025d60: 203d 3e20 3668 2020 2b20 3137 6820 202b   => 6h  + 17h  +
│ │ │ │ +00025d70: 2032 3868 2020 2b20 3237 6820 202b 2031   28h  + 27h  + 1
│ │ │ │ +00025d80: 3468 2020 2b20 3368 2020 2020 2020 2020  4h  + 3h        
│ │ │ │ +00025d90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025db0: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00025dc0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00025dd0: 2020 3120 2020 2020 3120 2020 2020 2020    1     1       
│ │ │ │ +00025de0: 2020 207c 0a2b 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 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
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│ │ │ │ +00026070: 696e 666f 726d 6174 696f 6e20 7468 6174  information that
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│ │ │ │ +000260f0: 7369 6e67 756c 6172 6974 7920 7375 6273  singularity subs
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│ │ │ │ +000261d0: 696e 7075 7420 6964 6561 6c2e 204e 6f74  input ideal. Not
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│ │ │ │ +00026270: 6964 6561 6c20 4920 6571 7561 6c73 2074  ideal I equals t
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│ │ │ │ +00026290: 7468 650a 7261 6469 6361 6c20 6f66 2049  the.radical of I
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│ │ │ │ +00026310: 5365 6772 6520 636c 6173 7365 7320 636f  Segre classes co
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│ │ │ │ +00026340: 6f66 2074 6865 2072 6164 6963 616c 206f  of the radical o
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│ │ │ │ +000263c0: 616e 0a65 7861 6d70 6c65 2062 656c 6f77  an.example below
│ │ │ │ +000263d0: 2e0a 0a2b 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 2d2d  ----------------
│ │ │ │  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)              
│ │ │ │ +00026420: 2d2b 0a7c 6931 3420 3a20 6373 6d58 4c68  -+.|i14 : csmXLh
│ │ │ │ +00026430: 6173 683d 4353 4d28 412c 492c 4f75 7470  ash=CSM(A,I,Outp
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│ │ │ │ +00026450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026470: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000264a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000264b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264f0: 2020 2020 7c0a 7c6f 3134 203d 204d 7574      |.|o14 = Mut
│ │ │ │ -00026500: 6162 6c65 4861 7368 5461 626c 657b 2e2e  ableHashTable{..
│ │ │ │ -00026510: 2e31 302e 2e2e 7d20 2020 2020 2020 2020  .10...}         
│ │ │ │ +000264c0: 207c 0a7c 6f31 3420 3d20 4d75 7461 626c   |.|o14 = Mutabl
│ │ │ │ +000264d0: 6548 6173 6854 6162 6c65 7b2e 2e2e 3130  eHashTable{...10
│ │ │ │ +000264e0: 2e2e 2e7d 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: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00026520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026540: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00026540: 2020 2020 2020 2020 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                  
│ │ │ │ +00026560: 207c 0a7c 6f31 3420 3a20 4d75 7461 626c   |.|o14 : Mutabl
│ │ │ │ +00026570: 6548 6173 6854 6162 6c65 2020 2020 2020  eHashTable      
│ │ │ │  00026580: 2020 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      |.+---------
│ │ │ │ +00026590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000265a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000265b0: 207c 0a2b 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 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                  
│ │ │ │ +00026600: 2d2b 0a7c 6931 3520 3a20 7065 656b 2063  -+.|i15 : peek c
│ │ │ │ +00026610: 736d 584c 6861 7368 2020 2020 2020 2020  smXLhash        
│ │ │ │ +00026620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026650: 207c 0a7c 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00026680: 2020 2020 2020 2020 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 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 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           
│ │ │ │ +000266a0: 207c 0a7c 6f31 3520 3d20 4d75 7461 626c   |.|o15 = Mutabl
│ │ │ │ +000266b0: 6548 6173 6854 6162 6c65 7b47 284a 6163  eHashTable{G(Jac
│ │ │ │ +000266c0: 6f62 6961 6e29 7b30 7d20 3d3e 2030 2020  obian){0} => 0  
│ │ │ │ +000266d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000266e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000266f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026700: 2020 2020 2020 2020 2020 2053 6567 7265             Segre
│ │ │ │ +00026710: 284a 6163 6f62 6961 6e29 7b30 7d20 3d3e  (Jacobian){0} =>
│ │ │ │ +00026720: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +00026730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026740: 207c 0a7c 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 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,
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│ │ │ │ -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 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 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 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      |.|         
│ │ │ │ +00026770: 2020 2020 2020 2020 3620 2020 2020 2020          6       
│ │ │ │ +00026780: 3520 2020 2020 2020 3420 2020 2020 2020  5       4       
│ │ │ │ +00026790: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000267a0: 2020 2020 2020 2020 2020 2053 6567 7265             Segre
│ │ │ │ +000267b0: 284a 6163 6f62 6961 6e29 7b30 2c20 317d  (Jacobian){0, 1}
│ │ │ │ +000267c0: 203d 3e20 3339 3068 2020 2d20 3338 3668   => 390h  - 386h
│ │ │ │ +000267d0: 2020 2b20 3135 3068 2020 2d20 2020 2020    + 150h  -     
│ │ │ │ +000267e0: 207c 0a7c 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 3120 2020 2020 2020          1       
│ │ │ │ +00026820: 3120 2020 2020 2020 3120 2020 2020 2020  1       1       
│ │ │ │ +00026830: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026860: 2020 2020 2020 2036 2020 2020 2020 3520         6      5 
│ │ │ │ +00026870: 2020 2020 3320 2020 2020 3220 2020 2020      3     2     
│ │ │ │ +00026880: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026890: 2020 2020 2020 2020 2020 2053 6567 7265             Segre
│ │ │ │ +000268a0: 284a 6163 6f62 6961 6e29 7b31 7d20 3d3e  (Jacobian){1} =>
│ │ │ │ +000268b0: 202d 2031 3630 6820 202b 2033 3268 2020   - 160h  + 32h  
│ │ │ │ +000268c0: 2d20 3468 2020 2b20 3268 2020 2020 2020  - 4h  + 2h      
│ │ │ │ +000268d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000268e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000268f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026900: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ +00026910: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ +00026920: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026950: 2020 2036 2020 2020 2020 3520 2020 2020     6      5     
│ │ │ │ +00026960: 2034 2020 2020 2020 3320 2020 2020 2020   4      3       
│ │ │ │ +00026970: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026980: 2020 2020 2020 2020 2020 2047 284a 6163             G(Jac
│ │ │ │ +00026990: 6f62 6961 6e29 7b30 2c20 317d 203d 3e20  obian){0, 1} => 
│ │ │ │ +000269a0: 3130 6820 202b 2031 3068 2020 2b20 3130  10h  + 10h  + 10
│ │ │ │ +000269b0: 6820 202b 2031 3068 2020 2b20 2020 2020  h  + 10h  +     
│ │ │ │ +000269c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000269d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000269e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000269f0: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00026a00: 2031 2020 2020 2020 3120 2020 2020 2020   1      1       
│ │ │ │ +00026a10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026a30: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00026a40: 2020 2020 2020 2020 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} => 
│ │ │ │ -00026ac0: 3268 2020 2b20 3268 2020 2b20 3120 2020  2h  + 2h  + 1   
│ │ │ │ -00026ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00026a60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026a70: 2020 2020 2020 2020 2020 2047 284a 6163             G(Jac
│ │ │ │ +00026a80: 6f62 6961 6e29 7b31 7d20 3d3e 2032 6820  obian){1} => 2h 
│ │ │ │ +00026a90: 202b 2032 6820 202b 2031 2020 2020 2020   + 2h  + 1      
│ │ │ │ +00026aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026ab0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026ad0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00026ae0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │  00026af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b10: 2020 3120 2020 2020 3120 2020 2020 2020    1     1       
│ │ │ │ -00026b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b50: 2020 2020 2020 2020 2020 3620 2020 2020            6     
│ │ │ │ -00026b60: 2035 2020 2020 2020 3420 2020 2020 2033   5      4      3
│ │ │ │ -00026b70: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -00026b80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026b90: 2020 2020 2020 2020 2020 2020 2020 7b30                {0
│ │ │ │ -00026ba0: 2c20 317d 203d 3e20 3268 2020 2b20 3233  , 1} => 2h  + 23
│ │ │ │ -00026bb0: 6820 202b 2033 3268 2020 2b20 3333 6820  h  + 32h  + 33h 
│ │ │ │ -00026bc0: 202b 2031 3868 2020 2b20 3568 2020 2020   + 18h  + 5h    
│ │ │ │ -00026bd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026bf0: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ -00026c00: 2031 2020 2020 2020 3120 2020 2020 2031   1      1      1
│ │ │ │ -00026c10: 2020 2020 2020 3120 2020 2020 3120 2020        1     1   
│ │ │ │ -00026c20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026c40: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ -00026c50: 2020 2020 2020 3420 2020 2020 2033 2020        4      3  
│ │ │ │ -00026c60: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00026c70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026c80: 2020 2020 2020 2020 2020 2020 2020 4353                CS
│ │ │ │ -00026c90: 4d20 3d3e 2031 3068 2020 2b20 3132 6820  M => 10h  + 12h 
│ │ │ │ -00026ca0: 202b 2032 3268 2020 2b20 3136 6820 202b   + 22h  + 16h  +
│ │ │ │ -00026cb0: 2036 6820 2020 2020 2020 2020 2020 2020   6h             
│ │ │ │ -00026cc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ce0: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ -00026cf0: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00026d00: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ -00026d10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026d30: 2020 2020 2020 2036 2020 2020 2020 3520         6      5 
│ │ │ │ -00026d40: 2020 2020 2034 2020 2020 2020 3320 2020       4      3   
│ │ │ │ -00026d50: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00026d60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026d70: 2020 2020 2020 2020 2020 2020 2020 7b30                {0
│ │ │ │ -00026d80: 7d20 3d3e 2036 6820 202b 2031 3868 2020  } => 6h  + 18h  
│ │ │ │ -00026d90: 2b20 3236 6820 202b 2032 3268 2020 2b20  + 26h  + 22h  + 
│ │ │ │ -00026da0: 3130 6820 202b 2032 6820 2020 2020 2020  10h  + 2h       
│ │ │ │ -00026db0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026dd0: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ -00026de0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00026df0: 2020 2031 2020 2020 2031 2020 2020 2020     1     1      
│ │ │ │ -00026e00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026e20: 2020 2020 2020 2036 2020 2020 2020 3520         6      5 
│ │ │ │ -00026e30: 2020 2020 2034 2020 2020 2020 3320 2020       4      3   
│ │ │ │ -00026e40: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00026e50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026e60: 2020 2020 2020 2020 2020 2020 2020 7b31                {1
│ │ │ │ -00026e70: 7d20 3d3e 2036 6820 202b 2031 3768 2020  } => 6h  + 17h  
│ │ │ │ -00026e80: 2b20 3238 6820 202b 2032 3768 2020 2b20  + 28h  + 27h  + 
│ │ │ │ -00026e90: 3134 6820 202b 2033 6820 2020 2020 2020  14h  + 3h       
│ │ │ │ -00026ea0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ec0: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ -00026ed0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00026ee0: 2020 2031 2020 2020 2031 2020 2020 2020     1     1      
│ │ │ │ -00026ef0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +00026b00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026b20: 2020 2020 2020 2036 2020 2020 2020 3520         6      5 
│ │ │ │ +00026b30: 2020 2020 2034 2020 2020 2020 3320 2020       4      3   
│ │ │ │ +00026b40: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00026b50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026b60: 2020 2020 2020 2020 2020 207b 302c 2031             {0, 1
│ │ │ │ +00026b70: 7d20 3d3e 2032 6820 202b 2032 3368 2020  } => 2h  + 23h  
│ │ │ │ +00026b80: 2b20 3332 6820 202b 2033 3368 2020 2b20  + 32h  + 33h  + 
│ │ │ │ +00026b90: 3138 6820 202b 2035 6820 2020 2020 2020  18h  + 5h       
│ │ │ │ +00026ba0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026bc0: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ +00026bd0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00026be0: 2020 2031 2020 2020 2031 2020 2020 2020     1     1      
│ │ │ │ +00026bf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026c10: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ +00026c20: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ +00026c30: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00026c40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026c50: 2020 2020 2020 2020 2020 2043 534d 203d             CSM =
│ │ │ │ +00026c60: 3e20 3130 6820 202b 2031 3268 2020 2b20  > 10h  + 12h  + 
│ │ │ │ +00026c70: 3232 6820 202b 2031 3668 2020 2b20 3668  22h  + 16h  + 6h
│ │ │ │ +00026c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026c90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026cb0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00026cc0: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00026cd0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00026ce0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026d00: 2020 2020 3620 2020 2020 2035 2020 2020      6      5    
│ │ │ │ +00026d10: 2020 3420 2020 2020 2033 2020 2020 2020    4      3      
│ │ │ │ +00026d20: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00026d30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026d40: 2020 2020 2020 2020 2020 207b 307d 203d             {0} =
│ │ │ │ +00026d50: 3e20 3668 2020 2b20 3138 6820 202b 2032  > 6h  + 18h  + 2
│ │ │ │ +00026d60: 3668 2020 2b20 3232 6820 202b 2031 3068  6h  + 22h  + 10h
│ │ │ │ +00026d70: 2020 2b20 3268 2020 2020 2020 2020 2020    + 2h          
│ │ │ │ +00026d80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026da0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00026db0: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ +00026dc0: 3120 2020 2020 3120 2020 2020 2020 2020  1     1         
│ │ │ │ +00026dd0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026df0: 2020 2020 3620 2020 2020 2035 2020 2020      6      5    
│ │ │ │ +00026e00: 2020 3420 2020 2020 2033 2020 2020 2020    4      3      
│ │ │ │ +00026e10: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00026e20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026e30: 2020 2020 2020 2020 2020 207b 317d 203d             {1} =
│ │ │ │ +00026e40: 3e20 3668 2020 2b20 3137 6820 202b 2032  > 6h  + 17h  + 2
│ │ │ │ +00026e50: 3868 2020 2b20 3237 6820 202b 2031 3468  8h  + 27h  + 14h
│ │ │ │ +00026e60: 2020 2b20 3368 2020 2020 2020 2020 2020    + 3h          
│ │ │ │ +00026e70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026e90: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00026ea0: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ +00026eb0: 3120 2020 2020 3120 2020 2020 2020 2020  1     1         
│ │ │ │ +00026ec0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ +00026ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026f40: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020  ----|.|         
│ │ │ │ -00026f50: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │ -00026f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026f10: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00026f20: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │ +00026f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026f60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00026f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026f90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00026f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026fb0: 207c 0a7c 2020 2033 2020 2020 2032 2020   |.|   3     2  
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│ │ │ │ -00027320: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00027380: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │  00027390: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000273b0: 616c 206f 6620 5220 2020 2020 2020 2020  al of R         
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│ │ │ │ -000273d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00027400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -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)         
│ │ │ │ +00027410: 2d2b 0a7c 6931 3720 3a20 4353 4d28 412c  -+.|i17 : CSM(A,
│ │ │ │ +00027420: 7261 6469 6361 6c20 4b29 3d3d 4353 4d28  radical K)==CSM(
│ │ │ │ +00027430: 412c 4b29 2020 2020 2020 2020 2020 2020  A,K)            
│ │ │ │ +00027440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027460: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00027470: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00027490: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000274b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000274b0: 207c 0a7c 6f31 3720 3d20 7472 7565 2020   |.|o17 = true  
│ │ │ │  000274c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000274d0: 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                  
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│ │ │ │ -00027530: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
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│ │ │ │ -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;        
│ │ │ │ +00027550: 2d2b 0a7c 6931 3820 3a20 4a3d 6964 6561  -+.|i18 : J=idea
│ │ │ │ +00027560: 6c20 6a61 636f 6269 616e 2072 6164 6963  l jacobian radic
│ │ │ │ +00027570: 616c 204b 3b20 2020 2020 2020 2020 2020  al K;           
│ │ │ │ +00027580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000275a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000275b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000275d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000275d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00027600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000275f0: 207c 0a7c 6f31 3820 3a20 4964 6561 6c20   |.|o18 : Ideal 
│ │ │ │ +00027600: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │  00027610: 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                  
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│ │ │ │ -00027670: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
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│ │ │ │ +00027670: 2d2d 2d2d 2d2d 2d2d 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)   
│ │ │ │ +00027690: 2d2b 0a7c 6931 3920 3a20 7365 674a 3d53  -+.|i19 : segJ=S
│ │ │ │ +000276a0: 6567 7265 2841 2c4a 2c4f 7574 7075 743d  egre(A,J,Output=
│ │ │ │ +000276b0: 3e48 6173 6846 6f72 6d29 2020 2020 2020  >HashForm)      
│ │ │ │ +000276c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000276d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000276e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000276f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00027750: 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...}          
│ │ │ │ +00027730: 207c 0a7c 6f31 3920 3d20 4d75 7461 626c   |.|o19 = Mutabl
│ │ │ │ +00027740: 6548 6173 6854 6162 6c65 7b2e 2e2e 342e  eHashTable{...4.
│ │ │ │ +00027750: 2e2e 7d20 2020 2020 2020 2020 2020 2020  ..}             
│ │ │ │ +00027760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027780: 207c 0a7c 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000277b0: 2020 2020 2020 2020 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                  
│ │ │ │ +000277d0: 207c 0a7c 6f31 3920 3a20 4d75 7461 626c   |.|o19 : Mutabl
│ │ │ │ +000277e0: 6548 6173 6854 6162 6c65 2020 2020 2020  eHashTable      
│ │ │ │  000277f0: 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                  
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│ │ │ │ +00027840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -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"
│ │ │ │ +00027870: 2d2b 0a7c 6932 3020 3a20 6373 6d58 4c68  -+.|i20 : csmXLh
│ │ │ │ +00027880: 6173 6823 2822 4728 4a61 636f 6269 616e  ash#("G(Jacobian
│ │ │ │ +00027890: 2922 7c74 6f53 7472 696e 6728 7b30 2c31  )"|toString({0,1
│ │ │ │ +000278a0: 7d29 293d 3d73 6567 4a23 2247 2220 2020  }))==segJ#"G"   
│ │ │ │ +000278b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000278c0: 207c 0a7c 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000278f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027900: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00027910: 207c 0a7c 6f32 3020 3d20 7472 7565 2020   |.|o20 = true  
│ │ │ │  00027920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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      |.+---------
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│ │ │ │ +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 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
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│ │ │ │ +000279c0: 6173 6823 2822 5365 6772 6528 4a61 636f  ash#("Segre(Jaco
│ │ │ │ +000279d0: 6269 616e 2922 7c74 6f53 7472 696e 6728  bian)"|toString(
│ │ │ │ +000279e0: 7b30 2c31 7d29 293d 3d73 6567 4a23 2253  {0,1}))==segJ#"S
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│ │ │ │ +00027a00: 207c 0a7c 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 2020                  
│ │ │ │  00027a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027a50: 207c 0a7c 6f32 3120 3d20 7472 7565 2020   |.|o21 = true  
│ │ │ │  00027a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a70: 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      |.+---------
│ │ │ │ +00027a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027aa0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
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│ │ │ │ +00027ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027ad0: 2d2d 2d2d 2d2d 2d2d 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,...--------
│ │ │ │ +00027af0: 2d2b 0a0a 4675 6e63 7469 6f6e 7320 7769  -+..Functions wi
│ │ │ │ +00027b00: 7468 206f 7074 696f 6e61 6c20 6172 6775  th optional argu
│ │ │ │ +00027b10: 6d65 6e74 206e 616d 6564 204f 7574 7075  ment named Outpu
│ │ │ │ +00027b20: 743a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  t:.=============
│ │ │ │ +00027b30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00027b40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00027b50: 3d0a 0a20 202a 2022 4368 6572 6e28 2e2e  =..  * "Chern(..
│ │ │ │ +00027b60: 2e2c 4f75 7470 7574 3d3e 2e2e 2e29 2220  .,Output=>...)" 
│ │ │ │ +00027b70: 2d2d 2073 6565 202a 6e6f 7465 2043 6865  -- see *note Che
│ │ │ │ +00027b80: 726e 3a20 4368 6572 6e2c 202d 2d20 5468  rn: Chern, -- Th
│ │ │ │ +00027b90: 6520 4368 6572 6e20 636c 6173 730a 2020  e Chern class.  
│ │ │ │ +00027ba0: 2a20 2243 534d 282e 2e2e 2c4f 7574 7075  * "CSM(...,Outpu
│ │ │ │ +00027bb0: 743d 3e2e 2e2e 2922 202d 2d20 7365 6520  t=>...)" -- see 
│ │ │ │ +00027bc0: 2a6e 6f74 6520 4353 4d3a 2043 534d 2c20  *note CSM: CSM, 
│ │ │ │ +00027bd0: 2d2d 2054 6865 0a20 2020 2043 6865 726e  -- The.    Chern
│ │ │ │ +00027be0: 2d53 6368 7761 7274 7a2d 4d61 6350 6865  -Schwartz-MacPhe
│ │ │ │ +00027bf0: 7273 6f6e 2063 6c61 7373 0a20 202a 2022  rson class.  * "
│ │ │ │ +00027c00: 4575 6c65 7228 2e2e 2e2c 4f75 7470 7574  Euler(...,Output
│ │ │ │ +00027c10: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a  =>...)" -- see *
│ │ │ │ +00027c20: 6e6f 7465 2045 756c 6572 3a20 4575 6c65  note Euler: Eule
│ │ │ │ +00027c30: 722c 202d 2d20 5468 6520 4575 6c65 720a  r, -- The Euler.
│ │ │ │ +00027c40: 2020 2020 4368 6172 6163 7465 7269 7374      Characterist
│ │ │ │ +00027c50: 6963 0a20 202a 2022 5365 6772 6528 2e2e  ic.  * "Segre(..
│ │ │ │ +00027c60: 2e2c 4f75 7470 7574 3d3e 2e2e 2e29 2220  .,Output=>...)" 
│ │ │ │ +00027c70: 2d2d 2073 6565 202a 6e6f 7465 2053 6567  -- see *note Seg
│ │ │ │ +00027c80: 7265 3a20 5365 6772 652c 202d 2d20 5468  re: Segre, -- Th
│ │ │ │ +00027c90: 6520 5365 6772 6520 636c 6173 7320 6f66  e Segre class of
│ │ │ │ +00027ca0: 2061 0a20 2020 2073 7562 7363 6865 6d65   a.    subscheme
│ │ │ │ +00027cb0: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ +00027cc0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +00027cd0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ +00027ce0: 6563 7420 2a6e 6f74 6520 4f75 7470 7574  ect *note Output
│ │ │ │ +00027cf0: 3a20 4f75 7470 7574 2c20 6973 2061 202a  : Output, is a *
│ │ │ │ +00027d00: 6e6f 7465 2073 796d 626f 6c3a 2028 4d61  note symbol: (Ma
│ │ │ │ +00027d10: 6361 756c 6179 3244 6f63 2953 796d 626f  caulay2Doc)Symbo
│ │ │ │ +00027d20: 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d  l,...-----------
│ │ │ │ +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: 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
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│ │ │ │ -00027dd0: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
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│ │ │ │ -00027e40: 6572 6973 7469 6343 6c61 7373 6573 2e69  eristicClasses.i
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│ │ │ │ -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  ****************
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│ │ │ │ -00027ed0: 6f72 6974 686d 7320 7573 6564 2066 6f72  orithms used for
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -000281a0: 2069 6e64 6963 6174 6520 7468 6174 2061   indicate that a
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│ │ │ │ -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
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│ │ │ │ -00028310: 206f 6620 7365 7665 7261 6c20 6469 6666   of several diff
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│ │ │ │ -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;          
│ │ │ │ +00027d70: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
│ │ │ │ +00027d80: 206f 6620 7468 6973 2064 6f63 756d 656e   of this documen
│ │ │ │ +00027d90: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ +00027da0: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ +00027db0: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e  /macaulay2-1.25.
│ │ │ │ +00027dc0: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61  11+ds/M2/Macaula
│ │ │ │ +00027dd0: 7932 2f70 6163 6b61 6765 732f 0a43 6861  y2/packages/.Cha
│ │ │ │ +00027de0: 7261 6374 6572 6973 7469 6343 6c61 7373  racteristicClass
│ │ │ │ +00027df0: 6573 2e6d 323a 3234 3639 3a30 2e0a 1f0a  es.m2:2469:0....
│ │ │ │ +00027e00: 4669 6c65 3a20 4368 6172 6163 7465 7269  File: Characteri
│ │ │ │ +00027e10: 7374 6963 436c 6173 7365 732e 696e 666f  sticClasses.info
│ │ │ │ +00027e20: 2c20 4e6f 6465 3a20 7072 6f62 6162 696c  , Node: probabil
│ │ │ │ +00027e30: 6973 7469 6320 616c 676f 7269 7468 6d2c  istic algorithm,
│ │ │ │ +00027e40: 204e 6578 743a 2053 6567 7265 2c20 5072   Next: Segre, Pr
│ │ │ │ +00027e50: 6576 3a20 4f75 7470 7574 2c20 5570 3a20  ev: Output, Up: 
│ │ │ │ +00027e60: 546f 700a 0a70 726f 6261 6269 6c69 7374  Top..probabilist
│ │ │ │ +00027e70: 6963 2061 6c67 6f72 6974 686d 0a2a 2a2a  ic algorithm.***
│ │ │ │ +00027e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00027e90: 2a2a 2a2a 0a0a 5468 6520 616c 676f 7269  ****..The algori
│ │ │ │ +00027ea0: 7468 6d73 2075 7365 6420 666f 7220 7468  thms used for th
│ │ │ │ +00027eb0: 6520 636f 6d70 7574 6174 696f 6e20 6f66  e computation of
│ │ │ │ +00027ec0: 2063 6861 7261 6374 6572 6973 7469 6320   characteristic 
│ │ │ │ +00027ed0: 636c 6173 7365 7320 6172 650a 7072 6f62  classes are.prob
│ │ │ │ +00027ee0: 6162 696c 6973 7469 632e 2054 6865 6f72  abilistic. Theor
│ │ │ │ +00027ef0: 6574 6963 616c 6c79 2c20 7468 6579 2063  etically, they c
│ │ │ │ +00027f00: 616c 6375 6c61 7465 2074 6865 2063 6c61  alculate the cla
│ │ │ │ +00027f10: 7373 6573 2063 6f72 7265 6374 6c79 2066  sses correctly f
│ │ │ │ +00027f20: 6f72 2061 0a67 656e 6572 616c 2063 686f  or a.general cho
│ │ │ │ +00027f30: 6963 6520 6f66 2063 6572 7461 696e 2070  ice of certain p
│ │ │ │ +00027f40: 6f6c 796e 6f6d 6961 6c73 2e20 5468 6174  olynomials. That
│ │ │ │ +00027f50: 2069 732c 2074 6865 7265 2069 7320 616e   is, there is an
│ │ │ │ +00027f60: 206f 7065 6e20 6465 6e73 6520 5a61 7269   open dense Zari
│ │ │ │ +00027f70: 736b 690a 7365 7420 666f 7220 7768 6963  ski.set for whic
│ │ │ │ +00027f80: 6820 7468 6520 616c 676f 7269 7468 6d20  h the algorithm 
│ │ │ │ +00027f90: 7969 656c 6473 2074 6865 2063 6f72 7265  yields the corre
│ │ │ │ +00027fa0: 6374 2063 6c61 7373 2c20 692e 652e 2c20  ct class, i.e., 
│ │ │ │ +00027fb0: 7468 6520 636f 7272 6563 7420 636c 6173  the correct clas
│ │ │ │ +00027fc0: 730a 6973 2063 616c 6375 6c61 7465 6420  s.is calculated 
│ │ │ │ +00027fd0: 7769 7468 2070 726f 6261 6269 6c69 7479  with probability
│ │ │ │ +00027fe0: 2031 2e20 486f 7765 7665 722c 2073 696e   1. However, sin
│ │ │ │ +00027ff0: 6365 2074 6865 2069 6d70 6c65 6d65 6e74  ce the implement
│ │ │ │ +00028000: 6174 696f 6e20 776f 726b 7320 6f76 6572  ation works over
│ │ │ │ +00028010: 0a61 2064 6973 6372 6574 6520 7072 6f62  .a discrete prob
│ │ │ │ +00028020: 6162 696c 6974 7920 7370 6163 6520 7468  ability space th
│ │ │ │ +00028030: 6572 6520 6973 2061 2076 6572 7920 736d  ere is a very sm
│ │ │ │ +00028040: 616c 6c2c 2062 7574 206e 6f6e 2d7a 6572  all, but non-zer
│ │ │ │ +00028050: 6f2c 2070 726f 6261 6269 6c69 7479 0a6f  o, probability.o
│ │ │ │ +00028060: 6620 6e6f 7420 636f 6d70 7574 696e 6720  f not computing 
│ │ │ │ +00028070: 7468 6520 636f 7272 6563 7420 636c 6173  the correct clas
│ │ │ │ +00028080: 732e 2053 6b65 7074 6963 616c 2075 7365  s. Skeptical use
│ │ │ │ +00028090: 7273 2073 686f 756c 6420 7265 7065 6174  rs should repeat
│ │ │ │ +000280a0: 2063 616c 6375 6c61 7469 6f6e 730a 7365   calculations.se
│ │ │ │ +000280b0: 7665 7261 6c20 7469 6d65 7320 746f 2069  veral times to i
│ │ │ │ +000280c0: 6e63 7265 6173 6520 7468 6520 7072 6f62  ncrease the prob
│ │ │ │ +000280d0: 6162 696c 6974 7920 6f66 2063 6f6d 7075  ability of compu
│ │ │ │ +000280e0: 7469 6e67 2074 6865 2063 6f72 7265 6374  ting the correct
│ │ │ │ +000280f0: 2063 6c61 7373 2e0a 0a49 6e20 7468 6520   class...In the 
│ │ │ │ +00028100: 6361 7365 206f 6620 7468 6520 7379 6d62  case of the symb
│ │ │ │ +00028110: 6f6c 6963 2069 6d70 6c65 6d65 6e74 6174  olic implementat
│ │ │ │ +00028120: 696f 6e20 6f66 2074 6865 2050 726f 6a65  ion of the Proje
│ │ │ │ +00028130: 6374 6976 6544 6567 7265 6520 6d65 7468  ctiveDegree meth
│ │ │ │ +00028140: 6f64 0a70 7261 6374 6963 616c 2065 7870  od.practical exp
│ │ │ │ +00028150: 6572 6965 6e63 6520 616e 6420 616c 676f  erience and algo
│ │ │ │ +00028160: 7269 7468 6d20 7465 7374 696e 6720 696e  rithm testing in
│ │ │ │ +00028170: 6469 6361 7465 2074 6861 7420 6120 6669  dicate that a fi
│ │ │ │ +00028180: 6e69 7465 2066 6965 6c64 2077 6974 680a  nite field with.
│ │ │ │ +00028190: 6f76 6572 2032 3530 3030 2065 6c65 6d65  over 25000 eleme
│ │ │ │ +000281a0: 6e74 7320 6973 206d 6f72 6520 7468 616e  nts is more than
│ │ │ │ +000281b0: 2073 7566 6669 6369 656e 7420 746f 2065   sufficient to e
│ │ │ │ +000281c0: 7870 6563 7420 6120 636f 7272 6563 7420  xpect a correct 
│ │ │ │ +000281d0: 7265 7375 6c74 2077 6974 680a 6869 6768  result with.high
│ │ │ │ +000281e0: 2070 726f 6261 6269 6c69 7479 2c20 692e   probability, i.
│ │ │ │ +000281f0: 652e 2075 7369 6e67 2074 6865 2066 696e  e. using the fin
│ │ │ │ +00028200: 6974 6520 6669 656c 6420 6b6b 3d5a 5a2f  ite field kk=ZZ/
│ │ │ │ +00028210: 3235 3037 3320 7468 6520 6578 7065 7269  25073 the experi
│ │ │ │ +00028220: 6d65 6e74 616c 0a63 6861 6e63 6520 6f66  mental.chance of
│ │ │ │ +00028230: 2066 6169 6c75 7265 2077 6974 6820 7468   failure with th
│ │ │ │ +00028240: 6520 5072 6f6a 6563 7469 7665 4465 6772  e ProjectiveDegr
│ │ │ │ +00028250: 6565 2061 6c67 6f72 6974 686d 206f 6e20  ee algorithm on 
│ │ │ │ +00028260: 6120 7661 7269 6574 7920 6f66 2065 7861  a variety of exa
│ │ │ │ +00028270: 6d70 6c65 730a 7761 7320 6c65 7373 2074  mples.was less t
│ │ │ │ +00028280: 6861 6e20 312f 3230 3030 2e20 5573 696e  han 1/2000. Usin
│ │ │ │ +00028290: 6720 7468 6520 6669 6e69 7465 2066 6965  g the finite fie
│ │ │ │ +000282a0: 6c64 206b 6b3d 5a5a 2f33 3237 3439 2072  ld kk=ZZ/32749 r
│ │ │ │ +000282b0: 6573 756c 7465 6420 696e 206e 6f0a 6661  esulted in no.fa
│ │ │ │ +000282c0: 696c 7572 6573 2069 6e20 6f76 6572 2031  ilures in over 1
│ │ │ │ +000282d0: 3030 3030 2061 7474 656d 7074 7320 6f66  0000 attempts of
│ │ │ │ +000282e0: 2073 6576 6572 616c 2064 6966 6665 7265   several differe
│ │ │ │ +000282f0: 6e74 2065 7861 6d70 6c65 732e 0a0a 5765  nt examples...We
│ │ │ │ +00028300: 2069 6c6c 7573 7472 6174 6520 7468 6520   illustrate the 
│ │ │ │ +00028310: 7072 6f62 6162 696c 6973 7469 6320 6265  probabilistic be
│ │ │ │ +00028320: 6861 7669 6f75 7220 7769 7468 2061 6e20  haviour with an 
│ │ │ │ +00028330: 6578 616d 706c 6520 7768 6572 6520 7468  example where th
│ │ │ │ +00028340: 6520 6368 6f73 656e 0a72 616e 646f 6d20  e chosen.random 
│ │ │ │ +00028350: 7365 6564 206c 6561 6473 2074 6f20 6120  seed leads to a 
│ │ │ │ +00028360: 7772 6f6e 6720 7265 7375 6c74 2069 6e20  wrong result in 
│ │ │ │ +00028370: 7468 6520 6669 7273 7420 6361 6c63 756c  the first calcul
│ │ │ │ +00028380: 6174 696f 6e2e 0a0a 2b2d 2d2d 2d2d 2d2d  ation...+-------
│ │ │ │ +00028390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000283a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000283b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ +000283c0: 2073 6574 5261 6e64 6f6d 5365 6564 2031   setRandomSeed 1
│ │ │ │ +000283d0: 3231 3b20 2020 2020 2020 2020 2020 2020  21;             
│ │ │ │ +000283e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000283f0: 2d2d 2073 6574 7469 6e67 2072 616e 646f  -- setting rando
│ │ │ │ +00028400: 6d20 7365 6564 2074 6f20 3132 3120 2020  m seed to 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]    
│ │ │ │ +00028420: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00028430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028450: 2d2d 2b0a 7c69 3220 3a20 5220 3d20 5151  --+.|i2 : R = QQ
│ │ │ │ +00028460: 5b78 2c79 2c7a 2c77 5d20 2020 2020 2020  [x,y,z,w]       
│ │ │ │ +00028470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028480: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +000284b0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ +000284c0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  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            
│ │ │ │ +000284e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000284f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +00028520: 7c6f 3220 3a20 506f 6c79 6e6f 6d69 616c  |o2 : Polynomial
│ │ │ │ +00028530: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │  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                  
│ │ │ │ +00028550: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00028560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028580: 2d2d 2d2d 2b0a 7c69 3320 3a20 4920 3d20  ----+.|i3 : I = 
│ │ │ │ +00028590: 6d69 6e6f 7273 2832 2c6d 6174 7269 787b  minors(2,matrix{
│ │ │ │ +000285a0: 7b78 2c79 2c7a 7d2c 7b79 2c7a 2c77 7d7d  {x,y,z},{y,z,w}}
│ │ │ │ +000285b0: 2920 2020 2020 207c 0a7c 2020 2020 2020  )      |.|      
│ │ │ │ +000285c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000285d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000285e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000285f0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  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                  
│ │ │ │ +00028610: 2020 2020 2032 2020 2020 2020 207c 0a7c       2       |.|
│ │ │ │ +00028620: 6f33 203d 2069 6465 616c 2028 2d20 7920  o3 = ideal (- y 
│ │ │ │ +00028630: 202b 2078 2a7a 2c20 2d20 792a 7a20 2b20   + x*z, - y*z + 
│ │ │ │ +00028640: 782a 772c 202d 207a 2020 2b20 792a 7729  x*w, - z  + y*w)
│ │ │ │ +00028650: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00028660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028680: 2020 207c 0a7c 6f33 203a 2049 6465 616c     |.|o3 : Ideal
│ │ │ │ +00028690: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │  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  .|              
│ │ │ │ +000286b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000286c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000286d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000286e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ +000286f0: 2043 6865 726e 2028 492c 436f 6d70 4d65   Chern (I,CompMe
│ │ │ │ +00028700: 7468 6f64 3d3e 506e 5265 7369 6475 616c  thod=>PnResidual
│ │ │ │ +00028710: 2920 2020 2020 2020 2020 2020 7c0a 7c20  )           |.| 
│ │ │ │ +00028720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028740: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028750: 0a7c 2020 2020 2020 2033 2020 2020 2032  .|       3     2
│ │ │ │  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             
│ │ │ │ +00028780: 2020 7c0a 7c6f 3420 3d20 3248 2020 2b20    |.|o4 = 2H  + 
│ │ │ │ +00028790: 3348 2020 2020 2020 2020 2020 2020 2020  3H              
│ │ │ │  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           
│ │ │ │ +000287b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000287c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +000287f0: 5a5a 5b48 5d20 2020 2020 2020 2020 2020  ZZ[H]           
│ │ │ │  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]        
│ │ │ │ +00028810: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +00028820: 203a 202d 2d2d 2d2d 2020 2020 2020 2020   : -----        
│ │ │ │  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 : -----     
│ │ │ │ +00028850: 7c20 2020 2020 2020 2034 2020 2020 2020  |        4      
│ │ │ │  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   
│ │ │ │ +00028880: 207c 0a7c 2020 2020 2020 2048 2020 2020   |.|       H    
│ │ │ │  00028890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +000288b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000288c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000288d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000288e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2043  -------+.|i5 : C
│ │ │ │ +000288f0: 6865 726e 2028 492c 436f 6d70 4d65 7468  hern (I,CompMeth
│ │ │ │ +00028900: 6f64 3d3e 506e 5265 7369 6475 616c 2920  od=>PnResidual) 
│ │ │ │ +00028910: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00028920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028940: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00028950: 2020 2020 2020 2033 2020 2020 2032 2020         3     2  
│ │ │ │  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               
│ │ │ │ +00028980: 7c0a 7c6f 3520 3d20 3248 2020 2b20 3348  |.|o5 = 2H  + 3H
│ │ │ │ +00028990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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             
│ │ │ │ +000289b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000289c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +000289e0: 2020 2020 2020 7c0a 7c20 2020 2020 5a5a        |.|     ZZ
│ │ │ │ +000289f0: 5b48 5d20 2020 2020 2020 2020 2020 2020  [H]             
│ │ │ │  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]          
│ │ │ │ +00028a10: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ +00028a20: 202d 2d2d 2d2d 2020 2020 2020 2020 2020   -----          
│ │ │ │  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 : -----       
│ │ │ │ +00028a40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00028a50: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │  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     
│ │ │ │ +00028a80: 0a7c 2020 2020 2020 2048 2020 2020 2020  .|       H      
│ │ │ │  00028a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ab0: 2020 7c0a 7c20 2020 2020 2020 4820 2020    |.|       H   
│ │ │ │ -00028ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ae0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00028af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028b10: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -00028b20: 4368 6572 6e20 2849 2c43 6f6d 704d 6574  Chern (I,CompMet
│ │ │ │ -00028b30: 686f 643d 3e50 6e52 6573 6964 7561 6c29  hod=>PnResidual)
│ │ │ │ +00028ab0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00028ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028ae0: 2d2d 2d2d 2d2b 0a7c 6936 203a 2043 6865  -----+.|i6 : Che
│ │ │ │ +00028af0: 726e 2028 492c 436f 6d70 4d65 7468 6f64  rn (I,CompMethod
│ │ │ │ +00028b00: 3d3e 506e 5265 7369 6475 616c 2920 2020  =>PnResidual)   
│ │ │ │ +00028b10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00028b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028b50: 2020 2020 2033 2020 2020 2032 2020 2020       3     2    
│ │ │ │  00028b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00028b80: 7c20 2020 2020 2020 3320 2020 2020 3220  |       3     2 
│ │ │ │ +00028b80: 7c6f 3620 3d20 3248 2020 2b20 3348 2020  |o6 = 2H  + 3H  
│ │ │ │  00028b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028bb0: 207c 0a7c 6f36 203d 2032 4820 202b 2033   |.|o6 = 2H  + 3
│ │ │ │ -00028bc0: 4820 2020 2020 2020 2020 2020 2020 2020  H               
│ │ │ │ +00028bb0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00028bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028be0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00028bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028be0: 2020 2020 7c0a 7c20 2020 2020 5a5a 5b48      |.|     ZZ[H
│ │ │ │ +00028bf0: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │  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]            
│ │ │ │ +00028c10: 2020 2020 2020 207c 0a7c 6f36 203a 202d         |.|o6 : -
│ │ │ │ +00028c20: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │  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  : -----         
│ │ │ │ +00028c40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00028c50: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │  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       
│ │ │ │ +00028c80: 2020 2020 2020 2048 2020 2020 2020 2020         H        
│ │ │ │  00028c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028cb0: 7c0a 7c20 2020 2020 2020 4820 2020 2020  |.|       H     
│ │ │ │ -00028cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ce0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00028cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028d10: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 4368  ------+.|i7 : Ch
│ │ │ │ -00028d20: 6572 6e28 492c 436f 6d70 4d65 7468 6f64  ern(I,CompMethod
│ │ │ │ -00028d30: 3d3e 5072 6f6a 6563 7469 7665 4465 6772  =>ProjectiveDegr
│ │ │ │ -00028d40: 6565 2920 2020 2020 207c 0a7c 2020 2020  ee)      |.|    
│ │ │ │ -00028d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028cb0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00028cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028ce0: 2d2d 2d2b 0a7c 6937 203a 2043 6865 726e  ---+.|i7 : Chern
│ │ │ │ +00028cf0: 2849 2c43 6f6d 704d 6574 686f 643d 3e50  (I,CompMethod=>P
│ │ │ │ +00028d00: 726f 6a65 6374 6976 6544 6567 7265 6529  rojectiveDegree)
│ │ │ │ +00028d10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00028d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028d40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00028d50: 2020 2033 2020 2020 2032 2020 2020 2020     3     2      
│ │ │ │  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   
│ │ │ │ +00028d70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00028d80: 3720 3d20 3268 2020 2b20 3368 2020 2020  7 = 2h  + 3h    
│ │ │ │  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 
│ │ │ │ +00028db0: 0a7c 2020 2020 2020 2031 2020 2020 2031  .|       1     1
│ │ │ │  00028dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028de0: 2020 7c0a 7c20 2020 2020 2020 3120 2020    |.|       1   
│ │ │ │ -00028df0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +00028de0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00028df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00028e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028e10: 2020 2020 207c 0a7c 2020 2020 205a 5a5b       |.|     ZZ[
│ │ │ │ +00028e20: 6820 5d20 2020 2020 2020 2020 2020 2020  h ]             
│ │ │ │  00028e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028e40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00028e50: 5a5a 5b68 205d 2020 2020 2020 2020 2020  ZZ[h ]          
│ │ │ │ +00028e50: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │  00028e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00028e80: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ +00028e70: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ +00028e80: 203a 202d 2d2d 2d2d 2d20 2020 2020 2020   : ------       
│ │ │ │  00028e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028ea0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00028eb0: 7c6f 3720 3a20 2d2d 2d2d 2d2d 2020 2020  |o7 : ------    
│ │ │ │ +00028eb0: 7c20 2020 2020 2020 2034 2020 2020 2020  |        4      
│ │ │ │  00028ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ee0: 207c 0a7c 2020 2020 2020 2020 3420 2020   |.|        4   
│ │ │ │ +00028ee0: 207c 0a7c 2020 2020 2020 2068 2020 2020   |.|       h    
│ │ │ │  00028ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f10: 2020 2020 7c0a 7c20 2020 2020 2020 6820      |.|       h 
│ │ │ │ +00028f10: 2020 2020 7c0a 7c20 2020 2020 2020 2031      |.|        1
│ │ │ │  00028f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028f50: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ -00028f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f70: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00028f40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00028f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 2d2d 2d2d  ----------+.----
│ │ │ │  00028f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a2d  -------------+.-
│ │ │ │ +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 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 352e 3131 2b64 732f  lay2-1.25.11+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:.     
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│ │ │ │ -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 
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│ │ │ │ -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
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│ │ │ │ -00029530: 2020 2020 2020 5374 616e 6c65 792d 5265        Stanley-Re
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│ │ │ │ -00029c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000290f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00029100: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +00029120: 2020 2020 2020 2020 5365 6772 6520 490a          Segre I.
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│ │ │ │ +000294e0: 4a29 2f28 5352 2b4c 5229 2077 6865 7265  J)/(SR+LR) where
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│ │ │ │ +00029500: 2020 2053 7461 6e6c 6579 2d52 6569 736e     Stanley-Reisn
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│ │ │ │ +00029520: 6661 6e20 6465 6669 6e69 6e67 2058 2061  fan defining X a
│ │ │ │ +00029530: 6e64 204c 5220 6973 2074 6865 206c 696e  nd LR is the lin
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│ │ │ │ +000295a0: 7269 6343 686f 7752 696e 672c 2063 6f6d  ricChowRing, com
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│ │ │ │ +00029770: 2020 2050 726f 6a65 6374 6976 6544 6567     ProjectiveDeg
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│ │ │ │ +000297b0: 7375 6273 6368 656d 6573 0a20 2020 2020  subschemes.     
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│ │ │ │ +000297f0: 7661 6c75 6520 4368 6f77 5269 6e67 456c  value ChowRingEl
│ │ │ │ +00029800: 656d 656e 742c 2043 686f 7752 696e 6745  ement, ChowRingE
│ │ │ │ +00029810: 6c65 6d65 6e74 2c20 7265 7475 726e 730a  lement, returns.
│ │ │ │ +00029820: 2020 2020 2020 2020 6120 5269 6e67 456c          a RingEl
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│ │ │ │ +00029870: 4f75 7470 7574 203d 3e20 2e2e 2e2c 2064  Output => ..., d
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│ │ │ │ +00029890: 7752 696e 6745 6c65 6d65 6e74 2c20 4861  wRingElement, Ha
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│ │ │ │ +00029900: 2020 2020 706f 6c79 6e6f 6d69 616c 2077      polynomial w
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│ │ │ │ +00029950: 2020 2020 7072 6f6a 6563 7469 7665 2064      projective d
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│ │ │ │ +00029a40: 2020 2020 2020 2020 7468 6520 746f 7461          the tota
│ │ │ │ +00029a50: 6c20 5365 6772 6520 636c 6173 7320 6f66  l Segre class of
│ │ │ │ +00029a60: 2074 6865 2073 6368 656d 6520 5620 6465   the scheme V de
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│ │ │ │ +00029a90: 2020 2020 2020 2020 6170 7072 6f70 7269          appropri
│ │ │ │ +00029aa0: 6174 6520 4368 6f77 2072 696e 670a 0a44  ate Chow ring..D
│ │ │ │ +00029ab0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00029ac0: 3d3d 3d3d 3d3d 0a0a 466f 7220 6120 7375  ======..For a su
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│ │ │ │ +00029ae0: 6170 706c 6963 6162 6c65 2074 6f72 6963  applicable toric
│ │ │ │ +00029af0: 2076 6172 6965 7479 2058 2074 6869 7320   variety X this 
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│ │ │ │ +00029b20: 6420 6f66 2074 6865 2074 6f74 616c 2053  d of the total S
│ │ │ │ +00029b30: 6567 7265 2063 6c61 7373 2073 2856 2c58  egre class s(V,X
│ │ │ │ +00029b40: 2920 6f66 2056 2069 6e20 5820 746f 2074  ) of V in X to t
│ │ │ │ +00029b50: 6865 2043 686f 7720 7269 6e67 206f 6620  he Chow ring of 
│ │ │ │ +00029b60: 582e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  X...+-----------
│ │ │ │ +00029b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029b90: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e  --+.|i1 : setRan
│ │ │ │ +00029ba0: 646f 6d53 6565 6420 3732 3b20 2020 2020  domSeed 72;     
│ │ │ │ +00029bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029bc0: 2020 7c0a 7c20 2d2d 2073 6574 7469 6e67    |.| -- setting
│ │ │ │ +00029bd0: 2072 616e 646f 6d20 7365 6564 2074 6f20   random seed to 
│ │ │ │ +00029be0: 3732 2020 2020 2020 2020 2020 2020 2020  72              
│ │ │ │ +00029bf0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00029c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029c20: 2d2d 2b0a 7c69 3220 3a20 5220 3d20 5a5a  --+.|i2 : R = ZZ
│ │ │ │ +00029c30: 2f33 3237 3439 5b77 2c79 2c7a 5d20 2020  /32749[w,y,z]   
│ │ │ │ +00029c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029c50: 2020 7c0a 7c20 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029c80: 2020 7c0a 7c6f 3220 3d20 5220 2020 2020    |.|o2 = R     
│ │ │ │  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  
│ │ │ │ +00029cb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  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                  
│ │ │ │ +00029ce0: 2020 7c0a 7c6f 3220 3a20 506f 6c79 6e6f    |.|o2 : Polyno
│ │ │ │ +00029cf0: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │  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)|.|        
│ │ │ │ +00029d10: 2020 7c0a 2b2d 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 2b0a 7c69 3320 3a20 5365 6772 6528  --+.|i3 : Segre(
│ │ │ │ +00029d50: 6964 6561 6c28 772a 7929 2c43 6f6d 704d  ideal(w*y),CompM
│ │ │ │ +00029d60: 6574 686f 643d 3e50 6e52 6573 6964 7561  ethod=>PnResidua
│ │ │ │ +00029d70: 6c29 7c0a 7c20 2020 2020 2020 2020 2020  l)|.|           
│ │ │ │ +00029d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029da0: 2020 7c0a 7c20 2020 2020 2020 2020 3220    |.|         2 
│ │ │ │  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              
│ │ │ │ +00029dd0: 2020 7c0a 7c6f 3320 3d20 2d20 3448 2020    |.|o3 = - 4H  
│ │ │ │ +00029de0: 2b20 3248 2020 2020 2020 2020 2020 2020  + 2H            
│ │ │ │  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         
│ │ │ │ +00029e00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00029e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029e30: 2020 7c0a 7c20 2020 2020 5a5a 5b48 5d20    |.|     ZZ[H] 
│ │ │ │  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]              
│ │ │ │ +00029e60: 2020 7c0a 7c6f 3320 3a20 2d2d 2d2d 2d20    |.|o3 : ----- 
│ │ │ │ +00029e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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  --              
│ │ │ │ +00029e90: 2020 7c0a 7c20 2020 2020 2020 2033 2020    |.|        3  
│ │ │ │ +00029ea0: 2020 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               
│ │ │ │ +00029ec0: 2020 7c0a 7c20 2020 2020 2020 4820 2020    |.|       H   
│ │ │ │ +00029ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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)      
│ │ │ │ +00029ef0: 2020 7c0a 2b2d 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 2b0a 7c69 3420 3a20 413d 4368 6f77  --+.|i4 : A=Chow
│ │ │ │ +00029f30: 5269 6e67 2852 2920 2020 2020 2020 2020  Ring(R)         
│ │ │ │ +00029f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029f50: 2020 7c0a 7c20 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029f80: 2020 7c0a 7c6f 3420 3d20 4120 2020 2020    |.|o4 = A     
│ │ │ │  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  
│ │ │ │ +00029fb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  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                  
│ │ │ │ +00029fe0: 2020 7c0a 7c6f 3420 3a20 5175 6f74 6965    |.|o4 : Quotie
│ │ │ │ +00029ff0: 6e74 5269 6e67 2020 2020 2020 2020 2020  ntRing          
│ │ │ │  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       |.|        
│ │ │ │ +0002a010: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a040: 2d2d 2b0a 7c69 3520 3a20 5365 6772 6528  --+.|i5 : Segre(
│ │ │ │ +0002a050: 412c 6964 6561 6c28 775e 322a 792c 772a  A,ideal(w^2*y,w*
│ │ │ │ +0002a060: 795e 3229 2920 2020 2020 2020 2020 2020  y^2))           
│ │ │ │ +0002a070: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a0a0: 2020 7c0a 7c20 2020 2020 2020 2020 3220    |.|         2 
│ │ │ │  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              
│ │ │ │ +0002a0d0: 2020 7c0a 7c6f 3520 3d20 2d20 3368 2020    |.|o5 = - 3h  
│ │ │ │ +0002a0e0: 2b20 3268 2020 2020 2020 2020 2020 2020  + 2h            
│ │ │ │  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         
│ │ │ │ +0002a100: 2020 7c0a 7c20 2020 2020 2020 2020 3120    |.|         1 
│ │ │ │ +0002a110: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │  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        
│ │ │ │ +0002a130: 2020 7c0a 7c20 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002a160: 2020 7c0a 7c6f 3520 3a20 4120 2020 2020    |.|o5 : A     
│ │ │ │  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...+---------
│ │ │ │ +0002a190: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a1c0: 2d2d 2b0a 0a4e 6f77 2063 6f6e 7369 6465  --+..Now conside
│ │ │ │ +0002a1d0: 7220 616e 2065 7861 6d70 6c65 2069 6e20  r an example in 
│ │ │ │ +0002a1e0: 5c50 505e 3220 5c74 696d 6573 205c 5050  \PP^2 \times \PP
│ │ │ │ +0002a1f0: 5e32 2c20 6966 2077 6520 696e 7075 7420  ^2, if we input 
│ │ │ │ +0002a200: 7468 6520 4368 6f77 2072 696e 6720 4120  the Chow ring A 
│ │ │ │ +0002a210: 7468 650a 6f75 7470 7574 2077 696c 6c20  the.output will 
│ │ │ │ +0002a220: 6265 2072 6574 7572 6e65 6420 696e 2074  be returned in t
│ │ │ │ +0002a230: 6865 2073 616d 6520 7269 6e67 2e20 546f  he same ring. To
│ │ │ │ +0002a240: 2065 6e73 7572 6520 7072 6f70 6572 2066   ensure proper f
│ │ │ │ +0002a250: 756e 6374 696f 6e20 6f66 2074 6865 0a6d  unction of the.m
│ │ │ │ +0002a260: 6574 686f 6473 2077 6520 6275 696c 6420  ethods we build 
│ │ │ │ +0002a270: 7468 6520 4368 6f77 2072 696e 6720 7573  the Chow ring us
│ │ │ │ +0002a280: 696e 6720 7468 6520 2a6e 6f74 6520 4368  ing the *note Ch
│ │ │ │ +0002a290: 6f77 5269 6e67 3a20 4368 6f77 5269 6e67  owRing: ChowRing
│ │ │ │ +0002a2a0: 2c20 636f 6d6d 616e 642e 2057 650a 6d61  , command. We.ma
│ │ │ │ +0002a2b0: 7920 616c 736f 2072 6574 7572 6e20 6120  y also return a 
│ │ │ │ +0002a2c0: 4d75 7461 626c 6548 6173 6854 6162 6c65  MutableHashTable
│ │ │ │ +0002a2d0: 2e0a 0a2b 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 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 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})         
│ │ │ │ +0002a320: 2d2b 0a7c 6936 203a 2052 3d4d 756c 7469  -+.|i6 : R=Multi
│ │ │ │ +0002a330: 5072 6f6a 436f 6f72 6452 696e 6728 7b32  ProjCoordRing({2
│ │ │ │ +0002a340: 2c32 7d29 2020 2020 2020 2020 2020 2020  ,2})            
│ │ │ │ +0002a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a370: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a3a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a3c0: 207c 0a7c 6f36 203d 2052 2020 2020 2020   |.|o6 = R      
│ │ │ │  0002a3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a3f0: 2020 2020 7c0a 7c6f 3620 3d20 5220 2020      |.|o6 = R   
│ │ │ │ +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                  
│ │ │ │ +0002a410: 207c 0a7c 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a440: 2020 2020 2020 2020 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                  
│ │ │ │ +0002a460: 207c 0a7c 6f36 203a 2050 6f6c 796e 6f6d   |.|o6 : Polynom
│ │ │ │ +0002a470: 6961 6c52 696e 6720 2020 2020 2020 2020  ialRing         
│ │ │ │  0002a480: 2020 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      |.+---------
│ │ │ │ +0002a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a4b0: 207c 0a2b 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 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                  
│ │ │ │ +0002a500: 2d2b 0a7c 6937 203a 2072 3d67 656e 7320  -+.|i7 : r=gens 
│ │ │ │ +0002a510: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +0002a520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a550: 207c 0a7c 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a580: 2020 2020 2020 2020 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 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                  
│ │ │ │ +0002a5a0: 207c 0a7c 6f37 203d 207b 7820 2c20 7820   |.|o7 = {x , x 
│ │ │ │ +0002a5b0: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ +0002a5c0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +0002a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a5f0: 207c 0a7c 2020 2020 2020 2030 2020 2031   |.|       0   1
│ │ │ │ +0002a600: 2020 2032 2020 2033 2020 2034 2020 2035     2   3   4   5
│ │ │ │  0002a610: 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             
│ │ │ │ +0002a620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a640: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a670: 2020 2020 7c0a 7c20 2020 2020 2020 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                  
│ │ │ │ +0002a690: 207c 0a7c 6f37 203a 204c 6973 7420 2020   |.|o7 : List   
│ │ │ │  0002a6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a6c0: 2020 2020 7c0a 7c6f 3720 3a20 4c69 7374      |.|o7 : List
│ │ │ │ +0002a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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      |.+---------
│ │ │ │ +0002a6e0: 207c 0a2b 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 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                  
│ │ │ │ +0002a730: 2d2b 0a7c 6938 203a 2041 3d43 686f 7752  -+.|i8 : A=ChowR
│ │ │ │ +0002a740: 696e 6728 5229 2020 2020 2020 2020 2020  ing(R)          
│ │ │ │ +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 2020 2020                  
│ │ │ │ +0002a780: 207c 0a7c 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a7d0: 207c 0a7c 6f38 203d 2041 2020 2020 2020   |.|o8 = A      
│ │ │ │  0002a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a800: 2020 2020 7c0a 7c6f 3820 3d20 4120 2020      |.|o8 = A   
│ │ │ │ +0002a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a820: 207c 0a7c 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 7c0a 7c20 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 2020 2020 2020                  
│ │ │ │ -0002a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a870: 207c 0a7c 6f38 203a 2051 756f 7469 656e   |.|o8 : Quotien
│ │ │ │ +0002a880: 7452 696e 6720 2020 2020 2020 2020 2020  tRing           
│ │ │ │  0002a890: 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      |.+---------
│ │ │ │ +0002a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a8c0: 207c 0a2b 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 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_
│ │ │ │ -0002a960: 342a 725f 312a 725f 322c 725f 325e 322a  4*r_1*r_2,r_2^2*
│ │ │ │ -0002a970: 725f 3529 2020 2020 2020 2020 2020 2020  r_5)            
│ │ │ │ +0002a910: 2d2b 0a7c 6939 203a 2049 3d69 6465 616c  -+.|i9 : I=ideal
│ │ │ │ +0002a920: 2872 5f30 5e32 2a72 5f33 2d72 5f34 2a72  (r_0^2*r_3-r_4*r
│ │ │ │ +0002a930: 5f31 2a72 5f32 2c72 5f32 5e32 2a72 5f35  _1*r_2,r_2^2*r_5
│ │ │ │ +0002a940: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0002a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a960: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a990: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a9e0: 2020 2020 7c0a 7c20 2020 2020 2020 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 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002aa90: 2020 2020 3020 3320 2020 2031 2032 2034      0 3    1 2 4
│ │ │ │ -0002aaa0: 2020 2032 2035 2020 2020 2020 2020 2020     2 5          
│ │ │ │ +0002a9b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002a9c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0002a9d0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0002a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aa00: 207c 0a7c 6f39 203d 2069 6465 616c 2028   |.|o9 = ideal (
│ │ │ │ +0002aa10: 7820 7820 202d 2078 2078 2078 202c 2078  x x  - x x x , x
│ │ │ │ +0002aa20: 2078 2029 2020 2020 2020 2020 2020 2020   x )            
│ │ │ │ +0002aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aa50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002aa60: 2030 2033 2020 2020 3120 3220 3420 2020   0 3    1 2 4   
│ │ │ │ +0002aa70: 3220 3520 2020 2020 2020 2020 2020 2020  2 5             
│ │ │ │ +0002aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aaa0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aad0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aaf0: 207c 0a7c 6f39 203a 2049 6465 616c 206f   |.|o9 : Ideal o
│ │ │ │ +0002ab00: 6620 5220 2020 2020 2020 2020 2020 2020  f R             
│ │ │ │  0002ab10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ab20: 2020 2020 7c0a 7c6f 3920 3a20 4964 6561      |.|o9 : Idea
│ │ │ │ -0002ab30: 6c20 6f66 2052 2020 2020 2020 2020 2020  l of R          
│ │ │ │ -0002ab40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ab50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ab60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ab70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002ab20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ab40: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002ab50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ab70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ab90: 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                  
│ │ │ │ +0002ab90: 2d2b 0a7c 6931 3020 3a20 5365 6772 6520  -+.|i10 : Segre 
│ │ │ │ +0002aba0: 4920 2020 2020 2020 2020 2020 2020 2020  I               
│ │ │ │ +0002abb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002abc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002abd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002abe0: 207c 0a7c 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ac10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 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        
│ │ │ │ +0002ac30: 207c 0a7c 2020 2020 2020 2020 2032 2032   |.|         2 2
│ │ │ │ +0002ac40: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0002ac50: 2032 2020 2020 2032 2020 2020 2020 2020   2     2        
│ │ │ │ +0002ac60: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002ac70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ac80: 207c 0a7c 6f31 3020 3d20 3732 6820 6820   |.|o10 = 72h h 
│ │ │ │ +0002ac90: 202d 2032 3468 2068 2020 2d20 3132 6820   - 24h h  - 12h 
│ │ │ │ +0002aca0: 6820 202b 2034 6820 202b 2034 6820 6820  h  + 4h  + 4h h 
│ │ │ │ +0002acb0: 202b 2068 2020 2020 2020 2020 2020 2020   + h            
│ │ │ │ +0002acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002acd0: 207c 0a7c 2020 2020 2020 2020 2031 2032   |.|         1 2
│ │ │ │ +0002ace0: 2020 2020 2020 3120 3220 2020 2020 2031        1 2      1
│ │ │ │ +0002acf0: 2032 2020 2020 2031 2020 2020 2031 2032   2     1     1 2
│ │ │ │ +0002ad00: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002ad10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ad20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ad40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ad50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002ad50: 2020 2020 2020 2020 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                  
│ │ │ │ +0002ad70: 207c 0a7c 2020 2020 2020 5a5a 5b68 202e   |.|      ZZ[h .
│ │ │ │ +0002ad80: 2e68 205d 2020 2020 2020 2020 2020 2020  .h ]            
│ │ │ │  0002ad90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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                  
│ │ │ │ +0002ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002adc0: 207c 0a7c 2020 2020 2020 2020 2020 3120   |.|          1 
│ │ │ │ +0002add0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0002ade0: 2020 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                  
│ │ │ │ +0002adf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ae10: 207c 0a7c 6f31 3020 3a20 2d2d 2d2d 2d2d   |.|o10 : ------
│ │ │ │ +0002ae20: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │  0002ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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                  
│ │ │ │ +0002ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ae60: 207c 0a7c 2020 2020 2020 2020 2033 2020   |.|         3  
│ │ │ │ +0002ae70: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  0002ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ae90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002aea0: 3320 2020 3320 2020 2020 2020 2020 2020  3   3           
│ │ │ │ -0002aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aeb0: 207c 0a7c 2020 2020 2020 2028 6820 2c20   |.|       (h , 
│ │ │ │ +0002aec0: 6820 2920 2020 2020 2020 2020 2020 2020  h )             
│ │ │ │  0002aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aee0: 2020 2020 7c0a 7c20 2020 2020 2020 2868      |.|       (h
│ │ │ │ -0002aef0: 202c 2068 2029 2020 2020 2020 2020 2020   , h )          
│ │ │ │ -0002af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002af00: 207c 0a7c 2020 2020 2020 2020 2031 2020   |.|         1  
│ │ │ │ +0002af10: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  0002af20: 2020 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      |.+---------
│ │ │ │ +0002af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002af50: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +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 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                  
│ │ │ │ +0002afa0: 2d2b 0a7c 6931 3120 3a20 7331 3d53 6567  -+.|i11 : s1=Seg
│ │ │ │ +0002afb0: 7265 2841 2c49 2920 2020 2020 2020 2020  re(A,I)         
│ │ │ │ +0002afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aff0: 207c 0a7c 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 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 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        
│ │ │ │ +0002b040: 207c 0a7c 2020 2020 2020 2020 2032 2032   |.|         2 2
│ │ │ │ +0002b050: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0002b060: 2032 2020 2020 2032 2020 2020 2020 2020   2     2        
│ │ │ │ +0002b070: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002b080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b090: 207c 0a7c 6f31 3120 3d20 3732 6820 6820   |.|o11 = 72h h 
│ │ │ │ +0002b0a0: 202d 2032 3468 2068 2020 2d20 3132 6820   - 24h h  - 12h 
│ │ │ │ +0002b0b0: 6820 202b 2034 6820 202b 2034 6820 6820  h  + 4h  + 4h h 
│ │ │ │ +0002b0c0: 202b 2068 2020 2020 2020 2020 2020 2020   + h            
│ │ │ │ +0002b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b0e0: 207c 0a7c 2020 2020 2020 2020 2031 2032   |.|         1 2
│ │ │ │ +0002b0f0: 2020 2020 2020 3120 3220 2020 2020 2031        1 2      1
│ │ │ │ +0002b100: 2032 2020 2020 2031 2020 2020 2031 2032   2     1     1 2
│ │ │ │ +0002b110: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b130: 207c 0a7c 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 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                  
│ │ │ │ +0002b180: 207c 0a7c 6f31 3120 3a20 4120 2020 2020   |.|o11 : A     
│ │ │ │  0002b190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b1b0: 2020 2020 7c0a 7c6f 3131 203a 2041 2020      |.|o11 : A  
│ │ │ │ +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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b200: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002b1d0: 207c 0a2b 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 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b250: 2d2d 2d2d 2b0a 7c69 3132 203a 2053 6567  ----+.|i12 : Seg
│ │ │ │ -0002b260: 4861 7368 3d53 6567 7265 2841 2c49 2c4f  Hash=Segre(A,I,O
│ │ │ │ -0002b270: 7574 7075 743d 3e48 6173 6846 6f72 6d29  utput=>HashForm)
│ │ │ │ +0002b220: 2d2b 0a7c 6931 3220 3a20 5365 6748 6173  -+.|i12 : SegHas
│ │ │ │ +0002b230: 683d 5365 6772 6528 412c 492c 4f75 7470  h=Segre(A,I,Outp
│ │ │ │ +0002b240: 7574 3d3e 4861 7368 466f 726d 2920 2020  ut=>HashForm)   
│ │ │ │ +0002b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b270: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2a0: 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2f0: 2020 2020 7c0a 7c6f 3132 203d 204d 7574      |.|o12 = Mut
│ │ │ │ -0002b300: 6162 6c65 4861 7368 5461 626c 657b 2e2e  ableHashTable{..
│ │ │ │ -0002b310: 2e34 2e2e 2e7d 2020 2020 2020 2020 2020  .4...}          
│ │ │ │ +0002b2c0: 207c 0a7c 6f31 3220 3d20 4d75 7461 626c   |.|o12 = Mutabl
│ │ │ │ +0002b2d0: 6548 6173 6854 6162 6c65 7b2e 2e2e 342e  eHashTable{...4.
│ │ │ │ +0002b2e0: 2e2e 7d20 2020 2020 2020 2020 2020 2020  ..}             
│ │ │ │ +0002b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b310: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b340: 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                  
│ │ │ │ +0002b360: 207c 0a7c 6f31 3220 3a20 4d75 7461 626c   |.|o12 : Mutabl
│ │ │ │ +0002b370: 6548 6173 6854 6162 6c65 2020 2020 2020  eHashTable      
│ │ │ │  0002b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b390: 2020 2020 7c0a 7c6f 3132 203a 204d 7574      |.|o12 : Mut
│ │ │ │ -0002b3a0: 6162 6c65 4861 7368 5461 626c 6520 2020  ableHashTable   
│ │ │ │ -0002b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b3b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002b3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b3d0: 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b430: 2d2d 2d2d 2b0a 7c69 3133 203a 2070 6565  ----+.|i13 : pee
│ │ │ │ -0002b440: 6b20 5365 6748 6173 6820 2020 2020 2020  k SegHash       
│ │ │ │ -0002b450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b400: 2d2b 0a7c 6931 3320 3a20 7065 656b 2053  -+.|i13 : peek S
│ │ │ │ +0002b410: 6567 4861 7368 2020 2020 2020 2020 2020  egHash          
│ │ │ │ +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 2020 2020                  
│ │ │ │ +0002b450: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b480: 2020 2020 7c0a 7c20 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 2020 2020 2020                  
│ │ │ │ -0002b4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4d0: 2020 2020 7c0a 7c6f 3133 203d 204d 7574      |.|o13 = Mut
│ │ │ │ -0002b4e0: 6162 6c65 4861 7368 5461 626c 657b 4720  ableHashTable{G 
│ │ │ │ -0002b4f0: 3d3e 2032 6820 202b 2068 2020 2b20 3120  => 2h  + h  + 1 
│ │ │ │ +0002b4a0: 207c 0a7c 6f31 3320 3d20 4d75 7461 626c   |.|o13 = Mutabl
│ │ │ │ +0002b4b0: 6548 6173 6854 6162 6c65 7b47 203d 3e20  eHashTable{G => 
│ │ │ │ +0002b4c0: 3268 2020 2b20 6820 202b 2031 2020 2020  2h  + h  + 1    
│ │ │ │ +0002b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b510: 2020 3120 2020 2032 2020 2020 2020 2020    1    2        
│ │ │ │ +0002b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b540: 2020 2020 2031 2020 2020 3220 2020 2020       1    2     
│ │ │ │ -0002b550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b570: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b580: 2020 2020 2020 2020 2020 2020 2020 476c                Gl
│ │ │ │ -0002b590: 6973 7420 3d3e 207b 312c 2032 6820 202b  ist => {1, 2h  +
│ │ │ │ -0002b5a0: 2068 202c 2030 2c20 302c 2030 7d20 2020   h , 0, 0, 0}   
│ │ │ │ -0002b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b540: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b550: 2020 2020 2020 2020 2020 2047 6c69 7374             Glist
│ │ │ │ +0002b560: 203d 3e20 7b31 2c20 3268 2020 2b20 6820   => {1, 2h  + h 
│ │ │ │ +0002b570: 2c20 302c 2030 2c20 307d 2020 2020 2020  , 0, 0, 0}      
│ │ │ │ +0002b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b590: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b5b0: 2020 2020 2020 2020 2020 3120 2020 2032            1    2
│ │ │ │ +0002b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5e0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -0002b5f0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0002b5e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b610: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b610: 2032 2020 2020 2020 2020 2020 2020 3220   2            2 
│ │ │ │  0002b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b640: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0002b650: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0002b660: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b670: 2020 2020 2020 2020 2020 2020 2020 5365                Se
│ │ │ │ -0002b680: 6772 654c 6973 7420 3d3e 207b 302c 2030  greList => {0, 0
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│ │ │ │ -0002b6a0: 6820 2c20 2d20 2020 2020 2020 2020 2020  h , -           
│ │ │ │ -0002b6b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b630: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b640: 2020 2020 2020 2020 2020 2053 6567 7265             Segre
│ │ │ │ +0002b650: 4c69 7374 203d 3e20 7b30 2c20 302c 2034  List => {0, 0, 4
│ │ │ │ +0002b660: 6820 202b 2034 6820 6820 202b 2068 202c  h  + 4h h  + h ,
│ │ │ │ +0002b670: 202d 2020 2020 2020 2020 2020 2020 2020   -              
│ │ │ │ +0002b680: 207c 0a7c 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: 2031 2020 2020 2031 2032 2020 2020 3220   1     1 2    2 
│ │ │ │  0002b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6e0: 2020 2020 3120 2020 2020 3120 3220 2020      1     1 2   
│ │ │ │ -0002b6f0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0002b700: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b720: 2020 2020 2020 2020 2020 3220 3220 2020            2 2   
│ │ │ │ -0002b730: 2020 2032 2020 2020 2020 2020 2020 3220     2          2 
│ │ │ │ -0002b740: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0002b750: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b760: 2020 2020 2020 2020 2020 2020 2020 5365                Se
│ │ │ │ -0002b770: 6772 6520 3d3e 2037 3268 2068 2020 2d20  gre => 72h h  - 
│ │ │ │ -0002b780: 3234 6820 6820 202d 2031 3268 2068 2020  24h h  - 12h h  
│ │ │ │ -0002b790: 2b20 3468 2020 2020 2020 2020 2020 2020  + 4h            
│ │ │ │ -0002b7a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b7c0: 2020 2020 2020 2020 2020 3120 3220 2020            1 2   
│ │ │ │ -0002b7d0: 2020 2031 2032 2020 2020 2020 3120 3220     1 2      1 2 
│ │ │ │ -0002b7e0: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ -0002b7f0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
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│ │ │ │ +0002b6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b6f0: 2020 2020 2020 2032 2032 2020 2020 2020         2 2      
│ │ │ │ +0002b700: 3220 2020 2020 2020 2020 2032 2020 2020  2          2    
│ │ │ │ +0002b710: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0002b720: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b730: 2020 2020 2020 2020 2020 2053 6567 7265             Segre
│ │ │ │ +0002b740: 203d 3e20 3732 6820 6820 202d 2032 3468   => 72h h  - 24h
│ │ │ │ +0002b750: 2068 2020 2d20 3132 6820 6820 202b 2034   h  - 12h h  + 4
│ │ │ │ +0002b760: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +0002b770: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b790: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ +0002b7a0: 3120 3220 2020 2020 2031 2032 2020 2020  1 2      1 2    
│ │ │ │ +0002b7b0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +0002b7c0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ +0002b7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b7e0: 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b840: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020  ----|.|         
│ │ │ │ -0002b850: 2020 2020 2020 2020 2020 2020 2020 207d                 }
│ │ │ │ -0002b860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b810: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0002b820: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ +0002b830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b860: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b890: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b8b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +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 2020 2020                  
│ │ │ │ +0002b900: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b930: 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                  
│ │ │ │ +0002b950: 207c 0a7c 2020 2032 2020 2020 2020 2020   |.|   2        
│ │ │ │ +0002b960: 2020 3220 2020 2020 3220 3220 2020 2020    2     2 2     
│ │ │ │  0002b970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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                  
│ │ │ │ +0002b980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b9a0: 207c 0a7c 3234 6820 6820 202d 2031 3268   |.|24h h  - 12h
│ │ │ │ +0002b9b0: 2068 202c 2037 3268 2068 207d 2020 2020   h , 72h h }    
│ │ │ │  0002b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b9f0: 207c 0a7c 2020 2031 2032 2020 2020 2020   |.|   1 2      
│ │ │ │ +0002ba00: 3120 3220 2020 2020 3120 3220 2020 2020  1 2     1 2     
│ │ │ │  0002ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba20: 2020 2020 7c0a 7c20 2020 3120 3220 2020      |.|   1 2   
│ │ │ │ -0002ba30: 2020 2031 2032 2020 2020 2031 2032 2020     1 2     1 2  
│ │ │ │ -0002ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba40: 207c 0a7c 2020 2020 2020 2020 2020 2032   |.|           2
│ │ │ │  0002ba50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002ba80: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0002ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba90: 207c 0a7c 2b20 3468 2068 2020 2b20 6820   |.|+ 4h h  + h 
│ │ │ │  0002baa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bac0: 2020 2020 7c0a 7c2b 2034 6820 6820 202b      |.|+ 4h h  +
│ │ │ │ -0002bad0: 2068 2020 2020 2020 2020 2020 2020 2020   h              
│ │ │ │ -0002bae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bae0: 207c 0a7c 2020 2020 3120 3220 2020 2032   |.|    1 2    2
│ │ │ │  0002baf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bb00: 2020 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      |.+---------
│ │ │ │ +0002bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bb30: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002bb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +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 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                  
│ │ │ │ +0002bb80: 2d2b 0a7c 6931 3420 3a20 7331 3d3d 5365  -+.|i14 : s1==Se
│ │ │ │ +0002bb90: 6748 6173 6823 2253 6567 7265 2220 2020  gHash#"Segre"   
│ │ │ │ +0002bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bbd0: 207c 0a7c 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bc20: 207c 0a7c 6f31 3420 3d20 7472 7565 2020   |.|o14 = true  
│ │ │ │  0002bc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bc40: 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      |.+---------
│ │ │ │ +0002bc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bc70: 207c 0a2b 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 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...+-
│ │ │ │ +0002bcc0: 2d2b 0a0a 496e 2074 6865 2063 6173 6520  -+..In the case 
│ │ │ │ +0002bcd0: 7768 6572 6520 7468 6520 616d 6269 656e  where the ambien
│ │ │ │ +0002bce0: 7420 7370 6163 6520 6973 2061 2074 6f72  t space is a tor
│ │ │ │ +0002bcf0: 6963 2076 6172 6965 7479 2077 6869 6368  ic variety which
│ │ │ │ +0002bd00: 2069 7320 6e6f 7420 6120 7072 6f64 7563   is not a produc
│ │ │ │ +0002bd10: 740a 6f66 2070 726f 6a65 6374 6976 6520  t.of projective 
│ │ │ │ +0002bd20: 7370 6163 6573 2077 6520 6d75 7374 206c  spaces we must l
│ │ │ │ +0002bd30: 6f61 6420 7468 6520 4e6f 726d 616c 546f  oad the NormalTo
│ │ │ │ +0002bd40: 7269 6356 6172 6965 7469 6573 2070 6163  ricVarieties pac
│ │ │ │ +0002bd50: 6b61 6765 2061 6e64 206d 7573 740a 616c  kage and must.al
│ │ │ │ +0002bd60: 736f 2069 6e70 7574 2074 6865 2074 6f72  so input the tor
│ │ │ │ +0002bd70: 6963 2076 6172 6965 7479 2e20 4966 2074  ic variety. If t
│ │ │ │ +0002bd80: 6865 2074 6f72 6963 2076 6172 6965 7479  he toric variety
│ │ │ │ +0002bd90: 2069 7320 6120 7072 6f64 7563 7420 6f66   is a product of
│ │ │ │ +0002bda0: 2070 726f 6a65 6374 6976 650a 7370 6163   projective.spac
│ │ │ │ +0002bdb0: 6520 6974 2069 7320 7265 636f 6d6d 656e  e it is recommen
│ │ │ │ +0002bdc0: 6465 6420 746f 2075 7365 2074 6865 2066  ded to use the f
│ │ │ │ +0002bdd0: 6f72 6d20 6162 6f76 6520 7261 7468 6572  orm above rather
│ │ │ │ +0002bde0: 2074 6861 6e20 696e 7075 7474 696e 6720   than inputting 
│ │ │ │ +0002bdf0: 7468 6520 746f 7269 630a 7661 7269 6574  the toric.variet
│ │ │ │ +0002be00: 7920 666f 7220 6566 6669 6369 656e 6379  y for efficiency
│ │ │ │ +0002be10: 2072 6561 736f 6e73 2e0a 0a2b 2d2d 2d2d   reasons...+----
│ │ │ │ +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: 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"             
│ │ │ │ +0002be60: 2d2d 2d2d 2b0a 7c69 3135 203a 206e 6565  ----+.|i15 : nee
│ │ │ │ +0002be70: 6473 5061 636b 6167 6520 224e 6f72 6d61  dsPackage "Norma
│ │ │ │ +0002be80: 6c54 6f72 6963 5661 7269 6574 6965 7322  lToricVarieties"
│ │ │ │ +0002be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bea0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002beb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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 207c 0a7c               |.|
│ │ │ │ -0002bf30: 6f31 3520 3d20 4e6f 726d 616c 546f 7269  o15 = NormalTori
│ │ │ │ -0002bf40: 6356 6172 6965 7469 6573 2020 2020 2020  cVarieties      
│ │ │ │ +0002bee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bef0: 2020 2020 2020 2020 2020 7c0a 7c6f 3135            |.|o15
│ │ │ │ +0002bf00: 203d 204e 6f72 6d61 6c54 6f72 6963 5661   = NormalToricVa
│ │ │ │ +0002bf10: 7269 6574 6965 7320 2020 2020 2020 2020  rieties         
│ │ │ │ +0002bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bf40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bf90: 7c0a 7c6f 3135 203a 2050 6163 6b61 6765  |.|o15 : Package
│ │ │ │  0002bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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  }}        |.|   
│ │ │ │ +0002bfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bfd0: 2020 2020 2020 2020 2020 207c 0a2b 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c020: 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a 2052  ------+.|i16 : R
│ │ │ │ +0002c030: 686f 203d 207b 7b31 2c30 2c30 7d2c 7b30  ho = {{1,0,0},{0
│ │ │ │ +0002c040: 2c31 2c30 7d2c 7b30 2c30 2c31 7d2c 7b2d  ,1,0},{0,0,1},{-
│ │ │ │ +0002c050: 312c 2d31 2c30 7d2c 7b30 2c30 2c2d 317d  1,-1,0},{0,0,-1}
│ │ │ │ +0002c060: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +0002c070: 207c 0a7c 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 2020 2020 2020                  
│ │ │ │ +0002c0b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0002c0c0: 3136 203d 207b 7b31 2c20 302c 2030 7d2c  16 = {{1, 0, 0},
│ │ │ │ +0002c0d0: 207b 302c 2031 2c20 307d 2c20 7b30 2c20   {0, 1, 0}, {0, 
│ │ │ │ +0002c0e0: 302c 2031 7d2c 207b 2d31 2c20 2d31 2c20  0, 1}, {-1, -1, 
│ │ │ │ +0002c0f0: 307d 2c20 7b30 2c20 302c 202d 317d 7d20  0}, {0, 0, -1}} 
│ │ │ │ +0002c100: 2020 2020 2020 207c 0a7c 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 2020                  
│ │ │ │  0002c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c150: 2020 7c0a 7c6f 3136 203a 204c 6973 7420    |.|o16 : List 
│ │ │ │  0002c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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}}|.| 
│ │ │ │ +0002c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c190: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +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 2d2d 2d2d 2b0a 7c69 3137 203a  --------+.|i17 :
│ │ │ │ +0002c1f0: 2053 6967 6d61 203d 207b 7b30 2c31 2c32   Sigma = {{0,1,2
│ │ │ │ +0002c200: 7d2c 7b31 2c32 2c33 7d2c 7b30 2c32 2c33  },{1,2,3},{0,2,3
│ │ │ │ +0002c210: 7d2c 7b30 2c31 2c34 7d2c 7b31 2c33 2c34  },{0,1,4},{1,3,4
│ │ │ │ +0002c220: 7d2c 7b30 2c33 2c34 7d7d 2020 2020 2020  },{0,3,4}}      
│ │ │ │ +0002c230: 2020 207c 0a7c 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 2020 2020 2020                  
│ │ │ │ +0002c270: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002c280: 7c6f 3137 203d 207b 7b30 2c20 312c 2032  |o17 = {{0, 1, 2
│ │ │ │ +0002c290: 7d2c 207b 312c 2032 2c20 337d 2c20 7b30  }, {1, 2, 3}, {0
│ │ │ │ +0002c2a0: 2c20 322c 2033 7d2c 207b 302c 2031 2c20  , 2, 3}, {0, 1, 
│ │ │ │ +0002c2b0: 347d 2c20 7b31 2c20 332c 2034 7d2c 207b  4}, {1, 3, 4}, {
│ │ │ │ +0002c2c0: 302c 2033 2c20 347d 7d7c 0a7c 2020 2020  0, 3, 4}}|.|    
│ │ │ │ +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: 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                  
│ │ │ │ +0002c310: 2020 2020 7c0a 7c6f 3137 203a 204c 6973      |.|o17 : Lis
│ │ │ │ +0002c320: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │  0002c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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)      |.|     
│ │ │ │ +0002c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c360: 0a2b 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 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138  ----------+.|i18
│ │ │ │ +0002c3b0: 203a 2058 203d 206e 6f72 6d61 6c54 6f72   : X = normalTor
│ │ │ │ +0002c3c0: 6963 5661 7269 6574 7928 5268 6f2c 5369  icVariety(Rho,Si
│ │ │ │ +0002c3d0: 676d 612c 436f 6566 6669 6369 656e 7452  gma,CoefficientR
│ │ │ │ +0002c3e0: 696e 6720 3d3e 5a5a 2f33 3237 3439 2920  ing =>ZZ/32749) 
│ │ │ │ +0002c3f0: 2020 2020 207c 0a7c 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 2020 2020                  
│ │ │ │  0002c430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c440: 7c0a 7c6f 3138 203d 2058 2020 2020 2020  |.|o18 = X      
│ │ │ │  0002c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c470: 2020 207c 0a7c 6f31 3820 3d20 5820 2020     |.|o18 = X   
│ │ │ │ -0002c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c480: 2020 2020 2020 2020 2020 207c 0a7c 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 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 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  )               
│ │ │ │ +0002c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c4d0: 2020 2020 2020 7c0a 7c6f 3138 203a 204e        |.|o18 : N
│ │ │ │ +0002c4e0: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ +0002c4f0: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │ +0002c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c520: 207c 0a2b 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 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0002c570: 3139 203a 2043 6865 636b 546f 7269 6356  19 : CheckToricV
│ │ │ │ +0002c580: 6172 6965 7479 5661 6c69 6428 5829 2020  arietyValid(X)  
│ │ │ │ +0002c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c5b0: 2020 2020 2020 207c 0a7c 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 7c0a 7c20 2020            |.|   
│ │ │ │ +0002c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c600: 2020 7c0a 7c6f 3139 203d 2074 7275 6520    |.|o19 = true 
│ │ │ │  0002c610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c620: 2020 2020 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                  
│ │ │ │ +0002c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c640: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c690: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a  --------+.|i20 :
│ │ │ │ +0002c6a0: 2052 3d72 696e 6728 5829 2020 2020 2020   R=ring(X)      
│ │ │ │ +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 207c 0a7c 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 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002c720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c720: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002c730: 7c6f 3230 203d 2052 2020 2020 2020 2020  |o20 = R        
│ │ │ │  0002c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c760: 207c 0a7c 6f32 3020 3d20 5220 2020 2020   |.|o20 = R     
│ │ │ │ -0002c770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c770: 2020 2020 2020 2020 207c 0a7c 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 7c0a 7c20              |.| 
│ │ │ │ +0002c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +0002c7c0: 2020 2020 7c0a 7c6f 3230 203a 2050 6f6c      |.|o20 : Pol
│ │ │ │ +0002c7d0: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │  0002c7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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          |.|     
│ │ │ │ +0002c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c800: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c810: 0a2b 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 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231  ----------+.|i21
│ │ │ │ +0002c860: 203a 2049 3d69 6465 616c 2852 5f30 5e34   : I=ideal(R_0^4
│ │ │ │ +0002c870: 2a52 5f31 2c52 5f30 2a52 5f33 2a52 5f34  *R_1,R_0*R_3*R_4
│ │ │ │ +0002c880: 2a52 5f32 2d52 5f32 5e32 2a52 5f30 5e32  *R_2-R_2^2*R_0^2
│ │ │ │ +0002c890: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0002c8a0: 2020 2020 207c 0a7c 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 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                  
│ │ │ │ +0002c8f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002c900: 2034 2020 2020 2020 2032 2032 2020 2020   4       2 2    
│ │ │ │  0002c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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 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 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
│ │ │ │ +0002c920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c930: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0002c940: 3120 3d20 6964 6561 6c20 2878 2078 202c  1 = ideal (x x ,
│ │ │ │ +0002c950: 202d 2078 2078 2020 2b20 7820 7820 7820   - x x  + x x x 
│ │ │ │ +0002c960: 7820 2920 2020 2020 2020 2020 2020 2020  x )             
│ │ │ │ +0002c970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c980: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002c990: 2020 2020 2020 2030 2031 2020 2020 2030         0 1     0
│ │ │ │ +0002c9a0: 2032 2020 2020 3020 3220 3320 3420 2020   2    0 2 3 4   
│ │ │ │ +0002c9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c9d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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                  
│ │ │ │ +0002ca00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ca10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0002ca20: 3231 203a 2049 6465 616c 206f 6620 5220  21 : Ideal of R 
│ │ │ │  0002ca30: 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            |.+---
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│ │ │ │ +0002ca60: 2020 2020 2020 207c 0a2b 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 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)        
│ │ │ │ +0002cab0: 2d2d 2b0a 7c69 3232 203a 2053 6567 7265  --+.|i22 : Segre
│ │ │ │ +0002cac0: 2858 2c49 2920 2020 2020 2020 2020 2020  (X,I)           
│ │ │ │ +0002cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002caf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0002cb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -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                  
│ │ │ │ +0002cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cb40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002cb50: 2020 2020 2020 3220 2020 2020 2020 3220        2       2 
│ │ │ │  0002cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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              
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│ │ │ │ -0002cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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: 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           
│ │ │ │ +0002cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cb90: 2020 207c 0a7c 6f32 3220 3d20 2d20 3732     |.|o22 = - 72
│ │ │ │ +0002cba0: 7820 7820 202b 2033 7820 202b 2038 7820  x x  + 3x  + 8x 
│ │ │ │ +0002cbb0: 7820 202b 2078 2020 2020 2020 2020 2020  x  + x          
│ │ │ │ +0002cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cbd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002cbe0: 7c20 2020 2020 2020 2020 2020 3320 3420  |           3 4 
│ │ │ │ +0002cbf0: 2020 2020 3320 2020 2020 3320 3420 2020      3     3 4   
│ │ │ │ +0002cc00: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +0002cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cc20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002cc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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 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 
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│ │ │ │ -0002cdc0: 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            
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│ │ │ │ -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  ----------------
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│ │ │ │ -0002d130: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0002d160: 2020 2020 2033 2034 2020 2020 2033 2020       3 4     3  
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│ │ │ │ +0002d0f0: 202b 2038 7820 7820 202b 2078 2020 2020   + 8x x  + x    
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│ │ │ │ -0002d4b0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2253  ========..  * "S
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│ │ │ │ -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
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│ │ │ │ -0002d540: 5365 6772 653a 2053 6567 7265 2c20 6973  Segre: Segre, is
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│ │ │ │ -0002d560: 6675 6e63 7469 6f6e 2077 6974 6820 6f70  function with op
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -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 
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│ │ │ │ -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
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│ │ │ │ +0002d270: 696e 6720 7379 6d62 6f6c 6963 2063 6f6d  ing symbolic com
│ │ │ │ +0002d280: 7075 7461 7469 6f6e 7320 7769 7468 2047  putations with G
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│ │ │ │ +0002d2a0: 4368 616e 6769 6e67 2074 6865 206f 7074  Changing the opt
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│ │ │ │ +0002d310: 2042 6572 7469 6e69 2069 7320 202a 6e6f   Bertini is  *no
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│ │ │ │ +0002d330: 2063 6f6e 6669 6775 7265 643a 0a63 6f6e   configured:.con
│ │ │ │ +0002d340: 6669 6775 7269 6e67 2042 6572 7469 6e69  figuring Bertini
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│ │ │ │ +0002d360: 2062 6572 7469 6e69 206f 7074 696f 6e20   bertini option 
│ │ │ │ +0002d370: 6973 206f 6e6c 7920 6176 6169 6c61 626c  is only availabl
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│ │ │ │ +0002d3a0: 7276 6520 7468 6174 2074 6865 2061 6c67  rve that the alg
│ │ │ │ +0002d3b0: 6f72 6974 686d 2069 7320 6120 7072 6f62  orithm is a prob
│ │ │ │ +0002d3c0: 6162 696c 6973 7469 6320 616c 676f 7269  abilistic algori
│ │ │ │ +0002d3d0: 7468 6d20 616e 6420 6d61 7920 6769 7665  thm and may give
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│ │ │ │ +0002d3f0: 7769 7468 2061 2073 6d61 6c6c 2062 7574  with a small but
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│ │ │ │ +0002d410: 6c69 7479 2e20 5265 6164 206d 6f72 6520  lity. Read more 
│ │ │ │ +0002d420: 756e 6465 7220 2a6e 6f74 650a 7072 6f62  under *note.prob
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│ │ │ │ +0002d440: 7468 6d3a 2070 726f 6261 6269 6c69 7374  thm: probabilist
│ │ │ │ +0002d450: 6963 2061 6c67 6f72 6974 686d 2c2e 0a0a  ic algorithm,...
│ │ │ │ +0002d460: 5761 7973 2074 6f20 7573 6520 5365 6772  Ways to use Segr
│ │ │ │ +0002d470: 653a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  e:.=============
│ │ │ │ +0002d480: 3d3d 3d3d 3d0a 0a20 202a 2022 5365 6772  =====..  * "Segr
│ │ │ │ +0002d490: 6528 4964 6561 6c29 220a 2020 2a20 2253  e(Ideal)".  * "S
│ │ │ │ +0002d4a0: 6567 7265 2849 6465 616c 2c53 796d 626f  egre(Ideal,Symbo
│ │ │ │ +0002d4b0: 6c29 220a 2020 2a20 2253 6567 7265 2851  l)".  * "Segre(Q
│ │ │ │ +0002d4c0: 756f 7469 656e 7452 696e 672c 4964 6561  uotientRing,Idea
│ │ │ │ +0002d4d0: 6c29 220a 0a46 6f72 2074 6865 2070 726f  l)"..For the pro
│ │ │ │ +0002d4e0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ +0002d4f0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ +0002d500: 6f62 6a65 6374 202a 6e6f 7465 2053 6567  object *note Seg
│ │ │ │ +0002d510: 7265 3a20 5365 6772 652c 2069 7320 6120  re: Segre, is a 
│ │ │ │ +0002d520: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e  *note method fun
│ │ │ │ +0002d530: 6374 696f 6e20 7769 7468 206f 7074 696f  ction with optio
│ │ │ │ +0002d540: 6e73 3a0a 284d 6163 6175 6c61 7932 446f  ns:.(Macaulay2Do
│ │ │ │ +0002d550: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ +0002d560: 5769 7468 4f70 7469 6f6e 732c 2e0a 0a2d  WithOptions,...-
│ │ │ │ +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 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002d5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +0002d5c0: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
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│ │ │ │ +0002d5e0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ +0002d5f0: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ +0002d600: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ +0002d610: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ +0002d620: 6b61 6765 732f 0a43 6861 7261 6374 6572  kages/.Character
│ │ │ │ +0002d630: 6973 7469 6343 6c61 7373 6573 2e6d 323a  isticClasses.m2:
│ │ │ │ +0002d640: 3137 3633 3a30 2e0a 1f0a 4669 6c65 3a20  1763:0....File: 
│ │ │ │ +0002d650: 4368 6172 6163 7465 7269 7374 6963 436c  CharacteristicCl
│ │ │ │ +0002d660: 6173 7365 732e 696e 666f 2c20 4e6f 6465  asses.info, Node
│ │ │ │ +0002d670: 3a20 546f 7269 6343 686f 7752 696e 672c  : ToricChowRing,
│ │ │ │ +0002d680: 2050 7265 763a 2053 6567 7265 2c20 5570   Prev: Segre, Up
│ │ │ │ +0002d690: 3a20 546f 700a 0a54 6f72 6963 4368 6f77  : Top..ToricChow
│ │ │ │ +0002d6a0: 5269 6e67 202d 2d20 436f 6d70 7574 6573  Ring -- Computes
│ │ │ │ +0002d6b0: 2074 6865 2043 686f 7720 7269 6e67 206f   the Chow ring o
│ │ │ │ +0002d6c0: 6620 6120 6e6f 726d 616c 2074 6f72 6963  f a normal toric
│ │ │ │ +0002d6d0: 2076 6172 6965 7479 0a2a 2a2a 2a2a 2a2a   variety.*******
│ │ │ │ +0002d6e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0002d6f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0002d700: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0002d710: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +0002d720: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +0002d730: 546f 7269 6343 686f 7752 696e 6720 580a  ToricChowRing X.
│ │ │ │ +0002d740: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +0002d750: 2020 2a20 522c 2061 202a 6e6f 7465 206e    * R, a *note n
│ │ │ │ +0002d760: 6f72 6d61 6c20 746f 7269 6320 7661 7269  ormal toric vari
│ │ │ │ +0002d770: 6574 793a 0a20 2020 2020 2020 2028 4e6f  ety:.        (No
│ │ │ │ +0002d780: 726d 616c 546f 7269 6356 6172 6965 7469  rmalToricVarieti
│ │ │ │ +0002d790: 6573 294e 6f72 6d61 6c54 6f72 6963 5661  es)NormalToricVa
│ │ │ │ +0002d7a0: 7269 6574 792c 2c20 4120 6e6f 726d 616c  riety,, A normal
│ │ │ │ +0002d7b0: 2074 6f72 6963 2076 6172 6965 7479 0a20   toric variety. 
│ │ │ │ +0002d7c0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +0002d7d0: 2020 2a20 6120 2a6e 6f74 6520 7175 6f74    * a *note quot
│ │ │ │ +0002d7e0: 6965 6e74 2072 696e 673a 2028 4d61 6361  ient ring: (Maca
│ │ │ │ +0002d7f0: 756c 6179 3244 6f63 2951 756f 7469 656e  ulay2Doc)Quotien
│ │ │ │ +0002d800: 7452 696e 672c 2c20 0a0a 4465 7363 7269  tRing,, ..Descri
│ │ │ │ +0002d810: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +0002d820: 3d0a 0a4c 6574 2058 2062 6520 6120 746f  =..Let X be a to
│ │ │ │ +0002d830: 7269 6320 7661 7269 6574 7920 7769 7468  ric variety with
│ │ │ │ +0002d840: 2074 6f74 616c 2063 6f6f 7264 696e 6174   total coordinat
│ │ │ │ +0002d850: 6520 7269 6e67 2028 436f 7820 7269 6e67  e ring (Cox ring
│ │ │ │ +0002d860: 2920 522e 2054 6869 7320 6d65 7468 6f64  ) R. This method
│ │ │ │ +0002d870: 0a63 6f6d 7075 7465 7320 7468 6520 4368  .computes the Ch
│ │ │ │ +0002d880: 6f77 2072 696e 6720 2043 686f 7720 7269  ow ring  Chow ri
│ │ │ │ +0002d890: 6e67 2043 683d 522f 2853 522b 4c52 2920  ng Ch=R/(SR+LR) 
│ │ │ │ +0002d8a0: 6f66 2058 3b20 6865 7265 2053 5220 6973  of X; here SR is
│ │ │ │ +0002d8b0: 2074 6865 0a53 7461 6e6c 6579 2d52 6569   the.Stanley-Rei
│ │ │ │ +0002d8c0: 736e 6572 2069 6465 616c 206f 6620 7468  sner ideal of th
│ │ │ │ +0002d8d0: 6520 636f 7272 6573 706f 6e64 696e 6720  e corresponding 
│ │ │ │ +0002d8e0: 6661 6e20 616e 6420 4c52 2069 7320 7468  fan and LR is th
│ │ │ │ +0002d8f0: 6520 6964 6561 6c20 6f66 206c 696e 6561  e ideal of linea
│ │ │ │ +0002d900: 720a 7265 6c61 7469 6f6e 7320 616d 6f75  r.relations amou
│ │ │ │ +0002d910: 6e74 2074 6865 2072 6179 732e 2049 7420  nt the rays. It 
│ │ │ │ +0002d920: 6973 206e 6565 6465 6420 666f 7220 696e  is needed for in
│ │ │ │ +0002d930: 7075 7420 696e 746f 2074 6865 206d 6574  put into the met
│ │ │ │ +0002d940: 686f 6473 202a 6e6f 7465 2053 6567 7265  hods *note Segre
│ │ │ │ +0002d950: 3a0a 5365 6772 652c 2c20 2a6e 6f74 6520  :.Segre,, *note 
│ │ │ │ +0002d960: 4368 6572 6e3a 2043 6865 726e 2c20 616e  Chern: Chern, an
│ │ │ │ +0002d970: 6420 2a6e 6f74 6520 4353 4d3a 2043 534d  d *note CSM: CSM
│ │ │ │ +0002d980: 2c20 696e 2074 6865 2063 6173 6573 2077  , in the cases w
│ │ │ │ +0002d990: 6865 7265 2061 2074 6f72 6963 0a76 6172  here a toric.var
│ │ │ │ +0002d9a0: 6965 7479 2069 7320 616c 736f 2069 6e70  iety is also inp
│ │ │ │ +0002d9b0: 7574 2074 6f20 656e 7375 7265 2074 6861  ut to ensure tha
│ │ │ │ +0002d9c0: 7420 7468 6573 6520 6d65 7468 6f64 7320  t these methods 
│ │ │ │ +0002d9d0: 7265 7475 726e 2072 6573 756c 7473 2069  return results i
│ │ │ │ +0002d9e0: 6e20 7468 6520 7361 6d65 0a72 696e 672e  n the same.ring.
│ │ │ │ +0002d9f0: 2057 6520 6769 7665 2061 6e20 6578 616d   We give an exam
│ │ │ │ +0002da00: 706c 6520 6f66 2074 6865 2075 7365 206f  ple of the use o
│ │ │ │ +0002da10: 6620 7468 6973 206d 6574 686f 6420 746f  f this method to
│ │ │ │ +0002da20: 2077 6f72 6b20 7769 7468 2065 6c65 6d65   work with eleme
│ │ │ │ +0002da30: 6e74 7320 6f66 2074 6865 0a43 686f 7720  nts of the.Chow 
│ │ │ │ +0002da40: 7269 6e67 206f 6620 6120 746f 7269 6320  ring of a toric 
│ │ │ │ +0002da50: 7661 7269 6574 790a 0a2b 2d2d 2d2d 2d2d  variety..+------
│ │ │ │ +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 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"           
│ │ │ │ +0002daa0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206e  -------+.|i1 : n
│ │ │ │ +0002dab0: 6565 6473 5061 636b 6167 6520 224e 6f72  eedsPackage "Nor
│ │ │ │ +0002dac0: 6d61 6c54 6f72 6963 5661 7269 6574 6965  malToricVarietie
│ │ │ │ +0002dad0: 7322 2020 2020 2020 2020 2020 2020 2020  s"              
│ │ │ │ +0002dae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002daf0: 2020 2020 2020 207c 0a7c 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 7c0a 7c20 2020            |.|   
│ │ │ │ +0002db20: 2020 2020 2020 2020 2020 2020 2020 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 7c0a 7c6f 3120            |.|o1 
│ │ │ │ -0002db80: 3d20 4e6f 726d 616c 546f 7269 6356 6172  = NormalToricVar
│ │ │ │ -0002db90: 6965 7469 6573 2020 2020 2020 2020 2020  ieties          
│ │ │ │ +0002db40: 2020 2020 2020 207c 0a7c 6f31 203d 204e         |.|o1 = N
│ │ │ │ +0002db50: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ +0002db60: 6965 7320 2020 2020 2020 2020 2020 2020  ies             
│ │ │ │ +0002db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002db90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002dba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dbc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002dbc0: 2020 2020 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                  
│ │ │ │ +0002dbe0: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ +0002dbf0: 6163 6b61 6765 2020 2020 2020 2020 2020  ackage          
│ │ │ │  0002dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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            |.+---
│ │ │ │ +0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dc30: 2020 2020 2020 207c 0a2b 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 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            |.|   
│ │ │ │ +0002dc80: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2052  -------+.|i2 : R
│ │ │ │ +0002dc90: 686f 203d 207b 7b31 2c30 2c30 7d2c 7b30  ho = {{1,0,0},{0
│ │ │ │ +0002dca0: 2c31 2c30 7d2c 7b30 2c30 2c31 7d2c 7b2d  ,1,0},{0,0,1},{-
│ │ │ │ +0002dcb0: 312c 2d31 2c30 7d2c 7b30 2c30 2c2d 317d  1,-1,0},{0,0,-1}
│ │ │ │ +0002dcc0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +0002dcd0: 2020 2020 2020 207c 0a7c 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 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            |.|   
│ │ │ │ +0002dd20: 2020 2020 2020 207c 0a7c 6f32 203d 207b         |.|o2 = {
│ │ │ │ +0002dd30: 7b31 2c20 302c 2030 7d2c 207b 302c 2031  {1, 0, 0}, {0, 1
│ │ │ │ +0002dd40: 2c20 307d 2c20 7b30 2c20 302c 2031 7d2c  , 0}, {0, 0, 1},
│ │ │ │ +0002dd50: 207b 2d31 2c20 2d31 2c20 307d 2c20 7b30   {-1, -1, 0}, {0
│ │ │ │ +0002dd60: 2c20 302c 202d 317d 7d20 2020 2020 2020  , 0, -1}}       
│ │ │ │ +0002dd70: 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ddc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ddc0: 2020 2020 2020 207c 0a7c 6f32 203a 204c         |.|o2 : L
│ │ │ │ +0002ddd0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │  0002dde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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            |.+---
│ │ │ │ +0002ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002de00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002de10: 2020 2020 2020 207c 0a2b 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 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            |.|   
│ │ │ │ +0002de60: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2053  -------+.|i3 : S
│ │ │ │ +0002de70: 6967 6d61 203d 207b 7b30 2c31 2c32 7d2c  igma = {{0,1,2},
│ │ │ │ +0002de80: 7b31 2c32 2c33 7d2c 7b30 2c32 2c33 7d2c  {1,2,3},{0,2,3},
│ │ │ │ +0002de90: 7b30 2c31 2c34 7d2c 7b31 2c33 2c34 7d2c  {0,1,4},{1,3,4},
│ │ │ │ +0002dea0: 7b30 2c33 2c34 7d7d 2020 2020 2020 2020  {0,3,4}}        
│ │ │ │ +0002deb0: 2020 2020 2020 207c 0a7c 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 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}}      |.|   
│ │ │ │ +0002df00: 2020 2020 2020 207c 0a7c 6f33 203d 207b         |.|o3 = {
│ │ │ │ +0002df10: 7b30 2c20 312c 2032 7d2c 207b 312c 2032  {0, 1, 2}, {1, 2
│ │ │ │ +0002df20: 2c20 337d 2c20 7b30 2c20 322c 2033 7d2c  , 3}, {0, 2, 3},
│ │ │ │ +0002df30: 207b 302c 2031 2c20 347d 2c20 7b31 2c20   {0, 1, 4}, {1, 
│ │ │ │ +0002df40: 332c 2034 7d2c 207b 302c 2033 2c20 347d  3, 4}, {0, 3, 4}
│ │ │ │ +0002df50: 7d20 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dfa0: 2020 2020 2020 207c 0a7c 6f33 203a 204c         |.|o3 : L
│ │ │ │ +0002dfb0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │  0002dfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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            |.+---
│ │ │ │ +0002dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dff0: 2020 2020 2020 207c 0a2b 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 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            |.|   
│ │ │ │ +0002e040: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2058  -------+.|i4 : X
│ │ │ │ +0002e050: 203d 206e 6f72 6d61 6c54 6f72 6963 5661   = normalToricVa
│ │ │ │ +0002e060: 7269 6574 7928 5268 6f2c 5369 676d 612c  riety(Rho,Sigma,
│ │ │ │ +0002e070: 436f 6566 6669 6369 656e 7452 696e 6720  CoefficientRing 
│ │ │ │ +0002e080: 3d3e 5a5a 2f33 3237 3439 2920 2020 2020  =>ZZ/32749)     
│ │ │ │ +0002e090: 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e0e0: 2020 2020 2020 207c 0a7c 6f34 203d 2058         |.|o4 = X
│ │ │ │  0002e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e100: 2020 2020 2020 2020 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                  
│ │ │ │ +0002e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e130: 2020 2020 2020 207c 0a7c 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 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e160: 2020 2020 2020 2020 2020 2020 2020 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 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            |.+---
│ │ │ │ +0002e180: 2020 2020 2020 207c 0a7c 6f34 203a 204e         |.|o4 : N
│ │ │ │ +0002e190: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ +0002e1a0: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │ +0002e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e1d0: 2020 2020 2020 207c 0a2b 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 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                  
│ │ │ │ +0002e220: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2052  -------+.|i5 : R
│ │ │ │ +0002e230: 3d72 696e 6720 5820 2020 2020 2020 2020  =ring X         
│ │ │ │ +0002e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e270: 2020 2020 2020 207c 0a7c 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 7c0a 7c20 2020            |.|   
│ │ │ │ +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                  
│ │ │ │ +0002e2c0: 2020 2020 2020 207c 0a7c 6f35 203d 2052         |.|o5 = R
│ │ │ │  0002e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e2e0: 2020 2020 2020 2020 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                  
│ │ │ │ +0002e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e310: 2020 2020 2020 207c 0a7c 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 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e340: 2020 2020 2020 2020 2020 2020 2020 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                  
│ │ │ │ +0002e360: 2020 2020 2020 207c 0a7c 6f35 203a 2050         |.|o5 : P
│ │ │ │ +0002e370: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │  0002e380: 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            |.+---
│ │ │ │ +0002e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e3b0: 2020 2020 2020 207c 0a2b 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 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)           
│ │ │ │ +0002e400: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2043  -------+.|i6 : C
│ │ │ │ +0002e410: 683d 546f 7269 6343 686f 7752 696e 6728  h=ToricChowRing(
│ │ │ │ +0002e420: 5829 2020 2020 2020 2020 2020 2020 2020  X)              
│ │ │ │ +0002e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e450: 2020 2020 2020 207c 0a7c 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 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e4a0: 2020 2020 2020 207c 0a7c 6f36 203d 2043         |.|o6 = C
│ │ │ │ +0002e4b0: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │  0002e4c0: 2020 2020 2020 2020 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                  
│ │ │ │ +0002e4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e4f0: 2020 2020 2020 207c 0a7c 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 7c0a 7c20 2020            |.|   
│ │ │ │ +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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e540: 2020 2020 2020 207c 0a7c 6f36 203a 2051         |.|o6 : Q
│ │ │ │ +0002e550: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │  0002e560: 2020 2020 2020 2020 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            |.+---
│ │ │ │ +0002e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e590: 2020 2020 2020 207c 0a2b 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 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                  
│ │ │ │ +0002e5e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2064  -------+.|i7 : d
│ │ │ │ +0002e5f0: 6573 6372 6962 6520 4368 2020 2020 2020  escribe Ch      
│ │ │ │ +0002e600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e630: 2020 2020 2020 207c 0a7c 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 7c0a 7c20 2020            |.|   
│ │ │ │ +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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e6b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e680: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002e690: 2020 2020 2020 2020 2020 2020 2020 205a                 Z
│ │ │ │ +0002e6a0: 5a5b 7820 2e2e 7820 5d20 2020 2020 2020  Z[x ..x ]       
│ │ │ │ +0002e6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e6d0: 2020 5a5a 5b78 202e 2e78 205d 2020 2020    ZZ[x ..x ]    
│ │ │ │ +0002e6d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e700: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e6f0: 2020 2030 2020 2034 2020 2020 2020 2020     0   4        
│ │ │ │ +0002e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e720: 2020 2020 2020 3020 2020 3420 2020 2020        0   4     
│ │ │ │ -0002e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e750: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ -0002e760: 3d20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  = --------------
│ │ │ │ -0002e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d20 2020 2020  -----------     
│ │ │ │ -0002e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e7a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002e7b0: 2020 2878 2078 202c 2078 2078 2078 202c    (x x , x x x ,
│ │ │ │ -0002e7c0: 2078 2020 2d20 7820 2c20 7820 202d 2078   x  - x , x  - x
│ │ │ │ -0002e7d0: 202c 2078 2020 2d20 7820 2920 2020 2020   , x  - x )     
│ │ │ │ -0002e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e7f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002e800: 2020 2020 3220 3420 2020 3020 3120 3320      2 4   0 1 3 
│ │ │ │ -0002e810: 2020 3020 2020 2033 2020 2031 2020 2020    0    3   1    
│ │ │ │ -0002e820: 3320 2020 3220 2020 2034 2020 2020 2020  3   2    4      
│ │ │ │ -0002e830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e840: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0002e720: 2020 2020 2020 207c 0a7c 6f37 203d 202d         |.|o7 = -
│ │ │ │ +0002e730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e750: 2d2d 2d2d 2d2d 2d2d 2020 2020 2020 2020  --------        
│ │ │ │ +0002e760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e770: 2020 2020 2020 207c 0a7c 2020 2020 2028         |.|     (
│ │ │ │ +0002e780: 7820 7820 2c20 7820 7820 7820 2c20 7820  x x , x x x , x 
│ │ │ │ +0002e790: 202d 2078 202c 2078 2020 2d20 7820 2c20   - x , x  - x , 
│ │ │ │ +0002e7a0: 7820 202d 2078 2029 2020 2020 2020 2020  x  - x )        
│ │ │ │ +0002e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e7c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002e7d0: 2032 2034 2020 2030 2031 2033 2020 2030   2 4   0 1 3   0
│ │ │ │ +0002e7e0: 2020 2020 3320 2020 3120 2020 2033 2020      3   1    3  
│ │ │ │ +0002e7f0: 2032 2020 2020 3420 2020 2020 2020 2020   2    4         
│ │ │ │ +0002e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e810: 2020 2020 2020 207c 0a2b 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e890: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ -0002e8a0: 3a20 723d 6765 6e73 2052 2020 2020 2020  : r=gens R      
│ │ │ │ -0002e8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e860: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2072  -------+.|i8 : r
│ │ │ │ +0002e870: 3d67 656e 7320 5220 2020 2020 2020 2020  =gens R         
│ │ │ │ +0002e880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e8b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e8e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e930: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ -0002e940: 3d20 7b78 202c 2078 202c 2078 202c 2078  = {x , x , x , x
│ │ │ │ -0002e950: 202c 2078 207d 2020 2020 2020 2020 2020   , x }          
│ │ │ │ -0002e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e980: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002e990: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ -0002e9a0: 3320 2020 3420 2020 2020 2020 2020 2020  3   4           
│ │ │ │ +0002e900: 2020 2020 2020 207c 0a7c 6f38 203d 207b         |.|o8 = {
│ │ │ │ +0002e910: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ +0002e920: 7820 7d20 2020 2020 2020 2020 2020 2020  x }             
│ │ │ │ +0002e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e950: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002e960: 2030 2020 2031 2020 2032 2020 2033 2020   0   1   2   3  
│ │ │ │ +0002e970: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +0002e980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e9a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e9d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ea00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e9f0: 2020 2020 2020 207c 0a7c 6f38 203a 204c         |.|o8 : L
│ │ │ │ +0002ea00: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │  0002ea10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ea20: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ -0002ea30: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ -0002ea40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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            |.+---
│ │ │ │ +0002ea20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ea30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ea40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002ea50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ea60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ea70: 2d2d 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 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))      
│ │ │ │ +0002ea90: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2049  -------+.|i9 : I
│ │ │ │ +0002eaa0: 3d69 6465 616c 2872 616e 646f 6d28 7b31  =ideal(random({1
│ │ │ │ +0002eab0: 2c30 7d2c 5229 2920 2020 2020 2020 2020  ,0},R))         
│ │ │ │ +0002eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002eae0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002eaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eb10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002eb10: 2020 2020 2020 2020 2020 2020 2020 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 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 7c0a 7c20 2020            |.|   
│ │ │ │ -0002ebc0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ -0002ebd0: 2020 2020 2031 2020 2020 2020 2020 3320       1        3 
│ │ │ │ +0002eb30: 2020 2020 2020 207c 0a7c 6f39 203d 2069         |.|o9 = i
│ │ │ │ +0002eb40: 6465 616c 2831 3037 7820 202b 2034 3337  deal(107x  + 437
│ │ │ │ +0002eb50: 3678 2020 2d20 3633 3136 7820 2920 2020  6x  - 6316x )   
│ │ │ │ +0002eb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002eb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002eb80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002eb90: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ +0002eba0: 2020 3120 2020 2020 2020 2033 2020 2020    1        3    
│ │ │ │ +0002ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ebc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ebd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002ebe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ebf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ec00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002ec00: 2020 2020 2020 2020 2020 2020 2020 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                  
│ │ │ │ +0002ec20: 2020 2020 2020 207c 0a7c 6f39 203a 2049         |.|o9 : I
│ │ │ │ +0002ec30: 6465 616c 206f 6620 5220 2020 2020 2020  deal of R       
│ │ │ │  0002ec40: 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            |.+---
│ │ │ │ +0002ec50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ec60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ec70: 2020 2020 2020 207c 0a2b 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 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))     
│ │ │ │ +0002ecc0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │ +0002ecd0: 4b3d 6964 6561 6c28 7261 6e64 6f6d 287b  K=ideal(random({
│ │ │ │ +0002ece0: 312c 317d 2c52 2929 2020 2020 2020 2020  1,1},R))        
│ │ │ │ +0002ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ed00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ed10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002ed20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ed30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ed40: 2020 2020 2020 2020 2020 7c0a 7c20 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 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 |.|   
│ │ │ │ +0002ed60: 2020 2020 2020 207c 0a7c 6f31 3020 3d20         |.|o10 = 
│ │ │ │ +0002ed70: 6964 6561 6c28 3331 3837 7820 7820 202d  ideal(3187x x  -
│ │ │ │ +0002ed80: 2036 3035 3378 2078 2020 2d20 3136 3039   6053x x  - 1609
│ │ │ │ +0002ed90: 3078 2078 2020 2b20 3337 3833 7820 7820  0x x  + 3783x x 
│ │ │ │ +0002eda0: 202b 2038 3537 3078 2078 2020 2b20 3834   + 8570x x  + 84
│ │ │ │ +0002edb0: 3434 7820 7820 297c 0a7c 2020 2020 2020  44x x )|.|      
│ │ │ │ +0002edc0: 2020 2020 2020 2020 2020 2030 2032 2020             0 2  
│ │ │ │ +0002edd0: 2020 2020 2020 3120 3220 2020 2020 2020        1 2       
│ │ │ │ +0002ede0: 2020 3220 3320 2020 2020 2020 2030 2034    2 3        0 4
│ │ │ │ +0002edf0: 2020 2020 2020 2020 3120 3420 2020 2020          1 4     
│ │ │ │ +0002ee00: 2020 2033 2034 207c 0a7c 2020 2020 2020     3 4 |.|      
│ │ │ │ +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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ee50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ee60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ee50: 2020 2020 2020 207c 0a7c 6f31 3020 3a20         |.|o10 : 
│ │ │ │ +0002ee60: 4964 6561 6c20 6f66 2052 2020 2020 2020  Ideal of R      
│ │ │ │  0002ee70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ee80: 2020 2020 2020 2020 2020 7c0a 7c6f 3130            |.|o10
│ │ │ │ -0002ee90: 203a 2049 6465 616c 206f 6620 5220 2020   : Ideal of R   
│ │ │ │ -0002eea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eed0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0002ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ee90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002eea0: 2020 2020 2020 207c 0a2b 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ef00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ef10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131  ----------+.|i11
│ │ │ │ -0002ef30: 203a 2063 3d43 6865 726e 2843 682c 582c   : c=Chern(Ch,X,
│ │ │ │ -0002ef40: 4929 2020 2020 2020 2020 2020 2020 2020  I)              
│ │ │ │ +0002eef0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20  -------+.|i11 : 
│ │ │ │ +0002ef00: 633d 4368 6572 6e28 4368 2c58 2c49 2920  c=Chern(Ch,X,I) 
│ │ │ │ +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 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ef70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ef90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002efa0: 2020 3220 2020 2020 2020 3220 2020 2020    2       2     
│ │ │ │  0002efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002efc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002efd0: 2020 2020 2032 2020 2020 2020 2032 2020       2       2  
│ │ │ │ -0002efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f010: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ -0002f020: 203d 2034 7820 7820 202b 2032 7820 202b   = 4x x  + 2x  +
│ │ │ │ -0002f030: 2032 7820 7820 202b 2078 2020 2020 2020   2x x  + x      
│ │ │ │ -0002f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f060: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f070: 2020 2020 2033 2034 2020 2020 2033 2020       3 4     3  
│ │ │ │ -0002f080: 2020 2033 2034 2020 2020 3320 2020 2020     3 4    3     
│ │ │ │ +0002efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002efe0: 2020 2020 2020 207c 0a7c 6f31 3120 3d20         |.|o11 = 
│ │ │ │ +0002eff0: 3478 2078 2020 2b20 3278 2020 2b20 3278  4x x  + 2x  + 2x
│ │ │ │ +0002f000: 2078 2020 2b20 7820 2020 2020 2020 2020   x  + x         
│ │ │ │ +0002f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f030: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f040: 2020 3320 3420 2020 2020 3320 2020 2020    3 4     3     
│ │ │ │ +0002f050: 3320 3420 2020 2033 2020 2020 2020 2020  3 4    3        
│ │ │ │ +0002f060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f080: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f0b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f0d0: 2020 2020 2020 207c 0a7c 6f31 3120 3a20         |.|o11 : 
│ │ │ │ +0002f0e0: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │  0002f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f100: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ -0002f110: 203a 2043 6820 2020 2020 2020 2020 2020   : Ch           
│ │ │ │ -0002f120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f150: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0002f100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f120: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002f130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132  ----------+.|i12
│ │ │ │ -0002f1b0: 203a 2073 3d53 6567 7265 2843 682c 582c   : s=Segre(Ch,X,
│ │ │ │ -0002f1c0: 4b29 2020 2020 2020 2020 2020 2020 2020  K)              
│ │ │ │ +0002f170: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20  -------+.|i12 : 
│ │ │ │ +0002f180: 733d 5365 6772 6528 4368 2c58 2c4b 2920  s=Segre(Ch,X,K) 
│ │ │ │ +0002f190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f1c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f1f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f210: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f220: 2020 3220 2020 2020 2032 2020 2020 2020    2      2      
│ │ │ │  0002f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f240: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f250: 2020 2020 2032 2020 2020 2020 3220 2020       2      2   
│ │ │ │ -0002f260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f290: 2020 2020 2020 2020 2020 7c0a 7c6f 3132            |.|o12
│ │ │ │ -0002f2a0: 203d 2033 7820 7820 202d 2078 2020 2d20   = 3x x  - x  - 
│ │ │ │ -0002f2b0: 3278 2078 2020 2b20 7820 202b 2078 2020  2x x  + x  + x  
│ │ │ │ -0002f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f2e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f2f0: 2020 2020 2033 2034 2020 2020 3320 2020       3 4    3   
│ │ │ │ -0002f300: 2020 3320 3420 2020 2033 2020 2020 3420    3 4    3    4 
│ │ │ │ +0002f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f260: 2020 2020 2020 207c 0a7c 6f31 3220 3d20         |.|o12 = 
│ │ │ │ +0002f270: 3378 2078 2020 2d20 7820 202d 2032 7820  3x x  - x  - 2x 
│ │ │ │ +0002f280: 7820 202b 2078 2020 2b20 7820 2020 2020  x  + x  + x     
│ │ │ │ +0002f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f2b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f2c0: 2020 3320 3420 2020 2033 2020 2020 2033    3 4    3     3
│ │ │ │ +0002f2d0: 2034 2020 2020 3320 2020 2034 2020 2020   4    3    4    
│ │ │ │ +0002f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f300: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f330: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f350: 2020 2020 2020 207c 0a7c 6f31 3220 3a20         |.|o12 : 
│ │ │ │ +0002f360: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │  0002f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f380: 2020 2020 2020 2020 2020 7c0a 7c6f 3132            |.|o12
│ │ │ │ -0002f390: 203a 2043 6820 2020 2020 2020 2020 2020   : Ch           
│ │ │ │ -0002f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f3d0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0002f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f3a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002f3b0: 2d2d 2d2d 2d2d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f420: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133  ----------+.|i13
│ │ │ │ -0002f430: 203a 2073 2d63 2020 2020 2020 2020 2020   : s-c          
│ │ │ │ -0002f440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f3f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20  -------+.|i13 : 
│ │ │ │ +0002f400: 732d 6320 2020 2020 2020 2020 2020 2020  s-c             
│ │ │ │ +0002f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f440: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f470: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002f470: 2020 2020 2020 2020 2020 2020 2020 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                  
│ │ │ │ +0002f490: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f4a0: 2020 2032 2020 2020 2020 2032 2020 2020     2       2    
│ │ │ │  0002f4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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 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 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f570: 2020 2020 2020 3320 3420 2020 2020 3320        3 4     3 
│ │ │ │ -0002f580: 2020 2020 3320 3420 2020 2034 2020 2020      3 4    4    
│ │ │ │ +0002f4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f4e0: 2020 2020 2020 207c 0a7c 6f31 3320 3d20         |.|o13 = 
│ │ │ │ +0002f4f0: 2d20 7820 7820 202d 2033 7820 202d 2034  - x x  - 3x  - 4
│ │ │ │ +0002f500: 7820 7820 202b 2078 2020 2020 2020 2020  x x  + x        
│ │ │ │ +0002f510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f530: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f540: 2020 2033 2034 2020 2020 2033 2020 2020     3 4     3    
│ │ │ │ +0002f550: 2033 2034 2020 2020 3420 2020 2020 2020   3 4    4       
│ │ │ │ +0002f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f580: 2020 2020 2020 207c 0a7c 2020 2020 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 7c0a 7c20 2020            |.|   
│ │ │ │ +0002f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f5d0: 2020 2020 2020 207c 0a7c 6f31 3320 3a20         |.|o13 : 
│ │ │ │ +0002f5e0: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │  0002f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f600: 2020 2020 2020 2020 2020 7c0a 7c6f 3133            |.|o13
│ │ │ │ -0002f610: 203a 2043 6820 2020 2020 2020 2020 2020   : Ch           
│ │ │ │ -0002f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f650: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0002f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f620: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002f630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f640: 2d2d 2d2d 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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134  ----------+.|i14
│ │ │ │ -0002f6b0: 203a 2073 2a63 2020 2020 2020 2020 2020   : s*c          
│ │ │ │ -0002f6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f670: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20  -------+.|i14 : 
│ │ │ │ +0002f680: 732a 6320 2020 2020 2020 2020 2020 2020  s*c             
│ │ │ │ +0002f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f6c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f6f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f710: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f720: 2020 3220 2020 2020 2032 2020 2020 2020    2      2      
│ │ │ │  0002f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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                  
│ │ │ │ +0002f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f760: 2020 2020 2020 207c 0a7c 6f31 3420 3d20         |.|o14 = 
│ │ │ │ +0002f770: 3278 2078 2020 2b20 7820 202b 2078 2078  2x x  + x  + x x
│ │ │ │  0002f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f7f0: 2020 2020 2033 2034 2020 2020 3320 2020       3 4    3   
│ │ │ │ -0002f800: 2033 2034 2020 2020 2020 2020 2020 2020   3 4            
│ │ │ │ +0002f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f7b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f7c0: 2020 3320 3420 2020 2033 2020 2020 3320    3 4    3    3 
│ │ │ │ +0002f7d0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +0002f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f800: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f830: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f850: 2020 2020 2020 207c 0a7c 6f31 3420 3a20         |.|o14 : 
│ │ │ │ +0002f860: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │  0002f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0002fca0: 2054 6167 2054 6162 6c65 0a               Tag Table.
│ │ ├── ./usr/share/info/Chordal.info.gz
│ │ │ ├── Chordal.info
│ │ │ │ @@ -3949,30 +3949,30 @@
│ │ │ │  0000f6c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f710: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │  0000f720: 203d 2045 6c69 6d54 7265 657b 6120 3d3e   = ElimTree{a =>
│ │ │ │ -0000f730: 2062 2020 207d 2020 2020 2020 2020 2020   b   }          
│ │ │ │ +0000f730: 2063 7d20 2020 2020 2020 2020 2020 2020   c}             
│ │ │ │  0000f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f760: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000f770: 2020 2020 2020 2020 2020 2020 6220 3d3e              b =>
│ │ │ │  0000f780: 2063 2020 2020 2020 2020 2020 2020 2020   c              
│ │ │ │  0000f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0000f7b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000f7c0: 2020 2020 2020 2020 2020 2020 6320 3d3e              c =>
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│ │ │ │  0000f810: 2020 2020 2020 2020 2020 2020 6420 3d3e              d =>
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│ │ │ │  0000f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4033,31 +4033,31 @@
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│ │ │ │  0000fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc60: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ -0000fc70: 203d 2043 686f 7264 616c 4e65 747b 2064   = ChordalNet{ d
│ │ │ │ -0000fc80: 203d 3e20 7b20 2c20 647d 2020 2020 7d20   => { , d}    } 
│ │ │ │ +0000fc70: 203d 2043 686f 7264 616c 4e65 747b 2061   = ChordalNet{ a
│ │ │ │ +0000fc80: 203d 3e20 7b61 2c20 207d 2020 2020 7d20   => {a,  }    } 
│ │ │ │  0000fc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fcb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0000fcc0: 2020 2020 2020 2020 2020 2020 2020 2062                 b
│ │ │ │ -0000fcd0: 203d 3e20 7b62 2c20 202c 2062 7d20 2020   => {b,  , b}   
│ │ │ │ +0000fcc0: 2020 2020 2020 2020 2020 2020 2020 2063                 c
│ │ │ │ +0000fcd0: 203d 3e20 7b20 2c20 637d 2020 2020 2020   => { , c}      
│ │ │ │  0000fce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0000fd10: 2020 2020 2020 2020 2020 2020 2020 2061                 a
│ │ │ │ -0000fd20: 203d 3e20 7b61 2c20 207d 2020 2020 2020   => {a,  }      
│ │ │ │ +0000fd10: 2020 2020 2020 2020 2020 2020 2020 2064                 d
│ │ │ │ +0000fd20: 203d 3e20 7b20 2c20 647d 2020 2020 2020   => { , d}      
│ │ │ │  0000fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0000fd60: 2020 2020 2020 2020 2020 2020 2020 2063                 c
│ │ │ │ -0000fd70: 203d 3e20 7b20 2c20 637d 2020 2020 2020   => { , c}      
│ │ │ │ +0000fd60: 2020 2020 2020 2020 2020 2020 2020 2062                 b
│ │ │ │ +0000fd70: 203d 3e20 7b62 2c20 202c 2062 7d20 2020   => {b,  , b}   
│ │ │ │  0000fd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fda0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fde0: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/CohomCalg.info.gz
│ │ │ ├── CohomCalg.info
│ │ │ │ @@ -1042,15 +1042,15 @@
│ │ │ │  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 2032 2e36 3734 3038 7320 656c  | -- 2.67408s el
│ │ │ │ +00004180: 7c20 2d2d 2033 2e33 3332 3537 7320 656c  | -- 3.33257s el
│ │ │ │  00004190: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  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                  
│ │ │ │ @@ -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 3330 3239 3135 7320 656c  | -- .302915s el
│ │ │ │ +00006930: 7c20 2d2d 202e 3533 3134 3634 7320 656c  | -- .531464s 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 3937 3932 7320 656c  | -- 10.9792s el
│ │ │ │ +00006b60: 7c20 2d2d 2031 302e 3237 3136 7320 656c  | -- 10.2716s 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                  
│ │ │ │ @@ -1797,16 +1797,16 @@
│ │ │ │  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 3334 3231 3236 7320 656c  | -- .342126s el
│ │ │ │ -000070c0: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │ +000070b0: 7c20 2d2d 202e 3536 3239 7320 656c 6170  | -- .5629s elap
│ │ │ │ +000070c0: 7365 6420 2020 2020 2020 2020 2020 2020  sed             
│ │ │ │  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                  
│ │ │ │  00007130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1832,20 +1832,20 @@
│ │ │ │  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 3531 3733 3436 7320 656c  | -- .517346s el
│ │ │ │ -000072f0: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │ +000072e0: 7c20 2d2d 202e 3436 3331 3873 2065 6c61  | -- .46318s ela
│ │ │ │ +000072f0: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  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 3531 3733 3733 7320 656c  | -- .517373s el
│ │ │ │ +00007330: 7c20 2d2d 202e 3436 3332 3131 7320 656c  | -- .463211s 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
│ │ ├── ./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: 2e36 3632 3932 7320 2863 7075 293b 2034  .66292s (cpu); 4
│ │ │ │ -00010ff0: 2e38 3937 3937 7320 2874 6872 6561 6429  .89797s (thread)
│ │ │ │ +00010fd0: 2020 207c 0a7c 202d 2d20 7573 6564 2038     |.| -- used 8
│ │ │ │ +00010fe0: 2e34 3933 3236 7320 2863 7075 293b 2036  .49326s (cpu); 6
│ │ │ │ +00010ff0: 2e34 3234 3833 7320 2874 6872 6561 6429  .42483s (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: 2e31 3639 3335 3673 2028 6370 7529 3b20  .169356s (cpu); 
│ │ │ │ -000131b0: 302e 3039 3630 3434 3573 2028 7468 7265  0.0960445s (thre
│ │ │ │ -000131c0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +000131a0: 2e32 3834 3236 3973 2028 6370 7529 3b20  .284269s (cpu); 
│ │ │ │ +000131b0: 302e 3138 3539 3034 7320 2874 6872 6561  0.185904s (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   
│ │ │ │ @@ -4925,17 +4925,17 @@
│ │ │ │  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 2030 2e39 3337 3435 3973 2028 6370  ed 0.937459s (cp
│ │ │ │ -00013440: 7529 3b20 302e 3732 3039 3132 7320 2874  u); 0.720912s (t
│ │ │ │ -00013450: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +00013430: 6564 2031 2e33 3039 3236 7320 2863 7075  ed 1.30926s (cpu
│ │ │ │ +00013440: 293b 2031 2e30 3134 3538 7320 2874 6872  ); 1.01458s (thr
│ │ │ │ +00013450: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 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     / 
│ │ │ │  000134c0: 525c 3620 2020 2020 2f20 525c 3138 2020  R\6     / R\18  
│ │ │ │ @@ -4982,16 +4982,16 @@
│ │ │ │  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 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 302e 3933 3032 3673 2028 6370 7529  d 0.93026s (cpu)
│ │ │ │ -000137d0: 3b20 302e 3730 3231 3332 7320 2874 6872  ; 0.702132s (thr
│ │ │ │ +000137c0: 6420 312e 3238 3135 3773 2028 6370 7529  d 1.28157s (cpu)
│ │ │ │ +000137d0: 3b20 302e 3936 3636 3534 7320 2874 6872  ; 0.966654s (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 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      
│ │ │ │ @@ -28230,45 +28230,45 @@
│ │ │ │  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 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 2030 2e33 3730 3233 3973 2028 6370  ed 0.370239s (cp
│ │ │ │ -0006e4d0: 7529 3b20 302e 3331 3937 3339 7320 2874  u); 0.319739s (t
│ │ │ │ +0006e4c0: 6564 2030 2e35 3736 3533 3473 2028 6370  ed 0.576534s (cp
│ │ │ │ +0006e4d0: 7529 3b20 302e 3435 3134 3039 7320 2874  u); 0.451409s (t
│ │ │ │  0006e4e0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 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 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 207c                 |
│ │ │ │ -0006e5a0: 0a7c 202d 2d20 7573 6564 2030 2e32 3130  .| -- used 0.210
│ │ │ │ -0006e5b0: 3130 3473 2028 6370 7529 3b20 302e 3133  104s (cpu); 0.13
│ │ │ │ -0006e5c0: 3831 3539 7320 2874 6872 6561 6429 3b20  8159s (thread); 
│ │ │ │ -0006e5d0: 3073 2028 6763 2920 207c 0a7c 3320 2020  0s (gc)  |.|3   
│ │ │ │ +0006e5a0: 0a7c 202d 2d20 7573 6564 2030 2e33 3039  .| -- used 0.309
│ │ │ │ +0006e5b0: 3233 7320 2863 7075 293b 2030 2e31 3639  23s (cpu); 0.169
│ │ │ │ +0006e5c0: 3537 3373 2028 7468 7265 6164 293b 2030  573s (thread); 0
│ │ │ │ +0006e5d0: 7320 2867 6329 2020 207c 0a7c 3320 2020  s (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 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 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -0006e690: 6564 2033 2e36 3937 652d 3036 7320 2863  ed 3.697e-06s (c
│ │ │ │ -0006e6a0: 7075 293b 2033 2e33 3236 652d 3036 7320  pu); 3.326e-06s 
│ │ │ │ +0006e690: 6564 2033 2e35 3035 652d 3036 7320 2863  ed 3.505e-06s (c
│ │ │ │ +0006e6a0: 7075 293b 2032 2e37 3433 652d 3036 7320  pu); 2.743e-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 207c 0a7c 5461             |.|Ta
│ │ │ │  0006e700: 6c6c 797b 7d20 2020 2020 2020 2020 2020  lly{}           
│ │ │ │  0006e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -28812,34 +28812,34 @@
│ │ │ │  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 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 2030     |.| -- used 0
│ │ │ │ -00070930: 2e38 3032 3531 3773 2028 6370 7529 3b20  .802517s (cpu); 
│ │ │ │ -00070940: 302e 3538 3533 3538 7320 2874 6872 6561  0.585358s (threa
│ │ │ │ -00070950: 6429 3b20 3073 2028 6763 297c 0a7c 3220  d); 0s (gc)|.|2 
│ │ │ │ +00070920: 2020 207c 0a7c 202d 2d20 7573 6564 2031     |.| -- used 1
│ │ │ │ +00070930: 2e32 3830 3638 7320 2863 7075 293b 2030  .28068s (cpu); 0
│ │ │ │ +00070940: 2e38 3032 3438 3873 2028 7468 7265 6164  .802488s (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 207c 0a7c 5461 6c6c 797b 7b7b 312c     |.|Tally{{{1,
│ │ │ │  000709a0: 2031 7d7d 203d 3e20 3220 2020 2020 2020   1}} => 2       
│ │ │ │  000709b0: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │  000709c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000709d0: 2020 2020 7b7b 322c 2032 7d2c 207b 312c      {{2, 2}, {1,
│ │ │ │  000709e0: 2032 7d7d 203d 3e20 3420 2020 2020 2020   2}} => 4       
│ │ │ │  000709f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070a00: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00070a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070a30: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00070a40: 2d20 7573 6564 2030 2e34 3031 3330 3373  - used 0.401303s
│ │ │ │ -00070a50: 2028 6370 7529 3b20 302e 3333 3530 3535   (cpu); 0.335055
│ │ │ │ +00070a40: 2d20 7573 6564 2030 2e36 3533 3230 3273  - used 0.653202s
│ │ │ │ +00070a50: 2028 6370 7529 3b20 302e 3433 3839 3036   (cpu); 0.438906
│ │ │ │  00070a60: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  00070a70: 6763 297c 0a7c 3320 2020 2020 2020 2020  gc)|.|3         
│ │ │ │  00070a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070aa0: 2020 2020 2020 2020 2020 207c 0a7c 5461             |.|Ta
│ │ │ │  00070ab0: 6c6c 797b 7b7b 322c 2032 7d2c 207b 312c  lly{{{2, 2}, {1,
│ │ │ │  00070ac0: 2032 7d7d 203d 3e20 327d 2020 2020 2020   2}} => 2}      
│ │ │ │ @@ -28848,16 +28848,16 @@
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │  00070b50: 2020 207c 0a7c 202d 2d20 7573 6564 2030     |.| -- used 0
│ │ │ │ -00070b60: 2e32 3032 3232 3173 2028 6370 7529 3b20  .202221s (cpu); 
│ │ │ │ -00070b70: 302e 3133 3830 3435 7320 2874 6872 6561  0.138045s (threa
│ │ │ │ +00070b60: 2e33 3031 3337 3473 2028 6370 7529 3b20  .301374s (cpu); 
│ │ │ │ +00070b70: 302e 3136 3035 3736 7320 2874 6872 6561  0.160576s (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}
│ │ ├── ./usr/share/info/ConnectionMatrices.info.gz
│ │ │ ├── ConnectionMatrices.info
│ │ │ │ @@ -2415,30 +2415,30 @@
│ │ │ │  000096e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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 2032 2e38 3130 3738 7320 656c  | -- 2.81078s el
│ │ │ │ +00009750: 7c20 2d2d 2032 2e36 3036 3239 7320 656c  | -- 2.60629s 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 2e39 3430 3133 7320 656c  | -- 5.94013s el
│ │ │ │ +00009840: 7c20 2d2d 2034 2e34 3734 3431 7320 656c  | -- 4.47441s 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  ----------------
│ │ │ │ @@ -4559,16 +4559,16 @@
│ │ │ │  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 3733 3532 3373 2065  |.| -- .73523s e
│ │ │ │ -00011d60: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │ +00011d50: 7c0a 7c20 2d2d 202e 3533 3839 3437 7320  |.| -- .538947s 
│ │ │ │ +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                  
│ │ │ │  00011dd0: 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 2e35 3730 3432 7320  |.| -- 1.57042s 
│ │ │ │ +00011f30: 7c0a 7c20 2d2d 2031 2e32 3130 3736 7320  |.| -- 1.21076s 
│ │ │ │  00011f40: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │  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 202e 3830 3836 3135 7320  |.| -- .808615s 
│ │ │ │ +00012020: 7c0a 7c20 2d2d 202e 3935 3530 3837 7320  |.| -- .955087s 
│ │ │ │  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 2e34 3936       |.| -- .496
│ │ │ │ -00013ad0: 3137 3373 2065 6c61 7073 6564 2020 2020  173s elapsed    
│ │ │ │ +00013ac0: 2020 2020 207c 0a7c 202d 2d20 2e33 3838       |.| -- .388
│ │ │ │ +00013ad0: 3134 3673 2065 6c61 7073 6564 2020 2020  146s 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 2e38 3330       |.| -- .830
│ │ │ │ -00013bc0: 3437 3973 2065 6c61 7073 6564 2020 2020  479s elapsed    
│ │ │ │ +00013bb0: 2020 2020 207c 0a7c 202d 2d20 2e37 3132       |.| -- .712
│ │ │ │ +00013bc0: 3832 3873 2065 6c61 7073 6564 2020 2020  828s 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 3637 3232 3673 2065 6c61   -- .267226s ela
│ │ │ │ +00015460: 202d 2d20 2e32 3139 3535 3373 2065 6c61   -- .219553s 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 3033 3838 3373 2065 6c61   -- .203883s ela
│ │ │ │ +00015550: 202d 2d20 2e31 3831 3532 3873 2065 6c61   -- .181528s 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 3034 3330 3231 3573 2028  ed 0.00430215s (
│ │ │ │ -000009a0: 6370 7529 3b20 302e 3030 3432 3938 3432  cpu); 0.00429842
│ │ │ │ +00000990: 6564 2030 2e30 3036 3531 3337 3173 2028  ed 0.00651371s (
│ │ │ │ +000009a0: 6370 7529 3b20 302e 3030 3635 3132 3237  cpu); 0.00651227
│ │ │ │  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                  
│ │ │ │ @@ -322,16 +322,16 @@
│ │ │ │  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 3337 3233 3173 2028 6370  ed 0.137231s (cp
│ │ │ │ -00001490: 7529 3b20 302e 3036 3939 3637 3973 2028  u); 0.0699679s (
│ │ │ │ +00001480: 6564 2030 2e31 3837 3339 3573 2028 6370  ed 0.187395s (cp
│ │ │ │ +00001490: 7529 3b20 302e 3039 3338 3835 3673 2028  u); 0.0938856s (
│ │ │ │  000014a0: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 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                  
│ │ │ │ @@ -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 3239 3434 7320 2863 7075  ed 0.02944s (cpu
│ │ │ │ -000018a0: 293b 2030 2e30 3239 3434 3435 7320 2874  ); 0.0294445s (t
│ │ │ │ -000018b0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ -000018c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00001890: 6564 2030 2e30 3538 3635 3032 7320 2863  ed 0.0586502s (c
│ │ │ │ +000018a0: 7075 293b 2030 2e30 3538 3634 3939 7320  pu); 0.0586499s 
│ │ │ │ +000018b0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +000018c0: 2920 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                  
│ │ │ │ @@ -412,16 +412,16 @@
│ │ │ │  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 2e36 3837 3536 7320 2863 7075  ed 0.68756s (cpu
│ │ │ │ -00001a30: 293b 2030 2e34 3837 3538 3673 2028 7468  ); 0.487586s (th
│ │ │ │ +00001a20: 6564 2030 2e37 3434 3331 3173 2028 6370  ed 0.744311s (cp
│ │ │ │ +00001a30: 7529 3b20 302e 3535 3835 3373 2028 7468  u); 0.55853s (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                  
│ │ │ │ @@ -447,18 +447,18 @@
│ │ │ │  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 2e30 3632 3332 3037 7320 2863  ed 0.0623207s (c
│ │ │ │ -00001c60: 7075 293b 2030 2e30 3632 3236 3533 7320  pu); 0.0622653s 
│ │ │ │ -00001c70: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -00001c80: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00001c50: 6564 2030 2e31 3033 3930 3373 2028 6370  ed 0.103903s (cp
│ │ │ │ +00001c60: 7529 3b20 302e 3130 3337 3034 7320 2874  u); 0.103704s (t
│ │ │ │ +00001c70: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 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 = {
│ │ │ │  00001cf0: 357d 2020 2020 2020 2020 2020 2020 2020  5}              
│ │ │ │ @@ -478,20 +478,20 @@
│ │ │ │  00001dd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  00001de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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 692c 2020 2020 2020 2020 2020 2020  phi,            
│ │ │ │ +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 3134 3836 7320 2863  ed 0.0021486s (c
│ │ │ │ -00001e90: 7075 293b 2020 2020 2020 2020 2020 2020  pu);            
│ │ │ │ +00001e80: 6564 2030 2e30 3032 3934 3032 3773 2028  ed 0.00294027s (
│ │ │ │ +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                  
│ │ │ │  00001f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -548,130 +548,130 @@
│ │ │ │  00002230: 2020 2020 2020 207c 0a7c 6f37 203a 2052         |.|o7 : R
│ │ │ │  00002240: 696e 674d 6170 202d 2d2d 2d2d 2d5b 7420  ingMap ------[t 
│ │ │ │  00002250: 2e2e 7420 2020 2020 2020 2020 2020 2020  ..t             
│ │ │ │  00002260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002280: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002290: 2020 2020 2020 2033 3030 3030 3720 2030         300007  0
│ │ │ │ -000022a0: 2020 2036 2020 2020 2020 2020 2020 2020     6            
│ │ │ │ +000022a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000022b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000022c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000022d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000022e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000022f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002320: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │  00002330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00002370: 2d2d 2d2d 2d2d 2d7c 0a7c 446f 6d69 6e61  -------|.|Domina
│ │ │ │ -00002380: 6e74 3d3e 4a29 2020 2020 2020 2020 2020  nt=>J)          
│ │ │ │ +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 2030 2e30 3032         |.| 0.002
│ │ │ │ -000023d0: 3134 3933 3973 2028 7468 7265 6164 293b  14939s (thread);
│ │ │ │ -000023e0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +000023c0: 2020 2020 2020 207c 0a7c 293b 2030 2e30         |.|); 0.0
│ │ │ │ +000023d0: 3032 3934 3733 3573 2028 7468 7265 6164  0294735s (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                  
│ │ │ │  00002450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002460: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002490: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │ +00002490: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │  000024a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000024b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000024c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000024d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000024e0: 2020 2020 202d 2d2d 2d2d 2d5b 7820 2e2e       ------[x ..
│ │ │ │ -000024f0: 7820 5d20 2020 2020 2020 2020 2020 2020  x ]             
│ │ │ │ +000024e0: 2020 2020 2020 2d2d 2d2d 2d2d 5b78 202e        ------[x .
│ │ │ │ +000024f0: 2e78 205d 2020 2020 2020 2020 2020 2020  .x ]            
│ │ │ │  00002500: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002530: 2020 2020 2033 3030 3030 3720 2030 2020       300007  0  
│ │ │ │ -00002540: 2039 2020 2020 2020 2020 2020 2020 2020   9              
│ │ │ │ -00002550: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +00002530: 2020 2020 2020 3330 3030 3037 2020 3020        300007  0 
│ │ │ │ +00002540: 2020 3920 2020 2020 2020 2020 2020 2020    9             
│ │ │ │ +00002550: 2020 2020 2020 207c 0a7c 202d 2d2d 2d2d         |.| -----
│ │ │ │  00002560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000025a0: 2d2d 2d2d 2d2d 2d7c 0a7c 2878 2078 2020  -------|.|(x x  
│ │ │ │ -000025b0: 2d20 7820 7820 202b 2078 2078 202c 2078  - x x  + x x , x
│ │ │ │ -000025c0: 2078 2020 2d20 7820 7820 202b 2078 2078   x  - x x  + x x
│ │ │ │ -000025d0: 202c 2078 2078 2020 2d20 7820 7820 202b   , x x  - x x  +
│ │ │ │ -000025e0: 2078 2078 202c 2078 2078 2020 2d20 7820   x x , x x  - x 
│ │ │ │ -000025f0: 7820 202b 2078 207c 0a7c 2020 3620 3720  x  + x |.|  6 7 
│ │ │ │ -00002600: 2020 2035 2038 2020 2020 3420 3920 2020     5 8    4 9   
│ │ │ │ -00002610: 3320 3720 2020 2032 2038 2020 2020 3120  3 7    2 8    1 
│ │ │ │ -00002620: 3920 2020 3320 3520 2020 2032 2036 2020  9   3 5    2 6  
│ │ │ │ -00002630: 2020 3020 3920 2020 3320 3420 2020 2031    0 9   3 4    1
│ │ │ │ -00002640: 2036 2020 2020 307c 0a7c 2020 2020 2020   6    0|.|      
│ │ │ │ +000025a0: 2d2d 2d2d 2d2d 2d7c 0a7c 2028 7820 7820  -------|.| (x x 
│ │ │ │ +000025b0: 202d 2078 2078 2020 2b20 7820 7820 2c20   - x x  + x x , 
│ │ │ │ +000025c0: 7820 7820 202d 2078 2078 2020 2b20 7820  x x  - x x  + x 
│ │ │ │ +000025d0: 7820 2c20 7820 7820 202d 2078 2078 2020  x , x x  - x x  
│ │ │ │ +000025e0: 2b20 7820 7820 2c20 7820 7820 202d 2078  + x x , x x  - x
│ │ │ │ +000025f0: 2078 2020 2b20 787c 0a7c 2020 2036 2037   x  + x|.|   6 7
│ │ │ │ +00002600: 2020 2020 3520 3820 2020 2034 2039 2020      5 8    4 9  
│ │ │ │ +00002610: 2033 2037 2020 2020 3220 3820 2020 2031   3 7    2 8    1
│ │ │ │ +00002620: 2039 2020 2033 2035 2020 2020 3220 3620   9   3 5    2 6 
│ │ │ │ +00002630: 2020 2030 2039 2020 2033 2034 2020 2020     0 9   3 4    
│ │ │ │ +00002640: 3120 3620 2020 207c 0a7c 2020 2020 2020  1 6    |.|      
│ │ │ │  00002650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002690: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000026a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000026b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000026c0: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
│ │ │ │ +000026c0: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │  000026d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000026e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000026f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002710: 2020 2020 2020 2020 2020 202d 2d2d 2d2d             -----
│ │ │ │ -00002720: 2d5b 7820 2e2e 7820 5d20 2020 2020 2020  -[x ..x ]       
│ │ │ │ +00002710: 2020 2020 2020 2020 2020 2020 2d2d 2d2d              ----
│ │ │ │ +00002720: 2d2d 5b78 202e 2e78 205d 2020 2020 2020  --[x ..x ]      
│ │ │ │  00002730: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002760: 2020 2020 2020 2020 2020 2033 3030 3030             30000
│ │ │ │ -00002770: 3720 2030 2020 2039 2020 2020 2020 2020  7  0   9        
│ │ │ │ -00002780: 2020 2020 2020 207c 0a7c 5d20 3c2d 2d20         |.|] <-- 
│ │ │ │ -00002790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002760: 2020 2020 2020 2020 2020 2020 3330 3030              3000
│ │ │ │ +00002770: 3037 2020 3020 2020 3920 2020 2020 2020  07  0   9       
│ │ │ │ +00002780: 2020 2020 2020 207c 0a7c 205d 203c 2d2d         |.| ] <--
│ │ │ │ +00002790: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   ---------------
│ │ │ │  000027a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000027b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000027c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000027d0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │ -000027e0: 2878 2078 2020 2d20 7820 7820 202b 2078  (x x  - x x  + x
│ │ │ │ -000027f0: 2078 202c 2078 2078 2020 2d20 7820 7820   x , x x  - x x 
│ │ │ │ -00002800: 202b 2078 2078 202c 2078 2078 2020 2d20   + x x , x x  - 
│ │ │ │ -00002810: 7820 7820 202b 2078 2078 202c 2078 2078  x x  + x x , x x
│ │ │ │ -00002820: 2020 2d20 7820 787c 0a7c 2020 2020 2020    - x x|.|      
│ │ │ │ -00002830: 2020 3620 3720 2020 2035 2038 2020 2020    6 7    5 8    
│ │ │ │ -00002840: 3420 3920 2020 3320 3720 2020 2032 2038  4 9   3 7    2 8
│ │ │ │ -00002850: 2020 2020 3120 3920 2020 3320 3520 2020      1 9   3 5   
│ │ │ │ -00002860: 2032 2036 2020 2020 3020 3920 2020 3320   2 6    0 9   3 
│ │ │ │ -00002870: 3420 2020 2031 207c 0a7c 2d2d 2d2d 2d2d  4    1 |.|------
│ │ │ │ +000027d0: 2d2d 2d2d 2d2d 2d7c 0a7c 3620 2020 2020  -------|.|6     
│ │ │ │ +000027e0: 2028 7820 7820 202d 2078 2078 2020 2b20   (x x  - x x  + 
│ │ │ │ +000027f0: 7820 7820 2c20 7820 7820 202d 2078 2078  x x , x x  - x x
│ │ │ │ +00002800: 2020 2b20 7820 7820 2c20 7820 7820 202d    + x x , x x  -
│ │ │ │ +00002810: 2078 2078 2020 2b20 7820 7820 2c20 7820   x x  + x x , x 
│ │ │ │ +00002820: 7820 202d 2078 207c 0a7c 2020 2020 2020  x  - x |.|      
│ │ │ │ +00002830: 2020 2036 2037 2020 2020 3520 3820 2020     6 7    5 8   
│ │ │ │ +00002840: 2034 2039 2020 2033 2037 2020 2020 3220   4 9   3 7    2 
│ │ │ │ +00002850: 3820 2020 2031 2039 2020 2033 2035 2020  8    1 9   3 5  
│ │ │ │ +00002860: 2020 3220 3620 2020 2030 2039 2020 2033    2 6    0 9   3
│ │ │ │ +00002870: 2034 2020 2020 317c 0a7c 2d2d 2d2d 2d2d   4    1|.|------
│ │ │ │  00002880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │  000028d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000028e0: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ -000028f0: 2020 2020 2020 2020 3220 2020 2032 2020          2    2  
│ │ │ │ -00002900: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +000028e0: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ +000028f0: 2020 2020 2020 2020 2032 2020 2020 3220           2    2 
│ │ │ │ +00002900: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  00002910: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │  00002920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00002930: 2d2c 207b 2d20 7420 202b 2032 7420 7420  -, {- t  + 2t t 
│ │ │ │ -00002940: 7420 202d 2074 2074 2020 2d20 7420 7420  t  - t t  - t t 
│ │ │ │ -00002950: 202b 2074 2074 2074 202c 202d 2074 2074   + t t t , - t t
│ │ │ │ -00002960: 2020 2b20 2020 207c 0a7c 7820 2c20 7820    +    |.|x , x 
│ │ │ │ -00002970: 7820 202d 2078 2078 2020 2b20 7820 7820  x  - x x  + x x 
│ │ │ │ -00002980: 2920 2020 2020 2032 2020 2020 2031 2032  )      2     1 2
│ │ │ │ -00002990: 2033 2020 2020 3020 3320 2020 2031 2034   3    0 3    1 4
│ │ │ │ -000029a0: 2020 2020 3020 3220 3420 2020 2020 3220      0 2 4     2 
│ │ │ │ -000029b0: 3320 2020 2020 207c 0a7c 2038 2020 2032  3      |.| 8   2
│ │ │ │ -000029c0: 2034 2020 2020 3120 3520 2020 2030 2037   4    1 5    0 7
│ │ │ │ -000029d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00002930: 2d2d 2c20 7b2d 2074 2020 2b20 3274 2074  --, {- t  + 2t t
│ │ │ │ +00002940: 2074 2020 2d20 7420 7420 202d 2074 2074   t  - t t  - t t
│ │ │ │ +00002950: 2020 2b20 7420 7420 7420 2c20 2d20 7420    + t t t , - t 
│ │ │ │ +00002960: 7420 202b 2020 207c 0a7c 2078 202c 2078  t  +   |.| x , x
│ │ │ │ +00002970: 2078 2020 2d20 7820 7820 202b 2078 2078   x  - x x  + x x
│ │ │ │ +00002980: 2029 2020 2020 2020 3220 2020 2020 3120   )      2     1 
│ │ │ │ +00002990: 3220 3320 2020 2030 2033 2020 2020 3120  2 3    0 3    1 
│ │ │ │ +000029a0: 3420 2020 2030 2032 2034 2020 2020 2032  4    0 2 4     2
│ │ │ │ +000029b0: 2033 2020 2020 207c 0a7c 3020 3820 2020   3     |.|0 8   
│ │ │ │ +000029c0: 3220 3420 2020 2031 2035 2020 2020 3020  2 4    1 5    0 
│ │ │ │ +000029d0: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │  000029e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000029f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002a00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -688,25 +688,25 @@
│ │ │ │  00002af0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b40: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │  00002b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00002b60: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020  -------         
│ │ │ │ +00002b60: 2d2d 2d2d 2d2d 2d2d 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 207c 0a7c 2020 2b20 7820         |.|  + x 
│ │ │ │ -00002ba0: 7820 2c20 7820 7820 202d 2078 2078 2020  x , x x  - x x  
│ │ │ │ -00002bb0: 2b20 7820 7820 2920 2020 2020 2020 2020  + x x )         
│ │ │ │ +00002b90: 2020 2020 2020 207c 0a7c 7820 202b 2078         |.|x  + x
│ │ │ │ +00002ba0: 2078 202c 2078 2078 2020 2d20 7820 7820   x , x x  - x x 
│ │ │ │ +00002bb0: 202b 2078 2078 2029 2020 2020 2020 2020   + x x )        
│ │ │ │  00002bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002be0: 2020 2020 2020 207c 0a7c 3620 2020 2030         |.|6    0
│ │ │ │ -00002bf0: 2038 2020 2032 2034 2020 2020 3120 3520   8   2 4    1 5 
│ │ │ │ -00002c00: 2020 2030 2037 2020 2020 2020 2020 2020     0 7          
│ │ │ │ +00002be0: 2020 2020 2020 207c 0a7c 2036 2020 2020         |.| 6    
│ │ │ │ +00002bf0: 3020 3820 2020 3220 3420 2020 2031 2035  0 8   2 4    1 5
│ │ │ │ +00002c00: 2020 2020 3020 3720 2020 2020 2020 2020      0 7         
│ │ │ │  00002c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002c30: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │  00002c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -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 3734 3235 3273 2028 6370  ed 0.474252s (cp
│ │ │ │ -00003470: 7529 3b20 302e 3339 3435 3334 7320 2874  u); 0.394534s (t
│ │ │ │ +00003460: 6564 2030 2e34 3437 3733 3973 2028 6370  ed 0.447739s (cp
│ │ │ │ +00003470: 7529 3b20 302e 3434 3737 3434 7320 2874  u); 0.447744s (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 3331 3630 3373 2028  ed 0.00931603s (
│ │ │ │ -00004640: 6370 7529 3b20 302e 3030 3933 3138 3535  cpu); 0.00931855
│ │ │ │ -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 3131 3332 3634 7320 2863  ed 0.0113264s (c
│ │ │ │ +00004640: 7075 293b 2030 2e30 3131 3333 3138 7320  pu); 0.0113318s 
│ │ │ │ +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             
│ │ │ │ @@ -1142,16 +1142,16 @@
│ │ │ │  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 2e34 3538 3439 3373 2028 6370  ed 0.458493s (cp
│ │ │ │ -000047d0: 7529 3b20 302e 3239 3432 3539 7320 2874  u); 0.294259s (t
│ │ │ │ +000047c0: 6564 2030 2e35 3633 3539 3573 2028 6370  ed 0.563595s (cp
│ │ │ │ +000047d0: 7529 3b20 302e 3239 3433 3038 7320 2874  u); 0.294308s (t
│ │ │ │  000047e0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 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                  
│ │ │ │ @@ -1167,16 +1167,16 @@
│ │ │ │  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 2e32 3930 3034 7320 2863 7075  ed 5.29004s (cpu
│ │ │ │ -00004960: 293b 2034 2e36 3339 3638 7320 2874 6872  ); 4.63968s (thr
│ │ │ │ +00004950: 6564 2036 2e34 3330 3134 7320 2863 7075  ed 6.43014s (cpu
│ │ │ │ +00004960: 293b 2035 2e39 3732 3935 7320 2874 6872  ); 5.97295s (thr
│ │ │ │  00004970: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 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                  
│ │ │ │ @@ -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: 3230 3233 3873 2028 6370 7529 3b20 302e  20238s (cpu); 0.
│ │ │ │ -00004c50: 3030 3232 3033 3133 7320 2874 6872 6561  00220313s (threa
│ │ │ │ +00004c40: 3530 3233 3873 2028 6370 7529 3b20 302e  50238s (cpu); 0.
│ │ │ │ +00004c50: 3030 3235 3038 3433 7320 2874 6872 6561  00250843s (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 2e31 3537  .| -- used 0.157
│ │ │ │ -00005dc0: 3331 7320 2863 7075 293b 2030 2e30 3834  31s (cpu); 0.084
│ │ │ │ -00005dd0: 3938 3639 7320 2874 6872 6561 6429 3b20  9869s (thread); 
│ │ │ │ +00005db0: 0a7c 202d 2d20 7573 6564 2030 2e31 3938  .| -- used 0.198
│ │ │ │ +00005dc0: 3238 3273 2028 6370 7529 3b20 302e 3130  282s (cpu); 0.10
│ │ │ │ +00005dd0: 3031 3638 7320 2874 6872 6561 6429 3b20  0168s (thread); 
│ │ │ │  00005de0: 3073 2028 6763 2920 2020 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 2e35 3132  .| -- used 0.512
│ │ │ │ -00008700: 3035 3973 2028 6370 7529 3b20 302e 3432  059s (cpu); 0.42
│ │ │ │ -00008710: 3638 3638 7320 2874 6872 6561 6429 3b20  6868s (thread); 
│ │ │ │ +000086f0: 0a7c 202d 2d20 7573 6564 2030 2e34 3933  .| -- used 0.493
│ │ │ │ +00008700: 3438 3973 2028 6370 7529 3b20 302e 3439  489s (cpu); 0.49
│ │ │ │ +00008710: 3331 3634 7320 2874 6872 6561 6429 3b20  3164s (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 2e33 3437  .| -- used 0.347
│ │ │ │ -0000a9b0: 3734 3973 2028 6370 7529 3b20 302e 3237  749s (cpu); 0.27
│ │ │ │ -0000a9c0: 3431 3431 7320 2874 6872 6561 6429 3b20  4141s (thread); 
│ │ │ │ +0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e34 3732  .| -- used 0.472
│ │ │ │ +0000a9b0: 3134 3473 2028 6370 7529 3b20 302e 3336  144s (cpu); 0.36
│ │ │ │ +0000a9c0: 3636 3038 7320 2874 6872 6561 6429 3b20  6608s (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 3138  .| -- used 0.018
│ │ │ │ -0000abe0: 3031 3133 7320 2863 7075 293b 2030 2e30  0113s (cpu); 0.0
│ │ │ │ -0000abf0: 3137 3639 3933 7320 2874 6872 6561 6429  176993s (thread)
│ │ │ │ +0000abd0: 0a7c 202d 2d20 7573 6564 2030 2e30 3837  .| -- used 0.087
│ │ │ │ +0000abe0: 3532 3732 7320 2863 7075 293b 2030 2e30  5272s (cpu); 0.0
│ │ │ │ +0000abf0: 3235 3133 3436 7320 2874 6872 6561 6429  251346s (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                 |
│ │ │ │ @@ -2778,17 +2778,17 @@
│ │ │ │  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 3033  .| -- used 0.003
│ │ │ │ -0000ae10: 3230 3731 3873 2028 6370 7529 3b20 302e  20718s (cpu); 0.
│ │ │ │ -0000ae20: 3030 3332 3037 3434 7320 2874 6872 6561  00320744s (threa
│ │ │ │ +0000ae00: 0a7c 202d 2d20 7573 6564 2030 2e30 3034  .| -- used 0.004
│ │ │ │ +0000ae10: 3231 3037 3573 2028 6370 7529 3b20 302e  21075s (cpu); 0.
│ │ │ │ +0000ae20: 3030 3432 3231 3139 7320 2874 6872 6561  00422119s (threa
│ │ │ │  0000ae30: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 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                 |
│ │ │ │ @@ -2843,18 +2843,18 @@
│ │ │ │  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: 3937 3939 3773 2028 6370 7529 3b20 302e  97997s (cpu); 0.
│ │ │ │ -0000b230: 3030 3939 3830 3773 2028 7468 7265 6164  0099807s (thread
│ │ │ │ -0000b240: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +0000b210: 0a7c 202d 2d20 7573 6564 2030 2e30 3131  .| -- used 0.011
│ │ │ │ +0000b220: 3933 3673 2028 6370 7529 3b20 302e 3031  936s (cpu); 0.01
│ │ │ │ +0000b230: 3139 3437 3373 2028 7468 7265 6164 293b  19473s (thread);
│ │ │ │ +0000b240: 2030 7320 2867 6329 2020 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                 |
│ │ │ │  0000b2b0: 0a7c 6f31 3820 3d20 7261 7469 6f6e 616c  .|o18 = rational
│ │ │ │ @@ -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: 3530 3537 3173 2028 6370 7529 3b20 302e  50571s (cpu); 0.
│ │ │ │ -0000b730: 3030 3935 3036 3539 7320 2874 6872 6561  00950659s (threa
│ │ │ │ -0000b740: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +0000b710: 0a7c 202d 2d20 7573 6564 2030 2e30 3132  .| -- used 0.012
│ │ │ │ +0000b720: 3036 3536 7320 2863 7075 293b 2030 2e30  0656s (cpu); 0.0
│ │ │ │ +0000b730: 3132 3037 3834 7320 2874 6872 6561 6429  120784s (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
│ │ │ │ @@ -2958,17 +2958,17 @@
│ │ │ │  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 3333  .| -- used 1.333
│ │ │ │ -0000b950: 3237 7320 2863 7075 293b 2030 2e39 3535  27s (cpu); 0.955
│ │ │ │ -0000b960: 3132 7320 2874 6872 6561 6429 3b20 3073  12s (thread); 0s
│ │ │ │ +0000b940: 0a7c 202d 2d20 7573 6564 2031 2e31 3938  .| -- used 1.198
│ │ │ │ +0000b950: 3131 7320 2863 7075 293b 2031 2e30 3239  11s (cpu); 1.029
│ │ │ │ +0000b960: 3036 7320 2874 6872 6561 6429 3b20 3073  06s (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                 |
│ │ │ │ @@ -2993,18 +2993,18 @@
│ │ │ │  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 3235  .| -- used 1.625
│ │ │ │ -0000bb80: 652d 3035 7320 2863 7075 293b 2031 2e35  e-05s (cpu); 1.5
│ │ │ │ -0000bb90: 3933 652d 3035 7320 2874 6872 6561 6429  93e-05s (thread)
│ │ │ │ -0000bba0: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +0000bb70: 0a7c 202d 2d20 7573 6564 2031 2e37 3638  .| -- used 1.768
│ │ │ │ +0000bb80: 3465 2d30 3573 2028 6370 7529 3b20 312e  4e-05s (cpu); 1.
│ │ │ │ +0000bb90: 3732 3036 652d 3035 7320 2874 6872 6561  7206e-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                 |
│ │ │ │  0000bc10: 0a7c 6f32 3220 3d20 3120 2020 2020 2020  .|o22 = 1       
│ │ │ │ @@ -3019,16 +3019,16 @@
│ │ │ │  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: 3631 3436 3573 2028 6370 7529 3b20 302e  61465s (cpu); 0.
│ │ │ │ -0000bd20: 3030 3136 3135 3535 7320 2874 6872 6561  00161555s (threa
│ │ │ │ +0000bd10: 3732 3234 3673 2028 6370 7529 3b20 302e  72246s (cpu); 0.
│ │ │ │ +0000bd20: 3030 3137 3237 3939 7320 2874 6872 6561  00172799s (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                 |
│ │ │ │ @@ -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 3130 3031 3973 2028 6370 7529 3b20  0410019s (cpu); 
│ │ │ │ -000124b0: 302e 3030 3034 3036 3239 3273 2028 7468  0.000406292s (th
│ │ │ │ +000124a0: 3034 3933 3339 3773 2028 6370 7529 3b20  0493397s (cpu); 
│ │ │ │ +000124b0: 302e 3030 3034 3835 3538 3573 2028 7468  0.000485585s (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 2e32 3938 3835 3373  - used 0.298853s
│ │ │ │ -00012910: 2028 6370 7529 3b20 302e 3138 3532 3773   (cpu); 0.18527s
│ │ │ │ -00012920: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -00012930: 6329 2020 2020 2020 207c 0a7c 2020 2020  c)       |.|    
│ │ │ │ +00012900: 2d20 7573 6564 2030 2e34 3134 3130 3773  - used 0.414107s
│ │ │ │ +00012910: 2028 6370 7529 3b20 302e 3234 3236 3331   (cpu); 0.242631
│ │ │ │ +00012920: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +00012930: 6763 2920 2020 2020 207c 0a7c 2020 2020  gc)      |.|    
│ │ │ │  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                  
│ │ │ │ @@ -4765,17 +4765,17 @@
│ │ │ │  000129c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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: 3034 3032 3673 2028 6370 7529 3b20 302e  04026s (cpu); 0.
│ │ │ │ -00012a50: 3336 3634 3939 7320 2874 6872 6561 6429  366499s (thread)
│ │ │ │ +00012a30: 207c 0a7c 202d 2d20 7573 6564 2030 2e36   |.| -- used 0.6
│ │ │ │ +00012a40: 3034 3931 3573 2028 6370 7529 3b20 302e  04915s (cpu); 0.
│ │ │ │ +00012a50: 3530 3438 3933 7320 2874 6872 6561 6429  504893s (thread)
│ │ │ │  00012a60: 3b20 3073 2028 6763 2920 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 --         
│ │ │ │ @@ -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 3134 3931 3835 7320 2863 7075  d 0.149185s (cpu
│ │ │ │ -000144c0: 293b 2030 2e30 3736 3932 3939 7320 2874  ); 0.0769299s (t
│ │ │ │ -000144d0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +000144b0: 6420 302e 3138 3236 3173 2028 6370 7529  d 0.18261s (cpu)
│ │ │ │ +000144c0: 3b20 302e 3038 3732 3631 3873 2028 7468  ; 0.0872618s (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
│ │ │ │ @@ -5261,48244 +5261,48218 @@
│ │ │ │  000148c0: 7468 6520 6162 7374 7261 6374 206d 6170  the abstract map
│ │ │ │  000148d0: 2054 2063 616e 2062 6520 6f62 7461 696e   T can be obtain
│ │ │ │  000148e0: 6564 2062 7920 7468 650a 666f 6c6c 6f77  ed by the.follow
│ │ │ │  000148f0: 696e 6720 636f 6d6d 616e 643a 0a0a 2b2d  ing command:..+-
│ │ │ │  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 342e 3037 3736 3473 2028 6370  sed 4.07764s (cp
│ │ │ │ -00014980: 7529 3b20 322e 3132 3032 3273 2028 7468  u); 2.12022s (th
│ │ │ │ -00014990: 7265 6164 293b 2030 7320 2867 6329 7c0a  read); 0s (gc)|.
│ │ │ │ -000149a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00014930: 2d2b 0a7c 6931 3520 3a20 7469 6d65 2070  -+.|i15 : time p
│ │ │ │ +00014940: 726f 6a65 6374 6976 6544 6567 7265 6573  rojectiveDegrees
│ │ │ │ +00014950: 2854 2c32 2920 2020 2020 2020 2020 2020  (T,2)           
│ │ │ │ +00014960: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ +00014970: 6420 352e 3334 3333 7320 2863 7075 293b  d 5.3433s (cpu);
│ │ │ │ +00014980: 2032 2e34 3636 3839 7320 2874 6872 6561   2.46689s (threa
│ │ │ │ +00014990: 6429 3b20 3073 2028 6763 297c 0a7c 2020  d); 0s (gc)|.|  
│ │ │ │ +000149a0: 2020 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  
│ │ │ │ +000149d0: 7c0a 7c6f 3135 203d 2033 2020 2020 2020  |.|o15 = 3      
│ │ │ │  000149e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000149f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014a00: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00014a00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00014a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014a40: 2b0a 0a57 6520 7665 7269 6679 2074 6861  +..We verify tha
│ │ │ │ -00014a50: 7420 7468 6520 636f 6d70 6f73 6974 696f  t the compositio
│ │ │ │ -00014a60: 6e20 6f66 2054 2077 6974 6820 6974 7365  n of T with itse
│ │ │ │ -00014a70: 6c66 2069 7320 6465 6669 6e65 6420 6279  lf is defined by
│ │ │ │ -00014a80: 206c 696e 6561 7220 666f 726d 733a 0a0a   linear forms:..
│ │ │ │ -00014a90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00014a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6520  ----------+..We 
│ │ │ │ +00014a40: 7665 7269 6679 2074 6861 7420 7468 6520  verify that the 
│ │ │ │ +00014a50: 636f 6d70 6f73 6974 696f 6e20 6f66 2054  composition of T
│ │ │ │ +00014a60: 2077 6974 6820 6974 7365 6c66 2069 7320   with itself is 
│ │ │ │ +00014a70: 6465 6669 6e65 6420 6279 206c 696e 6561  defined by linea
│ │ │ │ +00014a80: 7220 666f 726d 733a 0a0a 2b2d 2d2d 2d2d  r forms:..+-----
│ │ │ │ +00014a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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             
│ │ │ │ +00014ac0: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 7469  -----+.|i16 : ti
│ │ │ │ +00014ad0: 6d65 2054 3220 3d20 5420 2a20 5420 2020  me T2 = T * T   
│ │ │ │ +00014ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014b00: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -00014b10: 7365 6420 322e 3835 3634 652d 3035 7320  sed 2.8564e-05s 
│ │ │ │ -00014b20: 2863 7075 293b 2032 2e38 3237 3365 2d30  (cpu); 2.8273e-0
│ │ │ │ -00014b30: 3573 2028 7468 7265 6164 293b 2030 7320  5s (thread); 0s 
│ │ │ │ -00014b40: 2867 6329 207c 0a7c 2020 2020 2020 2020  (gc) |.|        
│ │ │ │ +00014b00: 2020 7c0a 7c20 2d2d 2075 7365 6420 322e    |.| -- used 2.
│ │ │ │ +00014b10: 3835 3436 652d 3035 7320 2863 7075 293b  8546e-05s (cpu);
│ │ │ │ +00014b20: 2032 2e36 3731 652d 3035 7320 2874 6872   2.671e-05s (thr
│ │ │ │ +00014b30: 6561 6429 3b20 3073 2028 6763 2920 207c  ead); 0s (gc)  |
│ │ │ │ +00014b40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 --   
│ │ │ │ +00014b70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00014b80: 3136 203d 202d 2d20 7261 7469 6f6e 616c  16 = -- rational
│ │ │ │ +00014b90: 206d 6170 202d 2d20 2020 2020 2020 2020   map --         
│ │ │ │  00014ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014bb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014bc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014bd0: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │ +00014bb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00014bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014bd0: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │  00014be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014bf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00014c00: 2020 2020 2073 6f75 7263 653a 2050 726f       source: Pro
│ │ │ │ -00014c10: 6a28 2d2d 2d2d 2d5b 7820 2c20 7820 2c20  j(-----[x , x , 
│ │ │ │ -00014c20: 7820 2c20 7820 5d29 2020 2020 2020 2020  x , x ])        
│ │ │ │ -00014c30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014c40: 2020 2020 2020 2020 2020 2020 2020 2036                 6
│ │ │ │ -00014c50: 3535 3231 2020 3020 2020 3120 2020 3220  5521  0   1   2 
│ │ │ │ -00014c60: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -00014c70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00014c80: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │ +00014bf0: 2020 2020 2020 7c0a 7c20 2020 2020 2073        |.|      s
│ │ │ │ +00014c00: 6f75 7263 653a 2050 726f 6a28 2d2d 2d2d  ource: Proj(----
│ │ │ │ +00014c10: 2d5b 7820 2c20 7820 2c20 7820 2c20 7820  -[x , x , x , x 
│ │ │ │ +00014c20: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ +00014c30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00014c40: 2020 2020 2020 2020 2036 3535 3231 2020           65521  
│ │ │ │ +00014c50: 3020 2020 3120 2020 3220 2020 3320 2020  0   1   2   3   
│ │ │ │ +00014c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014c70: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00014c80: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │  00014c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014cb0: 2020 207c 0a7c 2020 2020 2020 7461 7267     |.|      targ
│ │ │ │ -00014cc0: 6574 3a20 5072 6f6a 282d 2d2d 2d2d 5b78  et: Proj(-----[x
│ │ │ │ -00014cd0: 202c 2078 202c 2078 202c 2078 205d 2920   , x , x , x ]) 
│ │ │ │ -00014ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014cf0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00014d00: 2020 2020 2020 3635 3532 3120 2030 2020        65521  0  
│ │ │ │ -00014d10: 2031 2020 2032 2020 2033 2020 2020 2020   1   2   3      
│ │ │ │ -00014d20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00014d30: 2020 2020 2020 6465 6669 6e69 6e67 2066        defining f
│ │ │ │ -00014d40: 6f72 6d73 3a20 6769 7665 6e20 6279 2061  orms: given by a
│ │ │ │ -00014d50: 2066 756e 6374 696f 6e20 2020 2020 2020   function       
│ │ │ │ -00014d60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00014ca0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00014cb0: 2020 2020 2020 7461 7267 6574 3a20 5072        target: Pr
│ │ │ │ +00014cc0: 6f6a 282d 2d2d 2d2d 5b78 202c 2078 202c  oj(-----[x , x ,
│ │ │ │ +00014cd0: 2078 202c 2078 205d 2920 2020 2020 2020   x , x ])       
│ │ │ │ +00014ce0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00014cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014d00: 3635 3532 3120 2030 2020 2031 2020 2032  65521  0   1   2
│ │ │ │ +00014d10: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +00014d20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00014d30: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20  defining forms: 
│ │ │ │ +00014d40: 6769 7665 6e20 6279 2061 2066 756e 6374  given by a funct
│ │ │ │ +00014d50: 696f 6e20 2020 2020 2020 2020 2020 2020  ion             
│ │ │ │ +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 207c 0a7c 6f31 3620 3a20         |.|o16 : 
│ │ │ │ -00014db0: 4162 7374 7261 6374 5261 7469 6f6e 616c  AbstractRational
│ │ │ │ -00014dc0: 4d61 7020 2872 6174 696f 6e61 6c20 6d61  Map (rational ma
│ │ │ │ -00014dd0: 7020 6672 6f6d 2050 505e 3320 746f 2050  p from PP^3 to P
│ │ │ │ -00014de0: 505e 3329 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d  P^3)|.+---------
│ │ │ │ +00014da0: 207c 0a7c 6f31 3620 3a20 4162 7374 7261   |.|o16 : Abstra
│ │ │ │ +00014db0: 6374 5261 7469 6f6e 616c 4d61 7020 2872  ctRationalMap (r
│ │ │ │ +00014dc0: 6174 696f 6e61 6c20 6d61 7020 6672 6f6d  ational map from
│ │ │ │ +00014dd0: 2050 505e 3320 746f 2050 505e 3329 7c0a   PP^3 to PP^3)|.
│ │ │ │ +00014de0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  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 362e 3635 3930  | -- used 6.6590
│ │ │ │ -00014e70: 3173 2028 6370 7529 3b20 332e 3435 3639  1s (cpu); 3.4569
│ │ │ │ -00014e80: 3973 2028 7468 7265 6164 293b 2030 7320  9s (thread); 0s 
│ │ │ │ -00014e90: 2867 6329 2020 2020 2020 207c 0a7c 2020  (gc)       |.|  
│ │ │ │ +00014e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00014e20: 3720 3a20 7469 6d65 2070 726f 6a65 6374  7 : time project
│ │ │ │ +00014e30: 6976 6544 6567 7265 6573 2854 322c 3229  iveDegrees(T2,2)
│ │ │ │ +00014e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014e50: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ +00014e60: 7365 6420 372e 3939 3032 3973 2028 6370  sed 7.99029s (cp
│ │ │ │ +00014e70: 7529 3b20 332e 3932 3134 3473 2028 7468  u); 3.92144s (th
│ │ │ │ +00014e80: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +00014e90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  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              
│ │ │ │ +00014ed0: 2020 7c0a 7c6f 3137 203d 2031 2020 2020    |.|o17 = 1    
│ │ │ │ +00014ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014f10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00014f00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00014f10: 0a2b 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 2d2d  ----------------
│ │ │ │ -00014f50: 2d2d 2b0a 0a57 6520 7665 7269 6679 2074  --+..We verify t
│ │ │ │ -00014f60: 6861 7420 7468 6520 636f 6d70 6f73 6974  hat the composit
│ │ │ │ -00014f70: 696f 6e20 6f66 2054 2077 6974 6820 6974  ion of T with it
│ │ │ │ -00014f80: 7365 6c66 206c 6561 7665 7320 6120 7261  self leaves a ra
│ │ │ │ -00014f90: 6e64 6f6d 2070 6f69 6e74 2066 6978 6564  ndom point fixed
│ │ │ │ -00014fa0: 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  :..+------------
│ │ │ │ +00014f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57  ------------+..W
│ │ │ │ +00014f50: 6520 7665 7269 6679 2074 6861 7420 7468  e verify that th
│ │ │ │ +00014f60: 6520 636f 6d70 6f73 6974 696f 6e20 6f66  e composition of
│ │ │ │ +00014f70: 2054 2077 6974 6820 6974 7365 6c66 206c   T with itself l
│ │ │ │ +00014f80: 6561 7665 7320 6120 7261 6e64 6f6d 2070  eaves a random p
│ │ │ │ +00014f90: 6f69 6e74 2066 6978 6564 3a0a 0a2b 2d2d  oint fixed:..+--
│ │ │ │ +00014fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00014fd0: 7c69 3138 203a 2070 203d 2061 7070 6c79  |i18 : p = apply
│ │ │ │ -00014fe0: 2833 2c69 2d3e 7261 6e64 6f6d 285a 5a2f  (3,i->random(ZZ/
│ │ │ │ -00014ff0: 3635 3532 3129 297c 7b31 7d7c 0a7c 2020  65521))|{1}|.|  
│ │ │ │ +00014fc0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a  --------+.|i18 :
│ │ │ │ +00014fd0: 2070 203d 2061 7070 6c79 2833 2c69 2d3e   p = apply(3,i->
│ │ │ │ +00014fe0: 7261 6e64 6f6d 285a 5a2f 3635 3532 3129  random(ZZ/65521)
│ │ │ │ +00014ff0: 297c 7b31 7d7c 0a7c 2020 2020 2020 2020  )|{1}|.|        
│ │ │ │  00015000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015020: 2020 2020 2020 2020 7c0a 7c6f 3138 203d          |.|o18 =
│ │ │ │ -00015030: 207b 2d36 3634 382c 202d 3233 3339 362c   {-6648, -23396,
│ │ │ │ -00015040: 202d 3132 3331 312c 2031 7d20 2020 2020   -12311, 1}     
│ │ │ │ -00015050: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00015020: 2020 7c0a 7c6f 3138 203d 207b 2d36 3634    |.|o18 = {-664
│ │ │ │ +00015030: 382c 202d 3233 3339 362c 202d 3132 3331  8, -23396, -1231
│ │ │ │ +00015040: 312c 2031 7d20 2020 2020 2020 2020 207c  1, 1}          |
│ │ │ │ +00015050: 0a7c 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 7c0a 7c6f 3138 203a 204c 6973 7420    |.|o18 : List 
│ │ │ │ +00015070: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00015080: 3138 203a 204c 6973 7420 2020 2020 2020  18 : List       
│ │ │ │  00015090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000150a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000150b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000150a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000150b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000150c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000150d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000150e0: 3139 203a 2071 203d 2054 2070 2020 2020  19 : q = T p    
│ │ │ │ +000150d0: 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a 2071  ------+.|i19 : q
│ │ │ │ +000150e0: 203d 2054 2070 2020 2020 2020 2020 2020   = T p          
│ │ │ │  000150f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015100: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015100: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00015110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015130: 2020 2020 2020 7c0a 7c6f 3139 203d 207b        |.|o19 = {
│ │ │ │ -00015140: 2d39 3633 342c 2032 3037 3034 2c20 2d32  -9634, 20704, -2
│ │ │ │ -00015150: 3530 3134 2c20 317d 2020 2020 2020 2020  5014, 1}        
│ │ │ │ -00015160: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00015130: 7c0a 7c6f 3139 203d 207b 2d39 3633 342c  |.|o19 = {-9634,
│ │ │ │ +00015140: 2032 3037 3034 2c20 2d32 3530 3134 2c20   20704, -25014, 
│ │ │ │ +00015150: 317d 2020 2020 2020 2020 2020 207c 0a7c  1}           |.|
│ │ │ │ +00015160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015190: 7c0a 7c6f 3139 203a 204c 6973 7420 2020  |.|o19 : List   
│ │ │ │ +00015180: 2020 2020 2020 2020 2020 7c0a 7c6f 3139            |.|o19
│ │ │ │ +00015190: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │  000151a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000151b0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000151b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  000151c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000151d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000151e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230  ----------+.|i20
│ │ │ │ -000151f0: 203a 2054 2071 2020 2020 2020 2020 2020   : T q          
│ │ │ │ +000151e0: 2d2d 2d2d 2b0a 7c69 3230 203a 2054 2071  ----+.|i20 : T q
│ │ │ │ +000151f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015210: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00015210: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00015220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015240: 2020 2020 7c0a 7c6f 3230 203d 207b 2d36      |.|o20 = {-6
│ │ │ │ -00015250: 3634 382c 202d 3233 3339 362c 202d 3132  648, -23396, -12
│ │ │ │ -00015260: 3331 312c 2031 7d20 2020 2020 2020 2020  311, 1}         
│ │ │ │ -00015270: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00015230: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015240: 7c6f 3230 203d 207b 2d36 3634 382c 202d  |o20 = {-6648, -
│ │ │ │ +00015250: 3233 3339 362c 202d 3132 3331 312c 2031  23396, -12311, 1
│ │ │ │ +00015260: 7d20 2020 2020 2020 2020 207c 0a7c 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 7c0a                |.
│ │ │ │ -000152a0: 7c6f 3230 203a 204c 6973 7420 2020 2020  |o20 : List     
│ │ │ │ +00015290: 2020 2020 2020 2020 7c0a 7c6f 3230 203a          |.|o20 :
│ │ │ │ +000152a0: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │  000152b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000152c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000152c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  000152d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000152e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000152f0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6520 6e6f  --------+..We no
│ │ │ │ -00015300: 7720 636f 6d70 7574 6520 7468 6520 636f  w compute the co
│ │ │ │ -00015310: 6e63 7265 7465 2072 6174 696f 6e61 6c20  ncrete rational 
│ │ │ │ -00015320: 6d61 7020 636f 7272 6573 706f 6e64 696e  map correspondin
│ │ │ │ -00015330: 6720 746f 2054 3a0a 0a2b 2d2d 2d2d 2d2d  g to T:..+------
│ │ │ │ +000152f0: 2d2d 2b0a 0a57 6520 6e6f 7720 636f 6d70  --+..We now comp
│ │ │ │ +00015300: 7574 6520 7468 6520 636f 6e63 7265 7465  ute the concrete
│ │ │ │ +00015310: 2072 6174 696f 6e61 6c20 6d61 7020 636f   rational map co
│ │ │ │ +00015320: 7272 6573 706f 6e64 696e 6720 746f 2054  rresponding to T
│ │ │ │ +00015330: 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  :..+------------
│ │ │ │  00015340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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      
│ │ │ │ +00015370: 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 7469  -----+.|i21 : ti
│ │ │ │ +00015380: 6d65 2066 203d 2072 6174 696f 6e61 6c4d  me f = rationalM
│ │ │ │ +00015390: 6170 2054 2020 2020 2020 2020 2020 2020  ap 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 2035 2e33 3833  .| -- used 5.383
│ │ │ │ -000153d0: 3637 7320 2863 7075 293b 2032 2e39 3231  67s (cpu); 2.921
│ │ │ │ -000153e0: 3139 7320 2874 6872 6561 6429 3b20 3073  19s (thread); 0s
│ │ │ │ -000153f0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ -00015400: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000153b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ +000153c0: 7573 6564 2036 2e32 3039 3533 7320 2863  used 6.20953s (c
│ │ │ │ +000153d0: 7075 293b 2033 2e33 3130 3233 7320 2874  pu); 3.31023s (t
│ │ │ │ +000153e0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +000153f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00015400: 2020 2020 2020 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 
│ │ │ │ -00015460: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +00015440: 207c 0a7c 6f32 3120 3d20 2d2d 2072 6174   |.|o21 = -- rat
│ │ │ │ +00015450: 696f 6e61 6c20 6d61 7020 2d2d 2020 2020  ional map --    
│ │ │ │ +00015460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015480: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00015490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000154a0: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
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│ │ │ │ +00015490: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
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│ │ │ │ -000154c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000154d0: 0a7c 2020 2020 2020 736f 7572 6365 3a20  .|      source: 
│ │ │ │ -000154e0: 5072 6f6a 282d 2d2d 2d2d 5b78 202c 2078  Proj(-----[x , x
│ │ │ │ -000154f0: 202c 2078 202c 2078 205d 2920 2020 2020   , x , x ])     
│ │ │ │ -00015500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015510: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00015520: 2020 2020 2020 2020 2036 3535 3231 2020           65521  
│ │ │ │ -00015530: 3020 2020 3120 2020 3220 2020 3320 2020  0   1   2   3   
│ │ │ │ +000154c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000154d0: 2020 736f 7572 6365 3a20 5072 6f6a 282d    source: Proj(-
│ │ │ │ +000154e0: 2d2d 2d2d 5b78 202c 2078 202c 2078 202c  ----[x , x , x ,
│ │ │ │ +000154f0: 2078 205d 2920 2020 2020 2020 2020 2020   x ])           
│ │ │ │ +00015500: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00015510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015520: 2020 2036 3535 3231 2020 3020 2020 3120     65521  0   1 
│ │ │ │ +00015530: 2020 3220 2020 3320 2020 2020 2020 2020    2   3         
│ │ │ │  00015540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015550: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00015560: 2020 2020 2020 2020 2020 2020 2020 205a                 Z
│ │ │ │ -00015570: 5a20 2020 2020 2020 2020 2020 2020 2020  Z               
│ │ │ │ +00015550: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00015560: 2020 2020 2020 2020 205a 5a20 2020 2020           ZZ     
│ │ │ │ +00015570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015590: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000155a0: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
│ │ │ │ -000155b0: 282d 2d2d 2d2d 5b78 202c 2078 202c 2078  (-----[x , x , x
│ │ │ │ -000155c0: 202c 2078 205d 2920 2020 2020 2020 2020   , x ])         
│ │ │ │ -000155d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000155e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000155f0: 2020 2020 2036 3535 3231 2020 3020 2020       65521  0   
│ │ │ │ -00015600: 3120 2020 3220 2020 3320 2020 2020 2020  1   2   3       
│ │ │ │ -00015610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015620: 2020 207c 0a7c 2020 2020 2020 6465 6669     |.|      defi
│ │ │ │ -00015630: 6e69 6e67 2066 6f72 6d73 3a20 7b20 2020  ning forms: {   
│ │ │ │ +00015590: 2020 2020 207c 0a7c 2020 2020 2020 7461       |.|      ta
│ │ │ │ +000155a0: 7267 6574 3a20 5072 6f6a 282d 2d2d 2d2d  rget: Proj(-----
│ │ │ │ +000155b0: 5b78 202c 2078 202c 2078 202c 2078 205d  [x , x , x , x ]
│ │ │ │ +000155c0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +000155d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000155e0: 2020 2020 2020 2020 2020 2020 2020 2036                 6
│ │ │ │ +000155f0: 3535 3231 2020 3020 2020 3120 2020 3220  5521  0   1   2 
│ │ │ │ +00015600: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +00015610: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00015620: 2020 2020 2020 6465 6669 6e69 6e67 2066        defining f
│ │ │ │ +00015630: 6f72 6d73 3a20 7b20 2020 2020 2020 2020  orms: {         
│ │ │ │  00015640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015660: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00015670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015680: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ -00015690: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -000156a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ -000156c0: 2020 2020 202d 2078 2020 2d20 3332 3735       - x  - 3275
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│ │ │ │ -000156f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00015700: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ -00015710: 2020 2020 2020 3020 3120 3220 2020 2020        0 1 2     
│ │ │ │ -00015720: 2020 2020 3020 3320 2020 2020 2020 2020      0 3         
│ │ │ │ -00015730: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00015660: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00015670: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ +00015680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015690: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +000156a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000156b0: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ +000156c0: 2078 2020 2d20 3332 3735 3978 2078 2078   x  - 32759x x x
│ │ │ │ +000156d0: 2020 2b20 3332 3736 3078 2078 202c 2020    + 32760x x ,  
│ │ │ │ +000156e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000156f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015700: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +00015710: 3020 3120 3220 2020 2020 2020 2020 3020  0 1 2         0 
│ │ │ │ +00015720: 3320 2020 2020 2020 2020 2020 207c 0a7c  3            |.|
│ │ │ │ +00015730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015770: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00015770: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00015780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015790: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -000157a0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -000157b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ -000157d0: 2020 2020 2033 3237 3630 7820 7820 202b       32760x x  +
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│ │ │ │ +000157b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ +00015820: 3020 3220 2020 2020 2020 2020 3020 3120  0 2         0 1 
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│ │ │ │ +00015930: 2020 3120 3320 2020 2020 2020 2020 3020    1 3         0 
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│ │ │ │ +00015950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015980: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000159a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000159b0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -000159c0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -000159d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000159e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000159f0: 2020 2020 2078 2020 2b20 3332 3735 3978       x  + 32759x
│ │ │ │ -00015a00: 2078 2078 2020 2d20 3332 3736 3078 2078   x x  - 32760x x
│ │ │ │ -00015a10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015a20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00015a30: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -00015a40: 2020 2020 3120 3220 3320 2020 2020 2020      1 2 3       
│ │ │ │ -00015a50: 2020 3020 3320 2020 2020 2020 2020 2020    0 3           
│ │ │ │ -00015a60: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ +00015990: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000159a0: 2020 2020 2020 2020 2020 2020 3320 2020              3   
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│ │ │ │ +00015a00: 2d20 3332 3736 3078 2078 2020 2020 2020  - 32760x x      
│ │ │ │ +00015a10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015a30: 2020 2020 3220 2020 2020 2020 2020 3120      2         1 
│ │ │ │ +00015a40: 3220 3320 2020 2020 2020 2020 3020 3320  2 3         0 3 
│ │ │ │ +00015a50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00015a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015a70: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │  00015a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015aa0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00015aa0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00015ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ae0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00015af0: 3120 3a20 5261 7469 6f6e 616c 4d61 7020  1 : RationalMap 
│ │ │ │ -00015b00: 2863 7562 6963 2072 6174 696f 6e61 6c20  (cubic rational 
│ │ │ │ -00015b10: 6d61 7020 6672 6f6d 2050 505e 3320 746f  map from PP^3 to
│ │ │ │ -00015b20: 2050 505e 3329 2020 2020 2020 2020 207c   PP^3)         |
│ │ │ │ -00015b30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00015ae0: 2020 2020 207c 0a7c 6f32 3120 3a20 5261       |.|o21 : Ra
│ │ │ │ +00015af0: 7469 6f6e 616c 4d61 7020 2863 7562 6963  tionalMap (cubic
│ │ │ │ +00015b00: 2072 6174 696f 6e61 6c20 6d61 7020 6672   rational map fr
│ │ │ │ +00015b10: 6f6d 2050 505e 3320 746f 2050 505e 3329  om PP^3 to PP^3)
│ │ │ │ +00015b20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00015b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015b70: 2d2d 2d2b 0a7c 6932 3220 3a20 6465 7363  ---+.|i22 : desc
│ │ │ │ -00015b80: 7269 6265 2066 2120 2020 2020 2020 2020  ribe f!         
│ │ │ │ +00015b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00015b70: 6932 3220 3a20 6465 7363 7269 6265 2066  i22 : describe f
│ │ │ │ +00015b80: 2120 2020 2020 2020 2020 2020 2020 2020  !               
│ │ │ │  00015b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015bb0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00015bb0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00015bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015bf0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00015c00: 3220 3d20 7261 7469 6f6e 616c 206d 6170  2 = rational map
│ │ │ │ -00015c10: 2064 6566 696e 6564 2062 7920 666f 726d   defined by form
│ │ │ │ -00015c20: 7320 6f66 2064 6567 7265 6520 3320 2020  s of degree 3   
│ │ │ │ -00015c30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015c40: 0a7c 2020 2020 2020 736f 7572 6365 2076  .|      source v
│ │ │ │ -00015c50: 6172 6965 7479 3a20 5050 5e33 2020 2020  ariety: PP^3    
│ │ │ │ +00015bf0: 2020 2020 207c 0a7c 6f32 3220 3d20 7261       |.|o22 = ra
│ │ │ │ +00015c00: 7469 6f6e 616c 206d 6170 2064 6566 696e  tional map defin
│ │ │ │ +00015c10: 6564 2062 7920 666f 726d 7320 6f66 2064  ed by forms of d
│ │ │ │ +00015c20: 6567 7265 6520 3320 2020 2020 2020 2020  egree 3         
│ │ │ │ +00015c30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015c40: 2020 736f 7572 6365 2076 6172 6965 7479    source variety
│ │ │ │ +00015c50: 3a20 5050 5e33 2020 2020 2020 2020 2020  : PP^3          
│ │ │ │  00015c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c80: 2020 207c 0a7c 2020 2020 2020 7461 7267     |.|      targ
│ │ │ │ -00015c90: 6574 2076 6172 6965 7479 3a20 5050 5e33  et variety: PP^3
│ │ │ │ +00015c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00015c80: 2020 2020 2020 7461 7267 6574 2076 6172        target var
│ │ │ │ +00015c90: 6965 7479 3a20 5050 5e33 2020 2020 2020  iety: PP^3      
│ │ │ │  00015ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015cc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00015cd0: 646f 6d69 6e61 6e63 653a 2074 7275 6520  dominance: true 
│ │ │ │ +00015cc0: 207c 0a7c 2020 2020 2020 646f 6d69 6e61   |.|      domina
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│ │ │ │  00015cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00015d10: 2020 2020 6269 7261 7469 6f6e 616c 6974      birationalit
│ │ │ │ -00015d20: 793a 2074 7275 6520 2874 6865 2069 6e76  y: true (the inv
│ │ │ │ -00015d30: 6572 7365 206d 6170 2069 7320 616c 7265  erse map is alre
│ │ │ │ -00015d40: 6164 7920 6361 6c63 756c 6174 6564 297c  ady calculated)|
│ │ │ │ -00015d50: 0a7c 2020 2020 2020 7072 6f6a 6563 7469  .|      projecti
│ │ │ │ -00015d60: 7665 2064 6567 7265 6573 3a20 7b31 2c20  ve degrees: {1, 
│ │ │ │ -00015d70: 332c 2033 2c20 317d 2020 2020 2020 2020  3, 3, 1}        
│ │ │ │ -00015d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d90: 2020 207c 0a7c 2020 2020 2020 6e75 6d62     |.|      numb
│ │ │ │ -00015da0: 6572 206f 6620 6d69 6e69 6d61 6c20 7265  er of minimal re
│ │ │ │ -00015db0: 7072 6573 656e 7461 7469 7665 733a 2031  presentatives: 1
│ │ │ │ +00015d00: 2020 2020 207c 0a7c 2020 2020 2020 6269       |.|      bi
│ │ │ │ +00015d10: 7261 7469 6f6e 616c 6974 793a 2074 7275  rationality: tru
│ │ │ │ +00015d20: 6520 2874 6865 2069 6e76 6572 7365 206d  e (the inverse m
│ │ │ │ +00015d30: 6170 2069 7320 616c 7265 6164 7920 6361  ap is already ca
│ │ │ │ +00015d40: 6c63 756c 6174 6564 297c 0a7c 2020 2020  lculated)|.|    
│ │ │ │ +00015d50: 2020 7072 6f6a 6563 7469 7665 2064 6567    projective deg
│ │ │ │ +00015d60: 7265 6573 3a20 7b31 2c20 332c 2033 2c20  rees: {1, 3, 3, 
│ │ │ │ +00015d70: 317d 2020 2020 2020 2020 2020 2020 2020  1}              
│ │ │ │ +00015d80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00015d90: 2020 2020 2020 6e75 6d62 6572 206f 6620        number of 
│ │ │ │ +00015da0: 6d69 6e69 6d61 6c20 7265 7072 6573 656e  minimal represen
│ │ │ │ +00015db0: 7461 7469 7665 733a 2031 2020 2020 2020  tatives: 1      
│ │ │ │  00015dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015dd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00015de0: 6469 6d65 6e73 696f 6e20 6261 7365 206c  dimension base l
│ │ │ │ -00015df0: 6f63 7573 3a20 3120 2020 2020 2020 2020  ocus: 1         
│ │ │ │ +00015dd0: 207c 0a7c 2020 2020 2020 6469 6d65 6e73   |.|      dimens
│ │ │ │ +00015de0: 696f 6e20 6261 7365 206c 6f63 7573 3a20  ion base locus: 
│ │ │ │ +00015df0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  00015e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00015e20: 2020 2020 6465 6772 6565 2062 6173 6520      degree base 
│ │ │ │ -00015e30: 6c6f 6375 733a 2036 2020 2020 2020 2020  locus: 6        
│ │ │ │ +00015e10: 2020 2020 207c 0a7c 2020 2020 2020 6465       |.|      de
│ │ │ │ +00015e20: 6772 6565 2062 6173 6520 6c6f 6375 733a  gree base locus:
│ │ │ │ +00015e30: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │  00015e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015e60: 0a7c 2020 2020 2020 636f 6566 6669 6369  .|      coeffici
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│ │ │ │ -00015e80: 3231 2020 2020 2020 2020 2020 2020 2020  21              
│ │ │ │ -00015e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ea0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00015e50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015e60: 2020 636f 6566 6669 6369 656e 7420 7269    coefficient ri
│ │ │ │ +00015e70: 6e67 3a20 5a5a 2f36 3535 3231 2020 2020  ng: ZZ/65521    
│ │ │ │ +00015e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015e90: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00015ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ee0: 2d2d 2d2d 2d2d 2d2b 0a0a 4361 7665 6174  -------+..Caveat
│ │ │ │ -00015ef0: 0a3d 3d3d 3d3d 3d0a 0a54 6869 7320 6973  .======..This is
│ │ │ │ -00015f00: 2075 6e64 6572 2064 6576 656c 6f70 6d65   under developme
│ │ │ │ -00015f10: 6e74 2079 6574 2e0a 0a57 6179 7320 746f  nt yet...Ways to
│ │ │ │ -00015f20: 2075 7365 2061 6273 7472 6163 7452 6174   use abstractRat
│ │ │ │ -00015f30: 696f 6e61 6c4d 6170 3a0a 3d3d 3d3d 3d3d  ionalMap:.======
│ │ │ │ +00015ee0: 2d2b 0a0a 4361 7665 6174 0a3d 3d3d 3d3d  -+..Caveat.=====
│ │ │ │ +00015ef0: 3d0a 0a54 6869 7320 6973 2075 6e64 6572  =..This is under
│ │ │ │ +00015f00: 2064 6576 656c 6f70 6d65 6e74 2079 6574   development yet
│ │ │ │ +00015f10: 2e0a 0a57 6179 7320 746f 2075 7365 2061  ...Ways to use a
│ │ │ │ +00015f20: 6273 7472 6163 7452 6174 696f 6e61 6c4d  bstractRationalM
│ │ │ │ +00015f30: 6170 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ap:.============
│ │ │ │  00015f40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00015f50: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00015f60: 2261 6273 7472 6163 7452 6174 696f 6e61  "abstractRationa
│ │ │ │ -00015f70: 6c4d 6170 2850 6f6c 796e 6f6d 6961 6c52  lMap(PolynomialR
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│ │ │ │ -00015f90: 6e67 2c46 756e 6374 696f 6e43 6c6f 7375  ng,FunctionClosu
│ │ │ │ -00015fa0: 7265 2922 0a20 202a 2022 6162 7374 7261  re)".  * "abstra
│ │ │ │ -00015fb0: 6374 5261 7469 6f6e 616c 4d61 7028 506f  ctRationalMap(Po
│ │ │ │ -00015fc0: 6c79 6e6f 6d69 616c 5269 6e67 2c50 6f6c  lynomialRing,Pol
│ │ │ │ -00015fd0: 796e 6f6d 6961 6c52 696e 672c 4675 6e63  ynomialRing,Func
│ │ │ │ -00015fe0: 7469 6f6e 436c 6f73 7572 652c 5a5a 2922  tionClosure,ZZ)"
│ │ │ │ -00015ff0: 0a20 202a 2022 6162 7374 7261 6374 5261  .  * "abstractRa
│ │ │ │ -00016000: 7469 6f6e 616c 4d61 7028 5261 7469 6f6e  tionalMap(Ration
│ │ │ │ -00016010: 616c 4d61 7029 220a 0a46 6f72 2074 6865  alMap)"..For the
│ │ │ │ -00016020: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ -00016030: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -00016040: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ -00016050: 2061 6273 7472 6163 7452 6174 696f 6e61   abstractRationa
│ │ │ │ -00016060: 6c4d 6170 3a20 6162 7374 7261 6374 5261  lMap: abstractRa
│ │ │ │ -00016070: 7469 6f6e 616c 4d61 702c 2069 7320 6120  tionalMap, is a 
│ │ │ │ -00016080: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e  *note method.fun
│ │ │ │ -00016090: 6374 696f 6e3a 2028 4d61 6361 756c 6179  ction: (Macaulay
│ │ │ │ -000160a0: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ -000160b0: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d  ion,...---------
│ │ │ │ +00015f50: 3d3d 3d3d 0a0a 2020 2a20 2261 6273 7472  ====..  * "abstr
│ │ │ │ +00015f60: 6163 7452 6174 696f 6e61 6c4d 6170 2850  actRationalMap(P
│ │ │ │ +00015f70: 6f6c 796e 6f6d 6961 6c52 696e 672c 506f  olynomialRing,Po
│ │ │ │ +00015f80: 6c79 6e6f 6d69 616c 5269 6e67 2c46 756e  lynomialRing,Fun
│ │ │ │ +00015f90: 6374 696f 6e43 6c6f 7375 7265 2922 0a20  ctionClosure)". 
│ │ │ │ +00015fa0: 202a 2022 6162 7374 7261 6374 5261 7469   * "abstractRati
│ │ │ │ +00015fb0: 6f6e 616c 4d61 7028 506f 6c79 6e6f 6d69  onalMap(Polynomi
│ │ │ │ +00015fc0: 616c 5269 6e67 2c50 6f6c 796e 6f6d 6961  alRing,Polynomia
│ │ │ │ +00015fd0: 6c52 696e 672c 4675 6e63 7469 6f6e 436c  lRing,FunctionCl
│ │ │ │ +00015fe0: 6f73 7572 652c 5a5a 2922 0a20 202a 2022  osure,ZZ)".  * "
│ │ │ │ +00015ff0: 6162 7374 7261 6374 5261 7469 6f6e 616c  abstractRational
│ │ │ │ +00016000: 4d61 7028 5261 7469 6f6e 616c 4d61 7029  Map(RationalMap)
│ │ │ │ +00016010: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ +00016020: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ +00016030: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ +00016040: 6a65 6374 202a 6e6f 7465 2061 6273 7472  ject *note abstr
│ │ │ │ +00016050: 6163 7452 6174 696f 6e61 6c4d 6170 3a20  actRationalMap: 
│ │ │ │ +00016060: 6162 7374 7261 6374 5261 7469 6f6e 616c  abstractRational
│ │ │ │ +00016070: 4d61 702c 2069 7320 6120 2a6e 6f74 6520  Map, is a *note 
│ │ │ │ +00016080: 6d65 7468 6f64 0a66 756e 6374 696f 6e3a  method.function:
│ │ │ │ +00016090: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +000160a0: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ +000160b0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │  000160c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000160d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000160e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000160f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016100: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ -00016110: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
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│ │ │ │ -00016130: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ -00016140: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ -00016150: 352e 3131 2b64 732f 4d32 2f4d 6163 6175  5.11+ds/M2/Macau
│ │ │ │ -00016160: 6c61 7932 2f70 6163 6b61 6765 732f 4372  lay2/packages/Cr
│ │ │ │ -00016170: 656d 6f6e 612f 0a64 6f63 756d 656e 7461  emona/.documenta
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│ │ │ │ -00016190: 1f0a 4669 6c65 3a20 4372 656d 6f6e 612e  ..File: Cremona.
│ │ │ │ -000161a0: 696e 666f 2c20 4e6f 6465 3a20 6170 7072  info, Node: appr
│ │ │ │ -000161b0: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ -000161c0: 702c 204e 6578 743a 2042 6c6f 7755 7053  p, Next: BlowUpS
│ │ │ │ -000161d0: 7472 6174 6567 792c 2050 7265 763a 2061  trategy, Prev: a
│ │ │ │ -000161e0: 6273 7472 6163 7452 6174 696f 6e61 6c4d  bstractRationalM
│ │ │ │ -000161f0: 6170 2c20 5570 3a20 546f 700a 0a61 7070  ap, Up: Top..app
│ │ │ │ -00016200: 726f 7869 6d61 7465 496e 7665 7273 654d  roximateInverseM
│ │ │ │ -00016210: 6170 202d 2d20 7261 6e64 6f6d 206d 6170  ap -- random map
│ │ │ │ -00016220: 2072 656c 6174 6564 2074 6f20 7468 6520   related to the 
│ │ │ │ -00016230: 696e 7665 7273 6520 6f66 2061 2062 6972  inverse of a bir
│ │ │ │ -00016240: 6174 696f 6e61 6c20 6d61 700a 2a2a 2a2a  ational map.****
│ │ │ │ +00016100: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ +00016110: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
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│ │ │ │ +00016130: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ +00016140: 6175 6c61 7932 2d31 2e32 352e 3131 2b64  aulay2-1.25.11+d
│ │ │ │ +00016150: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ +00016160: 6163 6b61 6765 732f 4372 656d 6f6e 612f  ackages/Cremona/
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│ │ │ │ +000161a0: 4e6f 6465 3a20 6170 7072 6f78 696d 6174  Node: approximat
│ │ │ │ +000161b0: 6549 6e76 6572 7365 4d61 702c 204e 6578  eInverseMap, Nex
│ │ │ │ +000161c0: 743a 2042 6c6f 7755 7053 7472 6174 6567  t: BlowUpStrateg
│ │ │ │ +000161d0: 792c 2050 7265 763a 2061 6273 7472 6163  y, Prev: abstrac
│ │ │ │ +000161e0: 7452 6174 696f 6e61 6c4d 6170 2c20 5570  tRationalMap, Up
│ │ │ │ +000161f0: 3a20 546f 700a 0a61 7070 726f 7869 6d61  : Top..approxima
│ │ │ │ +00016200: 7465 496e 7665 7273 654d 6170 202d 2d20  teInverseMap -- 
│ │ │ │ +00016210: 7261 6e64 6f6d 206d 6170 2072 656c 6174  random map relat
│ │ │ │ +00016220: 6564 2074 6f20 7468 6520 696e 7665 7273  ed to the invers
│ │ │ │ +00016230: 6520 6f66 2061 2062 6972 6174 696f 6e61  e of a birationa
│ │ │ │ +00016240: 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a  l map.**********
│ │ │ │  00016250: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00016260: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00016270: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00016280: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00016290: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -000162a0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -000162b0: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ -000162c0: 7365 4d61 7020 7068 6920 0a20 2020 2020  seMap phi .     
│ │ │ │ -000162d0: 2020 2061 7070 726f 7869 6d61 7465 496e     approximateIn
│ │ │ │ -000162e0: 7665 7273 654d 6170 2870 6869 2c64 290a  verseMap(phi,d).
│ │ │ │ -000162f0: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -00016300: 2020 2a20 7068 692c 2061 202a 6e6f 7465    * phi, a *note
│ │ │ │ -00016310: 2072 6174 696f 6e61 6c20 6d61 703a 2052   rational map: R
│ │ │ │ -00016320: 6174 696f 6e61 6c4d 6170 2c2c 2061 2062  ationalMap,, a b
│ │ │ │ -00016330: 6972 6174 696f 6e61 6c20 6d61 700a 2020  irational map.  
│ │ │ │ -00016340: 2020 2020 2a20 642c 2061 6e20 2a6e 6f74      * d, an *not
│ │ │ │ -00016350: 6520 696e 7465 6765 723a 2028 4d61 6361  e integer: (Maca
│ │ │ │ -00016360: 756c 6179 3244 6f63 295a 5a2c 2c20 6f70  ulay2Doc)ZZ,, op
│ │ │ │ -00016370: 7469 6f6e 616c 2c20 6275 7420 6974 2073  tional, but it s
│ │ │ │ -00016380: 686f 756c 6420 6265 2074 6865 0a20 2020  hould be the.   
│ │ │ │ -00016390: 2020 2020 2064 6567 7265 6520 6f66 2074       degree of t
│ │ │ │ -000163a0: 6865 2066 6f72 6d73 2064 6566 696e 696e  he forms definin
│ │ │ │ -000163b0: 6720 7468 6520 696e 7665 7273 6520 6f66  g the inverse of
│ │ │ │ -000163c0: 2070 6869 0a20 202a 202a 6e6f 7465 204f   phi.  * *note O
│ │ │ │ -000163d0: 7074 696f 6e61 6c20 696e 7075 7473 3a20  ptional inputs: 
│ │ │ │ -000163e0: 284d 6163 6175 6c61 7932 446f 6329 7573  (Macaulay2Doc)us
│ │ │ │ -000163f0: 696e 6720 6675 6e63 7469 6f6e 7320 7769  ing functions wi
│ │ │ │ -00016400: 7468 206f 7074 696f 6e61 6c20 696e 7075  th optional inpu
│ │ │ │ -00016410: 7473 2c3a 0a20 2020 2020 202a 202a 6e6f  ts,:.      * *no
│ │ │ │ -00016420: 7465 2043 6572 7469 6679 3a20 4365 7274  te Certify: Cert
│ │ │ │ -00016430: 6966 792c 203d 3e20 2e2e 2e2c 2064 6566  ify, => ..., def
│ │ │ │ -00016440: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
│ │ │ │ -00016450: 2c20 7768 6574 6865 7220 746f 2065 6e73  , whether to ens
│ │ │ │ -00016460: 7572 650a 2020 2020 2020 2020 636f 7272  ure.        corr
│ │ │ │ -00016470: 6563 746e 6573 7320 6f66 206f 7574 7075  ectness of outpu
│ │ │ │ -00016480: 740a 2020 2020 2020 2a20 2a6e 6f74 6520  t.      * *note 
│ │ │ │ -00016490: 436f 6469 6d42 7349 6e76 3a20 436f 6469  CodimBsInv: Codi
│ │ │ │ -000164a0: 6d42 7349 6e76 2c20 3d3e 202e 2e2e 2c20  mBsInv, => ..., 
│ │ │ │ -000164b0: 6465 6661 756c 7420 7661 6c75 6520 6e75  default value nu
│ │ │ │ -000164c0: 6c6c 2c20 0a20 2020 2020 202a 202a 6e6f  ll, .      * *no
│ │ │ │ -000164d0: 7465 2056 6572 626f 7365 3a20 696e 7665  te Verbose: inve
│ │ │ │ -000164e0: 7273 654d 6170 5f6c 705f 7064 5f70 645f  rseMap_lp_pd_pd_
│ │ │ │ -000164f0: 7064 5f63 6d56 6572 626f 7365 3d3e 5f70  pd_cmVerbose=>_p
│ │ │ │ -00016500: 645f 7064 5f70 645f 7270 2c20 3d3e 202e  d_pd_pd_rp, => .
│ │ │ │ -00016510: 2e2e 2c0a 2020 2020 2020 2020 6465 6661  ..,.        defa
│ │ │ │ -00016520: 756c 7420 7661 6c75 6520 7472 7565 2c0a  ult value true,.
│ │ │ │ -00016530: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ -00016540: 2020 202a 2061 202a 6e6f 7465 2072 6174     * a *note rat
│ │ │ │ -00016550: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ -00016560: 6e61 6c4d 6170 2c2c 2061 2072 616e 646f  nalMap,, a rando
│ │ │ │ -00016570: 6d20 7261 7469 6f6e 616c 206d 6170 2077  m rational map w
│ │ │ │ -00016580: 6869 6368 2069 6e20 736f 6d65 0a20 2020  hich in some.   
│ │ │ │ -00016590: 2020 2020 2073 656e 7365 2069 7320 7265       sense is re
│ │ │ │ -000165a0: 6c61 7465 6420 746f 2074 6865 2069 6e76  lated to the inv
│ │ │ │ -000165b0: 6572 7365 206f 6620 7068 6920 2865 2e67  erse of phi (e.g
│ │ │ │ -000165c0: 2e2c 2074 6865 7920 7368 6f75 6c64 2068  ., they should h
│ │ │ │ -000165d0: 6176 6520 7468 6520 7361 6d65 0a20 2020  ave the same.   
│ │ │ │ -000165e0: 2020 2020 2062 6173 6520 6c6f 6375 7329       base locus)
│ │ │ │ -000165f0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ -00016600: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 2061  =========..The a
│ │ │ │ -00016610: 6c67 6f72 6974 686d 2069 7320 746f 2074  lgorithm is to t
│ │ │ │ -00016620: 7279 2074 6f20 636f 6e73 7472 7563 7420  ry to construct 
│ │ │ │ -00016630: 7468 6520 6964 6561 6c20 6f66 2074 6865  the ideal of the
│ │ │ │ -00016640: 2062 6173 6520 6c6f 6375 7320 6f66 2074   base locus of t
│ │ │ │ -00016650: 6865 2069 6e76 6572 7365 0a62 7920 6c6f  he inverse.by lo
│ │ │ │ -00016660: 6f6b 696e 6720 666f 7220 7468 6520 696d  oking for the im
│ │ │ │ -00016670: 6167 6573 2076 6961 2070 6869 206f 6620  ages via phi of 
│ │ │ │ -00016680: 7261 6e64 6f6d 206c 696e 6561 7220 7365  random linear se
│ │ │ │ -00016690: 6374 696f 6e73 206f 6620 7468 6520 736f  ctions of the so
│ │ │ │ -000166a0: 7572 6365 0a76 6172 6965 7479 2e20 4765  urce.variety. Ge
│ │ │ │ -000166b0: 6e65 7261 6c6c 792c 206f 6e65 2063 616e  nerally, one can
│ │ │ │ -000166c0: 2073 7065 6564 2075 7020 7468 6520 7072   speed up the pr
│ │ │ │ -000166d0: 6f63 6573 7320 6279 2070 6173 7369 6e67  ocess by passing
│ │ │ │ -000166e0: 2074 6872 6f75 6768 2074 6865 206f 7074   through the opt
│ │ │ │ -000166f0: 696f 6e0a 2a6e 6f74 6520 436f 6469 6d42  ion.*note CodimB
│ │ │ │ -00016700: 7349 6e76 3a20 436f 6469 6d42 7349 6e76  sInv: CodimBsInv
│ │ │ │ -00016710: 2c20 6120 676f 6f64 206c 6f77 6572 2062  , a good lower b
│ │ │ │ -00016720: 6f75 6e64 2066 6f72 2074 6865 2063 6f64  ound for the cod
│ │ │ │ -00016730: 696d 656e 7369 6f6e 206f 6620 7468 6973  imension of this
│ │ │ │ -00016740: 0a62 6173 6520 6c6f 6375 732e 0a0a 2b2d  .base locus...+-
│ │ │ │ +00016290: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ +000162a0: 200a 2020 2020 2020 2020 6170 7072 6f78   .        approx
│ │ │ │ +000162b0: 696d 6174 6549 6e76 6572 7365 4d61 7020  imateInverseMap 
│ │ │ │ +000162c0: 7068 6920 0a20 2020 2020 2020 2061 7070  phi .        app
│ │ │ │ +000162d0: 726f 7869 6d61 7465 496e 7665 7273 654d  roximateInverseM
│ │ │ │ +000162e0: 6170 2870 6869 2c64 290a 2020 2a20 496e  ap(phi,d).  * In
│ │ │ │ +000162f0: 7075 7473 3a0a 2020 2020 2020 2a20 7068  puts:.      * ph
│ │ │ │ +00016300: 692c 2061 202a 6e6f 7465 2072 6174 696f  i, a *note ratio
│ │ │ │ +00016310: 6e61 6c20 6d61 703a 2052 6174 696f 6e61  nal map: Rationa
│ │ │ │ +00016320: 6c4d 6170 2c2c 2061 2062 6972 6174 696f  lMap,, a biratio
│ │ │ │ +00016330: 6e61 6c20 6d61 700a 2020 2020 2020 2a20  nal map.      * 
│ │ │ │ +00016340: 642c 2061 6e20 2a6e 6f74 6520 696e 7465  d, an *note inte
│ │ │ │ +00016350: 6765 723a 2028 4d61 6361 756c 6179 3244  ger: (Macaulay2D
│ │ │ │ +00016360: 6f63 295a 5a2c 2c20 6f70 7469 6f6e 616c  oc)ZZ,, optional
│ │ │ │ +00016370: 2c20 6275 7420 6974 2073 686f 756c 6420  , but it should 
│ │ │ │ +00016380: 6265 2074 6865 0a20 2020 2020 2020 2064  be the.        d
│ │ │ │ +00016390: 6567 7265 6520 6f66 2074 6865 2066 6f72  egree of the for
│ │ │ │ +000163a0: 6d73 2064 6566 696e 696e 6720 7468 6520  ms defining the 
│ │ │ │ +000163b0: 696e 7665 7273 6520 6f66 2070 6869 0a20  inverse of phi. 
│ │ │ │ +000163c0: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ +000163d0: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ +000163e0: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ +000163f0: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ +00016400: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ +00016410: 2020 2020 202a 202a 6e6f 7465 2043 6572       * *note Cer
│ │ │ │ +00016420: 7469 6679 3a20 4365 7274 6966 792c 203d  tify: Certify, =
│ │ │ │ +00016430: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ +00016440: 616c 7565 2066 616c 7365 2c20 7768 6574  alue false, whet
│ │ │ │ +00016450: 6865 7220 746f 2065 6e73 7572 650a 2020  her to ensure.  
│ │ │ │ +00016460: 2020 2020 2020 636f 7272 6563 746e 6573        correctnes
│ │ │ │ +00016470: 7320 6f66 206f 7574 7075 740a 2020 2020  s of output.    
│ │ │ │ +00016480: 2020 2a20 2a6e 6f74 6520 436f 6469 6d42    * *note CodimB
│ │ │ │ +00016490: 7349 6e76 3a20 436f 6469 6d42 7349 6e76  sInv: CodimBsInv
│ │ │ │ +000164a0: 2c20 3d3e 202e 2e2e 2c20 6465 6661 756c  , => ..., defaul
│ │ │ │ +000164b0: 7420 7661 6c75 6520 6e75 6c6c 2c20 0a20  t value null, . 
│ │ │ │ +000164c0: 2020 2020 202a 202a 6e6f 7465 2056 6572       * *note Ver
│ │ │ │ +000164d0: 626f 7365 3a20 696e 7665 7273 654d 6170  bose: inverseMap
│ │ │ │ +000164e0: 5f6c 705f 7064 5f70 645f 7064 5f63 6d56  _lp_pd_pd_pd_cmV
│ │ │ │ +000164f0: 6572 626f 7365 3d3e 5f70 645f 7064 5f70  erbose=>_pd_pd_p
│ │ │ │ +00016500: 645f 7270 2c20 3d3e 202e 2e2e 2c0a 2020  d_rp, => ...,.  
│ │ │ │ +00016510: 2020 2020 2020 6465 6661 756c 7420 7661        default va
│ │ │ │ +00016520: 6c75 6520 7472 7565 2c0a 2020 2a20 4f75  lue true,.  * Ou
│ │ │ │ +00016530: 7470 7574 733a 0a20 2020 2020 202a 2061  tputs:.      * a
│ │ │ │ +00016540: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ +00016550: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ +00016560: 2c2c 2061 2072 616e 646f 6d20 7261 7469  ,, a random rati
│ │ │ │ +00016570: 6f6e 616c 206d 6170 2077 6869 6368 2069  onal map which i
│ │ │ │ +00016580: 6e20 736f 6d65 0a20 2020 2020 2020 2073  n some.        s
│ │ │ │ +00016590: 656e 7365 2069 7320 7265 6c61 7465 6420  ense is related 
│ │ │ │ +000165a0: 746f 2074 6865 2069 6e76 6572 7365 206f  to the inverse o
│ │ │ │ +000165b0: 6620 7068 6920 2865 2e67 2e2c 2074 6865  f phi (e.g., the
│ │ │ │ +000165c0: 7920 7368 6f75 6c64 2068 6176 6520 7468  y should have th
│ │ │ │ +000165d0: 6520 7361 6d65 0a20 2020 2020 2020 2062  e same.        b
│ │ │ │ +000165e0: 6173 6520 6c6f 6375 7329 0a0a 4465 7363  ase locus)..Desc
│ │ │ │ +000165f0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +00016600: 3d3d 3d0a 0a54 6865 2061 6c67 6f72 6974  ===..The algorit
│ │ │ │ +00016610: 686d 2069 7320 746f 2074 7279 2074 6f20  hm is to try to 
│ │ │ │ +00016620: 636f 6e73 7472 7563 7420 7468 6520 6964  construct the id
│ │ │ │ +00016630: 6561 6c20 6f66 2074 6865 2062 6173 6520  eal of the base 
│ │ │ │ +00016640: 6c6f 6375 7320 6f66 2074 6865 2069 6e76  locus of the inv
│ │ │ │ +00016650: 6572 7365 0a62 7920 6c6f 6f6b 696e 6720  erse.by looking 
│ │ │ │ +00016660: 666f 7220 7468 6520 696d 6167 6573 2076  for the images v
│ │ │ │ +00016670: 6961 2070 6869 206f 6620 7261 6e64 6f6d  ia phi of random
│ │ │ │ +00016680: 206c 696e 6561 7220 7365 6374 696f 6e73   linear sections
│ │ │ │ +00016690: 206f 6620 7468 6520 736f 7572 6365 0a76   of the source.v
│ │ │ │ +000166a0: 6172 6965 7479 2e20 4765 6e65 7261 6c6c  ariety. Generall
│ │ │ │ +000166b0: 792c 206f 6e65 2063 616e 2073 7065 6564  y, one can speed
│ │ │ │ +000166c0: 2075 7020 7468 6520 7072 6f63 6573 7320   up the process 
│ │ │ │ +000166d0: 6279 2070 6173 7369 6e67 2074 6872 6f75  by passing throu
│ │ │ │ +000166e0: 6768 2074 6865 206f 7074 696f 6e0a 2a6e  gh the option.*n
│ │ │ │ +000166f0: 6f74 6520 436f 6469 6d42 7349 6e76 3a20  ote CodimBsInv: 
│ │ │ │ +00016700: 436f 6469 6d42 7349 6e76 2c20 6120 676f  CodimBsInv, a go
│ │ │ │ +00016710: 6f64 206c 6f77 6572 2062 6f75 6e64 2066  od lower bound f
│ │ │ │ +00016720: 6f72 2074 6865 2063 6f64 696d 656e 7369  or the codimensi
│ │ │ │ +00016730: 6f6e 206f 6620 7468 6973 0a62 6173 6520  on of this.base 
│ │ │ │ +00016740: 6c6f 6375 732e 0a0a 2b2d 2d2d 2d2d 2d2d  locus...+-------
│ │ │ │  00016750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000167a0: 3120 3a20 5038 203d 205a 5a2f 3937 5b74  1 : P8 = ZZ/97[t
│ │ │ │ -000167b0: 5f30 2e2e 745f 385d 3b20 2020 2020 2020  _0..t_8];       
│ │ │ │ +00016790: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5038  ------+.|i1 : P8
│ │ │ │ +000167a0: 203d 205a 5a2f 3937 5b74 5f30 2e2e 745f   = ZZ/97[t_0..t_
│ │ │ │ +000167b0: 385d 3b20 2020 2020 2020 2020 2020 2020  8];             
│ │ │ │  000167c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000167d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000167e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000167e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  000167f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00016840: 3220 3a20 7068 6920 3d20 696e 7665 7273  2 : phi = invers
│ │ │ │ -00016850: 654d 6170 2072 6174 696f 6e61 6c4d 6170  eMap rationalMap
│ │ │ │ -00016860: 2874 7269 6d28 6d69 6e6f 7273 2832 2c67  (trim(minors(2,g
│ │ │ │ -00016870: 656e 6572 6963 4d61 7472 6978 2850 382c  enericMatrix(P8,
│ │ │ │ -00016880: 3320 2020 2020 2020 2020 2020 7c0a 7c20  3           |.| 
│ │ │ │ +00016830: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 7068  ------+.|i2 : ph
│ │ │ │ +00016840: 6920 3d20 696e 7665 7273 654d 6170 2072  i = inverseMap r
│ │ │ │ +00016850: 6174 696f 6e61 6c4d 6170 2874 7269 6d28  ationalMap(trim(
│ │ │ │ +00016860: 6d69 6e6f 7273 2832 2c67 656e 6572 6963  minors(2,generic
│ │ │ │ +00016870: 4d61 7472 6978 2850 382c 3320 2020 2020  Matrix(P8,3     
│ │ │ │ +00016880: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00016890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000168a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000168b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000168c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000168d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000168e0: 3220 3d20 2d2d 2072 6174 696f 6e61 6c20  2 = -- rational 
│ │ │ │ -000168f0: 6d61 7020 2d2d 2020 2020 2020 2020 2020  map --          
│ │ │ │ +000168d0: 2020 2020 2020 7c0a 7c6f 3220 3d20 2d2d        |.|o2 = --
│ │ │ │ +000168e0: 2072 6174 696f 6e61 6c20 6d61 7020 2d2d   rational map --
│ │ │ │ +000168f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016920: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00016920: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00016930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016940: 2020 2020 2020 2020 2020 2020 2020 205a                 Z
│ │ │ │ -00016950: 5a20 2020 2020 2020 2020 2020 2020 2020  Z               
│ │ │ │ +00016940: 2020 2020 2020 2020 205a 5a20 2020 2020           ZZ     
│ │ │ │ +00016950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016970: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00016980: 2020 2020 736f 7572 6365 3a20 7375 6276      source: subv
│ │ │ │ -00016990: 6172 6965 7479 206f 6620 5072 6f6a 282d  ariety of Proj(-
│ │ │ │ -000169a0: 2d5b 7820 2c20 7820 2c20 7820 2c20 7820  -[x , x , x , x 
│ │ │ │ -000169b0: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ -000169c0: 2c20 2020 2020 2020 2020 2020 7c0a 7c20  ,           |.| 
│ │ │ │ +00016970: 2020 2020 2020 7c0a 7c20 2020 2020 736f        |.|     so
│ │ │ │ +00016980: 7572 6365 3a20 7375 6276 6172 6965 7479  urce: subvariety
│ │ │ │ +00016990: 206f 6620 5072 6f6a 282d 2d5b 7820 2c20   of Proj(--[x , 
│ │ │ │ +000169a0: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ +000169b0: 7820 2c20 7820 2c20 7820 2c20 2020 2020  x , x , x ,     
│ │ │ │ +000169c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000169d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000169e0: 2020 2020 2020 2020 2020 2020 2020 2039                 9
│ │ │ │ -000169f0: 3720 2030 2020 2031 2020 2032 2020 2033  7  0   1   2   3
│ │ │ │ -00016a00: 2020 2034 2020 2035 2020 2036 2020 2037     4   5   6   7
│ │ │ │ -00016a10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00016a20: 2020 2020 2020 2020 2020 2020 7b20 2020              {   
│ │ │ │ +000169e0: 2020 2020 2020 2020 2039 3720 2030 2020           97  0  
│ │ │ │ +000169f0: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ +00016a00: 2035 2020 2036 2020 2037 2020 2020 2020   5   6   7      
│ │ │ │ +00016a10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016a20: 2020 2020 2020 7b20 2020 2020 2020 2020        {         
│ │ │ │  00016a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00016a70: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00016a80: 3220 2020 2020 2032 2020 2020 2020 2020  2      2        
│ │ │ │ -00016a90: 2020 3220 3220 2020 2020 2020 2020 2020    2 2           
│ │ │ │ +00016a60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016a70: 2020 2020 2020 2020 3220 3220 2020 2020          2 2     
│ │ │ │ +00016a80: 2032 2020 2020 2020 2020 2020 3220 3220   2          2 2 
│ │ │ │ +00016a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ab0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00016ac0: 2020 2020 2020 2020 2020 2020 2078 2078               x x
│ │ │ │ -00016ad0: 2020 2b20 3134 7820 7820 7820 202b 2032    + 14x x x  + 2
│ │ │ │ -00016ae0: 3678 2078 2020 2d20 3278 2078 2078 2078  6x x  - 2x x x x
│ │ │ │ -00016af0: 2020 2d20 3134 7820 7820 7820 7820 202b    - 14x x x x  +
│ │ │ │ -00016b00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00016b10: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00016b20: 3220 2020 2020 2031 2032 2033 2020 2020  2      1 2 3    
│ │ │ │ -00016b30: 2020 3120 3320 2020 2020 3020 3120 3220    1 3     0 1 2 
│ │ │ │ -00016b40: 3420 2020 2020 2030 2031 2033 2034 2020  4      0 1 3 4  
│ │ │ │ -00016b50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00016b60: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ +00016ab0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016ac0: 2020 2020 2020 2078 2078 2020 2b20 3134         x x  + 14
│ │ │ │ +00016ad0: 7820 7820 7820 202b 2032 3678 2078 2020  x x x  + 26x x  
│ │ │ │ +00016ae0: 2d20 3278 2078 2078 2078 2020 2d20 3134  - 2x x x x  - 14
│ │ │ │ +00016af0: 7820 7820 7820 7820 202b 2020 2020 2020  x x x x  +      
│ │ │ │ +00016b00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016b10: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ +00016b20: 2031 2032 2033 2020 2020 2020 3120 3320   1 2 3      1 3 
│ │ │ │ +00016b30: 2020 2020 3020 3120 3220 3420 2020 2020      0 1 2 4     
│ │ │ │ +00016b40: 2030 2031 2033 2034 2020 2020 2020 2020   0 1 3 4        
│ │ │ │ +00016b50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016b60: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │  00016b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ba0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00016bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016bc0: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │ +00016ba0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016bb0: 2020 2020 2020 2020 2020 205a 5a20 2020             ZZ   
│ │ │ │ +00016bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016bf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00016c00: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
│ │ │ │ -00016c10: 282d 2d5b 7420 2c20 7420 2c20 7420 2c20  (--[t , t , t , 
│ │ │ │ -00016c20: 7420 2c20 7420 2c20 7420 2c20 7420 2c20  t , t , t , t , 
│ │ │ │ -00016c30: 7420 2c20 7420 5d29 2020 2020 2020 2020  t , t ])        
│ │ │ │ -00016c40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00016c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c60: 2039 3720 2030 2020 2031 2020 2032 2020   97  0   1   2  
│ │ │ │ -00016c70: 2033 2020 2034 2020 2035 2020 2036 2020   3   4   5   6  
│ │ │ │ -00016c80: 2037 2020 2038 2020 2020 2020 2020 2020   7   8          
│ │ │ │ -00016c90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00016ca0: 2020 2020 6465 6669 6e69 6e67 2066 6f72      defining for
│ │ │ │ -00016cb0: 6d73 3a20 7b20 2020 2020 2020 2020 2020  ms: {           
│ │ │ │ +00016bf0: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ +00016c00: 7267 6574 3a20 5072 6f6a 282d 2d5b 7420  rget: Proj(--[t 
│ │ │ │ +00016c10: 2c20 7420 2c20 7420 2c20 7420 2c20 7420  , t , t , t , t 
│ │ │ │ +00016c20: 2c20 7420 2c20 7420 2c20 7420 2c20 7420  , t , t , t , t 
│ │ │ │ +00016c30: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ +00016c40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016c50: 2020 2020 2020 2020 2020 2039 3720 2030             97  0
│ │ │ │ +00016c60: 2020 2031 2020 2032 2020 2033 2020 2034     1   2   3   4
│ │ │ │ +00016c70: 2020 2035 2020 2036 2020 2037 2020 2038     5   6   7   8
│ │ │ │ +00016c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c90: 2020 2020 2020 7c0a 7c20 2020 2020 6465        |.|     de
│ │ │ │ +00016ca0: 6669 6e69 6e67 2066 6f72 6d73 3a20 7b20  fining forms: { 
│ │ │ │ +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 7c0a 7c20              |.| 
│ │ │ │ -00016cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d00: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ -00016d10: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00016ce0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016cf0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +00016d00: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
│ │ │ │ +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 7c0a 7c20              |.| 
│ │ │ │ +00016d30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00016d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d50: 2020 2020 2020 3520 3720 2020 2034 2038        5 7    4 8
│ │ │ │ +00016d50: 3520 3720 2020 2034 2038 2020 2020 2020  5 7    4 8      
│ │ │ │  00016d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00016d80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00016d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20              |.| 
│ │ │ │ -00016de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016df0: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ -00016e00: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00016dd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016de0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +00016df0: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
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│ │ │ │  00017ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017af0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017af0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00017b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017b40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017b40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00017b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017b90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017b90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00017ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017be0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017be0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00017bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017c30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00017c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017c80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00017c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00017cd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017cd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00017ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00017d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00017d70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  00017d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00017dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017e10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017e10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00017e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00017e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00017e60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00017e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017eb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00017ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00017f00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017f00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00017f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00017f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00017f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017fa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00017fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00017ff0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017ff0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00018000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018020: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00018050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018090: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00018090: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000180a0: 2020 2020 2020 2020 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 7c20              |.| 
│ │ │ │ +000180e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000180f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018110: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00018130: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00018130: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00018140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018180: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00018180: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00018190: 2020 2020 2020 2020 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              |.| 
│ │ │ │ +000181d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000181e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000181f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20              |.| 
│ │ │ │ +00018220: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00018230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018240: 2020 2020 2020 2020 2020 2020 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 7c0a 7c20              |.| 
│ │ │ │ +00018270: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00018280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c20              |.| 
│ │ │ │ +000182c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000182d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018310: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00018310: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00018320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018330: 2020 2020 2020 2020 2020 2020 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 7c0a 7c20              |.| 
│ │ │ │ +00018360: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00018370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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              |.| 
│ │ │ │ -000183c0: 746f 2050 505e 3829 2020 2020 2020 2020  to PP^8)        
│ │ │ │ +000183b0: 2020 2020 2020 7c0a 7c20 746f 2050 505e        |.| to PP^
│ │ │ │ +000183c0: 3829 2020 2020 2020 2020 2020 2020 2020  8)              
│ │ │ │  000183d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000183e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000183f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018400: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +00018400: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │  00018410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 7c0a 7c20  ------------|.| 
│ │ │ │ +00018450: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │  00018460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018470: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00018470: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00018480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000184a0: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ -000184b0: 3578 2078 2078 2078 2020 2d20 3139 7820  5x x x x  - 19x 
│ │ │ │ -000184c0: 7820 7820 7820 202b 2031 3478 2078 2078  x x x  + 14x x x
│ │ │ │ -000184d0: 2020 2b20 3131 7820 7820 7820 7820 202d    + 11x x x x  -
│ │ │ │ -000184e0: 2031 3978 2078 2078 2078 2020 2b20 3231   19x x x x  + 21
│ │ │ │ -000184f0: 7820 7820 7820 7820 202b 2020 7c0a 7c20  x x x x  +  |.| 
│ │ │ │ -00018500: 2020 3020 3120 3320 3520 2020 2020 2031    0 1 3 5      1
│ │ │ │ -00018510: 2032 2033 2035 2020 2020 2020 3020 3420   2 3 5      0 4 
│ │ │ │ -00018520: 3520 2020 2020 2030 2032 2034 2035 2020  5      0 2 4 5  
│ │ │ │ -00018530: 2020 2020 3020 3320 3420 3520 2020 2020      0 3 4 5     
│ │ │ │ -00018540: 2032 2033 2034 2035 2020 2020 7c0a 7c2d   2 3 4 5    |.|-
│ │ │ │ +000184a0: 2020 2020 2020 7c0a 7c34 3578 2078 2078        |.|45x x x
│ │ │ │ +000184b0: 2078 2020 2d20 3139 7820 7820 7820 7820   x  - 19x x x x 
│ │ │ │ +000184c0: 202b 2031 3478 2078 2078 2020 2b20 3131   + 14x x x  + 11
│ │ │ │ +000184d0: 7820 7820 7820 7820 202d 2031 3978 2078  x x x x  - 19x x
│ │ │ │ +000184e0: 2078 2078 2020 2b20 3231 7820 7820 7820   x x  + 21x x x 
│ │ │ │ +000184f0: 7820 202b 2020 7c0a 7c20 2020 3020 3120  x  +  |.|   0 1 
│ │ │ │ +00018500: 3320 3520 2020 2020 2031 2032 2033 2035  3 5      1 2 3 5
│ │ │ │ +00018510: 2020 2020 2020 3020 3420 3520 2020 2020        0 4 5     
│ │ │ │ +00018520: 2030 2032 2034 2035 2020 2020 2020 3020   0 2 4 5      0 
│ │ │ │ +00018530: 3320 3420 3520 2020 2020 2032 2033 2034  3 4 5      2 3 4
│ │ │ │ +00018540: 2035 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   5    |.|-------
│ │ │ │  00018550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -000185a0: 2020 3220 3220 2020 2020 2020 2020 2032    2 2          2
│ │ │ │ -000185b0: 2020 2020 2020 3220 3220 2020 2020 2020        2 2       
│ │ │ │ -000185c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -000185d0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -000185e0: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ -000185f0: 3678 2078 2020 2b20 3139 7820 7820 7820  6x x  + 19x x x 
│ │ │ │ -00018600: 202b 2033 3878 2078 2020 2d20 3330 7820   + 38x x  - 30x 
│ │ │ │ -00018610: 7820 7820 202d 2031 3178 2078 2078 2078  x x  - 11x x x x
│ │ │ │ -00018620: 2020 2d20 3131 7820 7820 7820 202b 2033    - 11x x x  + 3
│ │ │ │ -00018630: 3078 2078 2078 2078 2020 2b20 7c0a 7c20  0x x x x  + |.| 
│ │ │ │ -00018640: 2020 3020 3520 2020 2020 2030 2032 2035    0 5      0 2 5
│ │ │ │ -00018650: 2020 2020 2020 3220 3520 2020 2020 2031        2 5      1
│ │ │ │ -00018660: 2032 2036 2020 2020 2020 3120 3220 3320   2 6      1 2 3 
│ │ │ │ -00018670: 3620 2020 2020 2031 2033 2036 2020 2020  6      1 3 6    
│ │ │ │ -00018680: 2020 3020 3220 3420 3620 2020 7c0a 7c2d    0 2 4 6   |.|-
│ │ │ │ +00018590: 2d2d 2d2d 2d2d 7c0a 7c20 2020 3220 3220  ------|.|   2 2 
│ │ │ │ +000185a0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +000185b0: 3220 3220 2020 2020 2020 2032 2020 2020  2 2        2    
│ │ │ │ +000185c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000185d0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +000185e0: 2020 2020 2020 7c0a 7c32 3678 2078 2020        |.|26x x  
│ │ │ │ +000185f0: 2b20 3139 7820 7820 7820 202b 2033 3878  + 19x x x  + 38x
│ │ │ │ +00018600: 2078 2020 2d20 3330 7820 7820 7820 202d   x  - 30x x x  -
│ │ │ │ +00018610: 2031 3178 2078 2078 2078 2020 2d20 3131   11x x x x  - 11
│ │ │ │ +00018620: 7820 7820 7820 202b 2033 3078 2078 2078  x x x  + 30x x x
│ │ │ │ +00018630: 2078 2020 2b20 7c0a 7c20 2020 3020 3520   x  + |.|   0 5 
│ │ │ │ +00018640: 2020 2020 2030 2032 2035 2020 2020 2020       0 2 5      
│ │ │ │ +00018650: 3220 3520 2020 2020 2031 2032 2036 2020  2 5      1 2 6  
│ │ │ │ +00018660: 2020 2020 3120 3220 3320 3620 2020 2020      1 2 3 6     
│ │ │ │ +00018670: 2031 2033 2036 2020 2020 2020 3020 3220   1 3 6      0 2 
│ │ │ │ +00018680: 3420 3620 2020 7c0a 7c2d 2d2d 2d2d 2d2d  4 6   |.|-------
│ │ │ │  00018690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000186a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000186b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000186c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000186d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +000186d0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │  000186e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000186f0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +000186f0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  00018700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018720: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ -00018730: 3278 2078 2078 2078 2020 2b20 3131 7820  2x x x x  + 11x 
│ │ │ │ -00018740: 7820 7820 7820 202d 2032 3278 2078 2078  x x x  - 22x x x
│ │ │ │ -00018750: 2020 2b20 3130 7820 7820 7820 7820 202b    + 10x x x x  +
│ │ │ │ -00018760: 2031 3178 2078 2078 2078 2020 2d20 3432   11x x x x  - 42
│ │ │ │ -00018770: 7820 7820 7820 7820 202d 2020 7c0a 7c20  x x x x  -  |.| 
│ │ │ │ -00018780: 2020 3120 3220 3420 3620 2020 2020 2030    1 2 4 6      0
│ │ │ │ -00018790: 2033 2034 2036 2020 2020 2020 3020 3420   3 4 6      0 4 
│ │ │ │ -000187a0: 3620 2020 2020 2031 2032 2035 2036 2020  6      1 2 5 6  
│ │ │ │ -000187b0: 2020 2020 3020 3320 3520 3620 2020 2020      0 3 5 6     
│ │ │ │ -000187c0: 2031 2033 2035 2036 2020 2020 7c0a 7c2d   1 3 5 6    |.|-
│ │ │ │ +00018720: 2020 2020 2020 7c0a 7c32 3278 2078 2078        |.|22x x x
│ │ │ │ +00018730: 2078 2020 2b20 3131 7820 7820 7820 7820   x  + 11x x x x 
│ │ │ │ +00018740: 202d 2032 3278 2078 2078 2020 2b20 3130   - 22x x x  + 10
│ │ │ │ +00018750: 7820 7820 7820 7820 202b 2031 3178 2078  x x x x  + 11x x
│ │ │ │ +00018760: 2078 2078 2020 2d20 3432 7820 7820 7820   x x  - 42x x x 
│ │ │ │ +00018770: 7820 202d 2020 7c0a 7c20 2020 3120 3220  x  -  |.|   1 2 
│ │ │ │ +00018780: 3420 3620 2020 2020 2030 2033 2034 2036  4 6      0 3 4 6
│ │ │ │ +00018790: 2020 2020 2020 3020 3420 3620 2020 2020        0 4 6     
│ │ │ │ +000187a0: 2031 2032 2035 2036 2020 2020 2020 3020   1 2 5 6      0 
│ │ │ │ +000187b0: 3320 3520 3620 2020 2020 2031 2033 2035  3 5 6      1 3 5
│ │ │ │ +000187c0: 2036 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   6    |.|-------
│ │ │ │  000187d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000187e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000187f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000192b0: 7820 7820 7820 202d 2020 2020 7c0a 7c20  x x x  -    |.| 
│ │ │ │ -000192c0: 2020 3020 3120 3320 3820 2020 2020 2031    0 1 3 8      1
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│ │ │ │ +000192b0: 202d 2020 2020 7c0a 7c20 2020 3020 3120   -    |.|   0 1 
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│ │ │ │ -000193a0: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
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│ │ │ │ -000193e0: 3132 7820 7820 7820 202d 2031 3178 2078  12x x x  - 11x x
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│ │ │ │ +000193d0: 3678 2078 2078 2020 2d20 3132 7820 7820  6x x x  - 12x x 
│ │ │ │ +000193e0: 7820 202d 2031 3178 2078 2078 2020 2d20  x  - 11x x x  - 
│ │ │ │ +000193f0: 2020 2020 2020 7c0a 7c20 2020 3120 3220        |.|   1 2 
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│ │ │ │ -00019710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00019710: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │  00019720: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00019750: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00019790: 202b 2032 3978 2078 2078 2078 2020 2b20   + 29x x x x  + 
│ │ │ │ -000197a0: 3235 7820 7820 7820 7820 202b 2033 3378  25x x x x  + 33x
│ │ │ │ -000197b0: 2078 2078 2078 2020 2d20 2020 7c0a 7c20   x x x  -   |.| 
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│ │ │ │ -000197d0: 3420 3620 3820 2020 2020 2034 2036 2038  4 6 8      4 6 8
│ │ │ │ -000197e0: 2020 2020 2020 3020 3520 3620 3820 2020        0 5 6 8   
│ │ │ │ -000197f0: 2020 2031 2035 2036 2038 2020 2020 2020     1 5 6 8      
│ │ │ │ -00019800: 3420 3520 3620 3820 2020 2020 7c0a 7c2d  4 5 6 8     |.|-
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│ │ │ │ +000197a0: 7820 7820 202b 2033 3378 2078 2078 2078  x x  + 33x x x x
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│ │ │ │  00019830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00019870: 2020 2020 2020 2020 2032 2020 2020 2020           2      
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│ │ │ │ -000198f0: 2078 2020 2b20 2020 2020 2020 7c0a 7c20   x  +       |.| 
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│ │ │ │ -00019910: 3720 3820 2020 2020 2031 2037 2038 2020  7 8      1 7 8  
│ │ │ │ -00019920: 2020 2020 3020 3220 3720 3820 2020 2020      0 2 7 8     
│ │ │ │ -00019930: 2030 2033 2037 2038 2020 2020 3220 3320   0 3 7 8    2 3 
│ │ │ │ -00019940: 3720 3820 2020 2020 2020 2020 7c0a 7c2d  7 8         |.|-
│ │ │ │ +000198a0: 2020 2020 2020 7c0a 7c33 3778 2078 2078        |.|37x x x
│ │ │ │ +000198b0: 2020 2d20 3378 2078 2078 2078 2020 2d20    - 3x x x x  - 
│ │ │ │ +000198c0: 3437 7820 7820 7820 202b 2032 3178 2078  47x x x  + 21x x
│ │ │ │ +000198d0: 2078 2078 2020 2b20 3131 7820 7820 7820   x x  + 11x x x 
│ │ │ │ +000198e0: 7820 202b 2078 2078 2078 2078 2020 2b20  x  + x x x x  + 
│ │ │ │ +000198f0: 2020 2020 2020 7c0a 7c20 2020 3020 3720        |.|   0 7 
│ │ │ │ +00019900: 3820 2020 2020 3020 3120 3720 3820 2020  8     0 1 7 8   
│ │ │ │ +00019910: 2020 2031 2037 2038 2020 2020 2020 3020     1 7 8      0 
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│ │ │ │ +00019930: 2038 2020 2020 3220 3320 3720 3820 2020   8    2 3 7 8   
│ │ │ │ +00019940: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
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│ │ │ │  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 7c0a 7c20  ------------|.| 
│ │ │ │ +00019990: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │  000199a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000199b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000199c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000199d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000199e0: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ -000199f0: 3878 2078 2078 2078 2020 2d20 3435 7820  8x x x x  - 45x 
│ │ │ │ -00019a00: 7820 7820 7820 202b 2034 3778 2078 2078  x x x  + 47x x x
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│ │ │ │ -00019a20: 2b20 3333 7820 7820 7820 7820 202b 2034  + 33x x x x  + 4
│ │ │ │ -00019a30: 3778 2078 2078 2078 2020 2d20 7c0a 7c20  7x x x x  - |.| 
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│ │ │ │ -00019a50: 2034 2037 2038 2020 2020 2020 3320 3420   4 7 8      3 4 
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│ │ │ │ -00019a70: 2020 2020 2031 2035 2037 2038 2020 2020       1 5 7 8    
│ │ │ │ -00019a80: 2020 3220 3520 3720 3820 2020 7c0a 7c2d    2 5 7 8   |.|-
│ │ │ │ +000199e0: 2020 2020 2020 7c0a 7c34 3878 2078 2078        |.|48x x x
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│ │ │ │ +00019a00: 202b 2034 3778 2078 2078 2078 2020 2d20   + 47x x x x  - 
│ │ │ │ +00019a10: 3278 2078 2078 2078 2020 2b20 3333 7820  2x x x x  + 33x 
│ │ │ │ +00019a20: 7820 7820 7820 202b 2034 3778 2078 2078  x x x  + 47x x x
│ │ │ │ +00019a30: 2078 2020 2d20 7c0a 7c20 2020 3020 3420   x  - |.|   0 4 
│ │ │ │ +00019a40: 3720 3820 2020 2020 2031 2034 2037 2038  7 8      1 4 7 8
│ │ │ │ +00019a50: 2020 2020 2020 3320 3420 3720 3820 2020        3 4 7 8   
│ │ │ │ +00019a60: 2020 3020 3520 3720 3820 2020 2020 2031    0 5 7 8      1
│ │ │ │ +00019a70: 2035 2037 2038 2020 2020 2020 3220 3520   5 7 8      2 5 
│ │ │ │ +00019a80: 3720 3820 2020 7c0a 7c2d 2d2d 2d2d 2d2d  7 8   |.|-------
│ │ │ │  00019a90: 2d2d 2d2d 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 7c0a 7c20  ------------|.| 
│ │ │ │ -00019ae0: 2020 2020 2020 2020 2020 2020 3220 3220              2 2 
│ │ │ │ -00019af0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -00019b00: 3220 3220 2020 2020 2020 2020 2032 2020  2 2          2  
│ │ │ │ -00019b10: 2020 2020 3220 3220 2020 2020 2020 2020      2 2         
│ │ │ │ -00019b20: 3220 2020 2020 2020 2020 2032 7c0a 7c78  2          2|.|x
│ │ │ │ -00019b30: 2078 2078 2078 2020 2b20 3778 2078 2020   x x x  + 7x x  
│ │ │ │ -00019b40: 2d20 3239 7820 7820 7820 202b 2033 3678  - 29x x x  + 36x
│ │ │ │ -00019b50: 2078 2020 2d20 3131 7820 7820 7820 202b   x  - 11x x x  +
│ │ │ │ -00019b60: 2034 3878 2078 2020 2b20 3278 2078 2078   48x x  + 2x x x
│ │ │ │ -00019b70: 2020 2d20 3333 7820 7820 7820 7c0a 7c20    - 33x x x |.| 
│ │ │ │ -00019b80: 3420 3520 3720 3820 2020 2020 3020 3820  4 5 7 8     0 8 
│ │ │ │ -00019b90: 2020 2020 2030 2031 2038 2020 2020 2020       0 1 8      
│ │ │ │ -00019ba0: 3120 3820 2020 2020 2030 2032 2038 2020  1 8      0 2 8  
│ │ │ │ -00019bb0: 2020 2020 3220 3820 2020 2020 3020 3420      2 8     0 4 
│ │ │ │ -00019bc0: 3820 2020 2020 2031 2034 2038 7c0a 7c2d  8      1 4 8|.|-
│ │ │ │ +00019ad0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +00019ae0: 2020 2020 2020 3220 3220 2020 2020 2020        2 2       
│ │ │ │ +00019af0: 2020 2032 2020 2020 2020 3220 3220 2020     2      2 2   
│ │ │ │ +00019b00: 2020 2020 2020 2032 2020 2020 2020 3220         2      2 
│ │ │ │ +00019b10: 3220 2020 2020 2020 2020 3220 2020 2020  2         2     
│ │ │ │ +00019b20: 2020 2020 2032 7c0a 7c78 2078 2078 2078       2|.|x x x x
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│ │ │ │ +00019b40: 7820 7820 202b 2033 3678 2078 2020 2d20  x x  + 36x x  - 
│ │ │ │ +00019b50: 3131 7820 7820 7820 202b 2034 3878 2078  11x x x  + 48x x
│ │ │ │ +00019b60: 2020 2b20 3278 2078 2078 2020 2d20 3333    + 2x x x  - 33
│ │ │ │ +00019b70: 7820 7820 7820 7c0a 7c20 3420 3520 3720  x x x |.| 4 5 7 
│ │ │ │ +00019b80: 3820 2020 2020 3020 3820 2020 2020 2030  8     0 8      0
│ │ │ │ +00019b90: 2031 2038 2020 2020 2020 3120 3820 2020   1 8      1 8   
│ │ │ │ +00019ba0: 2020 2030 2032 2038 2020 2020 2020 3220     0 2 8      2 
│ │ │ │ +00019bb0: 3820 2020 2020 3020 3420 3820 2020 2020  8     0 4 8     
│ │ │ │ +00019bc0: 2031 2034 2038 7c0a 7c2d 2d2d 2d2d 2d2d   1 4 8|.|-------
│ │ │ │  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 7c0a 7c20  ------------|.| 
│ │ │ │ -00019c20: 2020 2020 2020 2020 3220 2020 2020 2032          2      2
│ │ │ │ -00019c30: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00019c10: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +00019c20: 2020 3220 2020 2020 2032 2032 2020 2020    2      2 2    
│ │ │ │ +00019c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c2d              |.|-
│ │ │ │ -00019c70: 2034 3778 2078 2078 2020 2d20 3438 7820   47x x x  - 48x 
│ │ │ │ -00019c80: 7820 202b 2034 3878 2078 2078 2078 2020  x  + 48x x x x  
│ │ │ │ -00019c90: 2d20 3438 7820 7820 7820 7820 202d 2034  - 48x x x x  - 4
│ │ │ │ -00019ca0: 3878 2078 2078 2078 2020 2b20 3438 7820  8x x x x  + 48x 
│ │ │ │ -00019cb0: 7820 7820 7820 202b 2020 2020 7c0a 7c20  x x x  +    |.| 
│ │ │ │ -00019cc0: 2020 2020 3220 3420 3820 2020 2020 2034      2 4 8      4
│ │ │ │ -00019cd0: 2038 2020 2020 2020 3320 3420 3620 3920   8      3 4 6 9 
│ │ │ │ -00019ce0: 2020 2020 2032 2035 2036 2039 2020 2020       2 5 6 9    
│ │ │ │ -00019cf0: 2020 3120 3320 3720 3920 2020 2020 2030    1 3 7 9      0
│ │ │ │ -00019d00: 2035 2037 2039 2020 2020 2020 7c0a 7c2d   5 7 9      |.|-
│ │ │ │ +00019c60: 2020 2020 2020 7c0a 7c2d 2034 3778 2078        |.|- 47x x
│ │ │ │ +00019c70: 2078 2020 2d20 3438 7820 7820 202b 2034   x  - 48x x  + 4
│ │ │ │ +00019c80: 3878 2078 2078 2078 2020 2d20 3438 7820  8x x x x  - 48x 
│ │ │ │ +00019c90: 7820 7820 7820 202d 2034 3878 2078 2078  x x x  - 48x x x
│ │ │ │ +00019ca0: 2078 2020 2b20 3438 7820 7820 7820 7820   x  + 48x x x x 
│ │ │ │ +00019cb0: 202b 2020 2020 7c0a 7c20 2020 2020 3220   +    |.|     2 
│ │ │ │ +00019cc0: 3420 3820 2020 2020 2034 2038 2020 2020  4 8      4 8    
│ │ │ │ +00019cd0: 2020 3320 3420 3620 3920 2020 2020 2032    3 4 6 9      2
│ │ │ │ +00019ce0: 2035 2036 2039 2020 2020 2020 3120 3320   5 6 9      1 3 
│ │ │ │ +00019cf0: 3720 3920 2020 2020 2030 2035 2037 2039  7 9      0 5 7 9
│ │ │ │ +00019d00: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │  00019d10: 2d2d 2d2d 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 7c0a 7c34  ------------|.|4
│ │ │ │ -00019d60: 3878 2078 2078 2078 2020 2d20 3438 7820  8x x x x  - 48x 
│ │ │ │ -00019d70: 7820 7820 7820 2020 2020 2020 2020 2020  x x x           
│ │ │ │ +00019d50: 2d2d 2d2d 2d2d 7c0a 7c34 3878 2078 2078  ------|.|48x x x
│ │ │ │ +00019d60: 2078 2020 2d20 3438 7820 7820 7820 7820   x  - 48x x x x 
│ │ │ │ +00019d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00019db0: 2020 3120 3220 3820 3920 2020 2020 2030    1 2 8 9      0
│ │ │ │ -00019dc0: 2034 2038 2039 2020 2020 2020 2020 2020   4 8 9          
│ │ │ │ +00019da0: 2020 2020 2020 7c0a 7c20 2020 3120 3220        |.|   1 2 
│ │ │ │ +00019db0: 3820 3920 2020 2020 2030 2034 2038 2039  8 9      0 4 8 9
│ │ │ │ +00019dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019df0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00019df0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  00019e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00019e50: 3320 3a20 7469 6d65 2070 7369 203d 2061  3 : time psi = a
│ │ │ │ -00019e60: 7070 726f 7869 6d61 7465 496e 7665 7273  pproximateInvers
│ │ │ │ -00019e70: 654d 6170 2070 6869 2020 2020 2020 2020  eMap phi        
│ │ │ │ +00019e40: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 7469  ------+.|i3 : ti
│ │ │ │ +00019e50: 6d65 2070 7369 203d 2061 7070 726f 7869  me psi = approxi
│ │ │ │ +00019e60: 6d61 7465 496e 7665 7273 654d 6170 2070  mateInverseMap p
│ │ │ │ +00019e70: 6869 2020 2020 2020 2020 2020 2020 2020  hi              
│ │ │ │  00019e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e90: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ -00019ea0: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ -00019eb0: 6572 7365 4d61 703a 2073 7465 7020 3120  erseMap: step 1 
│ │ │ │ -00019ec0: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │ +00019e90: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ +00019ea0: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +00019eb0: 703a 2073 7465 7020 3120 6f66 2031 3020  p: step 1 of 10 
│ │ │ │ +00019ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019ee0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ -00019ef0: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ -00019f00: 6572 7365 4d61 703a 2073 7465 7020 3220  erseMap: step 2 
│ │ │ │ -00019f10: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │ +00019ee0: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ +00019ef0: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +00019f00: 703a 2073 7465 7020 3220 6f66 2031 3020  p: step 2 of 10 
│ │ │ │ +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 7c0a 7c2d              |.|-
│ │ │ │ -00019f40: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ -00019f50: 6572 7365 4d61 703a 2073 7465 7020 3320  erseMap: step 3 
│ │ │ │ -00019f60: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │ +00019f30: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ +00019f40: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +00019f50: 703a 2073 7465 7020 3320 6f66 2031 3020  p: step 3 of 10 
│ │ │ │ +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 7c0a 7c2d              |.|-
│ │ │ │ -00019f90: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ -00019fa0: 6572 7365 4d61 703a 2073 7465 7020 3420  erseMap: step 4 
│ │ │ │ -00019fb0: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │ +00019f80: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ +00019f90: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +00019fa0: 703a 2073 7465 7020 3420 6f66 2031 3020  p: step 4 of 10 
│ │ │ │ +00019fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ -00019fe0: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ -00019ff0: 6572 7365 4d61 703a 2073 7465 7020 3520  erseMap: step 5 
│ │ │ │ -0001a000: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │ +00019fd0: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ +00019fe0: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +00019ff0: 703a 2073 7465 7020 3520 6f66 2031 3020  p: step 5 of 10 
│ │ │ │ +0001a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a020: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ -0001a030: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ -0001a040: 6572 7365 4d61 703a 2073 7465 7020 3620  erseMap: step 6 
│ │ │ │ -0001a050: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │ +0001a020: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ +0001a030: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +0001a040: 703a 2073 7465 7020 3620 6f66 2031 3020  p: step 6 of 10 
│ │ │ │ +0001a050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a070: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ -0001a080: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ -0001a090: 6572 7365 4d61 703a 2073 7465 7020 3720  erseMap: step 7 
│ │ │ │ -0001a0a0: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │ +0001a070: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ +0001a080: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +0001a090: 703a 2073 7465 7020 3720 6f66 2031 3020  p: step 7 of 10 
│ │ │ │ +0001a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a0c0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ -0001a0d0: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ -0001a0e0: 6572 7365 4d61 703a 2073 7465 7020 3820  erseMap: step 8 
│ │ │ │ -0001a0f0: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │ +0001a0c0: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ +0001a0d0: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +0001a0e0: 703a 2073 7465 7020 3820 6f66 2031 3020  p: step 8 of 10 
│ │ │ │ +0001a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a110: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ -0001a120: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ -0001a130: 6572 7365 4d61 703a 2073 7465 7020 3920  erseMap: step 9 
│ │ │ │ -0001a140: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │ +0001a110: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ +0001a120: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +0001a130: 703a 2073 7465 7020 3920 6f66 2031 3020  p: step 9 of 10 
│ │ │ │ +0001a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a160: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
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│ │ │ │ +0001a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a1b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001a1c0: 2d2d 2075 7365 6420 302e 3237 3230 3137  -- used 0.272017
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│ │ │ │  0001a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001a390: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  0001a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a3b0: 2020 2020 2020 2020 2020 2020 2020 205a                 Z
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│ │ │ │ -0001a3e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001a3f0: 2020 2020 7461 7267 6574 3a20 7375 6276      target: subv
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│ │ │ │  0001a440: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  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 7c0a 7c20              |.| 
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│ │ │ │ -0001a610: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001a620: 2020 2020 6465 6669 6e69 6e67 2066 6f72      defining for
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│ │ │ │  0001a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a840: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a860: 2020 2020 2074 2074 2020 2d20 7420 7420       t t  - t t 
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│ │ │ │ -0001a930: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  0001ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0001acd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ace0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001ad90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0001ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001ae30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae50: 2020 2020 2020 3120 3320 2020 2030 2034        1 3    0 4
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│ │ │ │  0001ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001ae80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020                  
│ │ │ │ -0001aed0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001aed0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aef0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -0001af00: 2020 2020 2020 3220 2020 2020 2020 2020        2         
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│ │ │ │ -0001af20: 2020 2020 2020 2020 2020 3220 7c0a 7c20            2 |.| 
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│ │ │ │ +0001af20: 2020 2020 3220 7c0a 7c20 2020 2020 2020      2 |.|       
│ │ │ │ +0001af30: 2020 2020 2020 2020 2020 2020 2020 2074                 t
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│ │ │ │ -0001afe0: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
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│ │ │ │ +0001afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b010: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b010: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b030: 2020 2020 2020 2020 2020 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 7c0a 7c6f              |.|o
│ │ │ │ -0001b070: 3320 3a20 5261 7469 6f6e 616c 4d61 7020  3 : RationalMap 
│ │ │ │ -0001b080: 2871 7561 6472 6174 6963 2072 6174 696f  (quadratic ratio
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│ │ │ │ +0001b060: 2020 2020 2020 7c0a 7c6f 3320 3a20 5261        |.|o3 : Ra
│ │ │ │ +0001b070: 7469 6f6e 616c 4d61 7020 2871 7561 6472  tionalMap (quadr
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│ │ │ │ +0001b0b0: 505e 3929 2020 7c0a 7c2d 2d2d 2d2d 2d2d  P^9)  |.|-------
│ │ │ │  0001b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001b110: 6566 696e 6564 2062 7920 2020 2020 2020  efined by       
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│ │ │ │ +0001b110: 2062 7920 2020 2020 2020 2020 2020 2020   by             
│ │ │ │  0001b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b150: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b150: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b1a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b1a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b1f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001b200: 2020 2020 2020 2032 2020 2020 2020 3220         2      2 
│ │ │ │ -0001b210: 3220 2020 2020 2020 2020 2032 2020 2020  2          2    
│ │ │ │ -0001b220: 2020 3220 3220 2020 2020 2020 2020 2020    2 2           
│ │ │ │ -0001b230: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0001b240: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001b250: 2b20 3139 7820 7820 7820 202b 2078 2078  + 19x x x  + x x
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│ │ │ │ -0001b2a0: 2020 2020 2031 2033 2034 2020 2020 3020       1 3 4    0 
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│ │ │ │ -0001b2d0: 2035 2020 2020 2020 3120 3220 3520 2020   5      1 2 5   
│ │ │ │ -0001b2e0: 2020 2030 2031 2033 2035 2020 7c0a 7c20     0 1 3 5  |.| 
│ │ │ │ +0001b1f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b200: 2032 2020 2020 2020 3220 3220 2020 2020   2      2 2     
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│ │ │ │ +0001b2e0: 2033 2035 2020 7c0a 7c20 2020 2020 2020   3 5  |.|       
│ │ │ │  0001b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b330: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b330: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b380: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b380: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001b3d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b3d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b420: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b420: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b450: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001b470: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b470: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b4c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b4c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b4d0: 2020 2020 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 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b510: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b560: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b560: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b5b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b600: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b600: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b650: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b650: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b6a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b6a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b6f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b6f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b740: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b740: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b790: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b790: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b7b0: 2020 2020 2020 2020 2020 2020 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 7c0a 7c20              |.| 
│ │ │ │ +0001b7e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b830: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b830: 2020 2020 2020 7c0a 7c20 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 2020                  
│ │ │ │ -0001b880: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b880: 2020 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │  0001b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b8d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b8d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b920: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b920: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  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 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b970: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b9c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001ba10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20              |.| 
│ │ │ │ +0001ba60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020 2020                  
│ │ │ │ -0001bab0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001bab0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001baf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bb00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001bb00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bb50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001bb50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001bb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bba0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001bba0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001bbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bbf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001bbf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bc10: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +0001bc10: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  0001bc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bc40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001bc50: 2b20 3474 2074 2020 2b20 3339 7420 7420  + 4t t  + 39t t 
│ │ │ │ -0001bc60: 202d 2032 3274 2074 2020 2b20 3134 7420   - 22t t  + 14t 
│ │ │ │ -0001bc70: 202d 2074 2074 2020 2d20 3674 2074 2020   - t t  - 6t t  
│ │ │ │ -0001bc80: 2d20 3431 7420 7420 202b 2034 3274 2074  - 41t t  + 42t t
│ │ │ │ -0001bc90: 2020 2b20 3233 7420 7420 202b 7c0a 7c20    + 23t t  +|.| 
│ │ │ │ -0001bca0: 2020 2020 3020 3420 2020 2020 2031 2034      0 4      1 4
│ │ │ │ -0001bcb0: 2020 2020 2020 3320 3420 2020 2020 2034        3 4      4
│ │ │ │ -0001bcc0: 2020 2020 3020 3520 2020 2020 3120 3520      0 5     1 5 
│ │ │ │ -0001bcd0: 2020 2020 2032 2035 2020 2020 2020 3320       2 5      3 
│ │ │ │ -0001bce0: 3520 2020 2020 2034 2035 2020 7c0a 7c2d  5      4 5  |.|-
│ │ │ │ +0001bc40: 2020 2020 2020 7c0a 7c20 2b20 3474 2074        |.| + 4t t
│ │ │ │ +0001bc50: 2020 2b20 3339 7420 7420 202d 2032 3274    + 39t t  - 22t
│ │ │ │ +0001bc60: 2074 2020 2b20 3134 7420 202d 2074 2074   t  + 14t  - t t
│ │ │ │ +0001bc70: 2020 2d20 3674 2074 2020 2d20 3431 7420    - 6t t  - 41t 
│ │ │ │ +0001bc80: 7420 202b 2034 3274 2074 2020 2b20 3233  t  + 42t t  + 23
│ │ │ │ +0001bc90: 7420 7420 202b 7c0a 7c20 2020 2020 3020  t t  +|.|     0 
│ │ │ │ +0001bca0: 3420 2020 2020 2031 2034 2020 2020 2020  4      1 4      
│ │ │ │ +0001bcb0: 3320 3420 2020 2020 2034 2020 2020 3020  3 4      4    0 
│ │ │ │ +0001bcc0: 3520 2020 2020 3120 3520 2020 2020 2032  5     1 5      2
│ │ │ │ +0001bcd0: 2035 2020 2020 2020 3320 3520 2020 2020   5      3 5     
│ │ │ │ +0001bce0: 2034 2035 2020 7c0a 7c2d 2d2d 2d2d 2d2d   4 5  |.|-------
│ │ │ │  0001bcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0001bd40: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +0001bd30: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +0001bd40: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  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: 2032 2032 2020 2020 2020 2020 7c0a 7c31   2 2        |.|1
│ │ │ │ -0001bd90: 3978 2078 2078 2078 2020 2b20 3134 7820  9x x x x  + 14x 
│ │ │ │ -0001bda0: 7820 7820 202b 2031 3178 2078 2078 2078  x x  + 11x x x x
│ │ │ │ -0001bdb0: 2020 2d20 3139 7820 7820 7820 7820 202b    - 19x x x x  +
│ │ │ │ -0001bdc0: 2032 3178 2078 2078 2078 2020 2b20 3236   21x x x x  + 26
│ │ │ │ -0001bdd0: 7820 7820 202b 2031 3978 2020 7c0a 7c20  x x  + 19x  |.| 
│ │ │ │ -0001bde0: 2020 3120 3220 3320 3520 2020 2020 2030    1 2 3 5      0
│ │ │ │ -0001bdf0: 2034 2035 2020 2020 2020 3020 3220 3420   4 5      0 2 4 
│ │ │ │ -0001be00: 3520 2020 2020 2030 2033 2034 2035 2020  5      0 3 4 5  
│ │ │ │ -0001be10: 2020 2020 3220 3320 3420 3520 2020 2020      2 3 4 5     
│ │ │ │ -0001be20: 2030 2035 2020 2020 2020 3020 7c0a 7c20   0 5      0 |.| 
│ │ │ │ +0001bd70: 2020 2020 2020 2020 2020 2032 2032 2020             2 2  
│ │ │ │ +0001bd80: 2020 2020 2020 7c0a 7c31 3978 2078 2078        |.|19x x x
│ │ │ │ +0001bd90: 2078 2020 2b20 3134 7820 7820 7820 202b   x  + 14x x x  +
│ │ │ │ +0001bda0: 2031 3178 2078 2078 2078 2020 2d20 3139   11x x x x  - 19
│ │ │ │ +0001bdb0: 7820 7820 7820 7820 202b 2032 3178 2078  x x x x  + 21x x
│ │ │ │ +0001bdc0: 2078 2078 2020 2b20 3236 7820 7820 202b   x x  + 26x x  +
│ │ │ │ +0001bdd0: 2031 3978 2020 7c0a 7c20 2020 3120 3220   19x  |.|   1 2 
│ │ │ │ +0001bde0: 3320 3520 2020 2020 2030 2034 2035 2020  3 5      0 4 5  
│ │ │ │ +0001bdf0: 2020 2020 3020 3220 3420 3520 2020 2020      0 2 4 5     
│ │ │ │ +0001be00: 2030 2033 2034 2035 2020 2020 2020 3220   0 3 4 5      2 
│ │ │ │ +0001be10: 3320 3420 3520 2020 2020 2030 2035 2020  3 4 5      0 5  
│ │ │ │ +0001be20: 2020 2020 3020 7c0a 7c20 2020 2020 2020      0 |.|       
│ │ │ │  0001be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001be70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001be80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bec0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001bec0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001bf10: 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001bf60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bfb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001bfb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001bfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c000: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c000: 2020 2020 2020 7c0a 7c20 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 2020                  
│ │ │ │  0001c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c050: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c050: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c0a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c0a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c0f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c0f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c140: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c140: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c190: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c190: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c1e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c1e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c230: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c230: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c280: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c280: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c2a0: 2020 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 7c0a 7c20              |.| 
│ │ │ │ +0001c2d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c320: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c320: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c370: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c3c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c3c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c410: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c410: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c460: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c460: 2020 2020 2020 7c0a 7c20 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 2020 2020 2020                  
│ │ │ │  0001c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c4b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c4b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20              |.| 
│ │ │ │ +0001c500: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c550: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c550: 2020 2020 2020 7c0a 7c20 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: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c5a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c5f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c5f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c640: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c640: 2020 2020 2020 7c0a 7c20 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c690: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c690: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c6e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c6e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c6f0: 2020 2020 2020 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 7c0a 7c20              |.| 
│ │ │ │ -0001c740: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0001c750: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +0001c730: 2020 2020 2020 7c0a 7c20 2020 3220 2020        |.|   2   
│ │ │ │ +0001c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c750: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  0001c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c780: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ -0001c790: 3174 2020 2b20 3234 7420 7420 202d 2038  1t  + 24t t  - 8
│ │ │ │ -0001c7a0: 7420 7420 202d 2032 3174 2020 2b20 3431  t t  - 21t  + 41
│ │ │ │ -0001c7b0: 7420 7420 202b 2031 3374 2074 2020 2b20  t t  + 13t t  + 
│ │ │ │ -0001c7c0: 3135 7420 7420 202d 2032 3274 2074 2020  15t t  - 22t t  
│ │ │ │ -0001c7d0: 2b20 3338 7420 7420 202d 2020 7c0a 7c20  + 38t t  -  |.| 
│ │ │ │ -0001c7e0: 2020 3520 2020 2020 2030 2036 2020 2020    5      0 6    
│ │ │ │ -0001c7f0: 2033 2036 2020 2020 2020 3620 2020 2020   3 6      6     
│ │ │ │ -0001c800: 2030 2037 2020 2020 2020 3120 3720 2020   0 7      1 7   
│ │ │ │ -0001c810: 2020 2033 2037 2020 2020 2020 3420 3720     3 7      4 7 
│ │ │ │ -0001c820: 2020 2020 2036 2037 2020 2020 7c0a 7c2d       6 7    |.|-
│ │ │ │ +0001c780: 2020 2020 2020 7c0a 7c32 3174 2020 2b20        |.|21t  + 
│ │ │ │ +0001c790: 3234 7420 7420 202d 2038 7420 7420 202d  24t t  - 8t t  -
│ │ │ │ +0001c7a0: 2032 3174 2020 2b20 3431 7420 7420 202b   21t  + 41t t  +
│ │ │ │ +0001c7b0: 2031 3374 2074 2020 2b20 3135 7420 7420   13t t  + 15t t 
│ │ │ │ +0001c7c0: 202d 2032 3274 2074 2020 2b20 3338 7420   - 22t t  + 38t 
│ │ │ │ +0001c7d0: 7420 202d 2020 7c0a 7c20 2020 3520 2020  t  -  |.|   5   
│ │ │ │ +0001c7e0: 2020 2030 2036 2020 2020 2033 2036 2020     0 6     3 6  
│ │ │ │ +0001c7f0: 2020 2020 3620 2020 2020 2030 2037 2020      6      0 7  
│ │ │ │ +0001c800: 2020 2020 3120 3720 2020 2020 2033 2037      1 7      3 7
│ │ │ │ +0001c810: 2020 2020 2020 3420 3720 2020 2020 2036        4 7      6
│ │ │ │ +0001c820: 2037 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   7    |.|-------
│ │ │ │  0001c830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0001c880: 2020 3220 2020 2020 2032 2032 2020 2020    2      2 2    
│ │ │ │ -0001c890: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0001c8a0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +0001c870: 2d2d 2d2d 2d2d 7c0a 7c20 2020 3220 2020  ------|.|   2   
│ │ │ │ +0001c880: 2020 2032 2032 2020 2020 2020 2020 3220     2 2        2 
│ │ │ │ +0001c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c8a0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  0001c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c8c0: 2020 2020 2020 2020 2020 2020 7c0a 7c78              |.|x
│ │ │ │ -0001c8d0: 2078 2020 2b20 3338 7820 7820 202d 2033   x  + 38x x  - 3
│ │ │ │ -0001c8e0: 3078 2078 2078 2020 2d20 3131 7820 7820  0x x x  - 11x x 
│ │ │ │ -0001c8f0: 7820 7820 202d 2031 3178 2078 2078 2020  x x  - 11x x x  
│ │ │ │ -0001c900: 2b20 3330 7820 7820 7820 7820 202b 2032  + 30x x x x  + 2
│ │ │ │ -0001c910: 3278 2078 2078 2078 2020 2b20 7c0a 7c20  2x x x x  + |.| 
│ │ │ │ -0001c920: 3220 3520 2020 2020 2032 2035 2020 2020  2 5      2 5    
│ │ │ │ -0001c930: 2020 3120 3220 3620 2020 2020 2031 2032    1 2 6      1 2
│ │ │ │ -0001c940: 2033 2036 2020 2020 2020 3120 3320 3620   3 6      1 3 6 
│ │ │ │ -0001c950: 2020 2020 2030 2032 2034 2036 2020 2020       0 2 4 6    
│ │ │ │ -0001c960: 2020 3120 3220 3420 3620 2020 7c0a 7c20    1 2 4 6   |.| 
│ │ │ │ +0001c8c0: 2020 2020 2020 7c0a 7c78 2078 2020 2b20        |.|x x  + 
│ │ │ │ +0001c8d0: 3338 7820 7820 202d 2033 3078 2078 2078  38x x  - 30x x x
│ │ │ │ +0001c8e0: 2020 2d20 3131 7820 7820 7820 7820 202d    - 11x x x x  -
│ │ │ │ +0001c8f0: 2031 3178 2078 2078 2020 2b20 3330 7820   11x x x  + 30x 
│ │ │ │ +0001c900: 7820 7820 7820 202b 2032 3278 2078 2078  x x x  + 22x x x
│ │ │ │ +0001c910: 2078 2020 2b20 7c0a 7c20 3220 3520 2020   x  + |.| 2 5   
│ │ │ │ +0001c920: 2020 2032 2035 2020 2020 2020 3120 3220     2 5      1 2 
│ │ │ │ +0001c930: 3620 2020 2020 2031 2032 2033 2036 2020  6      1 2 3 6  
│ │ │ │ +0001c940: 2020 2020 3120 3320 3620 2020 2020 2030      1 3 6      0
│ │ │ │ +0001c950: 2032 2034 2036 2020 2020 2020 3120 3220   2 4 6      1 2 
│ │ │ │ +0001c960: 3420 3620 2020 7c0a 7c20 2020 2020 2020  4 6   |.|       
│ │ │ │  0001c970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c980: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001c9b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c9b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001ca00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001ca10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001ca50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  0001ca60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ca70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001caa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001caa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001cab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001caf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001caf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001cb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cb40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001cb40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cb90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001cb90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001cba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cbe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001cbe0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001cbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20              |.| 
│ │ │ │ +0001cc30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001cc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001cc80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001cc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001ccd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0001cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001cd70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  0001cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001cdc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001cdc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001cdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001ce10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001ce20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ce30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ce40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ce50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001ce60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001ce70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ce80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ce90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ceb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001ceb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001cec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ced0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20              |.| 
│ │ │ │ +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 7c0a 7c20              |.| 
│ │ │ │ +0001cf50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  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 2020 7c0a 7c20              |.| 
│ │ │ │ +0001cfa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001cfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cff0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001cff0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d010: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0001d050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d090: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d090: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001d0e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d0e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d130: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d130: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d180: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d180: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d1a0: 2020 2020 2020 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 7c0a 7c20              |.| 
│ │ │ │ +0001d1d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d220: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d220: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d270: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001d280: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0001d270: 2020 2020 2020 7c0a 7c20 2020 3220 2020        |.|   2   
│ │ │ │ +0001d280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d2c0: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ -0001d2d0: 3574 2020 2d20 3435 7420 7420 202b 2032  5t  - 45t t  + 2
│ │ │ │ -0001d2e0: 3074 2074 2020 2b20 3434 7420 7420 202d  0t t  + 44t t  -
│ │ │ │ -0001d2f0: 2034 3074 2074 2020 2d20 3232 7420 7420   40t t  - 22t t 
│ │ │ │ -0001d300: 202b 2033 3774 2074 2020 2b20 3232 7420   + 37t t  + 22t 
│ │ │ │ -0001d310: 7420 202b 2032 3874 2074 2020 7c0a 7c20  t  + 28t t  |.| 
│ │ │ │ -0001d320: 2020 3720 2020 2020 2030 2038 2020 2020    7      0 8    
│ │ │ │ -0001d330: 2020 3120 3820 2020 2020 2032 2038 2020    1 8      2 8  
│ │ │ │ -0001d340: 2020 2020 3320 3820 2020 2020 2034 2038      3 8      4 8
│ │ │ │ -0001d350: 2020 2020 2020 3520 3820 2020 2020 2036        5 8      6
│ │ │ │ -0001d360: 2038 2020 2020 2020 3720 3820 7c0a 7c2d   8      7 8 |.|-
│ │ │ │ +0001d2c0: 2020 2020 2020 7c0a 7c34 3574 2020 2d20        |.|45t  - 
│ │ │ │ +0001d2d0: 3435 7420 7420 202b 2032 3074 2074 2020  45t t  + 20t t  
│ │ │ │ +0001d2e0: 2b20 3434 7420 7420 202d 2034 3074 2074  + 44t t  - 40t t
│ │ │ │ +0001d2f0: 2020 2d20 3232 7420 7420 202b 2033 3774    - 22t t  + 37t
│ │ │ │ +0001d300: 2074 2020 2b20 3232 7420 7420 202b 2032   t  + 22t t  + 2
│ │ │ │ +0001d310: 3874 2074 2020 7c0a 7c20 2020 3720 2020  8t t  |.|   7   
│ │ │ │ +0001d320: 2020 2030 2038 2020 2020 2020 3120 3820     0 8      1 8 
│ │ │ │ +0001d330: 2020 2020 2032 2038 2020 2020 2020 3320       2 8      3 
│ │ │ │ +0001d340: 3820 2020 2020 2034 2038 2020 2020 2020  8      4 8      
│ │ │ │ +0001d350: 3520 3820 2020 2020 2036 2038 2020 2020  5 8      6 8    
│ │ │ │ +0001d360: 2020 3720 3820 7c0a 7c2d 2d2d 2d2d 2d2d    7 8 |.|-------
│ │ │ │  0001d370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0001d3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d3d0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0001d3b0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +0001d3c0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0001d3d0: 2020 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 7c0a 7c31              |.|1
│ │ │ │ -0001d410: 3178 2078 2078 2078 2020 2d20 3232 7820  1x x x x  - 22x 
│ │ │ │ -0001d420: 7820 7820 202b 2031 3078 2078 2078 2078  x x  + 10x x x x
│ │ │ │ -0001d430: 2020 2b20 3131 7820 7820 7820 7820 202d    + 11x x x x  -
│ │ │ │ -0001d440: 2034 3278 2078 2078 2078 2020 2d20 3130   42x x x x  - 10
│ │ │ │ -0001d450: 7820 7820 7820 7820 202b 2020 7c0a 7c20  x x x x  +  |.| 
│ │ │ │ -0001d460: 2020 3020 3320 3420 3620 2020 2020 2030    0 3 4 6      0
│ │ │ │ -0001d470: 2034 2036 2020 2020 2020 3120 3220 3520   4 6      1 2 5 
│ │ │ │ -0001d480: 3620 2020 2020 2030 2033 2035 2036 2020  6      0 3 5 6  
│ │ │ │ -0001d490: 2020 2020 3120 3320 3520 3620 2020 2020      1 3 5 6     
│ │ │ │ -0001d4a0: 2030 2034 2035 2036 2020 2020 7c0a 7c20   0 4 5 6    |.| 
│ │ │ │ +0001d400: 2020 2020 2020 7c0a 7c31 3178 2078 2078        |.|11x x x
│ │ │ │ +0001d410: 2078 2020 2d20 3232 7820 7820 7820 202b   x  - 22x x x  +
│ │ │ │ +0001d420: 2031 3078 2078 2078 2078 2020 2b20 3131   10x x x x  + 11
│ │ │ │ +0001d430: 7820 7820 7820 7820 202d 2034 3278 2078  x x x x  - 42x x
│ │ │ │ +0001d440: 2078 2078 2020 2d20 3130 7820 7820 7820   x x  - 10x x x 
│ │ │ │ +0001d450: 7820 202b 2020 7c0a 7c20 2020 3020 3320  x  +  |.|   0 3 
│ │ │ │ +0001d460: 3420 3620 2020 2020 2030 2034 2036 2020  4 6      0 4 6  
│ │ │ │ +0001d470: 2020 2020 3120 3220 3520 3620 2020 2020      1 2 5 6     
│ │ │ │ +0001d480: 2030 2033 2035 2036 2020 2020 2020 3120   0 3 5 6      1 
│ │ │ │ +0001d490: 3320 3520 3620 2020 2020 2030 2034 2035  3 5 6      0 4 5
│ │ │ │ +0001d4a0: 2036 2020 2020 7c0a 7c20 2020 2020 2020   6    |.|       
│ │ │ │  0001d4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d4f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d4f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d540: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +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: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d590: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d5e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d5e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d630: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d630: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d680: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d680: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d6d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d6d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d720: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d720: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d770: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d770: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d7c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d7c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d810: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d810: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d860: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d860: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d8b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d8b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d8c0: 2020 2020 2020 2020 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 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d900: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d950: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d950: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d9a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d9a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001d9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d9f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d9f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001da00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001da20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001da30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001da40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001da40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001da50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001da60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001da80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001da90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001da90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001daa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dae0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001dae0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001daf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001db30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001db30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001db40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001db80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001db80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001db90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dbd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001dbd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001dbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dc20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001dc20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001dc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dc70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001dc70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001dc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dcc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001dcc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001dcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dd10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001dd10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001dd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dd60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001dd60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001dd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ddb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001ddc0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0001ddb0: 2020 2020 2020 7c0a 7c20 2020 2032 2020        |.|    2  
│ │ │ │ +0001ddc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001de00: 2020 2020 2020 2020 2020 2020 7c0a 7c2b              |.|+
│ │ │ │ -0001de10: 2032 7420 2020 2020 2020 2020 2020 2020   2t             
│ │ │ │ +0001de00: 2020 2020 2020 7c0a 7c2b 2032 7420 2020        |.|+ 2t   
│ │ │ │ +0001de10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001de40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001de50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001de60: 2020 2038 2020 2020 2020 2020 2020 2020     8            
│ │ │ │ +0001de50: 2020 2020 2020 7c0a 7c20 2020 2038 2020        |.|    8  
│ │ │ │ +0001de60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001de90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dea0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0001dea0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │  0001deb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001dec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ded0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001dee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001def0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0001df00: 2020 2020 3220 2020 2020 2020 2032 2032      2        2 2
│ │ │ │ -0001df10: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0001df20: 3220 3220 2020 2020 2020 2020 2032 2020  2 2          2  
│ │ │ │ -0001df30: 2020 2020 3220 3220 2020 2020 2020 2020      2 2         
│ │ │ │ -0001df40: 3220 2020 2020 2020 2020 2032 7c0a 7c34  2          2|.|4
│ │ │ │ -0001df50: 3278 2078 2078 2020 2d20 3338 7820 7820  2x x x  - 38x x 
│ │ │ │ -0001df60: 202d 2033 3778 2078 2078 2020 2b20 3778   - 37x x x  + 7x
│ │ │ │ -0001df70: 2078 2020 2b20 3238 7820 7820 7820 202d   x  + 28x x x  -
│ │ │ │ -0001df80: 2033 3878 2078 2020 2d20 3378 2078 2078   38x x  - 3x x x
│ │ │ │ -0001df90: 2020 2d20 3239 7820 7820 7820 7c0a 7c20    - 29x x x |.| 
│ │ │ │ -0001dfa0: 2020 3020 3520 3620 2020 2020 2032 2036    0 5 6      2 6
│ │ │ │ -0001dfb0: 2020 2020 2020 3220 3320 3620 2020 2020        2 3 6     
│ │ │ │ -0001dfc0: 3320 3620 2020 2020 2032 2034 2036 2020  3 6      2 4 6  
│ │ │ │ -0001dfd0: 2020 2020 3420 3620 2020 2020 3220 3520      4 6     2 5 
│ │ │ │ -0001dfe0: 3620 2020 2020 2033 2035 2036 7c0a 7c2d  6      3 5 6|.|-
│ │ │ │ +0001def0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 3220  ------|.|     2 
│ │ │ │ +0001df00: 2020 2020 2020 2032 2032 2020 2020 2020         2 2      
│ │ │ │ +0001df10: 2020 2020 3220 2020 2020 3220 3220 2020      2     2 2   
│ │ │ │ +0001df20: 2020 2020 2020 2032 2020 2020 2020 3220         2      2 
│ │ │ │ +0001df30: 3220 2020 2020 2020 2020 3220 2020 2020  2         2     
│ │ │ │ +0001df40: 2020 2020 2032 7c0a 7c34 3278 2078 2078       2|.|42x x x
│ │ │ │ +0001df50: 2020 2d20 3338 7820 7820 202d 2033 3778    - 38x x  - 37x
│ │ │ │ +0001df60: 2078 2078 2020 2b20 3778 2078 2020 2b20   x x  + 7x x  + 
│ │ │ │ +0001df70: 3238 7820 7820 7820 202d 2033 3878 2078  28x x x  - 38x x
│ │ │ │ +0001df80: 2020 2d20 3378 2078 2078 2020 2d20 3239    - 3x x x  - 29
│ │ │ │ +0001df90: 7820 7820 7820 7c0a 7c20 2020 3020 3520  x x x |.|   0 5 
│ │ │ │ +0001dfa0: 3620 2020 2020 2032 2036 2020 2020 2020  6      2 6      
│ │ │ │ +0001dfb0: 3220 3320 3620 2020 2020 3320 3620 2020  2 3 6     3 6   
│ │ │ │ +0001dfc0: 2020 2032 2034 2036 2020 2020 2020 3420     2 4 6      4 
│ │ │ │ +0001dfd0: 3620 2020 2020 3220 3520 3620 2020 2020  6     2 5 6     
│ │ │ │ +0001dfe0: 2033 2035 2036 7c0a 7c2d 2d2d 2d2d 2d2d   3 5 6|.|-------
│ │ │ │  0001dff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0001e040: 2020 2020 2020 2020 3220 2020 2020 2032          2      2
│ │ │ │ -0001e050: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0001e060: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0001e070: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0001e080: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ -0001e090: 2034 3778 2078 2078 2020 2b20 3336 7820   47x x x  + 36x 
│ │ │ │ -0001e0a0: 7820 202b 2033 3078 2078 2078 2078 2020  x  + 30x x x x  
│ │ │ │ -0001e0b0: 2d20 3232 7820 7820 7820 202d 2032 3078  - 22x x x  - 20x
│ │ │ │ -0001e0c0: 2078 2078 2078 2020 2d20 3431 7820 7820   x x x  - 41x x 
│ │ │ │ -0001e0d0: 7820 202d 2020 2020 2020 2020 7c0a 7c20  x  -        |.| 
│ │ │ │ -0001e0e0: 2020 2020 3420 3520 3620 2020 2020 2035      4 5 6      5
│ │ │ │ -0001e0f0: 2036 2020 2020 2020 3020 3120 3220 3720   6      0 1 2 7 
│ │ │ │ -0001e100: 2020 2020 2031 2032 2037 2020 2020 2020       1 2 7      
│ │ │ │ -0001e110: 3120 3220 3320 3720 2020 2020 2031 2033  1 2 3 7      1 3
│ │ │ │ -0001e120: 2037 2020 2020 2020 2020 2020 7c0a 7c2d   7          |.|-
│ │ │ │ +0001e030: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +0001e040: 2020 3220 2020 2020 2032 2032 2020 2020    2      2 2    
│ │ │ │ +0001e050: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +0001e060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e070: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0001e080: 2020 2020 2020 7c0a 7c2d 2034 3778 2078        |.|- 47x x
│ │ │ │ +0001e090: 2078 2020 2b20 3336 7820 7820 202b 2033   x  + 36x x  + 3
│ │ │ │ +0001e0a0: 3078 2078 2078 2078 2020 2d20 3232 7820  0x x x x  - 22x 
│ │ │ │ +0001e0b0: 7820 7820 202d 2032 3078 2078 2078 2078  x x  - 20x x x x
│ │ │ │ +0001e0c0: 2020 2d20 3431 7820 7820 7820 202d 2020    - 41x x x  -  
│ │ │ │ +0001e0d0: 2020 2020 2020 7c0a 7c20 2020 2020 3420        |.|     4 
│ │ │ │ +0001e0e0: 3520 3620 2020 2020 2035 2036 2020 2020  5 6      5 6    
│ │ │ │ +0001e0f0: 2020 3020 3120 3220 3720 2020 2020 2031    0 1 2 7      1
│ │ │ │ +0001e100: 2032 2037 2020 2020 2020 3120 3220 3320   2 7      1 2 3 
│ │ │ │ +0001e110: 3720 2020 2020 2031 2033 2037 2020 2020  7      1 3 7    
│ │ │ │ +0001e120: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │  0001e130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0001e180: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0001e170: 2d2d 2d2d 2d2d 7c0a 7c20 2020 3220 2020  ------|.|   2   
│ │ │ │ +0001e180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1a0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0001e1a0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0001e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1c0: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ -0001e1d0: 3078 2078 2078 2020 2b20 3232 7820 7820  0x x x  + 22x x 
│ │ │ │ -0001e1e0: 7820 7820 202b 2032 3078 2078 2078 2078  x x  + 20x x x x
│ │ │ │ -0001e1f0: 2020 2d20 3478 2078 2078 2020 2b20 3236    - 4x x x  + 26
│ │ │ │ -0001e200: 7820 7820 7820 7820 202b 2034 3178 2078  x x x x  + 41x x
│ │ │ │ -0001e210: 2078 2078 2020 2d20 2020 2020 7c0a 7c20   x x  -     |.| 
│ │ │ │ -0001e220: 2020 3020 3420 3720 2020 2020 2030 2031    0 4 7      0 1
│ │ │ │ -0001e230: 2034 2037 2020 2020 2020 3020 3320 3420   4 7      0 3 4 
│ │ │ │ -0001e240: 3720 2020 2020 3320 3420 3720 2020 2020  7     3 4 7     
│ │ │ │ -0001e250: 2031 2032 2035 2037 2020 2020 2020 3020   1 2 5 7      0 
│ │ │ │ -0001e260: 3320 3520 3720 2020 2020 2020 7c0a 7c2d  3 5 7       |.|-
│ │ │ │ +0001e1c0: 2020 2020 2020 7c0a 7c33 3078 2078 2078        |.|30x x x
│ │ │ │ +0001e1d0: 2020 2b20 3232 7820 7820 7820 7820 202b    + 22x x x x  +
│ │ │ │ +0001e1e0: 2032 3078 2078 2078 2078 2020 2d20 3478   20x x x x  - 4x
│ │ │ │ +0001e1f0: 2078 2078 2020 2b20 3236 7820 7820 7820   x x  + 26x x x 
│ │ │ │ +0001e200: 7820 202b 2034 3178 2078 2078 2078 2020  x  + 41x x x x  
│ │ │ │ +0001e210: 2d20 2020 2020 7c0a 7c20 2020 3020 3420  -     |.|   0 4 
│ │ │ │ +0001e220: 3720 2020 2020 2030 2031 2034 2037 2020  7      0 1 4 7  
│ │ │ │ +0001e230: 2020 2020 3020 3320 3420 3720 2020 2020      0 3 4 7     
│ │ │ │ +0001e240: 3320 3420 3720 2020 2020 2031 2032 2035  3 4 7      1 2 5
│ │ │ │ +0001e250: 2037 2020 2020 2020 3020 3320 3520 3720   7      0 3 5 7 
│ │ │ │ +0001e260: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │  0001e270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001e2b0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │  0001e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e2f0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0001e300: 2020 3220 2020 2020 2020 2020 7c0a 7c32    2         |.|2
│ │ │ │ -0001e310: 3878 2078 2078 2078 2020 2b20 3478 2078  8x x x x  + 4x x
│ │ │ │ -0001e320: 2078 2078 2020 2d20 3236 7820 7820 7820   x x  - 26x x x 
│ │ │ │ -0001e330: 7820 202b 2031 3278 2078 2078 2078 2020  x  + 12x x x x  
│ │ │ │ -0001e340: 2b20 3238 7820 7820 7820 202d 2031 3278  + 28x x x  - 12x
│ │ │ │ -0001e350: 2078 2078 2020 2d20 2020 2020 7c0a 7c20   x x  -     |.| 
│ │ │ │ -0001e360: 2020 3120 3320 3520 3720 2020 2020 3220    1 3 5 7     2 
│ │ │ │ -0001e370: 3320 3520 3720 2020 2020 2030 2034 2035  3 5 7      0 4 5
│ │ │ │ -0001e380: 2037 2020 2020 2020 3320 3420 3520 3720   7      3 4 5 7 
│ │ │ │ -0001e390: 2020 2020 2030 2035 2037 2020 2020 2020       0 5 7      
│ │ │ │ -0001e3a0: 3220 3520 3720 2020 2020 2020 7c0a 7c2d  2 5 7       |.|-
│ │ │ │ +0001e2f0: 2032 2020 2020 2020 2020 2020 3220 2020   2          2   
│ │ │ │ +0001e300: 2020 2020 2020 7c0a 7c32 3878 2078 2078        |.|28x x x
│ │ │ │ +0001e310: 2078 2020 2b20 3478 2078 2078 2078 2020   x  + 4x x x x  
│ │ │ │ +0001e320: 2d20 3236 7820 7820 7820 7820 202b 2031  - 26x x x x  + 1
│ │ │ │ +0001e330: 3278 2078 2078 2078 2020 2b20 3238 7820  2x x x x  + 28x 
│ │ │ │ +0001e340: 7820 7820 202d 2031 3278 2078 2078 2020  x x  - 12x x x  
│ │ │ │ +0001e350: 2d20 2020 2020 7c0a 7c20 2020 3120 3320  -     |.|   1 3 
│ │ │ │ +0001e360: 3520 3720 2020 2020 3220 3320 3520 3720  5 7     2 3 5 7 
│ │ │ │ +0001e370: 2020 2020 2030 2034 2035 2037 2020 2020       0 4 5 7    
│ │ │ │ +0001e380: 2020 3320 3420 3520 3720 2020 2020 2030    3 4 5 7      0
│ │ │ │ +0001e390: 2035 2037 2020 2020 2020 3220 3520 3720   5 7      2 5 7 
│ │ │ │ +0001e3a0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │  0001e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001e3f0: 2d2d 2d2d 2d2d 7c0a 7c20 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 3220 2020 2020 2020 2020        2         
│ │ │ │ -0001e440: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ -0001e450: 3178 2078 2078 2078 2020 2d20 3238 7820  1x x x x  - 28x 
│ │ │ │ -0001e460: 7820 7820 7820 202b 2033 3778 2078 2078  x x x  + 37x x x
│ │ │ │ -0001e470: 2078 2020 2b20 3231 7820 7820 7820 7820   x  + 21x x x x 
│ │ │ │ -0001e480: 202d 2031 3178 2078 2078 2020 2d20 3238   - 11x x x  - 28
│ │ │ │ -0001e490: 7820 7820 7820 7820 202d 2020 7c0a 7c20  x x x x  -  |.| 
│ │ │ │ -0001e4a0: 2020 3020 3220 3620 3720 2020 2020 2031    0 2 6 7      1
│ │ │ │ -0001e4b0: 2032 2036 2037 2020 2020 2020 3020 3320   2 6 7      0 3 
│ │ │ │ -0001e4c0: 3620 3720 2020 2020 2032 2033 2036 2037  6 7      2 3 6 7
│ │ │ │ -0001e4d0: 2020 2020 2020 3320 3620 3720 2020 2020        3 6 7     
│ │ │ │ -0001e4e0: 2030 2034 2036 2037 2020 2020 7c0a 7c2d   0 4 6 7    |.|-
│ │ │ │ +0001e430: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0001e440: 2020 2020 2020 7c0a 7c32 3178 2078 2078        |.|21x x x
│ │ │ │ +0001e450: 2078 2020 2d20 3238 7820 7820 7820 7820   x  - 28x x x x 
│ │ │ │ +0001e460: 202b 2033 3778 2078 2078 2078 2020 2b20   + 37x x x x  + 
│ │ │ │ +0001e470: 3231 7820 7820 7820 7820 202d 2031 3178  21x x x x  - 11x
│ │ │ │ +0001e480: 2078 2078 2020 2d20 3238 7820 7820 7820   x x  - 28x x x 
│ │ │ │ +0001e490: 7820 202d 2020 7c0a 7c20 2020 3020 3220  x  -  |.|   0 2 
│ │ │ │ +0001e4a0: 3620 3720 2020 2020 2031 2032 2036 2037  6 7      1 2 6 7
│ │ │ │ +0001e4b0: 2020 2020 2020 3020 3320 3620 3720 2020        0 3 6 7   
│ │ │ │ +0001e4c0: 2020 2032 2033 2036 2037 2020 2020 2020     2 3 6 7      
│ │ │ │ +0001e4d0: 3320 3620 3720 2020 2020 2030 2034 2036  3 6 7      0 4 6
│ │ │ │ +0001e4e0: 2037 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   7    |.|-------
│ │ │ │  0001e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001e530: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │  0001e540: 2020 2020 2020 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 7c0a 7c32              |.|2
│ │ │ │ -0001e590: 3178 2078 2078 2078 2020 2b20 3378 2078  1x x x x  + 3x x
│ │ │ │ -0001e5a0: 2078 2078 2020 2b20 3437 7820 7820 7820   x x  + 47x x x 
│ │ │ │ -0001e5b0: 7820 202b 2034 3878 2078 2078 2078 2020  x  + 48x x x x  
│ │ │ │ -0001e5c0: 2b20 3278 2078 2078 2078 2020 2d20 3435  + 2x x x x  - 45
│ │ │ │ -0001e5d0: 7820 7820 7820 7820 202d 2020 7c0a 7c20  x x x x  -  |.| 
│ │ │ │ -0001e5e0: 2020 3120 3420 3620 3720 2020 2020 3020    1 4 6 7     0 
│ │ │ │ -0001e5f0: 3520 3620 3720 2020 2020 2031 2035 2036  5 6 7      1 5 6
│ │ │ │ -0001e600: 2037 2020 2020 2020 3220 3520 3620 3720   7      2 5 6 7 
│ │ │ │ -0001e610: 2020 2020 3320 3520 3620 3720 2020 2020      3 5 6 7     
│ │ │ │ -0001e620: 2034 2035 2036 2037 2020 2020 7c0a 7c2d   4 5 6 7    |.|-
│ │ │ │ +0001e580: 2020 2020 2020 7c0a 7c32 3178 2078 2078        |.|21x x x
│ │ │ │ +0001e590: 2078 2020 2b20 3378 2078 2078 2078 2020   x  + 3x x x x  
│ │ │ │ +0001e5a0: 2b20 3437 7820 7820 7820 7820 202b 2034  + 47x x x x  + 4
│ │ │ │ +0001e5b0: 3878 2078 2078 2078 2020 2b20 3278 2078  8x x x x  + 2x x
│ │ │ │ +0001e5c0: 2078 2078 2020 2d20 3435 7820 7820 7820   x x  - 45x x x 
│ │ │ │ +0001e5d0: 7820 202d 2020 7c0a 7c20 2020 3120 3420  x  -  |.|   1 4 
│ │ │ │ +0001e5e0: 3620 3720 2020 2020 3020 3520 3620 3720  6 7     0 5 6 7 
│ │ │ │ +0001e5f0: 2020 2020 2031 2035 2036 2037 2020 2020       1 5 6 7    
│ │ │ │ +0001e600: 2020 3220 3520 3620 3720 2020 2020 3320    2 5 6 7     3 
│ │ │ │ +0001e610: 3520 3620 3720 2020 2020 2034 2035 2036  5 6 7      4 5 6
│ │ │ │ +0001e620: 2037 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   7    |.|-------
│ │ │ │  0001e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0001e680: 2020 3220 2020 2020 2020 2020 2032 2032    2          2 2
│ │ │ │ -0001e690: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0001e6a0: 2032 2032 2020 2020 2020 2020 2020 3220   2 2          2 
│ │ │ │ -0001e6b0: 2020 2020 2032 2032 2020 2020 2020 2020       2 2        
│ │ │ │ -0001e6c0: 2020 3220 2020 2020 2020 2020 7c0a 7c33    2         |.|3
│ │ │ │ -0001e6d0: 3378 2078 2078 2020 2d20 3338 7820 7820  3x x x  - 38x x 
│ │ │ │ -0001e6e0: 202b 2032 3878 2078 2078 2020 2d20 3338   + 28x x x  - 38
│ │ │ │ -0001e6f0: 7820 7820 202d 2032 3178 2078 2078 2020  x x  - 21x x x  
│ │ │ │ -0001e700: 2b20 3438 7820 7820 202d 2034 3878 2078  + 48x x  - 48x x
│ │ │ │ -0001e710: 2078 2020 2b20 2020 2020 2020 7c0a 7c20   x  +       |.| 
│ │ │ │ -0001e720: 2020 3520 3620 3720 2020 2020 2030 2037    5 6 7      0 7
│ │ │ │ -0001e730: 2020 2020 2020 3020 3120 3720 2020 2020        0 1 7     
│ │ │ │ -0001e740: 2031 2037 2020 2020 2020 3020 3320 3720   1 7      0 3 7 
│ │ │ │ -0001e750: 2020 2020 2033 2037 2020 2020 2020 3020       3 7      0 
│ │ │ │ -0001e760: 3520 3720 2020 2020 2020 2020 7c0a 7c2d  5 7         |.|-
│ │ │ │ +0001e670: 2d2d 2d2d 2d2d 7c0a 7c20 2020 3220 2020  ------|.|   2   
│ │ │ │ +0001e680: 2020 2020 2020 2032 2032 2020 2020 2020         2 2      
│ │ │ │ +0001e690: 2020 2020 3220 2020 2020 2032 2032 2020      2      2 2  
│ │ │ │ +0001e6a0: 2020 2020 2020 2020 3220 2020 2020 2032          2      2
│ │ │ │ +0001e6b0: 2032 2020 2020 2020 2020 2020 3220 2020   2          2   
│ │ │ │ +0001e6c0: 2020 2020 2020 7c0a 7c33 3378 2078 2078        |.|33x x x
│ │ │ │ +0001e6d0: 2020 2d20 3338 7820 7820 202b 2032 3878    - 38x x  + 28x
│ │ │ │ +0001e6e0: 2078 2078 2020 2d20 3338 7820 7820 202d   x x  - 38x x  -
│ │ │ │ +0001e6f0: 2032 3178 2078 2078 2020 2b20 3438 7820   21x x x  + 48x 
│ │ │ │ +0001e700: 7820 202d 2034 3878 2078 2078 2020 2b20  x  - 48x x x  + 
│ │ │ │ +0001e710: 2020 2020 2020 7c0a 7c20 2020 3520 3620        |.|   5 6 
│ │ │ │ +0001e720: 3720 2020 2020 2030 2037 2020 2020 2020  7      0 7      
│ │ │ │ +0001e730: 3020 3120 3720 2020 2020 2031 2037 2020  0 1 7      1 7  
│ │ │ │ +0001e740: 2020 2020 3020 3320 3720 2020 2020 2033      0 3 7      3
│ │ │ │ +0001e750: 2037 2020 2020 2020 3020 3520 3720 2020   7      0 5 7   
│ │ │ │ +0001e760: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │  0001e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0001e7c0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -0001e7d0: 2032 2020 2020 2020 3220 3220 2020 2020   2      2 2     
│ │ │ │ -0001e7e0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0001e7f0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0001e800: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ -0001e810: 3578 2078 2078 2020 2d20 3437 7820 7820  5x x x  - 47x x 
│ │ │ │ -0001e820: 7820 202d 2034 3878 2078 2020 2b20 3131  x  - 48x x  + 11
│ │ │ │ -0001e830: 7820 7820 7820 7820 202d 2031 3078 2078  x x x x  - 10x x
│ │ │ │ -0001e840: 2078 2020 2b20 3230 7820 7820 7820 202b   x  + 20x x x  +
│ │ │ │ -0001e850: 2031 3178 2078 2078 2078 2020 7c0a 7c20   11x x x x  |.| 
│ │ │ │ -0001e860: 2020 3120 3520 3720 2020 2020 2033 2035    1 5 7      3 5
│ │ │ │ -0001e870: 2037 2020 2020 2020 3520 3720 2020 2020   7      5 7     
│ │ │ │ -0001e880: 2030 2031 2032 2038 2020 2020 2020 3120   0 1 2 8      1 
│ │ │ │ -0001e890: 3220 3820 2020 2020 2031 2032 2038 2020  2 8      1 2 8  
│ │ │ │ -0001e8a0: 2020 2020 3020 3120 3320 3820 7c0a 7c2d      0 1 3 8 |.|-
│ │ │ │ +0001e7b0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +0001e7c0: 3220 2020 2020 2020 2020 2032 2020 2020  2          2    
│ │ │ │ +0001e7d0: 2020 3220 3220 2020 2020 2020 2020 2020    2 2           
│ │ │ │ +0001e7e0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0001e7f0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0001e800: 2020 2020 2020 7c0a 7c34 3578 2078 2078        |.|45x x x
│ │ │ │ +0001e810: 2020 2d20 3437 7820 7820 7820 202d 2034    - 47x x x  - 4
│ │ │ │ +0001e820: 3878 2078 2020 2b20 3131 7820 7820 7820  8x x  + 11x x x 
│ │ │ │ +0001e830: 7820 202d 2031 3078 2078 2078 2020 2b20  x  - 10x x x  + 
│ │ │ │ +0001e840: 3230 7820 7820 7820 202b 2031 3178 2078  20x x x  + 11x x
│ │ │ │ +0001e850: 2078 2078 2020 7c0a 7c20 2020 3120 3520   x x  |.|   1 5 
│ │ │ │ +0001e860: 3720 2020 2020 2033 2035 2037 2020 2020  7      3 5 7    
│ │ │ │ +0001e870: 2020 3520 3720 2020 2020 2030 2031 2032    5 7      0 1 2
│ │ │ │ +0001e880: 2038 2020 2020 2020 3120 3220 3820 2020   8      1 2 8   
│ │ │ │ +0001e890: 2020 2031 2032 2038 2020 2020 2020 3020     1 2 8      0 
│ │ │ │ +0001e8a0: 3120 3320 3820 7c0a 7c2d 2d2d 2d2d 2d2d  1 3 8 |.|-------
│ │ │ │  0001e8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0001e900: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0001e910: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +0001e8f0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 3220  ------|.|     2 
│ │ │ │ +0001e900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e910: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  0001e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e940: 2020 2020 2020 2020 2020 2020 7c0a 7c2b              |.|+
│ │ │ │ -0001e950: 2034 3278 2078 2078 2020 2b20 3431 7820   42x x x  + 41x 
│ │ │ │ -0001e960: 7820 7820 7820 202d 2031 3178 2078 2078  x x x  - 11x x x
│ │ │ │ -0001e970: 2020 2b20 3130 7820 7820 7820 7820 202d    + 10x x x x  -
│ │ │ │ -0001e980: 2032 3078 2078 2078 2078 2020 2d20 3236   20x x x x  - 26
│ │ │ │ -0001e990: 7820 7820 7820 7820 202b 2020 7c0a 7c20  x x x x  +  |.| 
│ │ │ │ -0001e9a0: 2020 2020 3120 3320 3820 2020 2020 2031      1 3 8      1
│ │ │ │ -0001e9b0: 2032 2033 2038 2020 2020 2020 3020 3420   2 3 8      0 4 
│ │ │ │ -0001e9c0: 3820 2020 2020 2030 2031 2034 2038 2020  8      0 1 4 8  
│ │ │ │ -0001e9d0: 2020 2020 3020 3220 3420 3820 2020 2020      0 2 4 8     
│ │ │ │ -0001e9e0: 2031 2032 2034 2038 2020 2020 7c0a 7c2d   1 2 4 8    |.|-
│ │ │ │ +0001e940: 2020 2020 2020 7c0a 7c2b 2034 3278 2078        |.|+ 42x x
│ │ │ │ +0001e950: 2078 2020 2b20 3431 7820 7820 7820 7820   x  + 41x x x x 
│ │ │ │ +0001e960: 202d 2031 3178 2078 2078 2020 2b20 3130   - 11x x x  + 10
│ │ │ │ +0001e970: 7820 7820 7820 7820 202d 2032 3078 2078  x x x x  - 20x x
│ │ │ │ +0001e980: 2078 2078 2020 2d20 3236 7820 7820 7820   x x  - 26x x x 
│ │ │ │ +0001e990: 7820 202b 2020 7c0a 7c20 2020 2020 3120  x  +  |.|     1 
│ │ │ │ +0001e9a0: 3320 3820 2020 2020 2031 2032 2033 2038  3 8      1 2 3 8
│ │ │ │ +0001e9b0: 2020 2020 2020 3020 3420 3820 2020 2020        0 4 8     
│ │ │ │ +0001e9c0: 2030 2031 2034 2038 2020 2020 2020 3020   0 1 4 8      0 
│ │ │ │ +0001e9d0: 3220 3420 3820 2020 2020 2031 2032 2034  2 4 8      1 2 4
│ │ │ │ +0001e9e0: 2038 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   8    |.|-------
│ │ │ │  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 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001ea30: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │  0001ea40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ea50: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0001ea60: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -0001ea70: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0001ea80: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ -0001ea90: 3878 2078 2078 2078 2020 2b20 3478 2078  8x x x x  + 4x x
│ │ │ │ -0001eaa0: 2078 2078 2020 2b20 3236 7820 7820 7820   x x  + 26x x x 
│ │ │ │ -0001eab0: 202d 2031 3278 2078 2078 2020 2d20 3131   - 12x x x  - 11
│ │ │ │ -0001eac0: 7820 7820 7820 202d 2034 3278 2078 2078  x x x  - 42x x x
│ │ │ │ -0001ead0: 2078 2020 2d20 2020 2020 2020 7c0a 7c20   x  -       |.| 
│ │ │ │ -0001eae0: 2020 3120 3320 3420 3820 2020 2020 3220    1 3 4 8     2 
│ │ │ │ -0001eaf0: 3320 3420 3820 2020 2020 2030 2034 2038  3 4 8      0 4 8
│ │ │ │ -0001eb00: 2020 2020 2020 3320 3420 3820 2020 2020        3 4 8     
│ │ │ │ -0001eb10: 2030 2035 2038 2020 2020 2020 3020 3120   0 5 8      0 1 
│ │ │ │ -0001eb20: 3520 3820 2020 2020 2020 2020 7c0a 7c2d  5 8         |.|-
│ │ │ │ +0001ea50: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0001ea60: 2020 3220 2020 2020 2020 2032 2020 2020    2        2    
│ │ │ │ +0001ea70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ea80: 2020 2020 2020 7c0a 7c32 3878 2078 2078        |.|28x x x
│ │ │ │ +0001ea90: 2078 2020 2b20 3478 2078 2078 2078 2020   x  + 4x x x x  
│ │ │ │ +0001eaa0: 2b20 3236 7820 7820 7820 202d 2031 3278  + 26x x x  - 12x
│ │ │ │ +0001eab0: 2078 2078 2020 2d20 3131 7820 7820 7820   x x  - 11x x x 
│ │ │ │ +0001eac0: 202d 2034 3278 2078 2078 2078 2020 2d20   - 42x x x x  - 
│ │ │ │ +0001ead0: 2020 2020 2020 7c0a 7c20 2020 3120 3320        |.|   1 3 
│ │ │ │ +0001eae0: 3420 3820 2020 2020 3220 3320 3420 3820  4 8     2 3 4 8 
│ │ │ │ +0001eaf0: 2020 2020 2030 2034 2038 2020 2020 2020       0 4 8      
│ │ │ │ +0001eb00: 3320 3420 3820 2020 2020 2030 2035 2038  3 4 8      0 5 8
│ │ │ │ +0001eb10: 2020 2020 2020 3020 3120 3520 3820 2020        0 1 5 8   
│ │ │ │ +0001eb20: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │  0001eb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001eb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0001eb80: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0001eb70: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +0001eb80: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0001eb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ebc0: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ -0001ebd0: 3178 2078 2078 2078 2020 2d20 3478 2078  1x x x x  - 4x x
│ │ │ │ -0001ebe0: 2078 2020 2d20 3238 7820 7820 7820 7820   x  - 28x x x x 
│ │ │ │ -0001ebf0: 202b 2031 3278 2078 2078 2078 2020 2b20   + 12x x x x  + 
│ │ │ │ -0001ec00: 3337 7820 7820 7820 7820 202b 2033 7820  37x x x x  + 3x 
│ │ │ │ -0001ec10: 7820 7820 7820 202d 2020 2020 7c0a 7c20  x x x  -    |.| 
│ │ │ │ -0001ec20: 2020 3020 3220 3520 3820 2020 2020 3220    0 2 5 8     2 
│ │ │ │ -0001ec30: 3520 3820 2020 2020 2030 2034 2035 2038  5 8      0 4 5 8
│ │ │ │ -0001ec40: 2020 2020 2020 3220 3420 3520 3820 2020        2 4 5 8   
│ │ │ │ -0001ec50: 2020 2030 2032 2036 2038 2020 2020 2031     0 2 6 8     1
│ │ │ │ -0001ec60: 2032 2036 2038 2020 2020 2020 7c0a 7c2d   2 6 8      |.|-
│ │ │ │ +0001ebc0: 2020 2020 2020 7c0a 7c34 3178 2078 2078        |.|41x x x
│ │ │ │ +0001ebd0: 2078 2020 2d20 3478 2078 2078 2020 2d20   x  - 4x x x  - 
│ │ │ │ +0001ebe0: 3238 7820 7820 7820 7820 202b 2031 3278  28x x x x  + 12x
│ │ │ │ +0001ebf0: 2078 2078 2078 2020 2b20 3337 7820 7820   x x x  + 37x x 
│ │ │ │ +0001ec00: 7820 7820 202b 2033 7820 7820 7820 7820  x x  + 3x x x x 
│ │ │ │ +0001ec10: 202d 2020 2020 7c0a 7c20 2020 3020 3220   -    |.|   0 2 
│ │ │ │ +0001ec20: 3520 3820 2020 2020 3220 3520 3820 2020  5 8     2 5 8   
│ │ │ │ +0001ec30: 2020 2030 2034 2035 2038 2020 2020 2020     0 4 5 8      
│ │ │ │ +0001ec40: 3220 3420 3520 3820 2020 2020 2030 2032  2 4 5 8      0 2
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│ │ │ │  0001ec80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0001ecc0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0001ecb0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 3220 2020  ------|.|   2   
│ │ │ │ +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 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ed00: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ -0001ed10: 3178 2078 2078 2020 2d20 3134 7820 7820  1x x x  - 14x x 
│ │ │ │ -0001ed20: 7820 7820 202b 2032 3978 2078 2078 2078  x x  + 29x x x x
│ │ │ │ -0001ed30: 2020 2b20 3131 7820 7820 7820 7820 202b    + 11x x x x  +
│ │ │ │ -0001ed40: 2034 3778 2078 2078 2078 2020 2d20 3438   47x x x x  - 48
│ │ │ │ -0001ed50: 7820 7820 7820 7820 202d 2020 7c0a 7c20  x x x x  -  |.| 
│ │ │ │ -0001ed60: 2020 3220 3620 3820 2020 2020 2030 2033    2 6 8      0 3
│ │ │ │ -0001ed70: 2036 2038 2020 2020 2020 3120 3320 3620   6 8      1 3 6 
│ │ │ │ -0001ed80: 3820 2020 2020 2032 2033 2036 2038 2020  8      2 3 6 8  
│ │ │ │ -0001ed90: 2020 2020 3120 3420 3620 3820 2020 2020      1 4 6 8     
│ │ │ │ -0001eda0: 2032 2034 2036 2038 2020 2020 7c0a 7c2d   2 4 6 8    |.|-
│ │ │ │ +0001ed00: 2020 2020 2020 7c0a 7c32 3178 2078 2078        |.|21x x x
│ │ │ │ +0001ed10: 2020 2d20 3134 7820 7820 7820 7820 202b    - 14x x x x  +
│ │ │ │ +0001ed20: 2032 3978 2078 2078 2078 2020 2b20 3131   29x x x x  + 11
│ │ │ │ +0001ed30: 7820 7820 7820 7820 202b 2034 3778 2078  x x x x  + 47x x
│ │ │ │ +0001ed40: 2078 2078 2020 2d20 3438 7820 7820 7820   x x  - 48x x x 
│ │ │ │ +0001ed50: 7820 202d 2020 7c0a 7c20 2020 3220 3620  x  -  |.|   2 6 
│ │ │ │ +0001ed60: 3820 2020 2020 2030 2033 2036 2038 2020  8      0 3 6 8  
│ │ │ │ +0001ed70: 2020 2020 3120 3320 3620 3820 2020 2020      1 3 6 8     
│ │ │ │ +0001ed80: 2032 2033 2036 2038 2020 2020 2020 3120   2 3 6 8      1 
│ │ │ │ +0001ed90: 3420 3620 3820 2020 2020 2032 2034 2036  4 6 8      2 4 6
│ │ │ │ +0001eda0: 2038 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   8    |.|-------
│ │ │ │  0001edb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001edc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001edd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ede0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001edf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0001ee00: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0001edf0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +0001ee00: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0001ee10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ee40: 3220 2020 2020 2020 2020 2020 7c0a 7c32  2           |.|2
│ │ │ │ -0001ee50: 7820 7820 7820 7820 202b 2034 3578 2078  x x x x  + 45x x
│ │ │ │ -0001ee60: 2078 2020 2b20 3239 7820 7820 7820 7820   x  + 29x x x x 
│ │ │ │ -0001ee70: 202b 2032 3578 2078 2078 2078 2020 2b20   + 25x x x x  + 
│ │ │ │ -0001ee80: 3333 7820 7820 7820 7820 202d 2033 3778  33x x x x  - 37x
│ │ │ │ -0001ee90: 2078 2078 2020 2d20 2020 2020 7c0a 7c20   x x  -     |.| 
│ │ │ │ -0001eea0: 2033 2034 2036 2038 2020 2020 2020 3420   3 4 6 8      4 
│ │ │ │ -0001eeb0: 3620 3820 2020 2020 2030 2035 2036 2038  6 8      0 5 6 8
│ │ │ │ -0001eec0: 2020 2020 2020 3120 3520 3620 3820 2020        1 5 6 8   
│ │ │ │ -0001eed0: 2020 2034 2035 2036 2038 2020 2020 2020     4 5 6 8      
│ │ │ │ -0001eee0: 3020 3720 3820 2020 2020 2020 7c0a 7c2d  0 7 8       |.|-
│ │ │ │ +0001ee30: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +0001ee40: 2020 2020 2020 7c0a 7c32 7820 7820 7820        |.|2x x x 
│ │ │ │ +0001ee50: 7820 202b 2034 3578 2078 2078 2020 2b20  x  + 45x x x  + 
│ │ │ │ +0001ee60: 3239 7820 7820 7820 7820 202b 2032 3578  29x x x x  + 25x
│ │ │ │ +0001ee70: 2078 2078 2078 2020 2b20 3333 7820 7820   x x x  + 33x x 
│ │ │ │ +0001ee80: 7820 7820 202d 2033 3778 2078 2078 2020  x x  - 37x x x  
│ │ │ │ +0001ee90: 2d20 2020 2020 7c0a 7c20 2033 2034 2036  -     |.|  3 4 6
│ │ │ │ +0001eea0: 2038 2020 2020 2020 3420 3620 3820 2020   8      4 6 8   
│ │ │ │ +0001eeb0: 2020 2030 2035 2036 2038 2020 2020 2020     0 5 6 8      
│ │ │ │ +0001eec0: 3120 3520 3620 3820 2020 2020 2034 2035  1 5 6 8      4 5
│ │ │ │ +0001eed0: 2036 2038 2020 2020 2020 3020 3720 3820   6 8      0 7 8 
│ │ │ │ +0001eee0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │  0001eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ef00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ef10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0001ef40: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0001ef30: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +0001ef40: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0001ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ef70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ef80: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ -0001ef90: 7820 7820 7820 7820 202d 2034 3778 2078  x x x x  - 47x x
│ │ │ │ -0001efa0: 2078 2020 2b20 3231 7820 7820 7820 7820   x  + 21x x x x 
│ │ │ │ -0001efb0: 202b 2031 3178 2078 2078 2078 2020 2b20   + 11x x x x  + 
│ │ │ │ -0001efc0: 7820 7820 7820 7820 202b 2034 3878 2078  x x x x  + 48x x
│ │ │ │ -0001efd0: 2078 2078 2020 2d20 2020 2020 7c0a 7c20   x x  -     |.| 
│ │ │ │ -0001efe0: 2030 2031 2037 2038 2020 2020 2020 3120   0 1 7 8      1 
│ │ │ │ -0001eff0: 3720 3820 2020 2020 2030 2032 2037 2038  7 8      0 2 7 8
│ │ │ │ -0001f000: 2020 2020 2020 3020 3320 3720 3820 2020        0 3 7 8   
│ │ │ │ -0001f010: 2032 2033 2037 2038 2020 2020 2020 3020   2 3 7 8      0 
│ │ │ │ -0001f020: 3420 3720 3820 2020 2020 2020 7c0a 7c2d  4 7 8       |.|-
│ │ │ │ +0001ef80: 2020 2020 2020 7c0a 7c33 7820 7820 7820        |.|3x x x 
│ │ │ │ +0001ef90: 7820 202d 2034 3778 2078 2078 2020 2b20  x  - 47x x x  + 
│ │ │ │ +0001efa0: 3231 7820 7820 7820 7820 202b 2031 3178  21x x x x  + 11x
│ │ │ │ +0001efb0: 2078 2078 2078 2020 2b20 7820 7820 7820   x x x  + x x x 
│ │ │ │ +0001efc0: 7820 202b 2034 3878 2078 2078 2078 2020  x  + 48x x x x  
│ │ │ │ +0001efd0: 2d20 2020 2020 7c0a 7c20 2030 2031 2037  -     |.|  0 1 7
│ │ │ │ +0001efe0: 2038 2020 2020 2020 3120 3720 3820 2020   8      1 7 8   
│ │ │ │ +0001eff0: 2020 2030 2032 2037 2038 2020 2020 2020     0 2 7 8      
│ │ │ │ +0001f000: 3020 3320 3720 3820 2020 2032 2033 2037  0 3 7 8    2 3 7
│ │ │ │ +0001f010: 2038 2020 2020 2020 3020 3420 3720 3820   8      0 4 7 8 
│ │ │ │ +0001f020: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
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│ │ │ │ -0001f070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001f070: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │  0001f080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f0c0: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ -0001f0d0: 3578 2078 2078 2078 2020 2b20 3437 7820  5x x x x  + 47x 
│ │ │ │ -0001f0e0: 7820 7820 7820 202d 2032 7820 7820 7820  x x x  - 2x x x 
│ │ │ │ -0001f0f0: 7820 202b 2033 3378 2078 2078 2078 2020  x  + 33x x x x  
│ │ │ │ -0001f100: 2b20 3437 7820 7820 7820 7820 202d 2078  + 47x x x x  - x
│ │ │ │ -0001f110: 2078 2078 2078 2020 2b20 2020 7c0a 7c20   x x x  +   |.| 
│ │ │ │ -0001f120: 2020 3120 3420 3720 3820 2020 2020 2033    1 4 7 8      3
│ │ │ │ -0001f130: 2034 2037 2038 2020 2020 2030 2035 2037   4 7 8     0 5 7
│ │ │ │ -0001f140: 2038 2020 2020 2020 3120 3520 3720 3820   8      1 5 7 8 
│ │ │ │ -0001f150: 2020 2020 2032 2035 2037 2038 2020 2020       2 5 7 8    
│ │ │ │ -0001f160: 3420 3520 3720 3820 2020 2020 7c0a 7c2d  4 5 7 8     |.|-
│ │ │ │ +0001f0c0: 2020 2020 2020 7c0a 7c34 3578 2078 2078        |.|45x x x
│ │ │ │ +0001f0d0: 2078 2020 2b20 3437 7820 7820 7820 7820   x  + 47x x x x 
│ │ │ │ +0001f0e0: 202d 2032 7820 7820 7820 7820 202b 2033   - 2x x x x  + 3
│ │ │ │ +0001f0f0: 3378 2078 2078 2078 2020 2b20 3437 7820  3x x x x  + 47x 
│ │ │ │ +0001f100: 7820 7820 7820 202d 2078 2078 2078 2078  x x x  - x x x x
│ │ │ │ +0001f110: 2020 2b20 2020 7c0a 7c20 2020 3120 3420    +   |.|   1 4 
│ │ │ │ +0001f120: 3720 3820 2020 2020 2033 2034 2037 2038  7 8      3 4 7 8
│ │ │ │ +0001f130: 2020 2020 2030 2035 2037 2038 2020 2020       0 5 7 8    
│ │ │ │ +0001f140: 2020 3120 3520 3720 3820 2020 2020 2032    1 5 7 8      2
│ │ │ │ +0001f150: 2035 2037 2038 2020 2020 3420 3520 3720   5 7 8    4 5 7 
│ │ │ │ +0001f160: 3820 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d  8     |.|-------
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│ │ │ │  0001f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0001f1c0: 2032 2032 2020 2020 2020 2020 2020 3220   2 2          2 
│ │ │ │ -0001f1d0: 2020 2020 2032 2032 2020 2020 2020 2020       2 2        
│ │ │ │ -0001f1e0: 2020 3220 2020 2020 2032 2032 2020 2020    2      2 2    
│ │ │ │ -0001f1f0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0001f200: 3220 2020 2020 2020 2020 2032 7c0a 7c37  2          2|.|7
│ │ │ │ -0001f210: 7820 7820 202d 2032 3978 2078 2078 2020  x x  - 29x x x  
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│ │ │ │ -0001f230: 2078 2020 2b20 3438 7820 7820 202b 2032   x  + 48x x  + 2
│ │ │ │ -0001f240: 7820 7820 7820 202d 2033 3378 2078 2078  x x x  - 33x x x
│ │ │ │ -0001f250: 2020 2d20 3437 7820 7820 7820 7c0a 7c20    - 47x x x |.| 
│ │ │ │ -0001f260: 2030 2038 2020 2020 2020 3020 3120 3820   0 8      0 1 8 
│ │ │ │ -0001f270: 2020 2020 2031 2038 2020 2020 2020 3020       1 8      0 
│ │ │ │ -0001f280: 3220 3820 2020 2020 2032 2038 2020 2020  2 8      2 8    
│ │ │ │ -0001f290: 2030 2034 2038 2020 2020 2020 3120 3420   0 4 8      1 4 
│ │ │ │ -0001f2a0: 3820 2020 2020 2032 2034 2038 7c0a 7c2d  8      2 4 8|.|-
│ │ │ │ +0001f1b0: 2d2d 2d2d 2d2d 7c0a 7c20 2032 2032 2020  ------|.|  2 2  
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│ │ │ │ +0001f1f0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +0001f200: 2020 2020 2032 7c0a 7c37 7820 7820 202d       2|.|7x x  -
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│ │ │ │ +0001f240: 202d 2033 3378 2078 2078 2020 2d20 3437   - 33x x x  - 47
│ │ │ │ +0001f250: 7820 7820 7820 7c0a 7c20 2030 2038 2020  x x x |.|  0 8  
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│ │ │ │ +0001f270: 2038 2020 2020 2020 3020 3220 3820 2020   8      0 2 8   
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│ │ │ │ +0001f290: 2020 2020 2020 3120 3420 3820 2020 2020        1 4 8     
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│ │ │ │  0001f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0001f300: 2020 2020 3220 3220 2020 2020 2020 2020      2 2         
│ │ │ │ +0001f2f0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 3220  ------|.|     2 
│ │ │ │ +0001f300: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  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: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ -0001f350: 2034 3878 2078 2020 2b20 3438 7820 7820   48x x  + 48x x 
│ │ │ │ -0001f360: 7820 7820 202d 2034 3878 2078 2078 2078  x x  - 48x x x x
│ │ │ │ -0001f370: 2020 2d20 3438 7820 7820 7820 7820 202b    - 48x x x x  +
│ │ │ │ -0001f380: 2034 3878 2078 2078 2078 2020 2b20 3438   48x x x x  + 48
│ │ │ │ -0001f390: 7820 7820 7820 7820 202d 2020 7c0a 7c20  x x x x  -  |.| 
│ │ │ │ -0001f3a0: 2020 2020 3420 3820 2020 2020 2033 2034      4 8      3 4
│ │ │ │ -0001f3b0: 2036 2039 2020 2020 2020 3220 3520 3620   6 9      2 5 6 
│ │ │ │ -0001f3c0: 3920 2020 2020 2031 2033 2037 2039 2020  9      1 3 7 9  
│ │ │ │ -0001f3d0: 2020 2020 3020 3520 3720 3920 2020 2020      0 5 7 9     
│ │ │ │ -0001f3e0: 2031 2032 2038 2039 2020 2020 7c0a 7c2d   1 2 8 9    |.|-
│ │ │ │ +0001f340: 2020 2020 2020 7c0a 7c2d 2034 3878 2078        |.|- 48x x
│ │ │ │ +0001f350: 2020 2b20 3438 7820 7820 7820 7820 202d    + 48x x x x  -
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│ │ │ │ +0001f390: 7820 202d 2020 7c0a 7c20 2020 2020 3420  x  -  |.|     4 
│ │ │ │ +0001f3a0: 3820 2020 2020 2033 2034 2036 2039 2020  8      3 4 6 9  
│ │ │ │ +0001f3b0: 2020 2020 3220 3520 3620 3920 2020 2020      2 5 6 9     
│ │ │ │ +0001f3c0: 2031 2033 2037 2039 2020 2020 2020 3020   1 3 7 9      0 
│ │ │ │ +0001f3d0: 3520 3720 3920 2020 2020 2031 2032 2038  5 7 9      1 2 8
│ │ │ │ +0001f3e0: 2039 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   9    |.|-------
│ │ │ │  0001f3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c34  ------------|.|4
│ │ │ │ -0001f440: 3878 2078 2078 2078 2020 2020 2020 2020  8x x x x        
│ │ │ │ +0001f430: 2d2d 2d2d 2d2d 7c0a 7c34 3878 2078 2078  ------|.|48x x x
│ │ │ │ +0001f440: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │  0001f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f480: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001f490: 2020 3020 3420 3820 3920 2020 2020 2020    0 4 8 9       
│ │ │ │ +0001f480: 2020 2020 2020 7c0a 7c20 2020 3020 3420        |.|   0 4 
│ │ │ │ +0001f490: 3820 3920 2020 2020 2020 2020 2020 2020  8 9             
│ │ │ │  0001f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f4d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001f4d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  0001f4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001f530: 3420 3a20 6173 7365 7274 2870 6869 202a  4 : assert(phi *
│ │ │ │ -0001f540: 2070 7369 203d 3d20 3120 616e 6420 7073   psi == 1 and ps
│ │ │ │ -0001f550: 6920 2a20 7068 6920 3d3d 2031 2920 2020  i * phi == 1)   
│ │ │ │ +0001f520: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 6173  ------+.|i4 : as
│ │ │ │ +0001f530: 7365 7274 2870 6869 202a 2070 7369 203d  sert(phi * psi =
│ │ │ │ +0001f540: 3d20 3120 616e 6420 7073 6920 2a20 7068  = 1 and psi * ph
│ │ │ │ +0001f550: 6920 3d3d 2031 2920 2020 2020 2020 2020  i == 1)         
│ │ │ │  0001f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f570: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001f570: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  0001f580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001f5d0: 3520 3a20 7469 6d65 2070 7369 2720 3d20  5 : time psi' = 
│ │ │ │ -0001f5e0: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ -0001f5f0: 7365 4d61 7028 7068 692c 436f 6469 6d42  seMap(phi,CodimB
│ │ │ │ -0001f600: 7349 6e76 3d3e 3529 3b20 2020 2020 2020  sInv=>5);       
│ │ │ │ -0001f610: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ -0001f620: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ -0001f630: 6572 7365 4d61 703a 2073 7465 7020 3120  erseMap: step 1 
│ │ │ │ -0001f640: 6f66 2033 2020 2020 2020 2020 2020 2020  of 3            
│ │ │ │ +0001f5c0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7469  ------+.|i5 : ti
│ │ │ │ +0001f5d0: 6d65 2070 7369 2720 3d20 6170 7072 6f78  me psi' = approx
│ │ │ │ +0001f5e0: 696d 6174 6549 6e76 6572 7365 4d61 7028  imateInverseMap(
│ │ │ │ +0001f5f0: 7068 692c 436f 6469 6d42 7349 6e76 3d3e  phi,CodimBsInv=>
│ │ │ │ +0001f600: 3529 3b20 2020 2020 2020 2020 2020 2020  5);             
│ │ │ │ +0001f610: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ +0001f620: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +0001f630: 703a 2073 7465 7020 3120 6f66 2033 2020  p: step 1 of 3  
│ │ │ │ +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 7c0a 7c2d              |.|-
│ │ │ │ -0001f670: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ -0001f680: 6572 7365 4d61 703a 2073 7465 7020 3220  erseMap: step 2 
│ │ │ │ -0001f690: 6f66 2033 2020 2020 2020 2020 2020 2020  of 3            
│ │ │ │ +0001f660: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ +0001f670: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +0001f680: 703a 2073 7465 7020 3220 6f66 2033 2020  p: step 2 of 3  
│ │ │ │ +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 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            
│ │ │ │ +0001f6b0: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ +0001f6c0: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +0001f6d0: 703a 2073 7465 7020 3320 6f66 2033 2020  p: step 3 of 3  
│ │ │ │ +0001f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f700: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001f710: 2d2d 2075 7365 6420 302e 3232 3739 3335  -- used 0.227935
│ │ │ │ -0001f720: 7320 2863 7075 293b 2030 2e31 3632 3333  s (cpu); 0.16233
│ │ │ │ -0001f730: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -0001f740: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ -0001f750: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001f700: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ +0001f710: 6420 302e 3332 3638 3836 7320 2863 7075  d 0.326886s (cpu
│ │ │ │ +0001f720: 293b 2030 2e32 3436 3931 3473 2028 7468  ); 0.246914s (th
│ │ │ │ +0001f730: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +0001f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f750: 2020 2020 2020 7c0a 7c20 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: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001f7b0: 3520 3a20 5261 7469 6f6e 616c 4d61 7020  5 : RationalMap 
│ │ │ │ -0001f7c0: 2871 7561 6472 6174 6963 2072 6174 696f  (quadratic ratio
│ │ │ │ -0001f7d0: 6e61 6c20 6d61 7020 6672 6f6d 2050 505e  nal map from PP^
│ │ │ │ -0001f7e0: 3820 746f 2068 7970 6572 7375 7266 6163  8 to hypersurfac
│ │ │ │ -0001f7f0: 6520 696e 2050 505e 3929 2020 7c0a 2b2d  e in PP^9)  |.+-
│ │ │ │ +0001f7a0: 2020 2020 2020 7c0a 7c6f 3520 3a20 5261        |.|o5 : Ra
│ │ │ │ +0001f7b0: 7469 6f6e 616c 4d61 7020 2871 7561 6472  tionalMap (quadr
│ │ │ │ +0001f7c0: 6174 6963 2072 6174 696f 6e61 6c20 6d61  atic rational ma
│ │ │ │ +0001f7d0: 7020 6672 6f6d 2050 505e 3820 746f 2068  p from PP^8 to h
│ │ │ │ +0001f7e0: 7970 6572 7375 7266 6163 6520 696e 2050  ypersurface in P
│ │ │ │ +0001f7f0: 505e 3929 2020 7c0a 2b2d 2d2d 2d2d 2d2d  P^9)  |.+-------
│ │ │ │  0001f800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001f850: 3620 3a20 6173 7365 7274 2870 7369 203d  6 : assert(psi =
│ │ │ │ -0001f860: 3d20 7073 6927 2920 2020 2020 2020 2020  = psi')         
│ │ │ │ +0001f840: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 6173  ------+.|i6 : as
│ │ │ │ +0001f850: 7365 7274 2870 7369 203d 3d20 7073 6927  sert(psi == psi'
│ │ │ │ +0001f860: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  0001f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f890: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001f890: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  0001f8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a41  ------------+..A
│ │ │ │ -0001f8f0: 206d 6f72 6520 636f 6d70 6c69 6361 7465   more complicate
│ │ │ │ -0001f900: 6420 6578 616d 706c 6520 6973 2074 6865  d example is the
│ │ │ │ -0001f910: 2066 6f6c 6c6f 7769 6e67 2028 6865 7265   following (here
│ │ │ │ -0001f920: 202a 6e6f 7465 2069 6e76 6572 7365 4d61   *note inverseMa
│ │ │ │ -0001f930: 703a 2069 6e76 6572 7365 4d61 702c 0a74  p: inverseMap,.t
│ │ │ │ -0001f940: 616b 6573 2061 206c 6f74 206f 6620 7469  akes a lot of ti
│ │ │ │ -0001f950: 6d65 2129 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  me!)...+--------
│ │ │ │ +0001f8e0: 2d2d 2d2d 2d2d 2b0a 0a41 206d 6f72 6520  ------+..A more 
│ │ │ │ +0001f8f0: 636f 6d70 6c69 6361 7465 6420 6578 616d  complicated exam
│ │ │ │ +0001f900: 706c 6520 6973 2074 6865 2066 6f6c 6c6f  ple is the follo
│ │ │ │ +0001f910: 7769 6e67 2028 6865 7265 202a 6e6f 7465  wing (here *note
│ │ │ │ +0001f920: 2069 6e76 6572 7365 4d61 703a 2069 6e76   inverseMap: inv
│ │ │ │ +0001f930: 6572 7365 4d61 702c 0a74 616b 6573 2061  erseMap,.takes a
│ │ │ │ +0001f940: 206c 6f74 206f 6620 7469 6d65 2129 2e0a   lot of time!)..
│ │ │ │ +0001f950: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0001f960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f9a0: 2d2d 2d2d 2d2b 0a7c 6937 203a 2070 6869  -----+.|i7 : phi
│ │ │ │ -0001f9b0: 203d 2072 6174 696f 6e61 6c4d 6170 206d   = rationalMap m
│ │ │ │ -0001f9c0: 6170 2850 382c 5a5a 2f39 375b 785f 302e  ap(P8,ZZ/97[x_0.
│ │ │ │ -0001f9d0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ -0001f9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f9f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001f9a0: 0a7c 6937 203a 2070 6869 203d 2072 6174  .|i7 : phi = rat
│ │ │ │ +0001f9b0: 696f 6e61 6c4d 6170 206d 6170 2850 382c  ionalMap map(P8,
│ │ │ │ +0001f9c0: 5a5a 2f39 375b 785f 302e 2e20 2020 2020  ZZ/97[x_0..     
│ │ │ │ +0001f9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f9e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f9f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fa40: 2020 2020 207c 0a7c 6f37 203d 202d 2d20       |.|o7 = -- 
│ │ │ │ -0001fa50: 7261 7469 6f6e 616c 206d 6170 202d 2d20  rational map -- 
│ │ │ │ +0001fa30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001fa40: 0a7c 6f37 203d 202d 2d20 7261 7469 6f6e  .|o7 = -- ration
│ │ │ │ +0001fa50: 616c 206d 6170 202d 2d20 2020 2020 2020  al map --       
│ │ │ │  0001fa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fa90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001faa0: 2020 2020 2020 2020 2020 5a5a 2020 2020            ZZ    
│ │ │ │ +0001fa80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001fa90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001faa0: 2020 2020 5a5a 2020 2020 2020 2020 2020      ZZ          
│ │ │ │  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 207c 0a7c 2020 2020 2073 6f75       |.|     sou
│ │ │ │ -0001faf0: 7263 653a 2050 726f 6a28 2d2d 5b74 202c  rce: Proj(--[t ,
│ │ │ │ -0001fb00: 2074 202c 2074 202c 2074 202c 2074 202c   t , t , t , t ,
│ │ │ │ +0001fad0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001fae0: 0a7c 2020 2020 2073 6f75 7263 653a 2050  .|     source: P
│ │ │ │ +0001faf0: 726f 6a28 2d2d 5b74 202c 2074 202c 2074  roj(--[t , t , t
│ │ │ │ +0001fb00: 202c 2074 202c 2074 202c 2020 2020 2020   , t , t ,      
│ │ │ │  0001fb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fb30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001fb40: 2020 2020 2020 2020 2020 3937 2020 3020            97  0 
│ │ │ │ -0001fb50: 2020 3120 2020 3220 2020 3320 2020 3420    1   2   3   4 
│ │ │ │ +0001fb20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001fb30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001fb40: 2020 2020 3937 2020 3020 2020 3120 2020      97  0   1   
│ │ │ │ +0001fb50: 3220 2020 3320 2020 3420 2020 2020 2020  2   3   4       
│ │ │ │  0001fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fb80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001fb70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001fb80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001fb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fba0: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │ +0001fba0: 2020 5a5a 2020 2020 2020 2020 2020 2020    ZZ            
│ │ │ │  0001fbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fbd0: 2020 2020 207c 0a7c 2020 2020 2074 6172       |.|     tar
│ │ │ │ -0001fbe0: 6765 743a 2073 7562 7661 7269 6574 7920  get: subvariety 
│ │ │ │ -0001fbf0: 6f66 2050 726f 6a28 2d2d 5b78 202c 2078  of Proj(--[x , x
│ │ │ │ +0001fbc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001fbd0: 0a7c 2020 2020 2074 6172 6765 743a 2073  .|     target: s
│ │ │ │ +0001fbe0: 7562 7661 7269 6574 7920 6f66 2050 726f  ubvariety of Pro
│ │ │ │ +0001fbf0: 6a28 2d2d 5b78 202c 2078 2020 2020 2020  j(--[x , x      
│ │ │ │  0001fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fc20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001fc10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001fc20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fc40: 2020 2020 2020 2020 3937 2020 3020 2020          97  0   
│ │ │ │ -0001fc50: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -0001fc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fc70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001fc80: 2020 2020 207b 2020 2020 2020 2020 2020       {          
│ │ │ │ +0001fc40: 2020 3937 2020 3020 2020 3120 2020 2020    97  0   1     
│ │ │ │ +0001fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fc60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001fc70: 0a7c 2020 2020 2020 2020 2020 2020 207b  .|             {
│ │ │ │ +0001fc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fcc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001fcb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001fcc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001fcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fce0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0001fce0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0001fcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fd10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001fd20: 2020 2020 2020 7820 7820 202d 2038 7820        x x  - 8x 
│ │ │ │ -0001fd30: 7820 202b 2032 3578 2020 2d20 3235 7820  x  + 25x  - 25x 
│ │ │ │ -0001fd40: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ -0001fd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fd60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001fd70: 2020 2020 2020 2031 2033 2020 2020 2032         1 3     2
│ │ │ │ -0001fd80: 2033 2020 2020 2020 3320 2020 2020 2032   3      3      2
│ │ │ │ +0001fd00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001fd10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001fd20: 7820 7820 202d 2038 7820 7820 202b 2032  x x  - 8x x  + 2
│ │ │ │ +0001fd30: 3578 2020 2d20 3235 7820 7820 2020 2020  5x  - 25x x     
│ │ │ │ +0001fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fd50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +00020980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00020990: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000209a0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000209f0: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ -00020a00: 202d 2034 3674 2074 2020 2d20 3333 7420   - 46t t  - 33t 
│ │ │ │ -00020a10: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ -00020a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00020a40: 2020 2020 2020 2020 2020 2020 2020 2032                 2
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│ │ │ │ +000209e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000209f0: 2020 2020 2020 2020 7420 202d 2034 3674          t  - 46t
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│ │ │ │ +00020a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020a20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +00020a40: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00020a50: 3120 3420 2020 2020 2030 2020 2020 2020  1 4      0      
│ │ │ │  00020a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │  00020a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ad0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00020ae0: 2020 2020 2020 2020 2020 2020 2020 7420                t 
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│ │ │ │ -00020b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ +00020ae0: 2020 2020 2020 2020 7420 7420 202b 2036          t t  + 6
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│ │ │ │ +00020b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00020b30: 2020 2020 2020 2020 2031 2032 2020 2020           1 2    
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│ │ │ │  00020b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020b60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00020b70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00020b80: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ +00020bc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00020bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │  00020c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00020c70: 2020 2020 2020 2020 2020 2020 2020 2030                 0
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│ │ │ │ +00020c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00020cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00020ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │  00020db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020df0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ +00020e50: 2020 2020 2020 2020 7420 202b 2031 3274          t  + 12t
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│ │ │ │  00020f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00020f80: 2020 2020 207c 0a7c 6f37 203a 2052 6174       |.|o7 : Rat
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│ │ │ │ +00020f80: 0a7c 6f37 203a 2052 6174 696f 6e61 6c4d  .|o7 : RationalM
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│ │ │ │  00021460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021480: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00021470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021480: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00021490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000214a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000214b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214d0: 2020 2020 207c 0a7c 202b 2033 3478 2078       |.| + 34x x
│ │ │ │ -000214e0: 2020 2b20 3434 7820 7820 202d 2033 3078    + 44x x  - 30x
│ │ │ │ -000214f0: 2078 2020 2b20 3237 7820 7820 202b 2033   x  + 27x x  + 3
│ │ │ │ -00021500: 3178 2078 2020 2d20 3336 7820 7820 202d  1x x  - 36x x  -
│ │ │ │ -00021510: 2078 2078 2020 2b20 3133 7820 7820 202b   x x  + 13x x  +
│ │ │ │ -00021520: 2038 7820 787c 0a7c 2020 2020 2020 3320   8x x|.|      3 
│ │ │ │ -00021530: 3420 2020 2020 2030 2035 2020 2020 2020  4      0 5      
│ │ │ │ -00021540: 3220 3520 2020 2020 2033 2035 2020 2020  2 5      3 5    
│ │ │ │ -00021550: 2020 3220 3620 2020 2020 2033 2036 2020    2 6      3 6  
│ │ │ │ -00021560: 2020 3020 3720 2020 2020 2031 2037 2020    0 7      1 7  
│ │ │ │ -00021570: 2020 2033 207c 0a7c 2020 2020 2020 2020     3 |.|        
│ │ │ │ +000214c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000214d0: 0a7c 202b 2033 3478 2078 2020 2b20 3434  .| + 34x x  + 44
│ │ │ │ +000214e0: 7820 7820 202d 2033 3078 2078 2020 2b20  x x  - 30x x  + 
│ │ │ │ +000214f0: 3237 7820 7820 202b 2033 3178 2078 2020  27x x  + 31x x  
│ │ │ │ +00021500: 2d20 3336 7820 7820 202d 2078 2078 2020  - 36x x  - x x  
│ │ │ │ +00021510: 2b20 3133 7820 7820 202b 2038 7820 787c  + 13x x  + 8x x|
│ │ │ │ +00021520: 0a7c 2020 2020 2020 3320 3420 2020 2020  .|      3 4     
│ │ │ │ +00021530: 2030 2035 2020 2020 2020 3220 3520 2020   0 5      2 5   
│ │ │ │ +00021540: 2020 2033 2035 2020 2020 2020 3220 3620     3 5      2 6 
│ │ │ │ +00021550: 2020 2020 2033 2036 2020 2020 3020 3720       3 6    0 7 
│ │ │ │ +00021560: 2020 2020 2031 2037 2020 2020 2033 207c       1 7     3 |
│ │ │ │ +00021570: 0a7c 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000215b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000215c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000215d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000215e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000215f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021610: 2020 2020 207c 0a7c 202b 2035 7820 7820       |.| + 5x x 
│ │ │ │ -00021620: 202d 2031 3678 2078 2020 2b20 3578 2078   - 16x x  + 5x x
│ │ │ │ -00021630: 2020 2d20 3336 7820 7820 202b 2033 3778    - 36x x  + 37x
│ │ │ │ -00021640: 2078 2020 2b20 3438 7820 7820 202d 2035   x  + 48x x  - 5
│ │ │ │ -00021650: 7820 7820 202d 2035 7820 7820 202b 2078  x x  - 5x x  + x
│ │ │ │ -00021660: 2078 2020 2b7c 0a7c 2020 2020 2032 2034   x  +|.|     2 4
│ │ │ │ -00021670: 2020 2020 2020 3320 3420 2020 2020 3020        3 4     0 
│ │ │ │ -00021680: 3520 2020 2020 2032 2035 2020 2020 2020  5      2 5      
│ │ │ │ -00021690: 3320 3520 2020 2020 2032 2036 2020 2020  3 5      2 6    
│ │ │ │ -000216a0: 2031 2037 2020 2020 2033 2037 2020 2020   1 7     3 7    
│ │ │ │ -000216b0: 3520 3720 207c 0a7c 2020 2020 2020 2020  5 7  |.|        
│ │ │ │ +00021600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021610: 0a7c 202b 2035 7820 7820 202d 2031 3678  .| + 5x x  - 16x
│ │ │ │ +00021620: 2078 2020 2b20 3578 2078 2020 2d20 3336   x  + 5x x  - 36
│ │ │ │ +00021630: 7820 7820 202b 2033 3778 2078 2020 2b20  x x  + 37x x  + 
│ │ │ │ +00021640: 3438 7820 7820 202d 2035 7820 7820 202d  48x x  - 5x x  -
│ │ │ │ +00021650: 2035 7820 7820 202b 2078 2078 2020 2b7c   5x x  + x x  +|
│ │ │ │ +00021660: 0a7c 2020 2020 2032 2034 2020 2020 2020  .|     2 4      
│ │ │ │ +00021670: 3320 3420 2020 2020 3020 3520 2020 2020  3 4     0 5     
│ │ │ │ +00021680: 2032 2035 2020 2020 2020 3320 3520 2020   2 5      3 5   
│ │ │ │ +00021690: 2020 2032 2036 2020 2020 2031 2037 2020     2 6     1 7  
│ │ │ │ +000216a0: 2020 2033 2037 2020 2020 3520 3720 207c     3 7    5 7  |
│ │ │ │ +000216b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000216c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000216f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021700: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000216f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021700: 0a7c 2020 2020 2020 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 2020 2020                  
│ │ │ │ -00021750: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00021760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021770: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00021740: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021750: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021760: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00021770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217a0: 2020 2020 207c 0a7c 3335 7420 7420 202b       |.|35t t  +
│ │ │ │ -000217b0: 2031 3074 2074 2020 2b20 3235 7420 7420   10t t  + 25t t 
│ │ │ │ -000217c0: 202d 2035 7420 202d 2031 3474 2074 2020   - 5t  - 14t t  
│ │ │ │ -000217d0: 2d20 3134 7420 7420 202d 2035 7420 7420  - 14t t  - 5t t 
│ │ │ │ -000217e0: 202d 2031 3374 2074 2020 2b20 3337 7420   - 13t t  + 37t 
│ │ │ │ -000217f0: 7420 202b 207c 0a7c 2020 2032 2035 2020  t  + |.|   2 5  
│ │ │ │ -00021800: 2020 2020 3320 3520 2020 2020 2034 2035      3 5      4 5
│ │ │ │ -00021810: 2020 2020 2035 2020 2020 2020 3020 3620       5      0 6 
│ │ │ │ -00021820: 2020 2020 2031 2036 2020 2020 2032 2036       1 6     2 6
│ │ │ │ -00021830: 2020 2020 2020 3420 3620 2020 2020 2035        4 6      5
│ │ │ │ -00021840: 2036 2020 207c 0a7c 2020 2020 2020 2020   6   |.|        
│ │ │ │ +00021790: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000217a0: 0a7c 3335 7420 7420 202b 2031 3074 2074  .|35t t  + 10t t
│ │ │ │ +000217b0: 2020 2b20 3235 7420 7420 202d 2035 7420    + 25t t  - 5t 
│ │ │ │ +000217c0: 202d 2031 3474 2074 2020 2d20 3134 7420   - 14t t  - 14t 
│ │ │ │ +000217d0: 7420 202d 2035 7420 7420 202d 2031 3374  t  - 5t t  - 13t
│ │ │ │ +000217e0: 2074 2020 2b20 3337 7420 7420 202b 207c   t  + 37t t  + |
│ │ │ │ +000217f0: 0a7c 2020 2032 2035 2020 2020 2020 3320  .|   2 5      3 
│ │ │ │ +00021800: 3520 2020 2020 2034 2035 2020 2020 2035  5      4 5     5
│ │ │ │ +00021810: 2020 2020 2020 3020 3620 2020 2020 2031        0 6      1
│ │ │ │ +00021820: 2036 2020 2020 2032 2036 2020 2020 2020   6     2 6      
│ │ │ │ +00021830: 3420 3620 2020 2020 2035 2036 2020 207c  4 6      5 6   |
│ │ │ │ +00021840: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020                  
│ │ │ │ -00021880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021890: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00021880: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021890: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000218a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000218b0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +000218b0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  000218c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000218d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000218e0: 2020 2020 207c 0a7c 7420 202d 2034 3074       |.|t  - 40t
│ │ │ │ -000218f0: 2074 2020 2b20 3430 7420 7420 202b 2032   t  + 40t t  + 2
│ │ │ │ -00021900: 3674 2074 2020 2d20 3230 7420 202b 2034  6t t  - 20t  + 4
│ │ │ │ -00021910: 3174 2074 2020 2b20 3336 7420 7420 202d  1t t  + 36t t  -
│ │ │ │ -00021920: 2032 3274 2074 2020 2b20 3336 7420 7420   22t t  + 36t t 
│ │ │ │ -00021930: 202d 2033 307c 0a7c 2035 2020 2020 2020   - 30|.| 5      
│ │ │ │ -00021940: 3220 3520 2020 2020 2033 2035 2020 2020  2 5      3 5    
│ │ │ │ -00021950: 2020 3420 3520 2020 2020 2035 2020 2020    4 5      5    
│ │ │ │ -00021960: 2020 3020 3620 2020 2020 2031 2036 2020    0 6      1 6  
│ │ │ │ -00021970: 2020 2020 3220 3620 2020 2020 2034 2036      2 6      4 6
│ │ │ │ -00021980: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000218d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000218e0: 0a7c 7420 202d 2034 3074 2074 2020 2b20  .|t  - 40t t  + 
│ │ │ │ +000218f0: 3430 7420 7420 202b 2032 3674 2074 2020  40t t  + 26t t  
│ │ │ │ +00021900: 2d20 3230 7420 202b 2034 3174 2074 2020  - 20t  + 41t t  
│ │ │ │ +00021910: 2b20 3336 7420 7420 202d 2032 3274 2074  + 36t t  - 22t t
│ │ │ │ +00021920: 2020 2b20 3336 7420 7420 202d 2033 307c    + 36t t  - 30|
│ │ │ │ +00021930: 0a7c 2035 2020 2020 2020 3220 3520 2020  .| 5      2 5   
│ │ │ │ +00021940: 2020 2033 2035 2020 2020 2020 3420 3520     3 5      4 5 
│ │ │ │ +00021950: 2020 2020 2035 2020 2020 2020 3020 3620       5      0 6 
│ │ │ │ +00021960: 2020 2020 2031 2036 2020 2020 2020 3220       1 6      2 
│ │ │ │ +00021970: 3620 2020 2020 2034 2036 2020 2020 207c  6      4 6     |
│ │ │ │ +00021980: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00021990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000219a0: 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000219c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000219d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000219e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000219f0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +000219f0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00021a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a20: 2020 2020 207c 0a7c 7420 202b 2031 3274       |.|t  + 12t
│ │ │ │ -00021a30: 2074 2020 2b20 3437 7420 7420 202b 2033   t  + 47t t  + 3
│ │ │ │ -00021a40: 3774 2074 2020 2b20 3235 7420 202d 2032  7t t  + 25t  - 2
│ │ │ │ -00021a50: 3774 2074 2020 2d20 3232 7420 7420 202b  7t t  - 22t t  +
│ │ │ │ -00021a60: 2032 3774 2074 2020 2d20 3233 7420 7420   27t t  - 23t t 
│ │ │ │ -00021a70: 202b 2035 747c 0a7c 2035 2020 2020 2020   + 5t|.| 5      
│ │ │ │ -00021a80: 3220 3520 2020 2020 2033 2035 2020 2020  2 5      3 5    
│ │ │ │ -00021a90: 2020 3420 3520 2020 2020 2035 2020 2020    4 5      5    
│ │ │ │ -00021aa0: 2020 3020 3620 2020 2020 2031 2036 2020    0 6      1 6  
│ │ │ │ -00021ab0: 2020 2020 3220 3620 2020 2020 2034 2036      2 6      4 6
│ │ │ │ -00021ac0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00021a10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021a20: 0a7c 7420 202b 2031 3274 2074 2020 2b20  .|t  + 12t t  + 
│ │ │ │ +00021a30: 3437 7420 7420 202b 2033 3774 2074 2020  47t t  + 37t t  
│ │ │ │ +00021a40: 2b20 3235 7420 202d 2032 3774 2074 2020  + 25t  - 27t t  
│ │ │ │ +00021a50: 2d20 3232 7420 7420 202b 2032 3774 2074  - 22t t  + 27t t
│ │ │ │ +00021a60: 2020 2d20 3233 7420 7420 202b 2035 747c    - 23t t  + 5t|
│ │ │ │ +00021a70: 0a7c 2035 2020 2020 2020 3220 3520 2020  .| 5      2 5   
│ │ │ │ +00021a80: 2020 2033 2035 2020 2020 2020 3420 3520     3 5      4 5 
│ │ │ │ +00021a90: 2020 2020 2035 2020 2020 2020 3020 3620       5      0 6 
│ │ │ │ +00021aa0: 2020 2020 2031 2036 2020 2020 2020 3220       1 6      2 
│ │ │ │ +00021ab0: 3620 2020 2020 2034 2036 2020 2020 207c  6      4 6     |
│ │ │ │ +00021ac0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00021ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00021b00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021b10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00021b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b30: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00021b30: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  00021b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b60: 2020 2020 207c 0a7c 202d 2033 3574 2074       |.| - 35t t
│ │ │ │ -00021b70: 2020 2d20 3130 7420 7420 202d 2033 3374    - 10t t  - 33t
│ │ │ │ -00021b80: 2074 2020 2b20 3574 2020 2b20 3135 7420   t  + 5t  + 15t 
│ │ │ │ -00021b90: 7420 202b 2031 3574 2074 2020 2b20 3574  t  + 15t t  + 5t
│ │ │ │ -00021ba0: 2074 2020 2b20 3135 7420 7420 202d 2033   t  + 15t t  - 3
│ │ │ │ -00021bb0: 3874 2074 207c 0a7c 2020 2020 2020 3220  8t t |.|      2 
│ │ │ │ -00021bc0: 3520 2020 2020 2033 2035 2020 2020 2020  5      3 5      
│ │ │ │ -00021bd0: 3420 3520 2020 2020 3520 2020 2020 2030  4 5     5      0
│ │ │ │ -00021be0: 2036 2020 2020 2020 3120 3620 2020 2020   6      1 6     
│ │ │ │ -00021bf0: 3220 3620 2020 2020 2034 2036 2020 2020  2 6      4 6    
│ │ │ │ -00021c00: 2020 3520 367c 0a7c 2020 2020 2020 2020    5 6|.|        
│ │ │ │ +00021b50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021b60: 0a7c 202d 2033 3574 2074 2020 2d20 3130  .| - 35t t  - 10
│ │ │ │ +00021b70: 7420 7420 202d 2033 3374 2074 2020 2b20  t t  - 33t t  + 
│ │ │ │ +00021b80: 3574 2020 2b20 3135 7420 7420 202b 2031  5t  + 15t t  + 1
│ │ │ │ +00021b90: 3574 2074 2020 2b20 3574 2074 2020 2b20  5t t  + 5t t  + 
│ │ │ │ +00021ba0: 3135 7420 7420 202d 2033 3874 2074 207c  15t t  - 38t t |
│ │ │ │ +00021bb0: 0a7c 2020 2020 2020 3220 3520 2020 2020  .|      2 5     
│ │ │ │ +00021bc0: 2033 2035 2020 2020 2020 3420 3520 2020   3 5      4 5   
│ │ │ │ +00021bd0: 2020 3520 2020 2020 2030 2036 2020 2020    5      0 6    
│ │ │ │ +00021be0: 2020 3120 3620 2020 2020 3220 3620 2020    1 6     2 6   
│ │ │ │ +00021bf0: 2020 2034 2036 2020 2020 2020 3520 367c     4 6      5 6|
│ │ │ │ +00021c00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00021c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00021c40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021c50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00021c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c70: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00021c70: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  00021c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ca0: 2020 2020 207c 0a7c 2d20 3335 7420 7420       |.|- 35t t 
│ │ │ │ -00021cb0: 202d 2031 3074 2074 2020 2d20 3333 7420   - 10t t  - 33t 
│ │ │ │ -00021cc0: 7420 202b 2035 7420 202b 2031 3474 2074  t  + 5t  + 14t t
│ │ │ │ -00021cd0: 2020 2b20 3134 7420 7420 202b 2035 7420    + 14t t  + 5t 
│ │ │ │ -00021ce0: 7420 202b 2031 3474 2074 2020 2d20 3331  t  + 14t t  - 31
│ │ │ │ -00021cf0: 7420 7420 207c 0a7c 2020 2020 2032 2035  t t  |.|     2 5
│ │ │ │ -00021d00: 2020 2020 2020 3320 3520 2020 2020 2034        3 5      4
│ │ │ │ -00021d10: 2035 2020 2020 2035 2020 2020 2020 3020   5     5      0 
│ │ │ │ -00021d20: 3620 2020 2020 2031 2036 2020 2020 2032  6      1 6     2
│ │ │ │ -00021d30: 2036 2020 2020 2020 3420 3620 2020 2020   6      4 6     
│ │ │ │ -00021d40: 2035 2036 207c 0a7c 2020 2020 2020 2020   5 6 |.|        
│ │ │ │ +00021c90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021ca0: 0a7c 2d20 3335 7420 7420 202d 2031 3074  .|- 35t t  - 10t
│ │ │ │ +00021cb0: 2074 2020 2d20 3333 7420 7420 202b 2035   t  - 33t t  + 5
│ │ │ │ +00021cc0: 7420 202b 2031 3474 2074 2020 2b20 3134  t  + 14t t  + 14
│ │ │ │ +00021cd0: 7420 7420 202b 2035 7420 7420 202b 2031  t t  + 5t t  + 1
│ │ │ │ +00021ce0: 3474 2074 2020 2d20 3331 7420 7420 207c  4t t  - 31t t  |
│ │ │ │ +00021cf0: 0a7c 2020 2020 2032 2035 2020 2020 2020  .|     2 5      
│ │ │ │ +00021d00: 3320 3520 2020 2020 2034 2035 2020 2020  3 5      4 5    
│ │ │ │ +00021d10: 2035 2020 2020 2020 3020 3620 2020 2020   5      0 6     
│ │ │ │ +00021d20: 2031 2036 2020 2020 2032 2036 2020 2020   1 6     2 6    
│ │ │ │ +00021d30: 2020 3420 3620 2020 2020 2035 2036 207c    4 6      5 6 |
│ │ │ │ +00021d40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00021d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021d90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00021da0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00021d80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021d90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021da0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00021db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021de0: 2020 2020 207c 0a7c 202b 2074 2074 2020       |.| + t t  
│ │ │ │ -00021df0: 2d20 3774 2074 2020 2b20 3274 2020 2d20  - 7t t  + 2t  - 
│ │ │ │ -00021e00: 7420 7420 2c20 2020 2020 2020 2020 2020  t t ,           
│ │ │ │ +00021dd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021de0: 0a7c 202b 2074 2074 2020 2d20 3774 2074  .| + t t  - 7t t
│ │ │ │ +00021df0: 2020 2b20 3274 2020 2d20 7420 7420 2c20    + 2t  - t t , 
│ │ │ │ +00021e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e30: 2020 2020 207c 0a7c 2020 2020 3420 3620       |.|    4 6 
│ │ │ │ -00021e40: 2020 2020 3520 3620 2020 2020 3620 2020      5 6     6   
│ │ │ │ -00021e50: 2033 2037 2020 2020 2020 2020 2020 2020   3 7            
│ │ │ │ +00021e20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021e30: 0a7c 2020 2020 3420 3620 2020 2020 3520  .|    4 6     5 
│ │ │ │ +00021e40: 3620 2020 2020 3620 2020 2033 2037 2020  6     6    3 7  
│ │ │ │ +00021e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00021e70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021e80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00021e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ed0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00021ec0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021ed0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00021ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ef0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00021ef0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00021f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f20: 2020 2020 207c 0a7c 2074 2020 2d20 3433       |.| t  - 43
│ │ │ │ -00021f30: 7420 7420 202d 2034 3174 2074 2020 2d20  t t  - 41t t  - 
│ │ │ │ -00021f40: 3236 7420 7420 202d 2032 3874 2020 2d20  26t t  - 28t  - 
│ │ │ │ -00021f50: 3335 7420 7420 202d 2033 3674 2074 2020  35t t  - 36t t  
│ │ │ │ -00021f60: 2b20 3230 7420 7420 202d 2033 3674 2074  + 20t t  - 36t t
│ │ │ │ -00021f70: 2020 2b20 397c 0a7c 3120 3520 2020 2020    + 9|.|1 5     
│ │ │ │ -00021f80: 2032 2035 2020 2020 2020 3320 3520 2020   2 5      3 5   
│ │ │ │ -00021f90: 2020 2034 2035 2020 2020 2020 3520 2020     4 5      5   
│ │ │ │ -00021fa0: 2020 2030 2036 2020 2020 2020 3120 3620     0 6      1 6 
│ │ │ │ -00021fb0: 2020 2020 2032 2036 2020 2020 2020 3420       2 6      4 
│ │ │ │ -00021fc0: 3620 2020 207c 0a7c 2020 2020 2020 2020  6    |.|        
│ │ │ │ +00021f10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021f20: 0a7c 2074 2020 2d20 3433 7420 7420 202d  .| t  - 43t t  -
│ │ │ │ +00021f30: 2034 3174 2074 2020 2d20 3236 7420 7420   41t t  - 26t t 
│ │ │ │ +00021f40: 202d 2032 3874 2020 2d20 3335 7420 7420   - 28t  - 35t t 
│ │ │ │ +00021f50: 202d 2033 3674 2074 2020 2b20 3230 7420   - 36t t  + 20t 
│ │ │ │ +00021f60: 7420 202d 2033 3674 2074 2020 2b20 397c  t  - 36t t  + 9|
│ │ │ │ +00021f70: 0a7c 3120 3520 2020 2020 2032 2035 2020  .|1 5      2 5  
│ │ │ │ +00021f80: 2020 2020 3320 3520 2020 2020 2034 2035      3 5      4 5
│ │ │ │ +00021f90: 2020 2020 2020 3520 2020 2020 2030 2036        5      0 6
│ │ │ │ +00021fa0: 2020 2020 2020 3120 3620 2020 2020 2032        1 6      2
│ │ │ │ +00021fb0: 2036 2020 2020 2020 3420 3620 2020 207c   6      4 6    |
│ │ │ │ +00021fc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00021fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022010: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00022000: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022010: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00022020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022040: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00022050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022060: 2020 2020 207c 0a7c 2020 2d20 3435 7420       |.|  - 45t 
│ │ │ │ -00022070: 7420 202d 2033 3974 2074 2020 2d20 3339  t  - 39t t  - 39
│ │ │ │ -00022080: 7420 7420 202d 2034 3674 2074 2020 2d20  t t  - 46t t  - 
│ │ │ │ -00022090: 3239 7420 202d 2034 3874 2074 2020 2d20  29t  - 48t t  - 
│ │ │ │ -000220a0: 3338 7420 7420 202d 2033 3074 2074 2020  38t t  - 30t t  
│ │ │ │ -000220b0: 2b20 3139 747c 0a7c 3520 2020 2020 2031  + 19t|.|5      1
│ │ │ │ -000220c0: 2035 2020 2020 2020 3220 3520 2020 2020   5      2 5     
│ │ │ │ -000220d0: 2033 2035 2020 2020 2020 3420 3520 2020   3 5      4 5   
│ │ │ │ -000220e0: 2020 2035 2020 2020 2020 3020 3620 2020     5      0 6   
│ │ │ │ -000220f0: 2020 2031 2036 2020 2020 2020 3220 3620     1 6      2 6 
│ │ │ │ -00022100: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00022030: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00022040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022050: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022060: 0a7c 2020 2d20 3435 7420 7420 202d 2033  .|  - 45t t  - 3
│ │ │ │ +00022070: 3974 2074 2020 2d20 3339 7420 7420 202d  9t t  - 39t t  -
│ │ │ │ +00022080: 2034 3674 2074 2020 2d20 3239 7420 202d   46t t  - 29t  -
│ │ │ │ +00022090: 2034 3874 2074 2020 2d20 3338 7420 7420   48t t  - 38t t 
│ │ │ │ +000220a0: 202d 2033 3074 2074 2020 2b20 3139 747c   - 30t t  + 19t|
│ │ │ │ +000220b0: 0a7c 3520 2020 2020 2031 2035 2020 2020  .|5      1 5    
│ │ │ │ +000220c0: 2020 3220 3520 2020 2020 2033 2035 2020    2 5      3 5  
│ │ │ │ +000220d0: 2020 2020 3420 3520 2020 2020 2035 2020      4 5      5  
│ │ │ │ +000220e0: 2020 2020 3020 3620 2020 2020 2031 2036      0 6      1 6
│ │ │ │ +000220f0: 2020 2020 2020 3220 3620 2020 2020 207c        2 6      |
│ │ │ │ +00022100: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00022110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022150: 2020 2020 207c 0a7c 2020 2d20 3274 2074       |.|  - 2t t
│ │ │ │ -00022160: 2020 2d20 7420 7420 202d 2074 2074 2020    - t t  - t t  
│ │ │ │ -00022170: 2b20 3574 2074 2020 2b20 7420 7420 202d  + 5t t  + t t  -
│ │ │ │ -00022180: 2032 7420 7420 202d 2037 7420 7420 202b   2t t  - 7t t  +
│ │ │ │ -00022190: 2032 7420 7420 202d 2032 7420 7420 202b   2t t  - 2t t  +
│ │ │ │ -000221a0: 2033 7420 747c 0a7c 3620 2020 2020 3120   3t t|.|6     1 
│ │ │ │ -000221b0: 3620 2020 2034 2036 2020 2020 3520 3620  6    4 6    5 6 
│ │ │ │ -000221c0: 2020 2020 3020 3720 2020 2031 2037 2020      0 7    1 7  
│ │ │ │ -000221d0: 2020 2032 2037 2020 2020 2035 2037 2020     2 7     5 7  
│ │ │ │ -000221e0: 2020 2036 2037 2020 2020 2031 2038 2020     6 7     1 8  
│ │ │ │ -000221f0: 2020 2037 207c 0a7c 2020 2020 2020 2020     7 |.|        
│ │ │ │ +00022140: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022150: 0a7c 2020 2d20 3274 2074 2020 2d20 7420  .|  - 2t t  - t 
│ │ │ │ +00022160: 7420 202d 2074 2074 2020 2b20 3574 2074  t  - t t  + 5t t
│ │ │ │ +00022170: 2020 2b20 7420 7420 202d 2032 7420 7420    + t t  - 2t t 
│ │ │ │ +00022180: 202d 2037 7420 7420 202b 2032 7420 7420   - 7t t  + 2t t 
│ │ │ │ +00022190: 202d 2032 7420 7420 202b 2033 7420 747c   - 2t t  + 3t t|
│ │ │ │ +000221a0: 0a7c 3620 2020 2020 3120 3620 2020 2034  .|6     1 6    4
│ │ │ │ +000221b0: 2036 2020 2020 3520 3620 2020 2020 3020   6    5 6     0 
│ │ │ │ +000221c0: 3720 2020 2031 2037 2020 2020 2032 2037  7    1 7     2 7
│ │ │ │ +000221d0: 2020 2020 2035 2037 2020 2020 2036 2037       5 7     6 7
│ │ │ │ +000221e0: 2020 2020 2031 2038 2020 2020 2037 207c       1 8     7 |
│ │ │ │ +000221f0: 0a7c 2020 2020 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00022230: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022240: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00022250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022270: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00022280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022290: 2020 2020 207c 0a7c 202b 2033 3274 2074       |.| + 32t t
│ │ │ │ -000222a0: 2020 2d20 3230 7420 7420 202d 2034 3774    - 20t t  - 47t
│ │ │ │ -000222b0: 2074 2020 2d20 3337 7420 7420 202d 2032   t  - 37t t  - 2
│ │ │ │ -000222c0: 3574 2020 2b20 3139 7420 7420 202b 2032  5t  + 19t t  + 2
│ │ │ │ -000222d0: 3274 2074 2020 2d20 3235 7420 7420 202b  2t t  - 25t t  +
│ │ │ │ -000222e0: 2032 3574 207c 0a7c 2020 2020 2020 3120   25t |.|      1 
│ │ │ │ -000222f0: 3520 2020 2020 2032 2035 2020 2020 2020  5      2 5      
│ │ │ │ -00022300: 3320 3520 2020 2020 2034 2035 2020 2020  3 5      4 5    
│ │ │ │ -00022310: 2020 3520 2020 2020 2030 2036 2020 2020    5      0 6    
│ │ │ │ -00022320: 2020 3120 3620 2020 2020 2032 2036 2020    1 6      2 6  
│ │ │ │ -00022330: 2020 2020 347c 0a7c 2020 2020 2020 2020      4|.|        
│ │ │ │ +00022260: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00022270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022280: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022290: 0a7c 202b 2033 3274 2074 2020 2d20 3230  .| + 32t t  - 20
│ │ │ │ +000222a0: 7420 7420 202d 2034 3774 2074 2020 2d20  t t  - 47t t  - 
│ │ │ │ +000222b0: 3337 7420 7420 202d 2032 3574 2020 2b20  37t t  - 25t  + 
│ │ │ │ +000222c0: 3139 7420 7420 202b 2032 3274 2074 2020  19t t  + 22t t  
│ │ │ │ +000222d0: 2d20 3235 7420 7420 202b 2032 3574 207c  - 25t t  + 25t |
│ │ │ │ +000222e0: 0a7c 2020 2020 2020 3120 3520 2020 2020  .|      1 5     
│ │ │ │ +000222f0: 2032 2035 2020 2020 2020 3320 3520 2020   2 5      3 5   
│ │ │ │ +00022300: 2020 2034 2035 2020 2020 2020 3520 2020     4 5      5   
│ │ │ │ +00022310: 2020 2030 2036 2020 2020 2020 3120 3620     0 6      1 6 
│ │ │ │ +00022320: 2020 2020 2032 2036 2020 2020 2020 347c       2 6      4|
│ │ │ │ +00022330: 0a7c 2020 2020 2020 2020 2020 2020 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 207c 0a7c 2074 2020 2d20 3337       |.| t  - 37
│ │ │ │ -00022390: 7420 7420 202d 2033 3974 2074 2020 2b20  t t  - 39t t  + 
│ │ │ │ -000223a0: 3139 7420 7420 202d 2033 3974 2074 2020  19t t  - 39t t  
│ │ │ │ -000223b0: 2d20 3338 7420 7420 202b 2033 3974 2074  - 38t t  + 39t t
│ │ │ │ -000223c0: 2020 2b20 3139 7420 7420 202b 2031 3874    + 19t t  + 18t
│ │ │ │ -000223d0: 2074 2020 2d7c 0a7c 3120 3520 2020 2020   t  -|.|1 5     
│ │ │ │ -000223e0: 2030 2036 2020 2020 2020 3120 3620 2020   0 6      1 6   
│ │ │ │ -000223f0: 2020 2034 2036 2020 2020 2020 3520 3620     4 6      5 6 
│ │ │ │ -00022400: 2020 2020 2030 2037 2020 2020 2020 3120       0 7      1 
│ │ │ │ -00022410: 3720 2020 2020 2032 2037 2020 2020 2020  7      2 7      
│ │ │ │ -00022420: 3520 3720 207c 0a7c 2020 2020 2020 2020  5 7  |.|        
│ │ │ │ +00022370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022380: 0a7c 2074 2020 2d20 3337 7420 7420 202d  .| t  - 37t t  -
│ │ │ │ +00022390: 2033 3974 2074 2020 2b20 3139 7420 7420   39t t  + 19t t 
│ │ │ │ +000223a0: 202d 2033 3974 2074 2020 2d20 3338 7420   - 39t t  - 38t 
│ │ │ │ +000223b0: 7420 202b 2033 3974 2074 2020 2b20 3139  t  + 39t t  + 19
│ │ │ │ +000223c0: 7420 7420 202b 2031 3874 2074 2020 2d7c  t t  + 18t t  -|
│ │ │ │ +000223d0: 0a7c 3120 3520 2020 2020 2030 2036 2020  .|1 5      0 6  
│ │ │ │ +000223e0: 2020 2020 3120 3620 2020 2020 2034 2036      1 6      4 6
│ │ │ │ +000223f0: 2020 2020 2020 3520 3620 2020 2020 2030        5 6      0
│ │ │ │ +00022400: 2037 2020 2020 2020 3120 3720 2020 2020   7      1 7     
│ │ │ │ +00022410: 2032 2037 2020 2020 2020 3520 3720 207c   2 7      5 7  |
│ │ │ │ +00022420: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020                  
│ │ │ │ -00022470: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00022460: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022470: 0a7c 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 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -000224b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000224c0: 2020 2020 207c 0a7c 2020 2b20 3237 7420       |.|  + 27t 
│ │ │ │ -000224d0: 7420 202d 2038 7420 7420 202b 2033 3774  t  - 8t t  + 37t
│ │ │ │ -000224e0: 2074 2020 2b20 3238 7420 7420 202b 2033   t  + 28t t  + 3
│ │ │ │ -000224f0: 3074 2020 2d20 3436 7420 7420 202b 2032  0t  - 46t t  + 2
│ │ │ │ -00022500: 3474 2074 2020 2d20 3430 7420 7420 202b  4t t  - 40t t  +
│ │ │ │ -00022510: 2032 3574 207c 0a7c 3520 2020 2020 2031   25t |.|5      1
│ │ │ │ -00022520: 2035 2020 2020 2032 2035 2020 2020 2020   5     2 5      
│ │ │ │ -00022530: 3320 3520 2020 2020 2034 2035 2020 2020  3 5      4 5    
│ │ │ │ -00022540: 2020 3520 2020 2020 2030 2036 2020 2020    5      0 6    
│ │ │ │ -00022550: 2020 3120 3620 2020 2020 2032 2036 2020    1 6      2 6  
│ │ │ │ -00022560: 2020 2020 347c 0a7c 2020 2020 2020 2020      4|.|        
│ │ │ │ +00022490: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +000224a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000224b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000224c0: 0a7c 2020 2b20 3237 7420 7420 202d 2038  .|  + 27t t  - 8
│ │ │ │ +000224d0: 7420 7420 202b 2033 3774 2074 2020 2b20  t t  + 37t t  + 
│ │ │ │ +000224e0: 3238 7420 7420 202b 2033 3074 2020 2d20  28t t  + 30t  - 
│ │ │ │ +000224f0: 3436 7420 7420 202b 2032 3474 2074 2020  46t t  + 24t t  
│ │ │ │ +00022500: 2d20 3430 7420 7420 202b 2032 3574 207c  - 40t t  + 25t |
│ │ │ │ +00022510: 0a7c 3520 2020 2020 2031 2035 2020 2020  .|5      1 5    
│ │ │ │ +00022520: 2032 2035 2020 2020 2020 3320 3520 2020   2 5      3 5   
│ │ │ │ +00022530: 2020 2034 2035 2020 2020 2020 3520 2020     4 5      5   
│ │ │ │ +00022540: 2020 2030 2036 2020 2020 2020 3120 3620     0 6      1 6 
│ │ │ │ +00022550: 2020 2020 2032 2036 2020 2020 2020 347c       2 6      4|
│ │ │ │ +00022560: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00022570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000225a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000225b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000225c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000225d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000225e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022600: 2020 2020 207c 0a7c 6672 6f6d 2050 505e       |.|from PP^
│ │ │ │ -00022610: 3820 746f 2038 2d64 696d 656e 7369 6f6e  8 to 8-dimension
│ │ │ │ -00022620: 616c 2073 7562 7661 7269 6574 7920 6f66  al subvariety of
│ │ │ │ -00022630: 2050 505e 3131 2920 2020 2020 2020 2020   PP^11)         
│ │ │ │ -00022640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022650: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +000225f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022600: 0a7c 6672 6f6d 2050 505e 3820 746f 2038  .|from PP^8 to 8
│ │ │ │ +00022610: 2d64 696d 656e 7369 6f6e 616c 2073 7562  -dimensional sub
│ │ │ │ +00022620: 7661 7269 6574 7920 6f66 2050 505e 3131  variety of PP^11
│ │ │ │ +00022630: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00022640: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022650: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00022660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000226a0: 2d2d 2d2d 2d7c 0a7c 5f35 2b34 312a 785f  -----|.|_5+41*x_
│ │ │ │ -000226b0: 332a 785f 352d 785f 302a 785f 362b 3132  3*x_5-x_0*x_6+12
│ │ │ │ -000226c0: 2a78 5f32 2a78 5f36 2b32 352a 785f 312a  *x_2*x_6+25*x_1*
│ │ │ │ -000226d0: 785f 3720 2020 2020 2020 2020 2020 2020  x_7             
│ │ │ │ -000226e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000226f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00022690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000226a0: 0a7c 5f35 2b34 312a 785f 332a 785f 352d  .|_5+41*x_3*x_5-
│ │ │ │ +000226b0: 785f 302a 785f 362b 3132 2a78 5f32 2a78  x_0*x_6+12*x_2*x
│ │ │ │ +000226c0: 5f36 2b32 352a 785f 312a 785f 3720 2020  _6+25*x_1*x_7   
│ │ │ │ +000226d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000226e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000226f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00022730: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022740: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022790: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00022780: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022790: 0a7c 2020 2020 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000227d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000227e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000227f0: 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │  00022840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022850: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00022870: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00022a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a10: 2020 2020 207c 0a7c 2020 2d20 3378 2078       |.|  - 3x x
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│ │ │ │ -00022a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a60: 2020 2020 207c 0a7c 3720 2020 2020 3620       |.|7     6 
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│ │ │ │ -00022a90: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -00022aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00022a10: 0a7c 2020 2d20 3378 2078 2020 2b20 3278  .|  - 3x x  + 2x
│ │ │ │ +00022a20: 2078 2020 2b20 3131 7820 7820 202d 2033   x  + 11x x  - 3
│ │ │ │ +00022a30: 3778 2078 2020 2d20 3233 7820 7820 2020  7x x  - 23x x   
│ │ │ │ +00022a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022a50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +00022a70: 3020 3820 2020 2020 2031 2038 2020 2020  0 8      1 8    
│ │ │ │ +00022a80: 2020 3320 3820 2020 2020 2034 2020 2020    3 8      4    
│ │ │ │ +00022a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022aa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022ab0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ -00022b00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00022af0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ -00022b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b50: 2020 2020 207c 0a7c 2020 2b20 3978 2078       |.|  + 9x x
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│ │ │ │ -00022b80: 7820 7820 2020 2020 2020 2020 2020 2020  x x             
│ │ │ │ -00022b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ba0: 2020 2020 207c 0a7c 3720 2020 2020 3520       |.|7     5 
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│ │ │ │ -00022bd0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -00022be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00022b50: 0a7c 2020 2b20 3978 2078 2020 2b20 3436  .|  + 9x x  + 46
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│ │ │ │ +00022b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022b90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022ba0: 0a7c 3720 2020 2020 3520 3720 2020 2020  .|7     5 7     
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│ │ │ │ +00022bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022be0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022bf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00022c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00022c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00022c30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │  00022c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00022c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c90: 2020 2020 207c 0a7c 2032 3078 2078 2020       |.| 20x x  
│ │ │ │ -00022ca0: 2b20 3130 7820 7820 202b 2033 3478 2078  + 10x x  + 34x x
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│ │ │ │ -00022cc0: 2020 2b20 2020 2020 2020 2020 2020 2020    +             
│ │ │ │ -00022cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00022d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00022c90: 0a7c 2032 3078 2078 2020 2b20 3130 7820  .| 20x x  + 10x 
│ │ │ │ +00022ca0: 7820 202b 2033 3478 2078 2020 2b20 3431  x  + 34x x  + 41
│ │ │ │ +00022cb0: 7820 7820 202d 2078 2078 2020 2b20 2020  x x  - x x  +   
│ │ │ │ +00022cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022cd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022ce0: 0a7c 2020 2020 3620 3720 2020 2020 2030  .|    6 7      0
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│ │ │ │ +00022d00: 2033 2038 2020 2020 3420 3820 2020 2020   3 8    4 8     
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│ │ │ │  00022da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00022dd0: 2020 2020 207c 0a7c 2020 2032 2020 2020       |.|   2    
│ │ │ │ +00022dc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022dd0: 0a7c 2020 2032 2020 2020 2020 2020 2020  .|   2          
│ │ │ │  00022de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00022e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e20: 2020 2020 207c 0a7c 3232 7420 202d 2033       |.|22t  - 3
│ │ │ │ -00022e30: 3174 2074 2020 2b20 3236 7420 7420 202b  1t t  + 26t t  +
│ │ │ │ -00022e40: 2031 3274 2074 2020 2d20 3435 7420 7420   12t t  - 45t t 
│ │ │ │ -00022e50: 202d 2020 2020 2020 2020 2020 2020 2020   -              
│ │ │ │ -00022e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e70: 2020 2020 207c 0a7c 2020 2036 2020 2020       |.|   6    
│ │ │ │ -00022e80: 2020 3320 3720 2020 2020 2034 2037 2020    3 7      4 7  
│ │ │ │ -00022e90: 2020 2020 3520 3720 2020 2020 2036 2037      5 7      6 7
│ │ │ │ +00022e10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022e20: 0a7c 3232 7420 202d 2033 3174 2074 2020  .|22t  - 31t t  
│ │ │ │ +00022e30: 2b20 3236 7420 7420 202b 2031 3274 2074  + 26t t  + 12t t
│ │ │ │ +00022e40: 2020 2d20 3435 7420 7420 202d 2020 2020    - 45t t  -    
│ │ │ │ +00022e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022e60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022e70: 0a7c 2020 2036 2020 2020 2020 3320 3720  .|   6      3 7 
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│ │ │ │ +00022e90: 3720 2020 2020 2036 2037 2020 2020 2020  7      6 7      
│ │ │ │  00022ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ec0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00022eb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022ec0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00022ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00022f20: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00022f00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022f10: 0a7c 2020 2020 2020 2020 2020 3220 2020  .|          2   
│ │ │ │ +00022f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020                  
│ │ │ │ -00022f60: 2020 2020 207c 0a7c 7420 7420 202d 2031       |.|t t  - 1
│ │ │ │ -00022f70: 3374 2020 2d20 3235 7420 7420 202b 2035  3t  - 25t t  + 5
│ │ │ │ -00022f80: 7420 7420 202d 2033 3574 2074 2020 2b20  t t  - 35t t  + 
│ │ │ │ -00022f90: 3130 7420 2020 2020 2020 2020 2020 2020  10t             
│ │ │ │ -00022fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022fb0: 2020 2020 207c 0a7c 2035 2036 2020 2020       |.| 5 6    
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│ │ │ │ +00022f50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022f60: 0a7c 7420 7420 202d 2031 3374 2020 2d20  .|t t  - 13t  - 
│ │ │ │ +00022f70: 3235 7420 7420 202b 2035 7420 7420 202d  25t t  + 5t t  -
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│ │ │ │ +00022f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022fa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +00022fd0: 2020 2020 3520 3720 2020 2020 2020 2020      5 7         
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│ │ │ │ -00022ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023000: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00022ff0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023000: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00023010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023050: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00023060: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00023040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023050: 0a7c 2020 2020 2020 2020 2032 2020 2020  .|         2    
│ │ │ │ +00023060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000230a0: 2020 2020 207c 0a7c 2074 2020 2d20 3133       |.| t  - 13
│ │ │ │ -000230b0: 7420 202d 2033 3974 2074 2020 2d20 3239  t  - 39t t  - 29
│ │ │ │ -000230c0: 7420 7420 202b 2039 7420 7420 202b 2033  t t  + 9t t  + 3
│ │ │ │ -000230d0: 3974 2020 2020 2020 2020 2020 2020 2020  9t              
│ │ │ │ -000230e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000230f0: 2020 2020 207c 0a7c 3520 3620 2020 2020       |.|5 6     
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│ │ │ │ -00023120: 2020 3620 2020 2020 2020 2020 2020 2020    6             
│ │ │ │ -00023130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023140: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023090: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000230a0: 0a7c 2074 2020 2d20 3133 7420 202d 2033  .| t  - 13t  - 3
│ │ │ │ +000230b0: 3974 2074 2020 2d20 3239 7420 7420 202b  9t t  - 29t t  +
│ │ │ │ +000230c0: 2039 7420 7420 202b 2033 3974 2020 2020   9t t  + 39t    
│ │ │ │ +000230d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000230e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000230f0: 0a7c 3520 3620 2020 2020 2036 2020 2020  .|5 6      6    
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│ │ │ │ +00023110: 2020 2035 2037 2020 2020 2020 3620 2020     5 7      6   
│ │ │ │ +00023120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023130: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023140: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00023150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2020 2020 2020 3220       |.|      2 
│ │ │ │ +00023180: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023190: 0a7c 2020 2020 2020 3220 2020 2020 2020  .|      2       
│ │ │ │  000231a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000231b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000231c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000231d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000231e0: 2020 2020 207c 0a7c 202d 2032 3274 2020       |.| - 22t  
│ │ │ │ -000231f0: 2b20 3331 7420 7420 202d 2032 3574 2074  + 31t t  - 25t t
│ │ │ │ -00023200: 2020 2d20 3139 7420 7420 202b 2034 3774    - 19t t  + 47t
│ │ │ │ -00023210: 2074 2020 2020 2020 2020 2020 2020 2020   t              
│ │ │ │ -00023220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023230: 2020 2020 207c 0a7c 2020 2020 2020 3620       |.|      6 
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│ │ │ │ -00023270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023280: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000231d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000231e0: 0a7c 202d 2032 3274 2020 2b20 3331 7420  .| - 22t  + 31t 
│ │ │ │ +000231f0: 7420 202d 2032 3574 2074 2020 2d20 3139  t  - 25t t  - 19
│ │ │ │ +00023200: 7420 7420 202b 2034 3774 2074 2020 2020  t t  + 47t t    
│ │ │ │ +00023210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023220: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023230: 0a7c 2020 2020 2020 3620 2020 2020 2033  .|      6      3
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│ │ │ │ +00023260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023270: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023280: 0a7c 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000232c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000232d0: 2020 2020 207c 0a7c 2020 2020 2032 2020       |.|     2  
│ │ │ │ +000232c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000232d0: 0a7c 2020 2020 2032 2020 2020 2020 2020  .|     2        
│ │ │ │  000232e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000232f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023320: 2020 2020 207c 0a7c 2d20 3234 7420 202b       |.|- 24t  +
│ │ │ │ -00023330: 2033 3274 2074 2020 2d20 3235 7420 7420   32t t  - 25t t 
│ │ │ │ -00023340: 202d 2031 3974 2074 2020 2b20 3437 7420   - 19t t  + 47t 
│ │ │ │ -00023350: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ -00023360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023370: 2020 2020 207c 0a7c 2020 2020 2036 2020       |.|     6  
│ │ │ │ -00023380: 2020 2020 3320 3720 2020 2020 2034 2037      3 7      4 7
│ │ │ │ -00023390: 2020 2020 2020 3520 3720 2020 2020 2036        5 7      6
│ │ │ │ -000233a0: 2037 2020 2020 2020 2020 2020 2020 2020   7              
│ │ │ │ -000233b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000233c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023310: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023320: 0a7c 2d20 3234 7420 202b 2033 3274 2074  .|- 24t  + 32t t
│ │ │ │ +00023330: 2020 2d20 3235 7420 7420 202d 2031 3974    - 25t t  - 19t
│ │ │ │ +00023340: 2074 2020 2b20 3437 7420 7420 2020 2020   t  + 47t t     
│ │ │ │ +00023350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023360: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023370: 0a7c 2020 2020 2036 2020 2020 2020 3320  .|     6      3 
│ │ │ │ +00023380: 3720 2020 2020 2034 2037 2020 2020 2020  7      4 7      
│ │ │ │ +00023390: 3520 3720 2020 2020 2036 2037 2020 2020  5 7      6 7    
│ │ │ │ +000233a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000233b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000233c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000233d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000233e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000233f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023410: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023400: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023410: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00023420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023460: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023450: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023460: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00023470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000234a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000234b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000234a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000234b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000234c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000234d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000234e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000234f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023500: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000234f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023500: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00023510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023550: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00023560: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00023540: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023550: 0a7c 2020 2020 2020 2020 2020 3220 2020  .|          2   
│ │ │ │ +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                  
│ │ │ │ -000235a0: 2020 2020 207c 0a7c 7420 7420 202b 2031       |.|t t  + 1
│ │ │ │ -000235b0: 3574 2020 2b20 3236 7420 7420 202d 2035  5t  + 26t t  - 5
│ │ │ │ -000235c0: 7420 7420 202b 2033 3574 2074 2020 2d20  t t  + 35t t  - 
│ │ │ │ -000235d0: 3130 7420 2020 2020 2020 2020 2020 2020  10t             
│ │ │ │ -000235e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000235f0: 2020 2020 207c 0a7c 2035 2036 2020 2020       |.| 5 6    
│ │ │ │ -00023600: 2020 3620 2020 2020 2033 2037 2020 2020    6      3 7    
│ │ │ │ -00023610: 2034 2037 2020 2020 2020 3520 3720 2020   4 7      5 7   
│ │ │ │ +00023590: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000235a0: 0a7c 7420 7420 202b 2031 3574 2020 2b20  .|t t  + 15t  + 
│ │ │ │ +000235b0: 3236 7420 7420 202d 2035 7420 7420 202b  26t t  - 5t t  +
│ │ │ │ +000235c0: 2033 3574 2074 2020 2d20 3130 7420 2020   35t t  - 10t   
│ │ │ │ +000235d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000235e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000235f0: 0a7c 2035 2036 2020 2020 2020 3620 2020  .| 5 6      6   
│ │ │ │ +00023600: 2020 2033 2037 2020 2020 2034 2037 2020     3 7     4 7  
│ │ │ │ +00023610: 2020 2020 3520 3720 2020 2020 2020 2020      5 7         
│ │ │ │  00023620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023640: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023630: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023640: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00023650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023690: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000236a0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00023680: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023690: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000236a0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  000236b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000236c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000236d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000236e0: 2020 2020 207c 0a7c 2074 2020 2d20 3434       |.| t  - 44
│ │ │ │ -000236f0: 7420 7420 202d 2034 3774 2020 2d20 3336  t t  - 47t  - 36
│ │ │ │ -00023700: 7420 7420 202d 2034 3674 2074 2020 2b20  t t  - 46t t  + 
│ │ │ │ -00023710: 7420 7420 2020 2020 2020 2020 2020 2020  t t             
│ │ │ │ -00023720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023730: 2020 2020 207c 0a7c 3420 3620 2020 2020       |.|4 6     
│ │ │ │ -00023740: 2035 2036 2020 2020 2020 3620 2020 2020   5 6      6     
│ │ │ │ -00023750: 2030 2037 2020 2020 2020 3120 3720 2020   0 7      1 7   
│ │ │ │ -00023760: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00023770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023780: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000236d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000236e0: 0a7c 2074 2020 2d20 3434 7420 7420 202d  .| t  - 44t t  -
│ │ │ │ +000236f0: 2034 3774 2020 2d20 3336 7420 7420 202d   47t  - 36t t  -
│ │ │ │ +00023700: 2034 3674 2074 2020 2b20 7420 7420 2020   46t t  + t t   
│ │ │ │ +00023710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023720: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023730: 0a7c 3420 3620 2020 2020 2035 2036 2020  .|4 6      5 6  
│ │ │ │ +00023740: 2020 2020 3620 2020 2020 2030 2037 2020      6      0 7  
│ │ │ │ +00023750: 2020 2020 3120 3720 2020 2032 2020 2020      1 7    2    
│ │ │ │ +00023760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023770: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023780: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00023790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000237a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000237b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000237c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000237d0: 2020 2020 207c 0a7c 202c 2020 2020 2020       |.| ,      
│ │ │ │ +000237c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000237d0: 0a7c 202c 2020 2020 2020 2020 2020 2020  .| ,            
│ │ │ │  000237e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000237f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023820: 2020 2020 207c 0a7c 3820 2020 2020 2020       |.|8       
│ │ │ │ +00023810: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023820: 0a7c 3820 2020 2020 2020 2020 2020 2020  .|8             
│ │ │ │  00023830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020 2020                  
│ │ │ │ -00023870: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023860: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023870: 0a7c 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 2020 2020 2020 2020                  
│ │ │ │ -000238b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000238c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000238d0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000238b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000238c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000238d0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  000238e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000238f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023910: 2020 2020 207c 0a7c 7420 202d 2035 7420       |.|t  - 5t 
│ │ │ │ -00023920: 7420 202b 2031 3374 2020 2b20 3574 2074  t  + 13t  + 5t t
│ │ │ │ -00023930: 2020 2b20 7420 7420 202b 2033 3974 2074    + t t  + 39t t
│ │ │ │ -00023940: 2020 2b20 2020 2020 2020 2020 2020 2020    +             
│ │ │ │ -00023950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023960: 2020 2020 207c 0a7c 2036 2020 2020 2035       |.| 6     5
│ │ │ │ -00023970: 2036 2020 2020 2020 3620 2020 2020 3020   6      6     0 
│ │ │ │ -00023980: 3720 2020 2031 2037 2020 2020 2020 3320  7    1 7      3 
│ │ │ │ -00023990: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │ -000239a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023900: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023910: 0a7c 7420 202d 2035 7420 7420 202b 2031  .|t  - 5t t  + 1
│ │ │ │ +00023920: 3374 2020 2b20 3574 2074 2020 2b20 7420  3t  + 5t t  + t 
│ │ │ │ +00023930: 7420 202b 2033 3974 2074 2020 2b20 2020  t  + 39t t  +   
│ │ │ │ +00023940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023950: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023960: 0a7c 2036 2020 2020 2035 2036 2020 2020  .| 6     5 6    
│ │ │ │ +00023970: 2020 3620 2020 2020 3020 3720 2020 2031    6     0 7    1
│ │ │ │ +00023980: 2037 2020 2020 2020 3320 3720 2020 2020   7      3 7     
│ │ │ │ +00023990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000239a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000239b0: 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a00: 2020 2020 207c 0a7c 2031 3974 2074 2020       |.| 19t t  
│ │ │ │ -00023a10: 2b20 3139 7420 7420 202b 2032 3074 2074  + 19t t  + 20t t
│ │ │ │ -00023a20: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │ +000239f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023a00: 0a7c 2031 3974 2074 2020 2b20 3139 7420  .| 19t t  + 19t 
│ │ │ │ +00023a10: 7420 202b 2032 3074 2074 202c 2020 2020  t  + 20t t ,    
│ │ │ │ +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 207c 0a7c 2020 2020 3620 3720       |.|    6 7 
│ │ │ │ -00023a60: 2020 2020 2031 2038 2020 2020 2020 3720       1 8      7 
│ │ │ │ -00023a70: 3820 2020 2020 2020 2020 2020 2020 2020  8               
│ │ │ │ +00023a40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023a50: 0a7c 2020 2020 3620 3720 2020 2020 2031  .|    6 7      1
│ │ │ │ +00023a60: 2038 2020 2020 2020 3720 3820 2020 2020   8      7 8     
│ │ │ │ +00023a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023aa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023a90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023aa0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00023ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023af0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00023b00: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00023ae0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023af0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023b00: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  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 207c 0a7c 7420 202b 2031 3674       |.|t  + 16t
│ │ │ │ -00023b50: 2074 2020 2d20 3335 7420 202b 2032 3974   t  - 35t  + 29t
│ │ │ │ -00023b60: 2074 2020 2b20 3132 7420 7420 202d 2033   t  + 12t t  - 3
│ │ │ │ -00023b70: 3574 2020 2020 2020 2020 2020 2020 2020  5t              
│ │ │ │ -00023b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b90: 2020 2020 207c 0a7c 2036 2020 2020 2020       |.| 6      
│ │ │ │ -00023ba0: 3520 3620 2020 2020 2036 2020 2020 2020  5 6      6      
│ │ │ │ -00023bb0: 3020 3720 2020 2020 2031 2037 2020 2020  0 7      1 7    
│ │ │ │ -00023bc0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00023bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023be0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00023b30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023b40: 0a7c 7420 202b 2031 3674 2074 2020 2d20  .|t  + 16t t  - 
│ │ │ │ +00023b50: 3335 7420 202b 2032 3974 2074 2020 2b20  35t  + 29t t  + 
│ │ │ │ +00023b60: 3132 7420 7420 202d 2033 3574 2020 2020  12t t  - 35t    
│ │ │ │ +00023b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023b80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023b90: 0a7c 2036 2020 2020 2020 3520 3620 2020  .| 6      5 6   
│ │ │ │ +00023ba0: 2020 2036 2020 2020 2020 3020 3720 2020     6      0 7   
│ │ │ │ +00023bb0: 2020 2031 2037 2020 2020 2020 3220 2020     1 7      2   
│ │ │ │ +00023bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023bd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023be0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00023bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023c30: 2d2d 2d2d 2d7c 0a7c 2b32 352a 785f 332a  -----|.|+25*x_3*
│ │ │ │ -00023c40: 785f 372b 3233 2a78 5f35 2a78 5f37 2d33  x_7+23*x_5*x_7-3
│ │ │ │ -00023c50: 2a78 5f36 2a78 5f37 2b32 2a78 5f30 2a78  *x_6*x_7+2*x_0*x
│ │ │ │ -00023c60: 5f38 2b31 312a 785f 312a 785f 382d 3337  _8+11*x_1*x_8-37
│ │ │ │ -00023c70: 2a78 5f33 2a78 5f38 2d32 332a 785f 342a  *x_3*x_8-23*x_4*
│ │ │ │ -00023c80: 7820 2020 207c 0a7c 2020 2020 2020 2020  x    |.|        
│ │ │ │ +00023c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00023c30: 0a7c 2b32 352a 785f 332a 785f 372b 3233  .|+25*x_3*x_7+23
│ │ │ │ +00023c40: 2a78 5f35 2a78 5f37 2d33 2a78 5f36 2a78  *x_5*x_7-3*x_6*x
│ │ │ │ +00023c50: 5f37 2b32 2a78 5f30 2a78 5f38 2b31 312a  _7+2*x_0*x_8+11*
│ │ │ │ +00023c60: 785f 312a 785f 382d 3337 2a78 5f33 2a78  x_1*x_8-37*x_3*x
│ │ │ │ +00023c70: 5f38 2d32 332a 785f 342a 7820 2020 207c  _8-23*x_4*x    |
│ │ │ │ +00023c80: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00023cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023cc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023cd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00023ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023d20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023d10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023d20: 0a7c 2020 2020 2020 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 2020 2020                  
│ │ │ │ -00023d70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023d60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023d70: 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023dc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023db0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023dc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00023dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023e00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023e10: 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023e50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023e60: 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023eb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023ea0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023eb0: 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023ef0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023f00: 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023f40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023f50: 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023fa0: 2020 2020 207c 0a7c 2020 2d20 3333 7820       |.|  - 33x 
│ │ │ │ -00023fb0: 7820 202b 2038 7820 7820 202b 2031 3078  x  + 8x x  + 10x
│ │ │ │ -00023fc0: 2078 2020 2d20 3235 7820 7820 202d 2039   x  - 25x x  - 9
│ │ │ │ -00023fd0: 7820 7820 202b 2033 7820 7820 202b 2032  x x  + 3x x  + 2
│ │ │ │ -00023fe0: 3478 2078 2020 2d20 3237 7820 7820 202d  4x x  - 27x x  -
│ │ │ │ -00023ff0: 2020 2020 207c 0a7c 3820 2020 2020 2036       |.|8      6
│ │ │ │ -00024000: 2038 2020 2020 2030 2039 2020 2020 2020   8     0 9      
│ │ │ │ -00024010: 3120 3920 2020 2020 2032 2039 2020 2020  1 9      2 9    
│ │ │ │ -00024020: 2033 2039 2020 2020 2034 2039 2020 2020   3 9     4 9    
│ │ │ │ -00024030: 2020 3520 3920 2020 2020 2036 2039 2020    5 9      6 9  
│ │ │ │ -00024040: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023fa0: 0a7c 2020 2d20 3333 7820 7820 202b 2038  .|  - 33x x  + 8
│ │ │ │ +00023fb0: 7820 7820 202b 2031 3078 2078 2020 2d20  x x  + 10x x  - 
│ │ │ │ +00023fc0: 3235 7820 7820 202d 2039 7820 7820 202b  25x x  - 9x x  +
│ │ │ │ +00023fd0: 2033 7820 7820 202b 2032 3478 2078 2020   3x x  + 24x x  
│ │ │ │ +00023fe0: 2d20 3237 7820 7820 202d 2020 2020 207c  - 27x x  -     |
│ │ │ │ +00023ff0: 0a7c 3820 2020 2020 2036 2038 2020 2020  .|8      6 8    
│ │ │ │ +00024000: 2030 2039 2020 2020 2020 3120 3920 2020   0 9      1 9   
│ │ │ │ +00024010: 2020 2032 2039 2020 2020 2033 2039 2020     2 9     3 9  
│ │ │ │ +00024020: 2020 2034 2039 2020 2020 2020 3520 3920     4 9      5 9 
│ │ │ │ +00024030: 2020 2020 2036 2039 2020 2020 2020 207c       6 9       |
│ │ │ │ +00024040: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024090: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024090: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000240a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000240b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000240c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000240d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000240e0: 2020 2020 207c 0a7c 2020 2d20 3978 2078       |.|  - 9x x
│ │ │ │ -000240f0: 2020 2d20 3436 7820 7820 202d 2031 3778    - 46x x  - 17x
│ │ │ │ -00024100: 2078 2020 2b20 3332 7820 7820 202d 2038   x  + 32x x  - 8
│ │ │ │ -00024110: 7820 7820 202d 2033 3578 2078 2020 2d20  x x  - 35x x  - 
│ │ │ │ -00024120: 3436 7820 7820 202b 2032 3678 2078 2020  46x x  + 26x x  
│ │ │ │ -00024130: 2b20 2020 207c 0a7c 3820 2020 2020 3420  +    |.|8     4 
│ │ │ │ -00024140: 3820 2020 2020 2036 2038 2020 2020 2020  8      6 8      
│ │ │ │ -00024150: 3020 3920 2020 2020 2031 2039 2020 2020  0 9      1 9    
│ │ │ │ -00024160: 2032 2039 2020 2020 2020 3320 3920 2020   2 9      3 9   
│ │ │ │ -00024170: 2020 2034 2039 2020 2020 2020 3520 3920     4 9      5 9 
│ │ │ │ -00024180: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000240d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000240e0: 0a7c 2020 2d20 3978 2078 2020 2d20 3436  .|  - 9x x  - 46
│ │ │ │ +000240f0: 7820 7820 202d 2031 3778 2078 2020 2b20  x x  - 17x x  + 
│ │ │ │ +00024100: 3332 7820 7820 202d 2038 7820 7820 202d  32x x  - 8x x  -
│ │ │ │ +00024110: 2033 3578 2078 2020 2d20 3436 7820 7820   35x x  - 46x x 
│ │ │ │ +00024120: 202b 2032 3678 2078 2020 2b20 2020 207c   + 26x x  +    |
│ │ │ │ +00024130: 0a7c 3820 2020 2020 3420 3820 2020 2020  .|8     4 8     
│ │ │ │ +00024140: 2036 2038 2020 2020 2020 3020 3920 2020   6 8      0 9   
│ │ │ │ +00024150: 2020 2031 2039 2020 2020 2032 2039 2020     1 9     2 9  
│ │ │ │ +00024160: 2020 2020 3320 3920 2020 2020 2034 2039      3 9      4 9
│ │ │ │ +00024170: 2020 2020 2020 3520 3920 2020 2020 207c        5 9      |
│ │ │ │ +00024180: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000241a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000241b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000241c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000241d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000241c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000241d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000241e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000241f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024220: 2020 2020 207c 0a7c 2078 2078 2020 2b20       |.| x x  + 
│ │ │ │ -00024230: 3430 7820 7820 202d 2033 3278 2078 2020  40x x  - 32x x  
│ │ │ │ -00024240: 2b20 3578 2078 2020 2d20 3131 7820 7820  + 5x x  - 11x x 
│ │ │ │ -00024250: 202d 2032 3078 2078 2020 2b20 3435 7820   - 20x x  + 45x 
│ │ │ │ -00024260: 7820 202d 2031 3478 2078 2020 2d20 3235  x  - 14x x  - 25
│ │ │ │ -00024270: 7820 2020 207c 0a7c 2020 3620 3820 2020  x    |.|  6 8   
│ │ │ │ -00024280: 2020 2030 2039 2020 2020 2020 3120 3920     0 9      1 9 
│ │ │ │ -00024290: 2020 2020 3220 3920 2020 2020 2033 2039      2 9      3 9
│ │ │ │ -000242a0: 2020 2020 2020 3420 3920 2020 2020 2035        4 9      5
│ │ │ │ -000242b0: 2039 2020 2020 2020 3620 3920 2020 2020   9      6 9     
│ │ │ │ -000242c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024220: 0a7c 2078 2078 2020 2b20 3430 7820 7820  .| x x  + 40x x 
│ │ │ │ +00024230: 202d 2033 3278 2078 2020 2b20 3578 2078   - 32x x  + 5x x
│ │ │ │ +00024240: 2020 2d20 3131 7820 7820 202d 2032 3078    - 11x x  - 20x
│ │ │ │ +00024250: 2078 2020 2b20 3435 7820 7820 202d 2031   x  + 45x x  - 1
│ │ │ │ +00024260: 3478 2078 2020 2d20 3235 7820 2020 207c  4x x  - 25x    |
│ │ │ │ +00024270: 0a7c 2020 3620 3820 2020 2020 2030 2039  .|  6 8      0 9
│ │ │ │ +00024280: 2020 2020 2020 3120 3920 2020 2020 3220        1 9     2 
│ │ │ │ +00024290: 3920 2020 2020 2033 2039 2020 2020 2020  9      3 9      
│ │ │ │ +000242a0: 3420 3920 2020 2020 2035 2039 2020 2020  4 9      5 9    
│ │ │ │ +000242b0: 2020 3620 3920 2020 2020 2020 2020 207c    6 9          |
│ │ │ │ +000242c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000242d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000242e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000242f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024310: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024300: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024310: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024360: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024360: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000243a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000243b0: 2020 2020 207c 0a7c 3436 7420 7420 202b       |.|46t t  +
│ │ │ │ -000243c0: 2033 3774 2074 2020 2b20 3238 7420 7420   37t t  + 28t t 
│ │ │ │ -000243d0: 202b 2033 3374 2074 202c 2020 2020 2020   + 33t t ,      
│ │ │ │ +000243a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000243b0: 0a7c 3436 7420 7420 202b 2033 3774 2074  .|46t t  + 37t t
│ │ │ │ +000243c0: 2020 2b20 3238 7420 7420 202b 2033 3374    + 28t t  + 33t
│ │ │ │ +000243d0: 2074 202c 2020 2020 2020 2020 2020 2020   t ,            
│ │ │ │  000243e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000243f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024400: 2020 2020 207c 0a7c 2020 2033 2038 2020       |.|   3 8  
│ │ │ │ -00024410: 2020 2020 3420 3820 2020 2020 2035 2038      4 8      5 8
│ │ │ │ -00024420: 2020 2020 2020 3620 3820 2020 2020 2020        6 8       
│ │ │ │ +000243f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024400: 0a7c 2020 2033 2038 2020 2020 2020 3420  .|   3 8      4 
│ │ │ │ +00024410: 3820 2020 2020 2035 2038 2020 2020 2020  8      5 8      
│ │ │ │ +00024420: 3620 3820 2020 2020 2020 2020 2020 2020  6 8             
│ │ │ │  00024430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024450: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024440: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024450: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000244a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024490: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000244a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000244b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000244c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000244d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000244e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000244f0: 2020 2020 207c 0a7c 2074 2020 2b20 3131       |.| t  + 11
│ │ │ │ -00024500: 7420 7420 202b 2034 3674 2074 2020 2b20  t t  + 46t t  + 
│ │ │ │ -00024510: 3239 7420 7420 202b 2032 3874 2074 202c  29t t  + 28t t ,
│ │ │ │ +000244e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000244f0: 0a7c 2074 2020 2b20 3131 7420 7420 202b  .| t  + 11t t  +
│ │ │ │ +00024500: 2034 3674 2074 2020 2b20 3239 7420 7420   46t t  + 29t t 
│ │ │ │ +00024510: 202b 2032 3874 2074 202c 2020 2020 2020   + 28t t ,      
│ │ │ │  00024520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024540: 2020 2020 207c 0a7c 3620 3720 2020 2020       |.|6 7     
│ │ │ │ -00024550: 2033 2038 2020 2020 2020 3420 3820 2020   3 8      4 8   
│ │ │ │ -00024560: 2020 2035 2038 2020 2020 2020 3620 3820     5 8      6 8 
│ │ │ │ +00024530: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024540: 0a7c 3620 3720 2020 2020 2033 2038 2020  .|6 7      3 8  
│ │ │ │ +00024550: 2020 2020 3420 3820 2020 2020 2035 2038      4 8      5 8
│ │ │ │ +00024560: 2020 2020 2020 3620 3820 2020 2020 2020        6 8       
│ │ │ │  00024570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024590: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024580: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024590: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000245a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000245b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000245c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000245d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000245e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000245d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000245e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000245f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024630: 2020 2020 207c 0a7c 7420 202b 2033 3674       |.|t  + 36t
│ │ │ │ -00024640: 2074 2020 2b20 3133 7420 7420 202b 2032   t  + 13t t  + 2
│ │ │ │ -00024650: 3674 2074 2020 2b20 3337 7420 7420 2c20  6t t  + 37t t , 
│ │ │ │ +00024620: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024630: 0a7c 7420 202b 2033 3674 2074 2020 2b20  .|t  + 36t t  + 
│ │ │ │ +00024640: 3133 7420 7420 202b 2032 3674 2074 2020  13t t  + 26t t  
│ │ │ │ +00024650: 2b20 3337 7420 7420 2c20 2020 2020 2020  + 37t t ,       
│ │ │ │  00024660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024680: 2020 2020 207c 0a7c 2037 2020 2020 2020       |.| 7      
│ │ │ │ -00024690: 3320 3820 2020 2020 2034 2038 2020 2020  3 8      4 8    
│ │ │ │ -000246a0: 2020 3520 3820 2020 2020 2036 2038 2020    5 8      6 8  
│ │ │ │ +00024670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024680: 0a7c 2037 2020 2020 2020 3320 3820 2020  .| 7      3 8   
│ │ │ │ +00024690: 2020 2034 2038 2020 2020 2020 3520 3820     4 8      5 8 
│ │ │ │ +000246a0: 2020 2020 2036 2038 2020 2020 2020 2020       6 8        
│ │ │ │  000246b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000246c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000246d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000246e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000246f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024720: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024710: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024720: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024770: 2020 2020 207c 0a7c 202b 2034 3674 2074       |.| + 46t t
│ │ │ │ -00024780: 2020 2d20 3336 7420 7420 202d 2033 3574    - 36t t  - 35t
│ │ │ │ -00024790: 2074 2020 2d20 3331 7420 7420 2c20 2020   t  - 31t t ,   
│ │ │ │ +00024760: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024770: 0a7c 202b 2034 3674 2074 2020 2d20 3336  .| + 46t t  - 36
│ │ │ │ +00024780: 7420 7420 202d 2033 3574 2074 2020 2d20  t t  - 35t t  - 
│ │ │ │ +00024790: 3331 7420 7420 2c20 2020 2020 2020 2020  31t t ,         
│ │ │ │  000247a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247c0: 2020 2020 207c 0a7c 2020 2020 2020 3320       |.|      3 
│ │ │ │ -000247d0: 3820 2020 2020 2034 2038 2020 2020 2020  8      4 8      
│ │ │ │ -000247e0: 3520 3820 2020 2020 2036 2038 2020 2020  5 8      6 8    
│ │ │ │ +000247b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000247c0: 0a7c 2020 2020 2020 3320 3820 2020 2020  .|      3 8     
│ │ │ │ +000247d0: 2034 2038 2020 2020 2020 3520 3820 2020   4 8      5 8   
│ │ │ │ +000247e0: 2020 2036 2038 2020 2020 2020 2020 2020     6 8          
│ │ │ │  000247f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024810: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024800: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024810: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024850: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024860: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000248a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000248b0: 2020 2020 207c 0a7c 2b20 3436 7420 7420       |.|+ 46t t 
│ │ │ │ -000248c0: 202d 2033 3674 2074 2020 2d20 3335 7420   - 36t t  - 35t 
│ │ │ │ -000248d0: 7420 202d 2033 3174 2074 202c 2020 2020  t  - 31t t ,    
│ │ │ │ +000248a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000248b0: 0a7c 2b20 3436 7420 7420 202d 2033 3674  .|+ 46t t  - 36t
│ │ │ │ +000248c0: 2074 2020 2d20 3335 7420 7420 202d 2033   t  - 35t t  - 3
│ │ │ │ +000248d0: 3174 2074 202c 2020 2020 2020 2020 2020  1t t ,          
│ │ │ │  000248e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000248f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024900: 2020 2020 207c 0a7c 2020 2020 2033 2038       |.|     3 8
│ │ │ │ -00024910: 2020 2020 2020 3420 3820 2020 2020 2035        4 8      5
│ │ │ │ -00024920: 2038 2020 2020 2020 3620 3820 2020 2020   8      6 8     
│ │ │ │ +000248f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024900: 0a7c 2020 2020 2033 2038 2020 2020 2020  .|     3 8      
│ │ │ │ +00024910: 3420 3820 2020 2020 2035 2038 2020 2020  4 8      5 8    
│ │ │ │ +00024920: 2020 3620 3820 2020 2020 2020 2020 2020    6 8           
│ │ │ │  00024930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024950: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024940: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024950: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024990: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000249a0: 0a7c 2020 2020 2020 2020 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 2020 2020 2020 2020                  
│ │ │ │ -000249f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000249e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000249f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024a30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024a40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024a80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024a90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ae0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024ad0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024ae0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024b30: 2020 2020 207c 0a7c 2074 2020 2d20 3130       |.| t  - 10
│ │ │ │ -00024b40: 7420 7420 202d 2034 3674 2074 2020 2b20  t t  - 46t t  + 
│ │ │ │ -00024b50: 3437 7420 7420 202d 2032 3574 2074 202c  47t t  - 25t t ,
│ │ │ │ +00024b20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024b30: 0a7c 2074 2020 2d20 3130 7420 7420 202d  .| t  - 10t t  -
│ │ │ │ +00024b40: 2034 3674 2074 2020 2b20 3437 7420 7420   46t t  + 47t t 
│ │ │ │ +00024b50: 202d 2032 3574 2074 202c 2020 2020 2020   - 25t t ,      
│ │ │ │  00024b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024b80: 2020 2020 207c 0a7c 3620 3720 2020 2020       |.|6 7     
│ │ │ │ -00024b90: 2033 2038 2020 2020 2020 3420 3820 2020   3 8      4 8   
│ │ │ │ -00024ba0: 2020 2035 2038 2020 2020 2020 3620 3820     5 8      6 8 
│ │ │ │ +00024b70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024b80: 0a7c 3620 3720 2020 2020 2033 2038 2020  .|6 7      3 8  
│ │ │ │ +00024b90: 2020 2020 3420 3820 2020 2020 2035 2038      4 8      5 8
│ │ │ │ +00024ba0: 2020 2020 2020 3620 3820 2020 2020 2020        6 8       
│ │ │ │  00024bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024bd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024bc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024bd0: 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024c10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024c20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c70: 2020 2020 207c 0a7c 2020 2d20 3434 7420       |.|  - 44t 
│ │ │ │ -00024c80: 7420 202b 2034 3874 2074 2020 2d20 3134  t  + 48t t  - 14
│ │ │ │ -00024c90: 7420 7420 202b 2034 7420 7420 202d 2033  t t  + 4t t  - 3
│ │ │ │ -00024ca0: 3674 2074 2020 2d20 3436 7420 7420 202b  6t t  - 46t t  +
│ │ │ │ -00024cb0: 2034 3774 2074 2020 2d20 3334 7420 7420   47t t  - 34t t 
│ │ │ │ -00024cc0: 2020 2020 207c 0a7c 3720 2020 2020 2033       |.|7      3
│ │ │ │ -00024cd0: 2037 2020 2020 2020 3420 3720 2020 2020   7      4 7     
│ │ │ │ -00024ce0: 2035 2037 2020 2020 2036 2037 2020 2020   5 7     6 7    
│ │ │ │ -00024cf0: 2020 3020 3820 2020 2020 2031 2038 2020    0 8      1 8  
│ │ │ │ -00024d00: 2020 2020 3220 3820 2020 2020 2033 2038      2 8      3 8
│ │ │ │ -00024d10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024c60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024c70: 0a7c 2020 2d20 3434 7420 7420 202b 2034  .|  - 44t t  + 4
│ │ │ │ +00024c80: 3874 2074 2020 2d20 3134 7420 7420 202b  8t t  - 14t t  +
│ │ │ │ +00024c90: 2034 7420 7420 202d 2033 3674 2074 2020   4t t  - 36t t  
│ │ │ │ +00024ca0: 2d20 3436 7420 7420 202b 2034 3774 2074  - 46t t  + 47t t
│ │ │ │ +00024cb0: 2020 2d20 3334 7420 7420 2020 2020 207c    - 34t t      |
│ │ │ │ +00024cc0: 0a7c 3720 2020 2020 2033 2037 2020 2020  .|7      3 7    
│ │ │ │ +00024cd0: 2020 3420 3720 2020 2020 2035 2037 2020    4 7      5 7  
│ │ │ │ +00024ce0: 2020 2036 2037 2020 2020 2020 3020 3820     6 7      0 8 
│ │ │ │ +00024cf0: 2020 2020 2031 2038 2020 2020 2020 3220       1 8      2 
│ │ │ │ +00024d00: 3820 2020 2020 2033 2038 2020 2020 207c  8      3 8     |
│ │ │ │ +00024d10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024d50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024d60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024db0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024db0: 0a7c 2020 2020 2020 2020 2020 2020 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 2020 2020                  
│ │ │ │ -00024e00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024df0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024e00: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024e40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024e50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ea0: 2020 2020 207c 0a7c 2032 3874 2074 2020       |.| 28t t  
│ │ │ │ -00024eb0: 2d20 3974 2074 2020 2d20 3339 7420 7420  - 9t t  - 39t t 
│ │ │ │ -00024ec0: 202b 2034 7420 7420 202b 2074 2074 2020   + 4t t  + t t  
│ │ │ │ -00024ed0: 2d20 3336 7420 7420 202d 2031 3474 2074  - 36t t  - 14t t
│ │ │ │ -00024ee0: 2020 2d20 3236 7420 7420 202d 2033 3774    - 26t t  - 37t
│ │ │ │ -00024ef0: 2020 2020 207c 0a7c 2020 2020 3420 3720       |.|    4 7 
│ │ │ │ -00024f00: 2020 2020 3520 3720 2020 2020 2036 2037      5 7      6 7
│ │ │ │ -00024f10: 2020 2020 2030 2038 2020 2020 3120 3820       0 8    1 8 
│ │ │ │ -00024f20: 2020 2020 2033 2038 2020 2020 2020 3420       3 8      4 
│ │ │ │ -00024f30: 3820 2020 2020 2035 2038 2020 2020 2020  8      5 8      
│ │ │ │ -00024f40: 3620 2020 207c 0a7c 2020 2020 2020 2020  6    |.|        
│ │ │ │ +00024e90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024ea0: 0a7c 2032 3874 2074 2020 2d20 3974 2074  .| 28t t  - 9t t
│ │ │ │ +00024eb0: 2020 2d20 3339 7420 7420 202b 2034 7420    - 39t t  + 4t 
│ │ │ │ +00024ec0: 7420 202b 2074 2074 2020 2d20 3336 7420  t  + t t  - 36t 
│ │ │ │ +00024ed0: 7420 202d 2031 3474 2074 2020 2d20 3236  t  - 14t t  - 26
│ │ │ │ +00024ee0: 7420 7420 202d 2033 3774 2020 2020 207c  t t  - 37t     |
│ │ │ │ +00024ef0: 0a7c 2020 2020 3420 3720 2020 2020 3520  .|    4 7     5 
│ │ │ │ +00024f00: 3720 2020 2020 2036 2037 2020 2020 2030  7      6 7     0
│ │ │ │ +00024f10: 2038 2020 2020 3120 3820 2020 2020 2033   8    1 8      3
│ │ │ │ +00024f20: 2038 2020 2020 2020 3420 3820 2020 2020   8      4 8     
│ │ │ │ +00024f30: 2035 2038 2020 2020 2020 3620 2020 207c   5 8      6    |
│ │ │ │ +00024f40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020                  
│ │ │ │ -00024f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024f90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024f80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024f90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024fe0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00024fd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024fe0: 0a7c 2020 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025030: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00025020: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00025030: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00025040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025080: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00025070: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00025080: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00025090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000250a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000250b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250d0: 2020 2020 207c 0a7c 7420 202d 2038 7420       |.|t  - 8t 
│ │ │ │ -000250e0: 7420 202d 2031 3874 2074 2020 2b20 3432  t  - 18t t  + 42
│ │ │ │ -000250f0: 7420 7420 202d 2031 3274 2074 2020 2d20  t t  - 12t t  - 
│ │ │ │ -00025100: 3674 2074 2020 2b20 3132 7420 7420 202d  6t t  + 12t t  -
│ │ │ │ -00025110: 2031 3574 2074 2020 2b20 3974 2074 2020   15t t  + 9t t  
│ │ │ │ -00025120: 2b20 2020 207c 0a7c 2037 2020 2020 2033  +    |.| 7     3
│ │ │ │ -00025130: 2037 2020 2020 2020 3420 3720 2020 2020   7      4 7     
│ │ │ │ -00025140: 2035 2037 2020 2020 2020 3620 3720 2020   5 7      6 7   
│ │ │ │ -00025150: 2020 3020 3820 2020 2020 2031 2038 2020    0 8      1 8  
│ │ │ │ -00025160: 2020 2020 3320 3820 2020 2020 3420 3820      3 8     4 8 
│ │ │ │ -00025170: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +000250c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000250d0: 0a7c 7420 202d 2038 7420 7420 202d 2031  .|t  - 8t t  - 1
│ │ │ │ +000250e0: 3874 2074 2020 2b20 3432 7420 7420 202d  8t t  + 42t t  -
│ │ │ │ +000250f0: 2031 3274 2074 2020 2d20 3674 2074 2020   12t t  - 6t t  
│ │ │ │ +00025100: 2b20 3132 7420 7420 202d 2031 3574 2074  + 12t t  - 15t t
│ │ │ │ +00025110: 2020 2b20 3974 2074 2020 2b20 2020 207c    + 9t t  +    |
│ │ │ │ +00025120: 0a7c 2037 2020 2020 2033 2037 2020 2020  .| 7     3 7    
│ │ │ │ +00025130: 2020 3420 3720 2020 2020 2035 2037 2020    4 7      5 7  
│ │ │ │ +00025140: 2020 2020 3620 3720 2020 2020 3020 3820      6 7     0 8 
│ │ │ │ +00025150: 2020 2020 2031 2038 2020 2020 2020 3320       1 8      3 
│ │ │ │ +00025160: 3820 2020 2020 3420 3820 2020 2020 207c  8     4 8      |
│ │ │ │ +00025170: 0a7c 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 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000251b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000251c0: 2d2d 2d2d 2d7c 0a7c 5f38 2d33 332a 785f  -----|.|_8-33*x_
│ │ │ │ -000251d0: 362a 785f 382b 382a 785f 302a 785f 392b  6*x_8+8*x_0*x_9+
│ │ │ │ -000251e0: 3130 2a78 5f31 2a78 5f39 2d32 352a 785f  10*x_1*x_9-25*x_
│ │ │ │ -000251f0: 322a 785f 392d 392a 785f 332a 785f 392b  2*x_9-9*x_3*x_9+
│ │ │ │ -00025200: 332a 785f 342a 785f 392b 3234 2a78 5f35  3*x_4*x_9+24*x_5
│ │ │ │ -00025210: 2a78 5f20 207c 0a7c 2020 2020 2020 2020  *x_  |.|        
│ │ │ │ +000251b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000251c0: 0a7c 5f38 2d33 332a 785f 362a 785f 382b  .|_8-33*x_6*x_8+
│ │ │ │ +000251d0: 382a 785f 302a 785f 392b 3130 2a78 5f31  8*x_0*x_9+10*x_1
│ │ │ │ +000251e0: 2a78 5f39 2d32 352a 785f 322a 785f 392d  *x_9-25*x_2*x_9-
│ │ │ │ +000251f0: 392a 785f 332a 785f 392b 332a 785f 342a  9*x_3*x_9+3*x_4*
│ │ │ │ +00025200: 785f 392b 3234 2a78 5f35 2a78 5f20 207c  x_9+24*x_5*x_  |
│ │ │ │ +00025210: 0a7c 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 2020 2020 2020 2020                  
│ │ │ │ -00025250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025260: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00025250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │  00025270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000252a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000252a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000252b0: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -000252f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025300: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000252f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00025300: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00025310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025320: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00025340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025350: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00025340: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00025350: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00025390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000253a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00025390: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000253a0: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -000253e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000253f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000253e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000253f0: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00025430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025440: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00025430: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00025440: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00025450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025490: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00025480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00025490: 0a7c 2020 2020 2020 2020 2020 2020 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 2020 2020                  
│ │ │ │ -000254e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000254d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000254e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000254f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025530: 2020 2020 207c 0a7c 3578 2078 2020 202b       |.|5x x   +
│ │ │ │ -00025540: 2032 3878 2078 2020 202b 2033 3778 2078   28x x   + 37x x
│ │ │ │ -00025550: 2020 202b 2039 7820 7820 2020 2b20 3237     + 9x x   + 27
│ │ │ │ -00025560: 7820 7820 2020 2d20 3235 7820 7820 2020  x x   - 25x x   
│ │ │ │ -00025570: 2b20 3978 2078 2020 202b 2032 3778 2078  + 9x x   + 27x x
│ │ │ │ -00025580: 2020 2020 207c 0a7c 2020 3020 3130 2020       |.|  0 10  
│ │ │ │ -00025590: 2020 2020 3120 3130 2020 2020 2020 3220      1 10      2 
│ │ │ │ -000255a0: 3130 2020 2020 2034 2031 3020 2020 2020  10     4 10     
│ │ │ │ -000255b0: 2036 2031 3020 2020 2020 2030 2031 3120   6 10      0 11 
│ │ │ │ -000255c0: 2020 2020 3220 3131 2020 2020 2020 3420      2 11      4 
│ │ │ │ -000255d0: 3131 2020 207c 0a7c 2020 2020 2020 2020  11   |.|        
│ │ │ │ +00025520: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00025530: 0a7c 3578 2078 2020 202b 2032 3878 2078  .|5x x   + 28x x
│ │ │ │ +00025540: 2020 202b 2033 3778 2078 2020 202b 2039     + 37x x   + 9
│ │ │ │ +00025550: 7820 7820 2020 2b20 3237 7820 7820 2020  x x   + 27x x   
│ │ │ │ +00025560: 2d20 3235 7820 7820 2020 2b20 3978 2078  - 25x x   + 9x x
│ │ │ │ +00025570: 2020 202b 2032 3778 2078 2020 2020 207c     + 27x x     |
│ │ │ │ +00025580: 0a7c 2020 3020 3130 2020 2020 2020 3120  .|  0 10      1 
│ │ │ │ +00025590: 3130 2020 2020 2020 3220 3130 2020 2020  10      2 10    
│ │ │ │ +000255a0: 2034 2031 3020 2020 2020 2036 2031 3020   4 10      6 10 
│ │ │ │ +000255b0: 2020 2020 2030 2031 3120 2020 2020 3220       0 11     2 
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│ │ │ │ +00026340: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00026350: 2020 2020 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00026380: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00026390: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000263a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000263d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000263e0: 0a7c 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026430: 2020 2020 207c 0a7c 7420 2c20 2020 2020       |.|t ,     
│ │ │ │ +00026420: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00026430: 0a7c 7420 2c20 2020 2020 2020 2020 2020  .|t ,           
│ │ │ │  00026440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026480: 2020 2020 207c 0a7c 2038 2020 2020 2020       |.| 8      
│ │ │ │ +00026470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00026480: 0a7c 2038 2020 2020 2020 2020 2020 2020  .| 8            
│ │ │ │  00026490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000264a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000264b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000264c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000264d0: 0a7c 2020 2020 2020 2020 2020 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 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00026510: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00026520: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00026530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026540: 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ +00026570: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00026580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000265a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000265b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000265c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000265b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000265c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000265d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000265e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000265f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026610: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00026600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00026610: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00026620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026640: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000266b0: 2020 2020 207c 0a7c 2020 2020 3520 3820       |.|    5 8 
│ │ │ │ -000266c0: 2020 2020 2036 2038 2020 2020 2037 2038       6 8     7 8
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│ │ │ │  000266d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000266e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000266f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026700: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
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│ │ │ │ +00026740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
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│ │ │ │ +000267a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000267b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000267c0: 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000267e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000267f0: 0a7c 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026830: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00026850: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00026870: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00026890: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │  000268a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000268c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000268d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000268e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000268d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000268e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000268f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026900: 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00026920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │  00026940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026950: 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00026970: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │  00026990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000269a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000269b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000269d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000269c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │  000269e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000269f0: 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │  00026a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00026a70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │  00026a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00026f10: 2a78 5f36 2d78 5f30 2a78 5f37 2b31 337c  *x_6-x_0*x_7+13|
│ │ │ │ +00026f20: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00026f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026f70: 2d2d 2d2d 2d7c 0a7c 2a78 5f31 2a78 5f37  -----|.|*x_1*x_7
│ │ │ │ -00026f80: 2b38 2a78 5f33 2a78 5f37 2b39 2a78 5f35  +8*x_3*x_7+9*x_5
│ │ │ │ -00026f90: 2a78 5f37 2b34 362a 785f 362a 785f 372b  *x_7+46*x_6*x_7+
│ │ │ │ -00026fa0: 3431 2a78 5f30 2a78 5f38 2d37 2a78 5f31  41*x_0*x_8-7*x_1
│ │ │ │ -00026fb0: 2a78 5f38 2d33 342a 785f 332a 785f 382d  *x_8-34*x_3*x_8-
│ │ │ │ -00026fc0: 392a 785f 347c 0a7c 2d2d 2d2d 2d2d 2d2d  9*x_4|.|--------
│ │ │ │ +00026f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00026f70: 0a7c 2a78 5f31 2a78 5f37 2b38 2a78 5f33  .|*x_1*x_7+8*x_3
│ │ │ │ +00026f80: 2a78 5f37 2b39 2a78 5f35 2a78 5f37 2b34  *x_7+9*x_5*x_7+4
│ │ │ │ +00026f90: 362a 785f 362a 785f 372b 3431 2a78 5f30  6*x_6*x_7+41*x_0
│ │ │ │ +00026fa0: 2a78 5f38 2d37 2a78 5f31 2a78 5f38 2d33  *x_8-7*x_1*x_8-3
│ │ │ │ +00026fb0: 342a 785f 332a 785f 382d 392a 785f 347c  4*x_3*x_8-9*x_4|
│ │ │ │ +00026fc0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00026fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027010: 2d2d 2d2d 2d7c 0a7c 2a78 5f38 2d34 362a  -----|.|*x_8-46*
│ │ │ │ -00027020: 785f 362a 785f 382d 3137 2a78 5f30 2a78  x_6*x_8-17*x_0*x
│ │ │ │ -00027030: 5f39 2b33 322a 785f 312a 785f 392d 382a  _9+32*x_1*x_9-8*
│ │ │ │ -00027040: 785f 322a 785f 392d 3335 2a78 5f33 2a78  x_2*x_9-35*x_3*x
│ │ │ │ -00027050: 5f39 2d34 362a 785f 342a 785f 392b 3236  _9-46*x_4*x_9+26
│ │ │ │ -00027060: 2a78 5f35 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d  *x_5*|.|--------
│ │ │ │ +00027000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027010: 0a7c 2a78 5f38 2d34 362a 785f 362a 785f  .|*x_8-46*x_6*x_
│ │ │ │ +00027020: 382d 3137 2a78 5f30 2a78 5f39 2b33 322a  8-17*x_0*x_9+32*
│ │ │ │ +00027030: 785f 312a 785f 392d 382a 785f 322a 785f  x_1*x_9-8*x_2*x_
│ │ │ │ +00027040: 392d 3335 2a78 5f33 2a78 5f39 2d34 362a  9-35*x_3*x_9-46*
│ │ │ │ +00027050: 785f 342a 785f 392b 3236 2a78 5f35 2a7c  x_4*x_9+26*x_5*|
│ │ │ │ +00027060: 0a7c 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 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000270a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000270b0: 2d2d 2d2d 2d7c 0a7c 785f 392b 3137 2a78  -----|.|x_9+17*x
│ │ │ │ -000270c0: 5f36 2a78 5f39 2b31 352a 785f 302a 785f  _6*x_9+15*x_0*x_
│ │ │ │ -000270d0: 3130 2b33 352a 785f 312a 785f 3130 2b33  10+35*x_1*x_10+3
│ │ │ │ -000270e0: 342a 785f 322a 785f 3130 2b32 302a 785f  4*x_2*x_10+20*x_
│ │ │ │ -000270f0: 342a 785f 3130 2b31 342a 785f 302a 785f  4*x_10+14*x_0*x_
│ │ │ │ -00027100: 3131 2b33 367c 0a7c 2d2d 2d2d 2d2d 2d2d  11+36|.|--------
│ │ │ │ +000270a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000270b0: 0a7c 785f 392b 3137 2a78 5f36 2a78 5f39  .|x_9+17*x_6*x_9
│ │ │ │ +000270c0: 2b31 352a 785f 302a 785f 3130 2b33 352a  +15*x_0*x_10+35*
│ │ │ │ +000270d0: 785f 312a 785f 3130 2b33 342a 785f 322a  x_1*x_10+34*x_2*
│ │ │ │ +000270e0: 785f 3130 2b32 302a 785f 342a 785f 3130  x_10+20*x_4*x_10
│ │ │ │ +000270f0: 2b31 342a 785f 302a 785f 3131 2b33 367c  +14*x_0*x_11+36|
│ │ │ │ +00027100: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00027110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027150: 2d2d 2d2d 2d7c 0a7c 2a78 5f31 2a78 5f31  -----|.|*x_1*x_1
│ │ │ │ -00027160: 312b 3335 2a78 5f32 2a78 5f31 312d 3137  1+35*x_2*x_11-17
│ │ │ │ -00027170: 2a78 5f34 2a78 5f31 312c 785f 312a 785f  *x_4*x_11,x_1*x_
│ │ │ │ -00027180: 322d 3430 2a78 5f32 2a78 5f33 2b32 382a  2-40*x_2*x_3+28*
│ │ │ │ -00027190: 785f 335e 322d 785f 302a 785f 342b 352a  x_3^2-x_0*x_4+5*
│ │ │ │ -000271a0: 785f 322a 787c 0a7c 2d2d 2d2d 2d2d 2d2d  x_2*x|.|--------
│ │ │ │ +00027140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027150: 0a7c 2a78 5f31 2a78 5f31 312b 3335 2a78  .|*x_1*x_11+35*x
│ │ │ │ +00027160: 5f32 2a78 5f31 312d 3137 2a78 5f34 2a78  _2*x_11-17*x_4*x
│ │ │ │ +00027170: 5f31 312c 785f 312a 785f 322d 3430 2a78  _11,x_1*x_2-40*x
│ │ │ │ +00027180: 5f32 2a78 5f33 2b32 382a 785f 335e 322d  _2*x_3+28*x_3^2-
│ │ │ │ +00027190: 785f 302a 785f 342b 352a 785f 322a 787c  x_0*x_4+5*x_2*x|
│ │ │ │ +000271a0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  000271b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000271c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000271d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000271e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000271f0: 2d2d 2d2d 2d7c 0a7c 5f34 2d31 362a 785f  -----|.|_4-16*x_
│ │ │ │ -00027200: 332a 785f 342b 352a 785f 302a 785f 352d  3*x_4+5*x_0*x_5-
│ │ │ │ -00027210: 3336 2a78 5f32 2a78 5f35 2b33 372a 785f  36*x_2*x_5+37*x_
│ │ │ │ -00027220: 332a 785f 352b 3438 2a78 5f32 2a78 5f36  3*x_5+48*x_2*x_6
│ │ │ │ -00027230: 2d35 2a78 5f31 2a78 5f37 2d35 2a78 5f33  -5*x_1*x_7-5*x_3
│ │ │ │ -00027240: 2a78 5f37 2b7c 0a7c 2d2d 2d2d 2d2d 2d2d  *x_7+|.|--------
│ │ │ │ +000271e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000271f0: 0a7c 5f34 2d31 362a 785f 332a 785f 342b  .|_4-16*x_3*x_4+
│ │ │ │ +00027200: 352a 785f 302a 785f 352d 3336 2a78 5f32  5*x_0*x_5-36*x_2
│ │ │ │ +00027210: 2a78 5f35 2b33 372a 785f 332a 785f 352b  *x_5+37*x_3*x_5+
│ │ │ │ +00027220: 3438 2a78 5f32 2a78 5f36 2d35 2a78 5f31  48*x_2*x_6-5*x_1
│ │ │ │ +00027230: 2a78 5f37 2d35 2a78 5f33 2a78 5f37 2b7c  *x_7-5*x_3*x_7+|
│ │ │ │ +00027240: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00027250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027290: 2d2d 2d2d 2d7c 0a7c 785f 352a 785f 372b  -----|.|x_5*x_7+
│ │ │ │ -000272a0: 3230 2a78 5f36 2a78 5f37 2b31 302a 785f  20*x_6*x_7+10*x_
│ │ │ │ -000272b0: 302a 785f 382b 3334 2a78 5f31 2a78 5f38  0*x_8+34*x_1*x_8
│ │ │ │ -000272c0: 2b34 312a 785f 332a 785f 382d 785f 342a  +41*x_3*x_8-x_4*
│ │ │ │ -000272d0: 785f 382b 785f 362a 785f 382b 3430 2a78  x_8+x_6*x_8+40*x
│ │ │ │ -000272e0: 5f30 2a78 5f7c 0a7c 2d2d 2d2d 2d2d 2d2d  _0*x_|.|--------
│ │ │ │ +00027280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027290: 0a7c 785f 352a 785f 372b 3230 2a78 5f36  .|x_5*x_7+20*x_6
│ │ │ │ +000272a0: 2a78 5f37 2b31 302a 785f 302a 785f 382b  *x_7+10*x_0*x_8+
│ │ │ │ +000272b0: 3334 2a78 5f31 2a78 5f38 2b34 312a 785f  34*x_1*x_8+41*x_
│ │ │ │ +000272c0: 332a 785f 382d 785f 342a 785f 382b 785f  3*x_8-x_4*x_8+x_
│ │ │ │ +000272d0: 362a 785f 382b 3430 2a78 5f30 2a78 5f7c  6*x_8+40*x_0*x_|
│ │ │ │ +000272e0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  000272f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027330: 2d2d 2d2d 2d7c 0a7c 392d 3332 2a78 5f31  -----|.|9-32*x_1
│ │ │ │ -00027340: 2a78 5f39 2b35 2a78 5f32 2a78 5f39 2d31  *x_9+5*x_2*x_9-1
│ │ │ │ -00027350: 312a 785f 332a 785f 392d 3230 2a78 5f34  1*x_3*x_9-20*x_4
│ │ │ │ -00027360: 2a78 5f39 2b34 352a 785f 352a 785f 392d  *x_9+45*x_5*x_9-
│ │ │ │ -00027370: 3134 2a78 5f36 2a78 5f39 2d32 352a 785f  14*x_6*x_9-25*x_
│ │ │ │ -00027380: 302a 785f 207c 0a7c 2d2d 2d2d 2d2d 2d2d  0*x_ |.|--------
│ │ │ │ +00027320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027330: 0a7c 392d 3332 2a78 5f31 2a78 5f39 2b35  .|9-32*x_1*x_9+5
│ │ │ │ +00027340: 2a78 5f32 2a78 5f39 2d31 312a 785f 332a  *x_2*x_9-11*x_3*
│ │ │ │ +00027350: 785f 392d 3230 2a78 5f34 2a78 5f39 2b34  x_9-20*x_4*x_9+4
│ │ │ │ +00027360: 352a 785f 352a 785f 392d 3134 2a78 5f36  5*x_5*x_9-14*x_6
│ │ │ │ +00027370: 2a78 5f39 2d32 352a 785f 302a 785f 207c  *x_9-25*x_0*x_ |
│ │ │ │ +00027380: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00027390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000273a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000273b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000273c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000273d0: 2d2d 2d2d 2d7c 0a7c 3130 2b34 352a 785f  -----|.|10+45*x_
│ │ │ │ -000273e0: 312a 785f 3130 2d34 312a 785f 322a 785f  1*x_10-41*x_2*x_
│ │ │ │ -000273f0: 3130 2d34 362a 785f 342a 785f 3130 2b38  10-46*x_4*x_10+8
│ │ │ │ -00027400: 2a78 5f36 2a78 5f31 302d 3238 2a78 5f30  *x_6*x_10-28*x_0
│ │ │ │ -00027410: 2a78 5f31 312b 3131 2a78 5f32 2a78 5f31  *x_11+11*x_2*x_1
│ │ │ │ -00027420: 312b 3134 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d  1+14*|.|--------
│ │ │ │ +000273c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000273d0: 0a7c 3130 2b34 352a 785f 312a 785f 3130  .|10+45*x_1*x_10
│ │ │ │ +000273e0: 2d34 312a 785f 322a 785f 3130 2d34 362a  -41*x_2*x_10-46*
│ │ │ │ +000273f0: 785f 342a 785f 3130 2b38 2a78 5f36 2a78  x_4*x_10+8*x_6*x
│ │ │ │ +00027400: 5f31 302d 3238 2a78 5f30 2a78 5f31 312b  _10-28*x_0*x_11+
│ │ │ │ +00027410: 3131 2a78 5f32 2a78 5f31 312b 3134 2a7c  11*x_2*x_11+14*|
│ │ │ │ +00027420: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00027430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027470: 2d2d 2d2d 2d7c 0a7c 785f 342a 785f 3131  -----|.|x_4*x_11
│ │ │ │ -00027480: 2d38 2a78 5f35 2a78 5f31 3129 2c7b 745f  -8*x_5*x_11),{t_
│ │ │ │ -00027490: 345e 322b 745f 302a 745f 352b 745f 312a  4^2+t_0*t_5+t_1*
│ │ │ │ -000274a0: 745f 352b 3335 2a74 5f32 2a74 5f35 2b31  t_5+35*t_2*t_5+1
│ │ │ │ -000274b0: 302a 745f 332a 745f 352b 3235 2a74 5f34  0*t_3*t_5+25*t_4
│ │ │ │ -000274c0: 2a74 5f35 2d7c 0a7c 2d2d 2d2d 2d2d 2d2d  *t_5-|.|--------
│ │ │ │ +00027460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027470: 0a7c 785f 342a 785f 3131 2d38 2a78 5f35  .|x_4*x_11-8*x_5
│ │ │ │ +00027480: 2a78 5f31 3129 2c7b 745f 345e 322b 745f  *x_11),{t_4^2+t_
│ │ │ │ +00027490: 302a 745f 352b 745f 312a 745f 352b 3335  0*t_5+t_1*t_5+35
│ │ │ │ +000274a0: 2a74 5f32 2a74 5f35 2b31 302a 745f 332a  *t_2*t_5+10*t_3*
│ │ │ │ +000274b0: 745f 352b 3235 2a74 5f34 2a74 5f35 2d7c  t_5+25*t_4*t_5-|
│ │ │ │ +000274c0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  000274d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000274e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000274f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027510: 2d2d 2d2d 2d7c 0a7c 352a 745f 355e 322d  -----|.|5*t_5^2-
│ │ │ │ -00027520: 3134 2a74 5f30 2a74 5f36 2d31 342a 745f  14*t_0*t_6-14*t_
│ │ │ │ -00027530: 312a 745f 362d 352a 745f 322a 745f 362d  1*t_6-5*t_2*t_6-
│ │ │ │ -00027540: 3133 2a74 5f34 2a74 5f36 2b33 372a 745f  13*t_4*t_6+37*t_
│ │ │ │ -00027550: 352a 745f 362b 3232 2a74 5f36 5e32 2d33  5*t_6+22*t_6^2-3
│ │ │ │ -00027560: 312a 745f 337c 0a7c 2d2d 2d2d 2d2d 2d2d  1*t_3|.|--------
│ │ │ │ +00027500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027510: 0a7c 352a 745f 355e 322d 3134 2a74 5f30  .|5*t_5^2-14*t_0
│ │ │ │ +00027520: 2a74 5f36 2d31 342a 745f 312a 745f 362d  *t_6-14*t_1*t_6-
│ │ │ │ +00027530: 352a 745f 322a 745f 362d 3133 2a74 5f34  5*t_2*t_6-13*t_4
│ │ │ │ +00027540: 2a74 5f36 2b33 372a 745f 352a 745f 362b  *t_6+37*t_5*t_6+
│ │ │ │ +00027550: 3232 2a74 5f36 5e32 2d33 312a 745f 337c  22*t_6^2-31*t_3|
│ │ │ │ +00027560: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000275b0: 2d2d 2d2d 2d7c 0a7c 2a74 5f37 2b32 362a  -----|.|*t_7+26*
│ │ │ │ -000275c0: 745f 342a 745f 372b 3132 2a74 5f35 2a74  t_4*t_7+12*t_5*t
│ │ │ │ -000275d0: 5f37 2d34 352a 745f 362a 745f 372d 3436  _7-45*t_6*t_7-46
│ │ │ │ -000275e0: 2a74 5f33 2a74 5f38 2b33 372a 745f 342a  *t_3*t_8+37*t_4*
│ │ │ │ -000275f0: 745f 382b 3238 2a74 5f35 2a74 5f38 2b33  t_8+28*t_5*t_8+3
│ │ │ │ -00027600: 332a 745f 367c 0a7c 2d2d 2d2d 2d2d 2d2d  3*t_6|.|--------
│ │ │ │ +000275a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000275b0: 0a7c 2a74 5f37 2b32 362a 745f 342a 745f  .|*t_7+26*t_4*t_
│ │ │ │ +000275c0: 372b 3132 2a74 5f35 2a74 5f37 2d34 352a  7+12*t_5*t_7-45*
│ │ │ │ +000275d0: 745f 362a 745f 372d 3436 2a74 5f33 2a74  t_6*t_7-46*t_3*t
│ │ │ │ +000275e0: 5f38 2b33 372a 745f 342a 745f 382b 3238  _8+37*t_4*t_8+28
│ │ │ │ +000275f0: 2a74 5f35 2a74 5f38 2b33 332a 745f 367c  *t_5*t_8+33*t_6|
│ │ │ │ +00027600: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00027610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027650: 2d2d 2d2d 2d7c 0a7c 2a74 5f38 2c74 5f33  -----|.|*t_8,t_3
│ │ │ │ -00027660: 2a74 5f34 2b34 2a74 5f30 2a74 5f35 2b33  *t_4+4*t_0*t_5+3
│ │ │ │ -00027670: 392a 745f 312a 745f 352d 3430 2a74 5f32  9*t_1*t_5-40*t_2
│ │ │ │ -00027680: 2a74 5f35 2b34 302a 745f 332a 745f 352b  *t_5+40*t_3*t_5+
│ │ │ │ -00027690: 3236 2a74 5f34 2a74 5f35 2d32 302a 745f  26*t_4*t_5-20*t_
│ │ │ │ -000276a0: 355e 322b 207c 0a7c 2d2d 2d2d 2d2d 2d2d  5^2+ |.|--------
│ │ │ │ +00027640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027650: 0a7c 2a74 5f38 2c74 5f33 2a74 5f34 2b34  .|*t_8,t_3*t_4+4
│ │ │ │ +00027660: 2a74 5f30 2a74 5f35 2b33 392a 745f 312a  *t_0*t_5+39*t_1*
│ │ │ │ +00027670: 745f 352d 3430 2a74 5f32 2a74 5f35 2b34  t_5-40*t_2*t_5+4
│ │ │ │ +00027680: 302a 745f 332a 745f 352b 3236 2a74 5f34  0*t_3*t_5+26*t_4
│ │ │ │ +00027690: 2a74 5f35 2d32 302a 745f 355e 322b 207c  *t_5-20*t_5^2+ |
│ │ │ │ +000276a0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000276f0: 2d2d 2d2d 2d7c 0a7c 3431 2a74 5f30 2a74  -----|.|41*t_0*t
│ │ │ │ -00027700: 5f36 2b33 362a 745f 312a 745f 362d 3232  _6+36*t_1*t_6-22
│ │ │ │ -00027710: 2a74 5f32 2a74 5f36 2b33 362a 745f 342a  *t_2*t_6+36*t_4*
│ │ │ │ -00027720: 745f 362d 3330 2a74 5f35 2a74 5f36 2d31  t_6-30*t_5*t_6-1
│ │ │ │ -00027730: 332a 745f 365e 322d 3235 2a74 5f33 2a74  3*t_6^2-25*t_3*t
│ │ │ │ -00027740: 5f37 2b35 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d  _7+5*|.|--------
│ │ │ │ +000276e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000276f0: 0a7c 3431 2a74 5f30 2a74 5f36 2b33 362a  .|41*t_0*t_6+36*
│ │ │ │ +00027700: 745f 312a 745f 362d 3232 2a74 5f32 2a74  t_1*t_6-22*t_2*t
│ │ │ │ +00027710: 5f36 2b33 362a 745f 342a 745f 362d 3330  _6+36*t_4*t_6-30
│ │ │ │ +00027720: 2a74 5f35 2a74 5f36 2d31 332a 745f 365e  *t_5*t_6-13*t_6^
│ │ │ │ +00027730: 322d 3235 2a74 5f33 2a74 5f37 2b35 2a7c  2-25*t_3*t_7+5*|
│ │ │ │ +00027740: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00027750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027790: 2d2d 2d2d 2d7c 0a7c 745f 342a 745f 372d  -----|.|t_4*t_7-
│ │ │ │ -000277a0: 3335 2a74 5f35 2a74 5f37 2b31 302a 745f  35*t_5*t_7+10*t_
│ │ │ │ -000277b0: 362a 745f 372b 3131 2a74 5f33 2a74 5f38  6*t_7+11*t_3*t_8
│ │ │ │ -000277c0: 2b34 362a 745f 342a 745f 382b 3239 2a74  +46*t_4*t_8+29*t
│ │ │ │ -000277d0: 5f35 2a74 5f38 2b32 382a 745f 362a 745f  _5*t_8+28*t_6*t_
│ │ │ │ -000277e0: 382c 745f 327c 0a7c 2d2d 2d2d 2d2d 2d2d  8,t_2|.|--------
│ │ │ │ +00027780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027790: 0a7c 745f 342a 745f 372d 3335 2a74 5f35  .|t_4*t_7-35*t_5
│ │ │ │ +000277a0: 2a74 5f37 2b31 302a 745f 362a 745f 372b  *t_7+10*t_6*t_7+
│ │ │ │ +000277b0: 3131 2a74 5f33 2a74 5f38 2b34 362a 745f  11*t_3*t_8+46*t_
│ │ │ │ +000277c0: 342a 745f 382b 3239 2a74 5f35 2a74 5f38  4*t_8+29*t_5*t_8
│ │ │ │ +000277d0: 2b32 382a 745f 362a 745f 382c 745f 327c  +28*t_6*t_8,t_2|
│ │ │ │ +000277e0: 0a7c 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027830: 2d2d 2d2d 2d7c 0a7c 2a74 5f34 2d35 2a74  -----|.|*t_4-5*t
│ │ │ │ -00027840: 5f30 2a74 5f35 2d34 302a 745f 312a 745f  _0*t_5-40*t_1*t_
│ │ │ │ -00027850: 352b 3132 2a74 5f32 2a74 5f35 2b34 372a  5+12*t_2*t_5+47*
│ │ │ │ -00027860: 745f 332a 745f 352b 3337 2a74 5f34 2a74  t_3*t_5+37*t_4*t
│ │ │ │ -00027870: 5f35 2b32 352a 745f 355e 322d 3237 2a74  _5+25*t_5^2-27*t
│ │ │ │ -00027880: 5f30 2a74 5f7c 0a7c 2d2d 2d2d 2d2d 2d2d  _0*t_|.|--------
│ │ │ │ +00027820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027830: 0a7c 2a74 5f34 2d35 2a74 5f30 2a74 5f35  .|*t_4-5*t_0*t_5
│ │ │ │ +00027840: 2d34 302a 745f 312a 745f 352b 3132 2a74  -40*t_1*t_5+12*t
│ │ │ │ +00027850: 5f32 2a74 5f35 2b34 372a 745f 332a 745f  _2*t_5+47*t_3*t_
│ │ │ │ +00027860: 352b 3337 2a74 5f34 2a74 5f35 2b32 352a  5+37*t_4*t_5+25*
│ │ │ │ +00027870: 745f 355e 322d 3237 2a74 5f30 2a74 5f7c  t_5^2-27*t_0*t_|
│ │ │ │ +00027880: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00027890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000278a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000278b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000278c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000278d0: 2d2d 2d2d 2d7c 0a7c 362d 3232 2a74 5f31  -----|.|6-22*t_1
│ │ │ │ -000278e0: 2a74 5f36 2b32 372a 745f 322a 745f 362d  *t_6+27*t_2*t_6-
│ │ │ │ -000278f0: 3233 2a74 5f34 2a74 5f36 2b35 2a74 5f35  23*t_4*t_6+5*t_5
│ │ │ │ -00027900: 2a74 5f36 2d31 332a 745f 365e 322d 3339  *t_6-13*t_6^2-39
│ │ │ │ -00027910: 2a74 5f33 2a74 5f37 2d32 392a 745f 342a  *t_3*t_7-29*t_4*
│ │ │ │ -00027920: 745f 372b 397c 0a7c 2d2d 2d2d 2d2d 2d2d  t_7+9|.|--------
│ │ │ │ +000278c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000278d0: 0a7c 362d 3232 2a74 5f31 2a74 5f36 2b32  .|6-22*t_1*t_6+2
│ │ │ │ +000278e0: 372a 745f 322a 745f 362d 3233 2a74 5f34  7*t_2*t_6-23*t_4
│ │ │ │ +000278f0: 2a74 5f36 2b35 2a74 5f35 2a74 5f36 2d31  *t_6+5*t_5*t_6-1
│ │ │ │ +00027900: 332a 745f 365e 322d 3339 2a74 5f33 2a74  3*t_6^2-39*t_3*t
│ │ │ │ +00027910: 5f37 2d32 392a 745f 342a 745f 372b 397c  _7-29*t_4*t_7+9|
│ │ │ │ +00027920: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00027930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027970: 2d2d 2d2d 2d7c 0a7c 2a74 5f35 2a74 5f37  -----|.|*t_5*t_7
│ │ │ │ -00027980: 2b33 392a 745f 362a 745f 372b 3336 2a74  +39*t_6*t_7+36*t
│ │ │ │ -00027990: 5f33 2a74 5f38 2b31 332a 745f 342a 745f  _3*t_8+13*t_4*t_
│ │ │ │ -000279a0: 382b 3236 2a74 5f35 2a74 5f38 2b33 372a  8+26*t_5*t_8+37*
│ │ │ │ -000279b0: 745f 362a 745f 382c 745f 302a 745f 342d  t_6*t_8,t_0*t_4-
│ │ │ │ -000279c0: 745f 302a 747c 0a7c 2d2d 2d2d 2d2d 2d2d  t_0*t|.|--------
│ │ │ │ +00027960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027970: 0a7c 2a74 5f35 2a74 5f37 2b33 392a 745f  .|*t_5*t_7+39*t_
│ │ │ │ +00027980: 362a 745f 372b 3336 2a74 5f33 2a74 5f38  6*t_7+36*t_3*t_8
│ │ │ │ +00027990: 2b31 332a 745f 342a 745f 382b 3236 2a74  +13*t_4*t_8+26*t
│ │ │ │ +000279a0: 5f35 2a74 5f38 2b33 372a 745f 362a 745f  _5*t_8+37*t_6*t_
│ │ │ │ +000279b0: 382c 745f 302a 745f 342d 745f 302a 747c  8,t_0*t_4-t_0*t|
│ │ │ │ +000279c0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  000279d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000279e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000279f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027a10: 2d2d 2d2d 2d7c 0a7c 5f35 2d38 2a74 5f31  -----|.|_5-8*t_1
│ │ │ │ -00027a20: 2a74 5f35 2d33 352a 745f 322a 745f 352d  *t_5-35*t_2*t_5-
│ │ │ │ -00027a30: 3130 2a74 5f33 2a74 5f35 2d33 332a 745f  10*t_3*t_5-33*t_
│ │ │ │ -00027a40: 342a 745f 352b 352a 745f 355e 322b 3135  4*t_5+5*t_5^2+15
│ │ │ │ -00027a50: 2a74 5f30 2a74 5f36 2b31 352a 745f 312a  *t_0*t_6+15*t_1*
│ │ │ │ -00027a60: 745f 362b 357c 0a7c 2d2d 2d2d 2d2d 2d2d  t_6+5|.|--------
│ │ │ │ +00027a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027a10: 0a7c 5f35 2d38 2a74 5f31 2a74 5f35 2d33  .|_5-8*t_1*t_5-3
│ │ │ │ +00027a20: 352a 745f 322a 745f 352d 3130 2a74 5f33  5*t_2*t_5-10*t_3
│ │ │ │ +00027a30: 2a74 5f35 2d33 332a 745f 342a 745f 352b  *t_5-33*t_4*t_5+
│ │ │ │ +00027a40: 352a 745f 355e 322b 3135 2a74 5f30 2a74  5*t_5^2+15*t_0*t
│ │ │ │ +00027a50: 5f36 2b31 352a 745f 312a 745f 362b 357c  _6+15*t_1*t_6+5|
│ │ │ │ +00027a60: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00027a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027ab0: 2d2d 2d2d 2d7c 0a7c 2a74 5f32 2a74 5f36  -----|.|*t_2*t_6
│ │ │ │ -00027ac0: 2b31 352a 745f 342a 745f 362d 3338 2a74  +15*t_4*t_6-38*t
│ │ │ │ -00027ad0: 5f35 2a74 5f36 2d32 322a 745f 365e 322b  _5*t_6-22*t_6^2+
│ │ │ │ -00027ae0: 3331 2a74 5f33 2a74 5f37 2d32 352a 745f  31*t_3*t_7-25*t_
│ │ │ │ -00027af0: 342a 745f 372d 3139 2a74 5f35 2a74 5f37  4*t_7-19*t_5*t_7
│ │ │ │ -00027b00: 2b34 372a 747c 0a7c 2d2d 2d2d 2d2d 2d2d  +47*t|.|--------
│ │ │ │ +00027aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027ab0: 0a7c 2a74 5f32 2a74 5f36 2b31 352a 745f  .|*t_2*t_6+15*t_
│ │ │ │ +00027ac0: 342a 745f 362d 3338 2a74 5f35 2a74 5f36  4*t_6-38*t_5*t_6
│ │ │ │ +00027ad0: 2d32 322a 745f 365e 322b 3331 2a74 5f33  -22*t_6^2+31*t_3
│ │ │ │ +00027ae0: 2a74 5f37 2d32 352a 745f 342a 745f 372d  *t_7-25*t_4*t_7-
│ │ │ │ +00027af0: 3139 2a74 5f35 2a74 5f37 2b34 372a 747c  19*t_5*t_7+47*t|
│ │ │ │ +00027b00: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00027b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027b50: 2d2d 2d2d 2d7c 0a7c 5f36 2a74 5f37 2b34  -----|.|_6*t_7+4
│ │ │ │ -00027b60: 362a 745f 332a 745f 382d 3336 2a74 5f34  6*t_3*t_8-36*t_4
│ │ │ │ -00027b70: 2a74 5f38 2d33 352a 745f 352a 745f 382d  *t_8-35*t_5*t_8-
│ │ │ │ -00027b80: 3331 2a74 5f36 2a74 5f38 2c74 5f32 2a74  31*t_6*t_8,t_2*t
│ │ │ │ -00027b90: 5f33 2d74 5f30 2a74 5f35 2d74 5f31 2a74  _3-t_0*t_5-t_1*t
│ │ │ │ -00027ba0: 5f35 2d33 357c 0a7c 2d2d 2d2d 2d2d 2d2d  _5-35|.|--------
│ │ │ │ +00027b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027b50: 0a7c 5f36 2a74 5f37 2b34 362a 745f 332a  .|_6*t_7+46*t_3*
│ │ │ │ +00027b60: 745f 382d 3336 2a74 5f34 2a74 5f38 2d33  t_8-36*t_4*t_8-3
│ │ │ │ +00027b70: 352a 745f 352a 745f 382d 3331 2a74 5f36  5*t_5*t_8-31*t_6
│ │ │ │ +00027b80: 2a74 5f38 2c74 5f32 2a74 5f33 2d74 5f30  *t_8,t_2*t_3-t_0
│ │ │ │ +00027b90: 2a74 5f35 2d74 5f31 2a74 5f35 2d33 357c  *t_5-t_1*t_5-35|
│ │ │ │ +00027ba0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00027bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027bf0: 2d2d 2d2d 2d7c 0a7c 2a74 5f32 2a74 5f35  -----|.|*t_2*t_5
│ │ │ │ -00027c00: 2d31 302a 745f 332a 745f 352d 3333 2a74  -10*t_3*t_5-33*t
│ │ │ │ -00027c10: 5f34 2a74 5f35 2b35 2a74 5f35 5e32 2b31  _4*t_5+5*t_5^2+1
│ │ │ │ -00027c20: 342a 745f 302a 745f 362b 3134 2a74 5f31  4*t_0*t_6+14*t_1
│ │ │ │ -00027c30: 2a74 5f36 2b35 2a74 5f32 2a74 5f36 2b31  *t_6+5*t_2*t_6+1
│ │ │ │ -00027c40: 342a 745f 347c 0a7c 2d2d 2d2d 2d2d 2d2d  4*t_4|.|--------
│ │ │ │ +00027be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027bf0: 0a7c 2a74 5f32 2a74 5f35 2d31 302a 745f  .|*t_2*t_5-10*t_
│ │ │ │ +00027c00: 332a 745f 352d 3333 2a74 5f34 2a74 5f35  3*t_5-33*t_4*t_5
│ │ │ │ +00027c10: 2b35 2a74 5f35 5e32 2b31 342a 745f 302a  +5*t_5^2+14*t_0*
│ │ │ │ +00027c20: 745f 362b 3134 2a74 5f31 2a74 5f36 2b35  t_6+14*t_1*t_6+5
│ │ │ │ +00027c30: 2a74 5f32 2a74 5f36 2b31 342a 745f 347c  *t_2*t_6+14*t_4|
│ │ │ │ +00027c40: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00027c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027c90: 2d2d 2d2d 2d7c 0a7c 2a74 5f36 2d33 312a  -----|.|*t_6-31*
│ │ │ │ -00027ca0: 745f 352a 745f 362d 3234 2a74 5f36 5e32  t_5*t_6-24*t_6^2
│ │ │ │ -00027cb0: 2b33 322a 745f 332a 745f 372d 3235 2a74  +32*t_3*t_7-25*t
│ │ │ │ -00027cc0: 5f34 2a74 5f37 2d31 392a 745f 352a 745f  _4*t_7-19*t_5*t_
│ │ │ │ -00027cd0: 372b 3437 2a74 5f36 2a74 5f37 2b34 362a  7+47*t_6*t_7+46*
│ │ │ │ -00027ce0: 745f 332a 747c 0a7c 2d2d 2d2d 2d2d 2d2d  t_3*t|.|--------
│ │ │ │ +00027c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027c90: 0a7c 2a74 5f36 2d33 312a 745f 352a 745f  .|*t_6-31*t_5*t_
│ │ │ │ +00027ca0: 362d 3234 2a74 5f36 5e32 2b33 322a 745f  6-24*t_6^2+32*t_
│ │ │ │ +00027cb0: 332a 745f 372d 3235 2a74 5f34 2a74 5f37  3*t_7-25*t_4*t_7
│ │ │ │ +00027cc0: 2d31 392a 745f 352a 745f 372b 3437 2a74  -19*t_5*t_7+47*t
│ │ │ │ +00027cd0: 5f36 2a74 5f37 2b34 362a 745f 332a 747c  _6*t_7+46*t_3*t|
│ │ │ │ +00027ce0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00027cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027d30: 2d2d 2d2d 2d7c 0a7c 5f38 2d33 362a 745f  -----|.|_8-36*t_
│ │ │ │ -00027d40: 342a 745f 382d 3335 2a74 5f35 2a74 5f38  4*t_8-35*t_5*t_8
│ │ │ │ -00027d50: 2d33 312a 745f 362a 745f 382c 745f 312a  -31*t_6*t_8,t_1*
│ │ │ │ -00027d60: 745f 332d 372a 745f 312a 745f 352b 745f  t_3-7*t_1*t_5+t_
│ │ │ │ -00027d70: 312a 745f 362b 745f 342a 745f 362d 372a  1*t_6+t_4*t_6-7*
│ │ │ │ -00027d80: 745f 352a 747c 0a7c 2d2d 2d2d 2d2d 2d2d  t_5*t|.|--------
│ │ │ │ +00027d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027d30: 0a7c 5f38 2d33 362a 745f 342a 745f 382d  .|_8-36*t_4*t_8-
│ │ │ │ +00027d40: 3335 2a74 5f35 2a74 5f38 2d33 312a 745f  35*t_5*t_8-31*t_
│ │ │ │ +00027d50: 362a 745f 382c 745f 312a 745f 332d 372a  6*t_8,t_1*t_3-7*
│ │ │ │ +00027d60: 745f 312a 745f 352b 745f 312a 745f 362b  t_1*t_5+t_1*t_6+
│ │ │ │ +00027d70: 745f 342a 745f 362d 372a 745f 352a 747c  t_4*t_6-7*t_5*t|
│ │ │ │ +00027d80: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00027d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027dd0: 2d2d 2d2d 2d7c 0a7c 5f36 2b32 2a74 5f36  -----|.|_6+2*t_6
│ │ │ │ -00027de0: 5e32 2d74 5f33 2a74 5f37 2c74 5f30 2a74  ^2-t_3*t_7,t_0*t
│ │ │ │ -00027df0: 5f33 2d34 362a 745f 302a 745f 352d 3339  _3-46*t_0*t_5-39
│ │ │ │ -00027e00: 2a74 5f31 2a74 5f35 2d34 332a 745f 322a  *t_1*t_5-43*t_2*
│ │ │ │ -00027e10: 745f 352d 3431 2a74 5f33 2a74 5f35 2d32  t_5-41*t_3*t_5-2
│ │ │ │ -00027e20: 362a 745f 347c 0a7c 2d2d 2d2d 2d2d 2d2d  6*t_4|.|--------
│ │ │ │ +00027dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027dd0: 0a7c 5f36 2b32 2a74 5f36 5e32 2d74 5f33  .|_6+2*t_6^2-t_3
│ │ │ │ +00027de0: 2a74 5f37 2c74 5f30 2a74 5f33 2d34 362a  *t_7,t_0*t_3-46*
│ │ │ │ +00027df0: 745f 302a 745f 352d 3339 2a74 5f31 2a74  t_0*t_5-39*t_1*t
│ │ │ │ +00027e00: 5f35 2d34 332a 745f 322a 745f 352d 3431  _5-43*t_2*t_5-41
│ │ │ │ +00027e10: 2a74 5f33 2a74 5f35 2d32 362a 745f 347c  *t_3*t_5-26*t_4|
│ │ │ │ +00027e20: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00027e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e70: 2d2d 2d2d 2d7c 0a7c 2a74 5f35 2d32 382a  -----|.|*t_5-28*
│ │ │ │ -00027e80: 745f 355e 322d 3335 2a74 5f30 2a74 5f36  t_5^2-35*t_0*t_6
│ │ │ │ -00027e90: 2d33 362a 745f 312a 745f 362b 3230 2a74  -36*t_1*t_6+20*t
│ │ │ │ -00027ea0: 5f32 2a74 5f36 2d33 362a 745f 342a 745f  _2*t_6-36*t_4*t_
│ │ │ │ -00027eb0: 362b 392a 745f 352a 745f 362b 3135 2a74  6+9*t_5*t_6+15*t
│ │ │ │ -00027ec0: 5f36 5e32 2b7c 0a7c 2d2d 2d2d 2d2d 2d2d  _6^2+|.|--------
│ │ │ │ +00027e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027e70: 0a7c 2a74 5f35 2d32 382a 745f 355e 322d  .|*t_5-28*t_5^2-
│ │ │ │ +00027e80: 3335 2a74 5f30 2a74 5f36 2d33 362a 745f  35*t_0*t_6-36*t_
│ │ │ │ +00027e90: 312a 745f 362b 3230 2a74 5f32 2a74 5f36  1*t_6+20*t_2*t_6
│ │ │ │ +00027ea0: 2d33 362a 745f 342a 745f 362b 392a 745f  -36*t_4*t_6+9*t_
│ │ │ │ +00027eb0: 352a 745f 362b 3135 2a74 5f36 5e32 2b7c  5*t_6+15*t_6^2+|
│ │ │ │ +00027ec0: 0a7c 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 2d2d 2d2d  ----------------
│ │ │ │ -00027f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027f10: 2d2d 2d2d 2d7c 0a7c 3236 2a74 5f33 2a74  -----|.|26*t_3*t
│ │ │ │ -00027f20: 5f37 2d35 2a74 5f34 2a74 5f37 2b33 352a  _7-5*t_4*t_7+35*
│ │ │ │ -00027f30: 745f 352a 745f 372d 3130 2a74 5f36 2a74  t_5*t_7-10*t_6*t
│ │ │ │ -00027f40: 5f37 2d31 302a 745f 332a 745f 382d 3436  _7-10*t_3*t_8-46
│ │ │ │ -00027f50: 2a74 5f34 2a74 5f38 2b34 372a 745f 352a  *t_4*t_8+47*t_5*
│ │ │ │ -00027f60: 745f 382d 207c 0a7c 2d2d 2d2d 2d2d 2d2d  t_8- |.|--------
│ │ │ │ +00027f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027f10: 0a7c 3236 2a74 5f33 2a74 5f37 2d35 2a74  .|26*t_3*t_7-5*t
│ │ │ │ +00027f20: 5f34 2a74 5f37 2b33 352a 745f 352a 745f  _4*t_7+35*t_5*t_
│ │ │ │ +00027f30: 372d 3130 2a74 5f36 2a74 5f37 2d31 302a  7-10*t_6*t_7-10*
│ │ │ │ +00027f40: 745f 332a 745f 382d 3436 2a74 5f34 2a74  t_3*t_8-46*t_4*t
│ │ │ │ +00027f50: 5f38 2b34 372a 745f 352a 745f 382d 207c  _8+47*t_5*t_8- |
│ │ │ │ +00027f60: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00027f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027fb0: 2d2d 2d2d 2d7c 0a7c 3235 2a74 5f36 2a74  -----|.|25*t_6*t
│ │ │ │ -00027fc0: 5f38 2c74 5f32 5e32 2d34 362a 745f 312a  _8,t_2^2-46*t_1*
│ │ │ │ -00027fd0: 745f 342d 3333 2a74 5f30 2a74 5f35 2d34  t_4-33*t_0*t_5-4
│ │ │ │ -00027fe0: 352a 745f 312a 745f 352d 3339 2a74 5f32  5*t_1*t_5-39*t_2
│ │ │ │ -00027ff0: 2a74 5f35 2d33 392a 745f 332a 745f 352d  *t_5-39*t_3*t_5-
│ │ │ │ -00028000: 3436 2a74 5f7c 0a7c 2d2d 2d2d 2d2d 2d2d  46*t_|.|--------
│ │ │ │ +00027fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00027fb0: 0a7c 3235 2a74 5f36 2a74 5f38 2c74 5f32  .|25*t_6*t_8,t_2
│ │ │ │ +00027fc0: 5e32 2d34 362a 745f 312a 745f 342d 3333  ^2-46*t_1*t_4-33
│ │ │ │ +00027fd0: 2a74 5f30 2a74 5f35 2d34 352a 745f 312a  *t_0*t_5-45*t_1*
│ │ │ │ +00027fe0: 745f 352d 3339 2a74 5f32 2a74 5f35 2d33  t_5-39*t_2*t_5-3
│ │ │ │ +00027ff0: 392a 745f 332a 745f 352d 3436 2a74 5f7c  9*t_3*t_5-46*t_|
│ │ │ │ +00028000: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00028010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028050: 2d2d 2d2d 2d7c 0a7c 342a 745f 352d 3239  -----|.|4*t_5-29
│ │ │ │ -00028060: 2a74 5f35 5e32 2d34 382a 745f 302a 745f  *t_5^2-48*t_0*t_
│ │ │ │ -00028070: 362d 3338 2a74 5f31 2a74 5f36 2d33 302a  6-38*t_1*t_6-30*
│ │ │ │ -00028080: 745f 322a 745f 362b 3139 2a74 5f34 2a74  t_2*t_6+19*t_4*t
│ │ │ │ -00028090: 5f36 2d34 342a 745f 352a 745f 362d 3437  _6-44*t_5*t_6-47
│ │ │ │ -000280a0: 2a74 5f36 5e7c 0a7c 2d2d 2d2d 2d2d 2d2d  *t_6^|.|--------
│ │ │ │ +00028040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00028050: 0a7c 342a 745f 352d 3239 2a74 5f35 5e32  .|4*t_5-29*t_5^2
│ │ │ │ +00028060: 2d34 382a 745f 302a 745f 362d 3338 2a74  -48*t_0*t_6-38*t
│ │ │ │ +00028070: 5f31 2a74 5f36 2d33 302a 745f 322a 745f  _1*t_6-30*t_2*t_
│ │ │ │ +00028080: 362b 3139 2a74 5f34 2a74 5f36 2d34 342a  6+19*t_4*t_6-44*
│ │ │ │ +00028090: 745f 352a 745f 362d 3437 2a74 5f36 5e7c  t_5*t_6-47*t_6^|
│ │ │ │ +000280a0: 0a7c 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 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000280e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000280f0: 2d2d 2d2d 2d7c 0a7c 322d 3336 2a74 5f30  -----|.|2-36*t_0
│ │ │ │ -00028100: 2a74 5f37 2d34 362a 745f 312a 745f 372b  *t_7-46*t_1*t_7+
│ │ │ │ -00028110: 745f 322a 745f 372d 3434 2a74 5f33 2a74  t_2*t_7-44*t_3*t
│ │ │ │ -00028120: 5f37 2b34 382a 745f 342a 745f 372d 3134  _7+48*t_4*t_7-14
│ │ │ │ -00028130: 2a74 5f35 2a74 5f37 2b34 2a74 5f36 2a74  *t_5*t_7+4*t_6*t
│ │ │ │ -00028140: 5f37 2d33 367c 0a7c 2d2d 2d2d 2d2d 2d2d  _7-36|.|--------
│ │ │ │ +000280e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000280f0: 0a7c 322d 3336 2a74 5f30 2a74 5f37 2d34  .|2-36*t_0*t_7-4
│ │ │ │ +00028100: 362a 745f 312a 745f 372b 745f 322a 745f  6*t_1*t_7+t_2*t_
│ │ │ │ +00028110: 372d 3434 2a74 5f33 2a74 5f37 2b34 382a  7-44*t_3*t_7+48*
│ │ │ │ +00028120: 745f 342a 745f 372d 3134 2a74 5f35 2a74  t_4*t_7-14*t_5*t
│ │ │ │ +00028130: 5f37 2b34 2a74 5f36 2a74 5f37 2d33 367c  _7+4*t_6*t_7-36|
│ │ │ │ +00028140: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00028150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00028190: 2d2d 2d2d 2d7c 0a7c 2a74 5f30 2a74 5f38  -----|.|*t_0*t_8
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│ │ │ │ -000281e0: 5f38 2b34 377c 0a7c 2d2d 2d2d 2d2d 2d2d  _8+47|.|--------
│ │ │ │ +00028180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
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│ │ │ │ -00028230: 2d2d 2d2d 2d7c 0a7c 2a74 5f37 2a74 5f38  -----|.|*t_7*t_8
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│ │ │ │ -00028280: 2b74 5f31 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d  +t_1*|.|--------
│ │ │ │ +00028220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
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│ │ │ │ +00028250: 5f30 2a74 5f36 2d32 2a74 5f31 2a74 5f36  _0*t_6-2*t_1*t_6
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│ │ │ │  000282b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000282c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000282d0: 2d2d 2d2d 2d7c 0a7c 745f 372d 322a 745f  -----|.|t_7-2*t_
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│ │ │ │ -00028300: 745f 382b 332a 745f 372a 745f 382c 745f  t_8+3*t_7*t_8,t_
│ │ │ │ -00028310: 302a 745f 322b 745f 312a 745f 342b 352a  0*t_2+t_1*t_4+5*
│ │ │ │ -00028320: 745f 302a 747c 0a7c 2d2d 2d2d 2d2d 2d2d  t_0*t|.|--------
│ │ │ │ +000282c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000282d0: 0a7c 745f 372d 322a 745f 322a 745f 372d  .|t_7-2*t_2*t_7-
│ │ │ │ +000282e0: 372a 745f 352a 745f 372b 322a 745f 362a  7*t_5*t_7+2*t_6*
│ │ │ │ +000282f0: 745f 372d 322a 745f 312a 745f 382b 332a  t_7-2*t_1*t_8+3*
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│ │ │ │  00028350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028370: 2d2d 2d2d 2d7c 0a7c 5f35 2b33 322a 745f  -----|.|_5+32*t_
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│ │ │ │ -000283a0: 5f34 2a74 5f35 2d32 352a 745f 355e 322b  _4*t_5-25*t_5^2+
│ │ │ │ -000283b0: 3139 2a74 5f30 2a74 5f36 2b32 322a 745f  19*t_0*t_6+22*t_
│ │ │ │ -000283c0: 312a 745f 367c 0a7c 2d2d 2d2d 2d2d 2d2d  1*t_6|.|--------
│ │ │ │ +00028360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00028370: 0a7c 5f35 2b33 322a 745f 312a 745f 352d  .|_5+32*t_1*t_5-
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│ │ │ │ +000283c0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
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│ │ │ │  000283e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000283f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028410: 2d2d 2d2d 2d7c 0a7c 2d32 352a 745f 322a  -----|.|-25*t_2*
│ │ │ │ -00028420: 745f 362b 3235 2a74 5f34 2a74 5f36 2d35  t_6+25*t_4*t_6-5
│ │ │ │ -00028430: 2a74 5f35 2a74 5f36 2b31 332a 745f 365e  *t_5*t_6+13*t_6^
│ │ │ │ -00028440: 322b 352a 745f 302a 745f 372b 745f 312a  2+5*t_0*t_7+t_1*
│ │ │ │ -00028450: 745f 372b 3339 2a74 5f33 2a74 5f37 2b32  t_7+39*t_3*t_7+2
│ │ │ │ -00028460: 382a 745f 347c 0a7c 2d2d 2d2d 2d2d 2d2d  8*t_4|.|--------
│ │ │ │ +00028400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00028410: 0a7c 2d32 352a 745f 322a 745f 362b 3235  .|-25*t_2*t_6+25
│ │ │ │ +00028420: 2a74 5f34 2a74 5f36 2d35 2a74 5f35 2a74  *t_4*t_6-5*t_5*t
│ │ │ │ +00028430: 5f36 2b31 332a 745f 365e 322b 352a 745f  _6+13*t_6^2+5*t_
│ │ │ │ +00028440: 302a 745f 372b 745f 312a 745f 372b 3339  0*t_7+t_1*t_7+39
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│ │ │ │ +00028460: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
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│ │ │ │  00028480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000284a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000284b0: 2d2d 2d2d 2d7c 0a7c 2a74 5f37 2d39 2a74  -----|.|*t_7-9*t
│ │ │ │ -000284c0: 5f35 2a74 5f37 2d33 392a 745f 362a 745f  _5*t_7-39*t_6*t_
│ │ │ │ -000284d0: 372b 342a 745f 302a 745f 382b 745f 312a  7+4*t_0*t_8+t_1*
│ │ │ │ -000284e0: 745f 382d 3336 2a74 5f33 2a74 5f38 2d31  t_8-36*t_3*t_8-1
│ │ │ │ -000284f0: 342a 745f 342a 745f 382d 3236 2a74 5f35  4*t_4*t_8-26*t_5
│ │ │ │ -00028500: 2a74 5f38 2d7c 0a7c 2d2d 2d2d 2d2d 2d2d  *t_8-|.|--------
│ │ │ │ +000284a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000284b0: 0a7c 2a74 5f37 2d39 2a74 5f35 2a74 5f37  .|*t_7-9*t_5*t_7
│ │ │ │ +000284c0: 2d33 392a 745f 362a 745f 372b 342a 745f  -39*t_6*t_7+4*t_
│ │ │ │ +000284d0: 302a 745f 382b 745f 312a 745f 382d 3336  0*t_8+t_1*t_8-36
│ │ │ │ +000284e0: 2a74 5f33 2a74 5f38 2d31 342a 745f 342a  *t_3*t_8-14*t_4*
│ │ │ │ +000284f0: 745f 382d 3236 2a74 5f35 2a74 5f38 2d7c  t_8-26*t_5*t_8-|
│ │ │ │ +00028500: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
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│ │ │ │  00028530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028550: 2d2d 2d2d 2d7c 0a7c 3337 2a74 5f36 2a74  -----|.|37*t_6*t
│ │ │ │ -00028560: 5f38 2c74 5f30 2a74 5f31 2d33 392a 745f  _8,t_0*t_1-39*t_
│ │ │ │ -00028570: 312a 745f 342b 3430 2a74 5f31 2a74 5f35  1*t_4+40*t_1*t_5
│ │ │ │ -00028580: 2d33 372a 745f 302a 745f 362d 3339 2a74  -37*t_0*t_6-39*t
│ │ │ │ -00028590: 5f31 2a74 5f36 2b31 392a 745f 342a 745f  _1*t_6+19*t_4*t_
│ │ │ │ -000285a0: 362d 3339 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d  6-39*|.|--------
│ │ │ │ +00028540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00028550: 0a7c 3337 2a74 5f36 2a74 5f38 2c74 5f30  .|37*t_6*t_8,t_0
│ │ │ │ +00028560: 2a74 5f31 2d33 392a 745f 312a 745f 342b  *t_1-39*t_1*t_4+
│ │ │ │ +00028570: 3430 2a74 5f31 2a74 5f35 2d33 372a 745f  40*t_1*t_5-37*t_
│ │ │ │ +00028580: 302a 745f 362d 3339 2a74 5f31 2a74 5f36  0*t_6-39*t_1*t_6
│ │ │ │ +00028590: 2b31 392a 745f 342a 745f 362d 3339 2a7c  +19*t_4*t_6-39*|
│ │ │ │ +000285a0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
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│ │ │ │  000285c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000285d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000285e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000285f0: 2d2d 2d2d 2d7c 0a7c 745f 352a 745f 362d  -----|.|t_5*t_6-
│ │ │ │ -00028600: 3338 2a74 5f30 2a74 5f37 2b33 392a 745f  38*t_0*t_7+39*t_
│ │ │ │ -00028610: 312a 745f 372b 3139 2a74 5f32 2a74 5f37  1*t_7+19*t_2*t_7
│ │ │ │ -00028620: 2b31 382a 745f 352a 745f 372d 3139 2a74  +18*t_5*t_7-19*t
│ │ │ │ -00028630: 5f36 2a74 5f37 2b31 392a 745f 312a 745f  _6*t_7+19*t_1*t_
│ │ │ │ -00028640: 382b 3230 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d  8+20*|.|--------
│ │ │ │ +000285e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000285f0: 0a7c 745f 352a 745f 362d 3338 2a74 5f30  .|t_5*t_6-38*t_0
│ │ │ │ +00028600: 2a74 5f37 2b33 392a 745f 312a 745f 372b  *t_7+39*t_1*t_7+
│ │ │ │ +00028610: 3139 2a74 5f32 2a74 5f37 2b31 382a 745f  19*t_2*t_7+18*t_
│ │ │ │ +00028620: 352a 745f 372d 3139 2a74 5f36 2a74 5f37  5*t_7-19*t_6*t_7
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│ │ │ │ +00028640: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
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│ │ │ │  00028660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028690: 2d2d 2d2d 2d7c 0a7c 745f 372a 745f 382c  -----|.|t_7*t_8,
│ │ │ │ -000286a0: 745f 305e 322b 3132 2a74 5f31 2a74 5f34  t_0^2+12*t_1*t_4
│ │ │ │ -000286b0: 2b32 302a 745f 302a 745f 352b 3237 2a74  +20*t_0*t_5+27*t
│ │ │ │ -000286c0: 5f31 2a74 5f35 2d38 2a74 5f32 2a74 5f35  _1*t_5-8*t_2*t_5
│ │ │ │ -000286d0: 2b33 372a 745f 332a 745f 352b 3238 2a74  +37*t_3*t_5+28*t
│ │ │ │ -000286e0: 5f34 2a74 5f7c 0a7c 2d2d 2d2d 2d2d 2d2d  _4*t_|.|--------
│ │ │ │ +00028680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00028690: 0a7c 745f 372a 745f 382c 745f 305e 322b  .|t_7*t_8,t_0^2+
│ │ │ │ +000286a0: 3132 2a74 5f31 2a74 5f34 2b32 302a 745f  12*t_1*t_4+20*t_
│ │ │ │ +000286b0: 302a 745f 352b 3237 2a74 5f31 2a74 5f35  0*t_5+27*t_1*t_5
│ │ │ │ +000286c0: 2d38 2a74 5f32 2a74 5f35 2b33 372a 745f  -8*t_2*t_5+37*t_
│ │ │ │ +000286d0: 332a 745f 352b 3238 2a74 5f34 2a74 5f7c  3*t_5+28*t_4*t_|
│ │ │ │ +000286e0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d  ----------------
│ │ │ │ -00028720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028730: 2d2d 2d2d 2d7c 0a7c 352b 3330 2a74 5f35  -----|.|5+30*t_5
│ │ │ │ -00028740: 5e32 2d34 362a 745f 302a 745f 362b 3234  ^2-46*t_0*t_6+24
│ │ │ │ -00028750: 2a74 5f31 2a74 5f36 2d34 302a 745f 322a  *t_1*t_6-40*t_2*
│ │ │ │ -00028760: 745f 362b 3235 2a74 5f34 2a74 5f36 2b31  t_6+25*t_4*t_6+1
│ │ │ │ -00028770: 362a 745f 352a 745f 362d 3335 2a74 5f36  6*t_5*t_6-35*t_6
│ │ │ │ -00028780: 5e32 2b32 397c 0a7c 2d2d 2d2d 2d2d 2d2d  ^2+29|.|--------
│ │ │ │ +00028720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00028730: 0a7c 352b 3330 2a74 5f35 5e32 2d34 362a  .|5+30*t_5^2-46*
│ │ │ │ +00028740: 745f 302a 745f 362b 3234 2a74 5f31 2a74  t_0*t_6+24*t_1*t
│ │ │ │ +00028750: 5f36 2d34 302a 745f 322a 745f 362b 3235  _6-40*t_2*t_6+25
│ │ │ │ +00028760: 2a74 5f34 2a74 5f36 2b31 362a 745f 352a  *t_4*t_6+16*t_5*
│ │ │ │ +00028770: 745f 362d 3335 2a74 5f36 5e32 2b32 397c  t_6-35*t_6^2+29|
│ │ │ │ +00028780: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00028790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000287a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000287b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000287c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000287d0: 2d2d 2d2d 2d7c 0a7c 2a74 5f30 2a74 5f37  -----|.|*t_0*t_7
│ │ │ │ -000287e0: 2b31 322a 745f 312a 745f 372d 3335 2a74  +12*t_1*t_7-35*t
│ │ │ │ -000287f0: 5f32 2a74 5f37 2d38 2a74 5f33 2a74 5f37  _2*t_7-8*t_3*t_7
│ │ │ │ -00028800: 2d31 382a 745f 342a 745f 372b 3432 2a74  -18*t_4*t_7+42*t
│ │ │ │ -00028810: 5f35 2a74 5f37 2d31 322a 745f 362a 745f  _5*t_7-12*t_6*t_
│ │ │ │ -00028820: 372d 362a 747c 0a7c 2d2d 2d2d 2d2d 2d2d  7-6*t|.|--------
│ │ │ │ +000287c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000287d0: 0a7c 2a74 5f30 2a74 5f37 2b31 322a 745f  .|*t_0*t_7+12*t_
│ │ │ │ +000287e0: 312a 745f 372d 3335 2a74 5f32 2a74 5f37  1*t_7-35*t_2*t_7
│ │ │ │ +000287f0: 2d38 2a74 5f33 2a74 5f37 2d31 382a 745f  -8*t_3*t_7-18*t_
│ │ │ │ +00028800: 342a 745f 372b 3432 2a74 5f35 2a74 5f37  4*t_7+42*t_5*t_7
│ │ │ │ +00028810: 2d31 322a 745f 362a 745f 372d 362a 747c  -12*t_6*t_7-6*t|
│ │ │ │ +00028820: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00028830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028870: 2d2d 2d2d 2d7c 0a7c 5f30 2a74 5f38 2b31  -----|.|_0*t_8+1
│ │ │ │ -00028880: 322a 745f 312a 745f 382d 3135 2a74 5f33  2*t_1*t_8-15*t_3
│ │ │ │ -00028890: 2a74 5f38 2b39 2a74 5f34 2a74 5f38 2b32  *t_8+9*t_4*t_8+2
│ │ │ │ -000288a0: 302a 745f 352a 745f 382d 3330 2a74 5f36  0*t_5*t_8-30*t_6
│ │ │ │ -000288b0: 2a74 5f38 2b34 2a74 5f37 2a74 5f38 7d29  *t_8+4*t_7*t_8})
│ │ │ │ -000288c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00028860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00028870: 0a7c 5f30 2a74 5f38 2b31 322a 745f 312a  .|_0*t_8+12*t_1*
│ │ │ │ +00028880: 745f 382d 3135 2a74 5f33 2a74 5f38 2b39  t_8-15*t_3*t_8+9
│ │ │ │ +00028890: 2a74 5f34 2a74 5f38 2b32 302a 745f 352a  *t_4*t_8+20*t_5*
│ │ │ │ +000288a0: 745f 382d 3330 2a74 5f36 2a74 5f38 2b34  t_8-30*t_6*t_8+4
│ │ │ │ +000288b0: 2a74 5f37 2a74 5f38 7d29 2020 2020 207c  *t_7*t_8})     |
│ │ │ │ +000288c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000288d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000288e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000288f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028910: 2d2d 2d2d 2d2b 0a7c 6938 203a 202d 2d20  -----+.|i8 : -- 
│ │ │ │ -00028920: 7769 7468 6f75 7420 7468 6520 6f70 7469  without the opti
│ │ │ │ -00028930: 6f6e 2027 436f 6469 6d42 7349 6e76 3d3e  on 'CodimBsInv=>
│ │ │ │ -00028940: 3427 2c20 6974 2074 616b 6573 2061 626f  4', it takes abo
│ │ │ │ -00028950: 7574 2020 2020 2020 2020 2020 2020 2020  ut              
│ │ │ │ -00028960: 2020 2020 207c 0a7c 2020 2020 2074 696d       |.|     tim
│ │ │ │ -00028970: 6520 7073 693d 6170 7072 6f78 696d 6174  e psi=approximat
│ │ │ │ -00028980: 6549 6e76 6572 7365 4d61 7028 7068 692c  eInverseMap(phi,
│ │ │ │ -00028990: 436f 6469 6d42 7349 6e76 3d3e 3429 2020  CodimBsInv=>4)  
│ │ │ │ -000289a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000289b0: 2020 2020 207c 0a7c 2d2d 2061 7070 726f       |.|-- appro
│ │ │ │ -000289c0: 7869 6d61 7465 496e 7665 7273 654d 6170  ximateInverseMap
│ │ │ │ -000289d0: 3a20 7374 6570 2031 206f 6620 3320 2020  : step 1 of 3   
│ │ │ │ +00028900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00028910: 0a7c 6938 203a 202d 2d20 7769 7468 6f75  .|i8 : -- withou
│ │ │ │ +00028920: 7420 7468 6520 6f70 7469 6f6e 2027 436f  t the option 'Co
│ │ │ │ +00028930: 6469 6d42 7349 6e76 3d3e 3427 2c20 6974  dimBsInv=>4', it
│ │ │ │ +00028940: 2074 616b 6573 2061 626f 7574 2020 2020   takes about    
│ │ │ │ +00028950: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028960: 0a7c 2020 2020 2074 696d 6520 7073 693d  .|     time psi=
│ │ │ │ +00028970: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ +00028980: 7365 4d61 7028 7068 692c 436f 6469 6d42  seMap(phi,CodimB
│ │ │ │ +00028990: 7349 6e76 3d3e 3429 2020 2020 2020 2020  sInv=>4)        
│ │ │ │ +000289a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000289b0: 0a7c 2d2d 2061 7070 726f 7869 6d61 7465  .|-- approximate
│ │ │ │ +000289c0: 496e 7665 7273 654d 6170 3a20 7374 6570  InverseMap: step
│ │ │ │ +000289d0: 2031 206f 6620 3320 2020 2020 2020 2020   1 of 3         
│ │ │ │  000289e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000289f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a00: 2020 2020 207c 0a7c 2d2d 2061 7070 726f       |.|-- appro
│ │ │ │ -00028a10: 7869 6d61 7465 496e 7665 7273 654d 6170  ximateInverseMap
│ │ │ │ -00028a20: 3a20 7374 6570 2032 206f 6620 3320 2020  : step 2 of 3   
│ │ │ │ +000289f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028a00: 0a7c 2d2d 2061 7070 726f 7869 6d61 7465  .|-- approximate
│ │ │ │ +00028a10: 496e 7665 7273 654d 6170 3a20 7374 6570  InverseMap: step
│ │ │ │ +00028a20: 2032 206f 6620 3320 2020 2020 2020 2020   2 of 3         
│ │ │ │  00028a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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   
│ │ │ │ +00028a40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028a50: 0a7c 2d2d 2061 7070 726f 7869 6d61 7465  .|-- approximate
│ │ │ │ +00028a60: 496e 7665 7273 654d 6170 3a20 7374 6570  InverseMap: step
│ │ │ │ +00028a70: 2033 206f 6620 3320 2020 2020 2020 2020   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 2e33 3134 3637 7320 2863 7075 293b   2.31467s (cpu);
│ │ │ │ -00028ac0: 2031 2e37 3837 3336 7320 2874 6872 6561   1.78736s (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       |.|        
│ │ │ │ +00028a90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028aa0: 0a7c 202d 2d20 7573 6564 2032 2e32 3238  .| -- used 2.228
│ │ │ │ +00028ab0: 7320 2863 7075 293b 2031 2e38 3839 3137  s (cpu); 1.88917
│ │ │ │ +00028ac0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +00028ad0: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +00028ae0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028af0: 0a7c 2020 2020 2020 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                  
│ │ │ │ -00028b40: 2020 2020 207c 0a7c 6f38 203d 202d 2d20       |.|o8 = -- 
│ │ │ │ -00028b50: 7261 7469 6f6e 616c 206d 6170 202d 2d20  rational map -- 
│ │ │ │ +00028b30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028b40: 0a7c 6f38 203d 202d 2d20 7261 7469 6f6e  .|o8 = -- ration
│ │ │ │ +00028b50: 616c 206d 6170 202d 2d20 2020 2020 2020  al map --       
│ │ │ │  00028b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028b80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028b90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00028ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028bb0: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │ +00028bb0: 2020 5a5a 2020 2020 2020 2020 2020 2020    ZZ            
│ │ │ │  00028bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028be0: 2020 2020 207c 0a7c 2020 2020 2073 6f75       |.|     sou
│ │ │ │ -00028bf0: 7263 653a 2073 7562 7661 7269 6574 7920  rce: subvariety 
│ │ │ │ -00028c00: 6f66 2050 726f 6a28 2d2d 5b78 202c 2078  of Proj(--[x , x
│ │ │ │ -00028c10: 202c 2078 202c 2078 202c 2078 202c 2078   , x , x , x , x
│ │ │ │ -00028c20: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │ -00028c30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028bd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028be0: 0a7c 2020 2020 2073 6f75 7263 653a 2073  .|     source: s
│ │ │ │ +00028bf0: 7562 7661 7269 6574 7920 6f66 2050 726f  ubvariety of Pro
│ │ │ │ +00028c00: 6a28 2d2d 5b78 202c 2078 202c 2078 202c  j(--[x , x , x ,
│ │ │ │ +00028c10: 2078 202c 2078 202c 2078 202c 2020 2020   x , x , x ,    
│ │ │ │ +00028c20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028c30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00028c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c50: 2020 2020 2020 2020 3937 2020 3020 2020          97  0   
│ │ │ │ -00028c60: 3120 2020 3220 2020 3320 2020 3420 2020  1   2   3   4   
│ │ │ │ -00028c70: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │ -00028c80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00028c90: 2020 2020 207b 2020 2020 2020 2020 2020       {          
│ │ │ │ +00028c50: 2020 3937 2020 3020 2020 3120 2020 3220    97  0   1   2 
│ │ │ │ +00028c60: 2020 3320 2020 3420 2020 3520 2020 2020    3   4   5     
│ │ │ │ +00028c70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028c80: 0a7c 2020 2020 2020 2020 2020 2020 207b  .|             {
│ │ │ │ +00028c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028cc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028cd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00028ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028cf0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00028cf0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  00028d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00028d30: 2020 2020 2020 7820 7820 202d 2038 7820        x x  - 8x 
│ │ │ │ -00028d40: 7820 202b 2032 3578 2020 2d20 3235 7820  x  + 25x  - 25x 
│ │ │ │ -00028d50: 7820 202d 2032 3278 2078 2020 2b20 7820  x  - 22x x  + x 
│ │ │ │ -00028d60: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ -00028d70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00028d80: 2020 2020 2020 2031 2033 2020 2020 2032         1 3     2
│ │ │ │ -00028d90: 2033 2020 2020 2020 3320 2020 2020 2032   3      3      2
│ │ │ │ -00028da0: 2034 2020 2020 2020 3320 3420 2020 2030   4      3 4    0
│ │ │ │ -00028db0: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ -00028dc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028d10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028d20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00028d30: 7820 7820 202d 2038 7820 7820 202b 2032  x x  - 8x x  + 2
│ │ │ │ +00028d40: 3578 2020 2d20 3235 7820 7820 202d 2032  5x  - 25x x  - 2
│ │ │ │ +00028d50: 3278 2078 2020 2b20 7820 7820 2020 2020  2x x  + x x     
│ │ │ │ +00028d60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028d70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00028d80: 2031 2033 2020 2020 2032 2033 2020 2020   1 3     2 3    
│ │ │ │ +00028d90: 2020 3320 2020 2020 2032 2034 2020 2020    3      2 4    
│ │ │ │ +00028da0: 2020 3320 3420 2020 2030 2035 2020 2020    3 4    0 5    
│ │ │ │ +00028db0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028dc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00028dd0: 2020 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00028e20: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -00028e30: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00028e00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028e10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00028e20: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00028e30: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  00028e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00028e70: 2020 2020 2020 7820 202b 2031 3778 2078        x  + 17x x
│ │ │ │ -00028e80: 2020 2d20 3134 7820 202d 2031 3378 2078    - 14x  - 13x x
│ │ │ │ -00028e90: 2020 2b20 3334 7820 7820 202b 2034 3478    + 34x x  + 44x
│ │ │ │ -00028ea0: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │ -00028eb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00028ec0: 2020 2020 2020 2032 2020 2020 2020 3220         2      2 
│ │ │ │ -00028ed0: 3320 2020 2020 2033 2020 2020 2020 3220  3      3      2 
│ │ │ │ -00028ee0: 3420 2020 2020 2033 2034 2020 2020 2020  4      3 4      
│ │ │ │ -00028ef0: 3020 3520 2020 2020 2020 2020 2020 2020  0 5             
│ │ │ │ -00028f00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028e50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028e60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00028e70: 7820 202b 2031 3778 2078 2020 2d20 3134  x  + 17x x  - 14
│ │ │ │ +00028e80: 7820 202d 2031 3378 2078 2020 2b20 3334  x  - 13x x  + 34
│ │ │ │ +00028e90: 7820 7820 202b 2034 3478 2078 2020 2020  x x  + 44x x    
│ │ │ │ +00028ea0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028eb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00028ec0: 2032 2020 2020 2020 3220 3320 2020 2020   2      2 3     
│ │ │ │ +00028ed0: 2033 2020 2020 2020 3220 3420 2020 2020   3      2 4     
│ │ │ │ +00028ee0: 2033 2034 2020 2020 2020 3020 3520 2020   3 4      0 5   
│ │ │ │ +00028ef0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028f00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00028f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028f40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028f50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00028f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f70: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00028f70: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  00028f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028fa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00028fb0: 2020 2020 2020 7820 7820 202d 2034 3078        x x  - 40x
│ │ │ │ -00028fc0: 2078 2020 2b20 3238 7820 202d 2078 2078   x  + 28x  - x x
│ │ │ │ -00028fd0: 2020 2b20 3578 2078 2020 2d20 3136 7820    + 5x x  - 16x 
│ │ │ │ -00028fe0: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ -00028ff0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00029000: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ -00029010: 3220 3320 2020 2020 2033 2020 2020 3020  2 3      3    0 
│ │ │ │ -00029020: 3420 2020 2020 3220 3420 2020 2020 2033  4     2 4      3
│ │ │ │ -00029030: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -00029040: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00029050: 2020 2020 207d 2020 2020 2020 2020 2020       }          
│ │ │ │ +00028f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028fa0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00028fb0: 7820 7820 202d 2034 3078 2078 2020 2b20  x x  - 40x x  + 
│ │ │ │ +00028fc0: 3238 7820 202d 2078 2078 2020 2b20 3578  28x  - x x  + 5x
│ │ │ │ +00028fd0: 2078 2020 2d20 3136 7820 7820 2020 2020   x  - 16x x     
│ │ │ │ +00028fe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028ff0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029000: 2031 2032 2020 2020 2020 3220 3320 2020   1 2      2 3   
│ │ │ │ +00029010: 2020 2033 2020 2020 3020 3420 2020 2020     3    0 4     
│ │ │ │ +00029020: 3220 3420 2020 2020 2033 2034 2020 2020  2 4      3 4    
│ │ │ │ +00029030: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +00029a50: 2020 2020 2020 2020 2030 2031 2020 2020           0 1    
│ │ │ │ +00029a60: 2030 2032 2020 2020 2030 2033 2020 2020   0 2     0 3    
│ │ │ │ +00029a70: 2020 3020 3420 2020 2020 2030 2020 2020    0 4      0    
│ │ │ │ +00029a80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029a90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029aa0: 2020 2020 2020 207d 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 2020 2020 2020                  
│ │ │ │ -00029ae0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029ad0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029ae0: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029b30: 2020 2020 207c 0a7c 6f38 203a 2052 6174       |.|o8 : Rat
│ │ │ │ -00029b40: 696f 6e61 6c4d 6170 2028 7175 6164 7261  ionalMap (quadra
│ │ │ │ -00029b50: 7469 6320 7261 7469 6f6e 616c 206d 6170  tic rational map
│ │ │ │ -00029b60: 2066 726f 6d20 382d 6469 6d65 6e73 696f   from 8-dimensio
│ │ │ │ -00029b70: 6e61 6c20 2020 2020 2020 2020 2020 2020  nal             
│ │ │ │ -00029b80: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00029b20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029b30: 0a7c 6f38 203a 2052 6174 696f 6e61 6c4d  .|o8 : RationalM
│ │ │ │ +00029b40: 6170 2028 7175 6164 7261 7469 6320 7261  ap (quadratic ra
│ │ │ │ +00029b50: 7469 6f6e 616c 206d 6170 2066 726f 6d20  tional map from 
│ │ │ │ +00029b60: 382d 6469 6d65 6e73 696f 6e61 6c20 2020  8-dimensional   
│ │ │ │ +00029b70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029b80: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00029b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029bd0: 2d2d 2d2d 2d7c 0a7c 7472 6970 6c65 2074  -----|.|triple t
│ │ │ │ -00029be0: 696d 6520 2020 2020 2020 2020 2020 2020  ime             
│ │ │ │ +00029bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00029bd0: 0a7c 7472 6970 6c65 2074 696d 6520 2020  .|triple time   
│ │ │ │ +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 2020 2020 2020                  
│ │ │ │ -00029c20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029c10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029c20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020                  
│ │ │ │ -00029c70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029c60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029c70: 0a7c 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 2020 2020 2020                  
│ │ │ │ -00029cc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029cb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029cc0: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029d00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029d10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00029d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029d50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +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 2020                  
│ │ │ │  00029d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029db0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029db0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00029dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029df0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029e00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00029e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029e40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029e50: 0a7c 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 2020 2020 2020                  
│ │ │ │ -00029ea0: 2020 2020 207c 0a7c 7820 2c20 7820 2c20       |.|x , x , 
│ │ │ │ -00029eb0: 7820 2c20 7820 2c20 7820 202c 2078 2020  x , x , x  , x  
│ │ │ │ -00029ec0: 5d29 2064 6566 696e 6564 2062 7920 2020  ]) defined by   
│ │ │ │ +00029e90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029ea0: 0a7c 7820 2c20 7820 2c20 7820 2c20 7820  .|x , x , x , x 
│ │ │ │ +00029eb0: 2c20 7820 202c 2078 2020 5d29 2064 6566  , x  , x  ]) def
│ │ │ │ +00029ec0: 696e 6564 2062 7920 2020 2020 2020 2020  ined by         
│ │ │ │  00029ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ef0: 2020 2020 207c 0a7c 2036 2020 2037 2020       |.| 6   7  
│ │ │ │ -00029f00: 2038 2020 2039 2020 2031 3020 2020 3131   8   9   10   11
│ │ │ │ +00029ee0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029ef0: 0a7c 2036 2020 2037 2020 2038 2020 2039  .| 6   7   8   9
│ │ │ │ +00029f00: 2020 2031 3020 2020 3131 2020 2020 2020     10   11      
│ │ │ │  00029f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029f40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +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 2020                  
│ │ │ │  00029f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029f90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029f80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029f90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020                  
│ │ │ │ -00029fe0: 2020 2020 207c 0a7c 2b20 3133 7820 7820       |.|+ 13x x 
│ │ │ │ -00029ff0: 202b 2034 3178 2078 2020 2d20 7820 7820   + 41x x  - x x 
│ │ │ │ -0002a000: 202b 2031 3278 2078 2020 2b20 3235 7820   + 12x x  + 25x 
│ │ │ │ -0002a010: 7820 202b 2032 3578 2078 2020 2b20 3233  x  + 25x x  + 23
│ │ │ │ -0002a020: 7820 7820 202d 2033 7820 7820 202b 2032  x x  - 3x x  + 2
│ │ │ │ -0002a030: 7820 7820 207c 0a7c 2020 2020 2032 2035  x x  |.|     2 5
│ │ │ │ -0002a040: 2020 2020 2020 3320 3520 2020 2030 2036        3 5    0 6
│ │ │ │ -0002a050: 2020 2020 2020 3220 3620 2020 2020 2031        2 6      1
│ │ │ │ -0002a060: 2037 2020 2020 2020 3320 3720 2020 2020   7      3 7     
│ │ │ │ -0002a070: 2035 2037 2020 2020 2036 2037 2020 2020   5 7     6 7    
│ │ │ │ -0002a080: 2030 2020 207c 0a7c 2020 2020 2020 2020   0   |.|        
│ │ │ │ +00029fd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029fe0: 0a7c 2b20 3133 7820 7820 202b 2034 3178  .|+ 13x x  + 41x
│ │ │ │ +00029ff0: 2078 2020 2d20 7820 7820 202b 2031 3278   x  - x x  + 12x
│ │ │ │ +0002a000: 2078 2020 2b20 3235 7820 7820 202b 2032   x  + 25x x  + 2
│ │ │ │ +0002a010: 3578 2078 2020 2b20 3233 7820 7820 202d  5x x  + 23x x  -
│ │ │ │ +0002a020: 2033 7820 7820 202b 2032 7820 7820 207c   3x x  + 2x x  |
│ │ │ │ +0002a030: 0a7c 2020 2020 2032 2035 2020 2020 2020  .|     2 5      
│ │ │ │ +0002a040: 3320 3520 2020 2030 2036 2020 2020 2020  3 5    0 6      
│ │ │ │ +0002a050: 3220 3620 2020 2020 2031 2037 2020 2020  2 6      1 7    
│ │ │ │ +0002a060: 2020 3320 3720 2020 2020 2035 2037 2020    3 7      5 7  
│ │ │ │ +0002a070: 2020 2036 2037 2020 2020 2030 2020 207c     6 7     0   |
│ │ │ │ +0002a080: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020                  
│ │ │ │ -0002a0d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002a0c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a0d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a120: 2020 2020 207c 0a7c 202d 2033 3078 2078       |.| - 30x x
│ │ │ │ -0002a130: 2020 2b20 3237 7820 7820 202b 2033 3178    + 27x x  + 31x
│ │ │ │ -0002a140: 2078 2020 2d20 3336 7820 7820 202d 2078   x  - 36x x  - x
│ │ │ │ -0002a150: 2078 2020 2b20 3133 7820 7820 202b 2038   x  + 13x x  + 8
│ │ │ │ -0002a160: 7820 7820 202b 2039 7820 7820 202b 2034  x x  + 9x x  + 4
│ │ │ │ -0002a170: 3678 2020 207c 0a7c 2020 2020 2020 3220  6x   |.|      2 
│ │ │ │ -0002a180: 3520 2020 2020 2033 2035 2020 2020 2020  5      3 5      
│ │ │ │ -0002a190: 3220 3620 2020 2020 2033 2036 2020 2020  2 6      3 6    
│ │ │ │ -0002a1a0: 3020 3720 2020 2020 2031 2037 2020 2020  0 7      1 7    
│ │ │ │ -0002a1b0: 2033 2037 2020 2020 2035 2037 2020 2020   3 7     5 7    
│ │ │ │ -0002a1c0: 2020 3620 207c 0a7c 2020 2020 2020 2020    6  |.|        
│ │ │ │ +0002a110: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a120: 0a7c 202d 2033 3078 2078 2020 2b20 3237  .| - 30x x  + 27
│ │ │ │ +0002a130: 7820 7820 202b 2033 3178 2078 2020 2d20  x x  + 31x x  - 
│ │ │ │ +0002a140: 3336 7820 7820 202d 2078 2078 2020 2b20  36x x  - x x  + 
│ │ │ │ +0002a150: 3133 7820 7820 202b 2038 7820 7820 202b  13x x  + 8x x  +
│ │ │ │ +0002a160: 2039 7820 7820 202b 2034 3678 2020 207c   9x x  + 46x   |
│ │ │ │ +0002a170: 0a7c 2020 2020 2020 3220 3520 2020 2020  .|      2 5     
│ │ │ │ +0002a180: 2033 2035 2020 2020 2020 3220 3620 2020   3 5      2 6   
│ │ │ │ +0002a190: 2020 2033 2036 2020 2020 3020 3720 2020     3 6    0 7   
│ │ │ │ +0002a1a0: 2020 2031 2037 2020 2020 2033 2037 2020     1 7     3 7  
│ │ │ │ +0002a1b0: 2020 2035 2037 2020 2020 2020 3620 207c     5 7      6  |
│ │ │ │ +0002a1c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a210: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002a200: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a210: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002a220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a260: 2020 2020 207c 0a7c 2b20 3578 2078 2020       |.|+ 5x x  
│ │ │ │ -0002a270: 2d20 3336 7820 7820 202b 2033 3778 2078  - 36x x  + 37x x
│ │ │ │ -0002a280: 2020 2b20 3438 7820 7820 202d 2035 7820    + 48x x  - 5x 
│ │ │ │ -0002a290: 7820 202d 2035 7820 7820 202b 2078 2078  x  - 5x x  + x x
│ │ │ │ -0002a2a0: 2020 2b20 3230 7820 7820 202b 2031 3078    + 20x x  + 10x
│ │ │ │ -0002a2b0: 2078 2020 207c 0a7c 2020 2020 3020 3520   x   |.|    0 5 
│ │ │ │ -0002a2c0: 2020 2020 2032 2035 2020 2020 2020 3320       2 5      3 
│ │ │ │ -0002a2d0: 3520 2020 2020 2032 2036 2020 2020 2031  5      2 6     1
│ │ │ │ -0002a2e0: 2037 2020 2020 2033 2037 2020 2020 3520   7     3 7    5 
│ │ │ │ -0002a2f0: 3720 2020 2020 2036 2037 2020 2020 2020  7      6 7      
│ │ │ │ -0002a300: 3020 3820 207c 0a7c 2020 2020 2020 2020  0 8  |.|        
│ │ │ │ +0002a250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a260: 0a7c 2b20 3578 2078 2020 2d20 3336 7820  .|+ 5x x  - 36x 
│ │ │ │ +0002a270: 7820 202b 2033 3778 2078 2020 2b20 3438  x  + 37x x  + 48
│ │ │ │ +0002a280: 7820 7820 202d 2035 7820 7820 202d 2035  x x  - 5x x  - 5
│ │ │ │ +0002a290: 7820 7820 202b 2078 2078 2020 2b20 3230  x x  + x x  + 20
│ │ │ │ +0002a2a0: 7820 7820 202b 2031 3078 2078 2020 207c  x x  + 10x x   |
│ │ │ │ +0002a2b0: 0a7c 2020 2020 3020 3520 2020 2020 2032  .|    0 5      2
│ │ │ │ +0002a2c0: 2035 2020 2020 2020 3320 3520 2020 2020   5      3 5     
│ │ │ │ +0002a2d0: 2032 2036 2020 2020 2031 2037 2020 2020   2 6     1 7    
│ │ │ │ +0002a2e0: 2033 2037 2020 2020 3520 3720 2020 2020   3 7    5 7     
│ │ │ │ +0002a2f0: 2036 2037 2020 2020 2020 3020 3820 207c   6 7      0 8  |
│ │ │ │ +0002a300: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020                  
│ │ │ │ -0002a350: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002a340: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a350: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a3a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002a390: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a3a0: 0a7c 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002a3e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a3f0: 0a7c 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 2020 2020 2020                  
│ │ │ │ -0002a440: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002a430: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a440: 0a7c 2020 2020 2020 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 207c 0a7c 2032 7820 7820 202b       |.| 2x x  +
│ │ │ │ -0002a4a0: 2078 2078 2020 2d20 3678 2078 2020 2d20   x x  - 6x x  - 
│ │ │ │ -0002a4b0: 3578 2078 2020 202d 2031 3078 2078 2020  5x x   - 10x x  
│ │ │ │ -0002a4c0: 202b 2032 3578 2078 2020 202b 2035 7820   + 25x x   + 5x 
│ │ │ │ -0002a4d0: 7820 2020 2d20 3578 2078 2020 2c20 2020  x   - 5x x  ,   
│ │ │ │ -0002a4e0: 2020 2020 207c 0a7c 2020 2034 2038 2020       |.|   4 8  
│ │ │ │ -0002a4f0: 2020 3520 3820 2020 2020 3520 3920 2020    5 8     5 9   
│ │ │ │ -0002a500: 2020 3120 3130 2020 2020 2020 3220 3130    1 10      2 10
│ │ │ │ -0002a510: 2020 2020 2020 3320 3130 2020 2020 2034        3 10     4
│ │ │ │ -0002a520: 2031 3020 2020 2020 3520 3130 2020 2020   10     5 10    
│ │ │ │ -0002a530: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002a480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a490: 0a7c 2032 7820 7820 202b 2078 2078 2020  .| 2x x  + x x  
│ │ │ │ +0002a4a0: 2d20 3678 2078 2020 2d20 3578 2078 2020  - 6x x  - 5x x  
│ │ │ │ +0002a4b0: 202d 2031 3078 2078 2020 202b 2032 3578   - 10x x   + 25x
│ │ │ │ +0002a4c0: 2078 2020 202b 2035 7820 7820 2020 2d20   x   + 5x x   - 
│ │ │ │ +0002a4d0: 3578 2078 2020 2c20 2020 2020 2020 207c  5x x  ,        |
│ │ │ │ +0002a4e0: 0a7c 2020 2034 2038 2020 2020 3520 3820  .|   4 8    5 8 
│ │ │ │ +0002a4f0: 2020 2020 3520 3920 2020 2020 3120 3130      5 9     1 10
│ │ │ │ +0002a500: 2020 2020 2020 3220 3130 2020 2020 2020        2 10      
│ │ │ │ +0002a510: 3320 3130 2020 2020 2034 2031 3020 2020  3 10     4 10   
│ │ │ │ +0002a520: 2020 3520 3130 2020 2020 2020 2020 207c    5 10         |
│ │ │ │ +0002a530: 0a7c 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 2020 2020 2020                  
│ │ │ │ -0002a580: 2020 2020 207c 0a7c 202d 2036 7820 7820       |.| - 6x x 
│ │ │ │ -0002a590: 202d 2034 3678 2078 2020 2d20 3434 7820   - 46x x  - 44x 
│ │ │ │ -0002a5a0: 7820 202d 2032 3278 2078 2020 2b20 3336  x  - 22x x  + 36
│ │ │ │ -0002a5b0: 7820 7820 2020 2d20 3235 7820 7820 2020  x x   - 25x x   
│ │ │ │ -0002a5c0: 2b20 3134 7820 7820 2020 2b20 3135 7820  + 14x x   + 15x 
│ │ │ │ -0002a5d0: 7820 2020 207c 0a7c 2020 2020 2034 2038  x    |.|     4 8
│ │ │ │ -0002a5e0: 2020 2020 2020 3520 3820 2020 2020 2036        5 8      6
│ │ │ │ -0002a5f0: 2038 2020 2020 2020 3520 3920 2020 2020   8      5 9     
│ │ │ │ -0002a600: 2031 2031 3020 2020 2020 2032 2031 3020   1 10      2 10 
│ │ │ │ -0002a610: 2020 2020 2033 2031 3020 2020 2020 2034       3 10      4
│ │ │ │ -0002a620: 2031 3020 207c 0a7c 2020 2020 2020 2020   10  |.|        
│ │ │ │ +0002a570: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a580: 0a7c 202d 2036 7820 7820 202d 2034 3678  .| - 6x x  - 46x
│ │ │ │ +0002a590: 2078 2020 2d20 3434 7820 7820 202d 2032   x  - 44x x  - 2
│ │ │ │ +0002a5a0: 3278 2078 2020 2b20 3336 7820 7820 2020  2x x  + 36x x   
│ │ │ │ +0002a5b0: 2d20 3235 7820 7820 2020 2b20 3134 7820  - 25x x   + 14x 
│ │ │ │ +0002a5c0: 7820 2020 2b20 3135 7820 7820 2020 207c  x   + 15x x    |
│ │ │ │ +0002a5d0: 0a7c 2020 2020 2034 2038 2020 2020 2020  .|     4 8      
│ │ │ │ +0002a5e0: 3520 3820 2020 2020 2036 2038 2020 2020  5 8      6 8    
│ │ │ │ +0002a5f0: 2020 3520 3920 2020 2020 2031 2031 3020    5 9      1 10 
│ │ │ │ +0002a600: 2020 2020 2032 2031 3020 2020 2020 2033       2 10      3
│ │ │ │ +0002a610: 2031 3020 2020 2020 2034 2031 3020 207c   10      4 10  |
│ │ │ │ +0002a620: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020                  
│ │ │ │ -0002a670: 2020 2020 207c 0a7c 2d20 3235 7820 7820       |.|- 25x x 
│ │ │ │ -0002a680: 202d 2038 7820 7820 202b 2035 7820 7820   - 8x x  + 5x x 
│ │ │ │ -0002a690: 202d 2032 3578 2078 2020 2d20 3230 7820   - 25x x  - 20x 
│ │ │ │ -0002a6a0: 7820 2020 2d20 3435 7820 7820 2020 2b20  x   - 45x x   + 
│ │ │ │ -0002a6b0: 3238 7820 7820 2020 2b20 3230 7820 7820  28x x   + 20x x 
│ │ │ │ -0002a6c0: 2020 2d20 207c 0a7c 2020 2020 2033 2038    -  |.|     3 8
│ │ │ │ -0002a6d0: 2020 2020 2034 2038 2020 2020 2035 2038       4 8     5 8
│ │ │ │ -0002a6e0: 2020 2020 2020 3520 3920 2020 2020 2031        5 9      1
│ │ │ │ -0002a6f0: 2031 3020 2020 2020 2032 2031 3020 2020   10      2 10   
│ │ │ │ -0002a700: 2020 2033 2031 3020 2020 2020 2034 2031     3 10      4 1
│ │ │ │ -0002a710: 3020 2020 207c 0a7c 2020 2020 2020 2020  0    |.|        
│ │ │ │ +0002a660: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a670: 0a7c 2d20 3235 7820 7820 202d 2038 7820  .|- 25x x  - 8x 
│ │ │ │ +0002a680: 7820 202b 2035 7820 7820 202d 2032 3578  x  + 5x x  - 25x
│ │ │ │ +0002a690: 2078 2020 2d20 3230 7820 7820 2020 2d20   x  - 20x x   - 
│ │ │ │ +0002a6a0: 3435 7820 7820 2020 2b20 3238 7820 7820  45x x   + 28x x 
│ │ │ │ +0002a6b0: 2020 2b20 3230 7820 7820 2020 2d20 207c    + 20x x   -  |
│ │ │ │ +0002a6c0: 0a7c 2020 2020 2033 2038 2020 2020 2034  .|     3 8     4
│ │ │ │ +0002a6d0: 2038 2020 2020 2035 2038 2020 2020 2020   8     5 8      
│ │ │ │ +0002a6e0: 3520 3920 2020 2020 2031 2031 3020 2020  5 9      1 10   
│ │ │ │ +0002a6f0: 2020 2032 2031 3020 2020 2020 2033 2031     2 10      3 1
│ │ │ │ +0002a700: 3020 2020 2020 2034 2031 3020 2020 207c  0      4 10    |
│ │ │ │ +0002a710: 0a7c 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 2020 2020 2020 2020                  
│ │ │ │ -0002a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a760: 2020 2020 207c 0a7c 7820 202d 2033 3978       |.|x  - 39x
│ │ │ │ -0002a770: 2078 2020 2d20 3431 7820 7820 202b 2031   x  - 41x x  + 1
│ │ │ │ -0002a780: 3978 2078 2020 2b20 3430 7820 7820 202b  9x x  + 40x x  +
│ │ │ │ -0002a790: 2034 3378 2078 2020 2d20 3339 7820 7820   43x x  - 39x x 
│ │ │ │ -0002a7a0: 202d 2033 3778 2078 2020 2b20 3378 2078   - 37x x  + 3x x
│ │ │ │ -0002a7b0: 2020 2b20 207c 0a7c 2037 2020 2020 2020    +  |.| 7      
│ │ │ │ -0002a7c0: 3620 3720 2020 2020 2031 2038 2020 2020  6 7      1 8    
│ │ │ │ -0002a7d0: 2020 3220 3820 2020 2020 2033 2038 2020    2 8      3 8  
│ │ │ │ -0002a7e0: 2020 2020 3420 3820 2020 2020 2035 2038      4 8      5 8
│ │ │ │ -0002a7f0: 2020 2020 2020 3620 3820 2020 2020 3120        6 8     1 
│ │ │ │ -0002a800: 3920 2020 207c 0a7c 2020 2020 2020 2020  9    |.|        
│ │ │ │ +0002a750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a760: 0a7c 7820 202d 2033 3978 2078 2020 2d20  .|x  - 39x x  - 
│ │ │ │ +0002a770: 3431 7820 7820 202b 2031 3978 2078 2020  41x x  + 19x x  
│ │ │ │ +0002a780: 2b20 3430 7820 7820 202b 2034 3378 2078  + 40x x  + 43x x
│ │ │ │ +0002a790: 2020 2d20 3339 7820 7820 202d 2033 3778    - 39x x  - 37x
│ │ │ │ +0002a7a0: 2078 2020 2b20 3378 2078 2020 2b20 207c   x  + 3x x  +  |
│ │ │ │ +0002a7b0: 0a7c 2037 2020 2020 2020 3620 3720 2020  .| 7      6 7   
│ │ │ │ +0002a7c0: 2020 2031 2038 2020 2020 2020 3220 3820     1 8      2 8 
│ │ │ │ +0002a7d0: 2020 2020 2033 2038 2020 2020 2020 3420       3 8      4 
│ │ │ │ +0002a7e0: 3820 2020 2020 2035 2038 2020 2020 2020  8      5 8      
│ │ │ │ +0002a7f0: 3620 3820 2020 2020 3120 3920 2020 207c  6 8     1 9    |
│ │ │ │ +0002a800: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 207c 0a7c 2020 2b20 3436 7820       |.|  + 46x 
│ │ │ │ -0002a860: 7820 202d 2031 3678 2078 2020 2d20 3336  x  - 16x x  - 36
│ │ │ │ -0002a870: 7820 7820 202b 2039 7820 7820 202d 2032  x x  + 9x x  - 2
│ │ │ │ -0002a880: 3778 2078 2020 2d20 3436 7820 7820 202d  7x x  - 46x x  -
│ │ │ │ -0002a890: 2034 3478 2078 2020 2d20 3239 7820 7820   44x x  - 29x x 
│ │ │ │ -0002a8a0: 202d 2020 207c 0a7c 3720 2020 2020 2035   -   |.|7      5
│ │ │ │ -0002a8b0: 2037 2020 2020 2020 3620 3720 2020 2020   7      6 7     
│ │ │ │ -0002a8c0: 2031 2038 2020 2020 2032 2038 2020 2020   1 8     2 8    
│ │ │ │ -0002a8d0: 2020 3320 3820 2020 2020 2034 2038 2020    3 8      4 8  
│ │ │ │ -0002a8e0: 2020 2020 3520 3820 2020 2020 2036 2038      5 8      6 8
│ │ │ │ -0002a8f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002a840: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a850: 0a7c 2020 2b20 3436 7820 7820 202d 2031  .|  + 46x x  - 1
│ │ │ │ +0002a860: 3678 2078 2020 2d20 3336 7820 7820 202b  6x x  - 36x x  +
│ │ │ │ +0002a870: 2039 7820 7820 202d 2032 3778 2078 2020   9x x  - 27x x  
│ │ │ │ +0002a880: 2d20 3436 7820 7820 202d 2034 3478 2078  - 46x x  - 44x x
│ │ │ │ +0002a890: 2020 2d20 3239 7820 7820 202d 2020 207c    - 29x x  -   |
│ │ │ │ +0002a8a0: 0a7c 3720 2020 2020 2035 2037 2020 2020  .|7      5 7    
│ │ │ │ +0002a8b0: 2020 3620 3720 2020 2020 2031 2038 2020    6 7      1 8  
│ │ │ │ +0002a8c0: 2020 2032 2038 2020 2020 2020 3320 3820     2 8      3 8 
│ │ │ │ +0002a8d0: 2020 2020 2034 2038 2020 2020 2020 3520       4 8      5 
│ │ │ │ +0002a8e0: 3820 2020 2020 2036 2038 2020 2020 207c  8      6 8     |
│ │ │ │ +0002a8f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a940: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002a930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a940: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002a950: 2020 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                  
│ │ │ │ -0002a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a990: 2020 2020 207c 0a7c 2020 2b20 3333 7820       |.|  + 33x 
│ │ │ │ -0002a9a0: 7820 202b 2033 3378 2078 2020 2d20 3231  x  + 33x x  - 21
│ │ │ │ -0002a9b0: 7820 7820 202d 2032 3578 2078 2020 2b20  x x  - 25x x  + 
│ │ │ │ -0002a9c0: 3435 7820 7820 202b 2032 3378 2078 2020  45x x  + 23x x  
│ │ │ │ -0002a9d0: 2b20 3130 7820 7820 202d 2037 7820 7820  + 10x x  - 7x x 
│ │ │ │ -0002a9e0: 202b 2020 207c 0a7c 3620 2020 2020 2030   +   |.|6      0
│ │ │ │ -0002a9f0: 2037 2020 2020 2020 3220 3720 2020 2020   7      2 7     
│ │ │ │ -0002aa00: 2033 2037 2020 2020 2020 3520 3720 2020   3 7      5 7   
│ │ │ │ -0002aa10: 2020 2036 2037 2020 2020 2020 3020 3820     6 7      0 8 
│ │ │ │ -0002aa20: 2020 2020 2031 2038 2020 2020 2032 2038       1 8     2 8
│ │ │ │ -0002aa30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002a980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a990: 0a7c 2020 2b20 3333 7820 7820 202b 2033  .|  + 33x x  + 3
│ │ │ │ +0002a9a0: 3378 2078 2020 2d20 3231 7820 7820 202d  3x x  - 21x x  -
│ │ │ │ +0002a9b0: 2032 3578 2078 2020 2b20 3435 7820 7820   25x x  + 45x x 
│ │ │ │ +0002a9c0: 202b 2032 3378 2078 2020 2b20 3130 7820   + 23x x  + 10x 
│ │ │ │ +0002a9d0: 7820 202d 2037 7820 7820 202b 2020 207c  x  - 7x x  +   |
│ │ │ │ +0002a9e0: 0a7c 3620 2020 2020 2030 2037 2020 2020  .|6      0 7    
│ │ │ │ +0002a9f0: 2020 3220 3720 2020 2020 2033 2037 2020    2 7      3 7  
│ │ │ │ +0002aa00: 2020 2020 3520 3720 2020 2020 2036 2037      5 7      6 7
│ │ │ │ +0002aa10: 2020 2020 2020 3020 3820 2020 2020 2031        0 8      1
│ │ │ │ +0002aa20: 2038 2020 2020 2032 2038 2020 2020 207c   8     2 8     |
│ │ │ │ +0002aa30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa80: 2020 2020 207c 0a7c 202b 2032 7820 7820       |.| + 2x x 
│ │ │ │ -0002aa90: 202b 2033 3478 2078 2020 2b20 3232 7820   + 34x x  + 22x 
│ │ │ │ -0002aaa0: 7820 202b 2033 3878 2078 2020 2b20 7820  x  + 38x x  + x 
│ │ │ │ -0002aab0: 7820 202b 2033 3078 2078 2020 2d20 3231  x  + 30x x  - 21
│ │ │ │ -0002aac0: 7820 7820 202d 2033 3478 2078 2020 2d20  x x  - 34x x  - 
│ │ │ │ -0002aad0: 3231 7820 207c 0a7c 2020 2020 2032 2037  21x  |.|     2 7
│ │ │ │ -0002aae0: 2020 2020 2020 3320 3720 2020 2020 2035        3 7      5
│ │ │ │ -0002aaf0: 2037 2020 2020 2020 3620 3720 2020 2030   7      6 7    0
│ │ │ │ -0002ab00: 2038 2020 2020 2020 3120 3820 2020 2020   8      1 8     
│ │ │ │ -0002ab10: 2032 2038 2020 2020 2020 3320 3820 2020   2 8      3 8   
│ │ │ │ -0002ab20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002aa70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002aa80: 0a7c 202b 2032 7820 7820 202b 2033 3478  .| + 2x x  + 34x
│ │ │ │ +0002aa90: 2078 2020 2b20 3232 7820 7820 202b 2033   x  + 22x x  + 3
│ │ │ │ +0002aaa0: 3878 2078 2020 2b20 7820 7820 202b 2033  8x x  + x x  + 3
│ │ │ │ +0002aab0: 3078 2078 2020 2d20 3231 7820 7820 202d  0x x  - 21x x  -
│ │ │ │ +0002aac0: 2033 3478 2078 2020 2d20 3231 7820 207c   34x x  - 21x  |
│ │ │ │ +0002aad0: 0a7c 2020 2020 2032 2037 2020 2020 2020  .|     2 7      
│ │ │ │ +0002aae0: 3320 3720 2020 2020 2035 2037 2020 2020  3 7      5 7    
│ │ │ │ +0002aaf0: 2020 3620 3720 2020 2030 2038 2020 2020    6 7    0 8    
│ │ │ │ +0002ab00: 2020 3120 3820 2020 2020 2032 2038 2020    1 8      2 8  
│ │ │ │ +0002ab10: 2020 2020 3320 3820 2020 2020 2020 207c      3 8        |
│ │ │ │ +0002ab20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ab40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ab50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ab60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ab70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002ab60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002ab70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002abc0: 2020 2020 207c 0a7c 3231 7820 7820 202b       |.|21x x  +
│ │ │ │ -0002abd0: 2033 3078 2078 2020 2d20 7820 7820 202d   30x x  - x x  -
│ │ │ │ -0002abe0: 2035 7820 7820 202d 2032 3978 2078 2020   5x x  - 29x x  
│ │ │ │ -0002abf0: 2b20 3278 2078 2020 2d20 3239 7820 7820  + 2x x  - 29x x 
│ │ │ │ -0002ac00: 202d 2032 3778 2078 2020 2d20 3578 2078   - 27x x  - 5x x
│ │ │ │ -0002ac10: 2020 2b20 207c 0a7c 2020 2032 2036 2020    +  |.|   2 6  
│ │ │ │ -0002ac20: 2020 2020 3320 3620 2020 2034 2036 2020      3 6    4 6  
│ │ │ │ -0002ac30: 2020 2035 2036 2020 2020 2020 3020 3720     5 6      0 7 
│ │ │ │ -0002ac40: 2020 2020 3120 3720 2020 2020 2032 2037      1 7      2 7
│ │ │ │ -0002ac50: 2020 2020 2020 3320 3720 2020 2020 3520        3 7     5 
│ │ │ │ -0002ac60: 3720 2020 207c 0a7c 2020 2020 2020 2020  7    |.|        
│ │ │ │ +0002abb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002abc0: 0a7c 3231 7820 7820 202b 2033 3078 2078  .|21x x  + 30x x
│ │ │ │ +0002abd0: 2020 2d20 7820 7820 202d 2035 7820 7820    - x x  - 5x x 
│ │ │ │ +0002abe0: 202d 2032 3978 2078 2020 2b20 3278 2078   - 29x x  + 2x x
│ │ │ │ +0002abf0: 2020 2d20 3239 7820 7820 202d 2032 3778    - 29x x  - 27x
│ │ │ │ +0002ac00: 2078 2020 2d20 3578 2078 2020 2b20 207c   x  - 5x x  +  |
│ │ │ │ +0002ac10: 0a7c 2020 2032 2036 2020 2020 2020 3320  .|   2 6      3 
│ │ │ │ +0002ac20: 3620 2020 2034 2036 2020 2020 2035 2036  6    4 6     5 6
│ │ │ │ +0002ac30: 2020 2020 2020 3020 3720 2020 2020 3120        0 7     1 
│ │ │ │ +0002ac40: 3720 2020 2020 2032 2037 2020 2020 2020  7      2 7      
│ │ │ │ +0002ac50: 3320 3720 2020 2020 3520 3720 2020 207c  3 7     5 7    |
│ │ │ │ +0002ac60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002ac70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ac80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002acb0: 2020 2020 207c 0a7c 2020 2b20 3331 7820       |.|  + 31x 
│ │ │ │ -0002acc0: 7820 202b 2032 3978 2078 2020 2b20 3130  x  + 29x x  + 10
│ │ │ │ -0002acd0: 7820 7820 202d 2032 3778 2078 2020 2d20  x x  - 27x x  - 
│ │ │ │ -0002ace0: 3331 7820 7820 202d 2031 3778 2078 2020  31x x  - 17x x  
│ │ │ │ -0002acf0: 2d20 3238 7820 7820 202b 2033 3978 2078  - 28x x  + 39x x
│ │ │ │ -0002ad00: 2020 2d20 207c 0a7c 3620 2020 2020 2030    -  |.|6      0
│ │ │ │ -0002ad10: 2037 2020 2020 2020 3220 3720 2020 2020   7      2 7     
│ │ │ │ -0002ad20: 2033 2037 2020 2020 2020 3520 3720 2020   3 7      5 7   
│ │ │ │ -0002ad30: 2020 2036 2037 2020 2020 2020 3020 3820     6 7      0 8 
│ │ │ │ -0002ad40: 2020 2020 2031 2038 2020 2020 2020 3220       1 8      2 
│ │ │ │ -0002ad50: 3820 2020 207c 0a7c 2020 2020 2020 2020  8    |.|        
│ │ │ │ +0002aca0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002acb0: 0a7c 2020 2b20 3331 7820 7820 202b 2032  .|  + 31x x  + 2
│ │ │ │ +0002acc0: 3978 2078 2020 2b20 3130 7820 7820 202d  9x x  + 10x x  -
│ │ │ │ +0002acd0: 2032 3778 2078 2020 2d20 3331 7820 7820   27x x  - 31x x 
│ │ │ │ +0002ace0: 202d 2031 3778 2078 2020 2d20 3238 7820   - 17x x  - 28x 
│ │ │ │ +0002acf0: 7820 202b 2033 3978 2078 2020 2d20 207c  x  + 39x x  -  |
│ │ │ │ +0002ad00: 0a7c 3620 2020 2020 2030 2037 2020 2020  .|6      0 7    
│ │ │ │ +0002ad10: 2020 3220 3720 2020 2020 2033 2037 2020    2 7      3 7  
│ │ │ │ +0002ad20: 2020 2020 3520 3720 2020 2020 2036 2037      5 7      6 7
│ │ │ │ +0002ad30: 2020 2020 2020 3020 3820 2020 2020 2031        0 8      1
│ │ │ │ +0002ad40: 2038 2020 2020 2020 3220 3820 2020 207c   8      2 8    |
│ │ │ │ +0002ad50: 0a7c 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002ad90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002ada0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2073 7562 7661 7269       |.| subvari
│ │ │ │ -0002ae00: 6574 7920 6f66 2050 505e 3131 2074 6f20  ety of PP^11 to 
│ │ │ │ -0002ae10: 5050 5e38 2920 2020 2020 2020 2020 2020  PP^8)           
│ │ │ │ +0002ade0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002adf0: 0a7c 2073 7562 7661 7269 6574 7920 6f66  .| subvariety of
│ │ │ │ +0002ae00: 2050 505e 3131 2074 6f20 5050 5e38 2920   PP^11 to PP^8) 
│ │ │ │ +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 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +0002ae30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002ae40: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0002ae50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ae70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ae90: 2d2d 2d2d 2d7c 0a7c 2020 2b20 3131 7820  -----|.|  + 11x 
│ │ │ │ -0002aea0: 7820 202d 2033 3778 2078 2020 2d20 3233  x  - 37x x  - 23
│ │ │ │ -0002aeb0: 7820 7820 202d 2033 3378 2078 2020 2b20  x x  - 33x x  + 
│ │ │ │ -0002aec0: 3878 2078 2020 2b20 3130 7820 7820 202d  8x x  + 10x x  -
│ │ │ │ -0002aed0: 2032 3578 2078 2020 2d20 3978 2078 2020   25x x  - 9x x  
│ │ │ │ -0002aee0: 2b20 3378 207c 0a7c 3820 2020 2020 2031  + 3x |.|8      1
│ │ │ │ -0002aef0: 2038 2020 2020 2020 3320 3820 2020 2020   8      3 8     
│ │ │ │ -0002af00: 2034 2038 2020 2020 2020 3620 3820 2020   4 8      6 8   
│ │ │ │ -0002af10: 2020 3020 3920 2020 2020 2031 2039 2020    0 9      1 9  
│ │ │ │ -0002af20: 2020 2020 3220 3920 2020 2020 3320 3920      2 9     3 9 
│ │ │ │ -0002af30: 2020 2020 347c 0a7c 2020 2020 2020 2020      4|.|        
│ │ │ │ +0002ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0002ae90: 0a7c 2020 2b20 3131 7820 7820 202d 2033  .|  + 11x x  - 3
│ │ │ │ +0002aea0: 3778 2078 2020 2d20 3233 7820 7820 202d  7x x  - 23x x  -
│ │ │ │ +0002aeb0: 2033 3378 2078 2020 2b20 3878 2078 2020   33x x  + 8x x  
│ │ │ │ +0002aec0: 2b20 3130 7820 7820 202d 2032 3578 2078  + 10x x  - 25x x
│ │ │ │ +0002aed0: 2020 2d20 3978 2078 2020 2b20 3378 207c    - 9x x  + 3x |
│ │ │ │ +0002aee0: 0a7c 3820 2020 2020 2031 2038 2020 2020  .|8      1 8    
│ │ │ │ +0002aef0: 2020 3320 3820 2020 2020 2034 2038 2020    3 8      4 8  
│ │ │ │ +0002af00: 2020 2020 3620 3820 2020 2020 3020 3920      6 8     0 9 
│ │ │ │ +0002af10: 2020 2020 2031 2039 2020 2020 2020 3220       1 9      2 
│ │ │ │ +0002af20: 3920 2020 2020 3320 3920 2020 2020 347c  9     3 9     4|
│ │ │ │ +0002af30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002af70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002af80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002afd0: 2020 2020 207c 0a7c 7820 202b 2034 3178       |.|x  + 41x
│ │ │ │ -0002afe0: 2078 2020 2d20 3778 2078 2020 2d20 3334   x  - 7x x  - 34
│ │ │ │ -0002aff0: 7820 7820 202d 2039 7820 7820 202d 2034  x x  - 9x x  - 4
│ │ │ │ -0002b000: 3678 2078 2020 2d20 3137 7820 7820 202b  6x x  - 17x x  +
│ │ │ │ -0002b010: 2033 3278 2078 2020 2d20 3878 2078 2020   32x x  - 8x x  
│ │ │ │ -0002b020: 2d20 3335 787c 0a7c 2037 2020 2020 2020  - 35x|.| 7      
│ │ │ │ -0002b030: 3020 3820 2020 2020 3120 3820 2020 2020  0 8     1 8     
│ │ │ │ -0002b040: 2033 2038 2020 2020 2034 2038 2020 2020   3 8     4 8    
│ │ │ │ -0002b050: 2020 3620 3820 2020 2020 2030 2039 2020    6 8      0 9  
│ │ │ │ -0002b060: 2020 2020 3120 3920 2020 2020 3220 3920      1 9     2 9 
│ │ │ │ -0002b070: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002afc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002afd0: 0a7c 7820 202b 2034 3178 2078 2020 2d20  .|x  + 41x x  - 
│ │ │ │ +0002afe0: 3778 2078 2020 2d20 3334 7820 7820 202d  7x x  - 34x x  -
│ │ │ │ +0002aff0: 2039 7820 7820 202d 2034 3678 2078 2020   9x x  - 46x x  
│ │ │ │ +0002b000: 2d20 3137 7820 7820 202b 2033 3278 2078  - 17x x  + 32x x
│ │ │ │ +0002b010: 2020 2d20 3878 2078 2020 2d20 3335 787c    - 8x x  - 35x|
│ │ │ │ +0002b020: 0a7c 2037 2020 2020 2020 3020 3820 2020  .| 7      0 8   
│ │ │ │ +0002b030: 2020 3120 3820 2020 2020 2033 2038 2020    1 8      3 8  
│ │ │ │ +0002b040: 2020 2034 2038 2020 2020 2020 3620 3820     4 8      6 8 
│ │ │ │ +0002b050: 2020 2020 2030 2039 2020 2020 2020 3120       0 9      1 
│ │ │ │ +0002b060: 3920 2020 2020 3220 3920 2020 2020 207c  9     2 9      |
│ │ │ │ +0002b070: 0a7c 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 2020 2020 2020                  
│ │ │ │ -0002b0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b0c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002b0b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b0c0: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b110: 2020 2020 207c 0a7c 202b 2033 3478 2078       |.| + 34x x
│ │ │ │ -0002b120: 2020 2b20 3431 7820 7820 202d 2078 2078    + 41x x  - x x
│ │ │ │ -0002b130: 2020 2b20 7820 7820 202b 2034 3078 2078    + x x  + 40x x
│ │ │ │ -0002b140: 2020 2d20 3332 7820 7820 202b 2035 7820    - 32x x  + 5x 
│ │ │ │ -0002b150: 7820 202d 2031 3178 2078 2020 2d20 3230  x  - 11x x  - 20
│ │ │ │ -0002b160: 7820 7820 207c 0a7c 2020 2020 2020 3120  x x  |.|      1 
│ │ │ │ -0002b170: 3820 2020 2020 2033 2038 2020 2020 3420  8      3 8    4 
│ │ │ │ -0002b180: 3820 2020 2036 2038 2020 2020 2020 3020  8    6 8      0 
│ │ │ │ -0002b190: 3920 2020 2020 2031 2039 2020 2020 2032  9      1 9     2
│ │ │ │ -0002b1a0: 2039 2020 2020 2020 3320 3920 2020 2020   9      3 9     
│ │ │ │ -0002b1b0: 2034 2039 207c 0a7c 2020 2020 2020 2020   4 9 |.|        
│ │ │ │ +0002b100: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b110: 0a7c 202b 2033 3478 2078 2020 2b20 3431  .| + 34x x  + 41
│ │ │ │ +0002b120: 7820 7820 202d 2078 2078 2020 2b20 7820  x x  - x x  + x 
│ │ │ │ +0002b130: 7820 202b 2034 3078 2078 2020 2d20 3332  x  + 40x x  - 32
│ │ │ │ +0002b140: 7820 7820 202b 2035 7820 7820 202d 2031  x x  + 5x x  - 1
│ │ │ │ +0002b150: 3178 2078 2020 2d20 3230 7820 7820 207c  1x x  - 20x x  |
│ │ │ │ +0002b160: 0a7c 2020 2020 2020 3120 3820 2020 2020  .|      1 8     
│ │ │ │ +0002b170: 2033 2038 2020 2020 3420 3820 2020 2036   3 8    4 8    6
│ │ │ │ +0002b180: 2038 2020 2020 2020 3020 3920 2020 2020   8      0 9     
│ │ │ │ +0002b190: 2031 2039 2020 2020 2032 2039 2020 2020   1 9     2 9    
│ │ │ │ +0002b1a0: 2020 3320 3920 2020 2020 2034 2039 207c    3 9      4 9 |
│ │ │ │ +0002b1b0: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b200: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002b1f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b200: 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b250: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002b240: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b250: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002b290: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b2a0: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002b2e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b2f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b340: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002b330: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b340: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b390: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002b380: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b390: 0a7c 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                  
│ │ │ │  0002b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002b3d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b3e0: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b430: 2020 2020 207c 0a7c 202b 2033 3678 2078       |.| + 36x x
│ │ │ │ -0002b440: 2020 202b 2033 3678 2078 2020 202d 2033     + 36x x   - 3
│ │ │ │ -0002b450: 3678 2078 2020 2c20 2020 2020 2020 2020  6x x  ,         
│ │ │ │ +0002b420: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b430: 0a7c 202b 2033 3678 2078 2020 202b 2033  .| + 36x x   + 3
│ │ │ │ +0002b440: 3678 2078 2020 202d 2033 3678 2078 2020  6x x   - 36x x  
│ │ │ │ +0002b450: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  0002b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b480: 2020 2020 207c 0a7c 2020 2020 2020 3520       |.|      5 
│ │ │ │ -0002b490: 3130 2020 2020 2020 3620 3130 2020 2020  10      6 10    
│ │ │ │ -0002b4a0: 2020 3520 3131 2020 2020 2020 2020 2020    5 11          
│ │ │ │ +0002b470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b480: 0a7c 2020 2020 2020 3520 3130 2020 2020  .|      5 10    
│ │ │ │ +0002b490: 2020 3620 3130 2020 2020 2020 3520 3131    6 10      5 11
│ │ │ │ +0002b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002b4c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b4d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b4f0: 2020 2020 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 207c 0a7c 2032 3578 2078 2020       |.| 25x x  
│ │ │ │ -0002b530: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +0002b510: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b520: 0a7c 2032 3578 2078 2020 2c20 2020 2020  .| 25x x  ,     
│ │ │ │ +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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b570: 2020 2020 207c 0a7c 2020 2020 3520 3130       |.|    5 10
│ │ │ │ +0002b560: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b570: 0a7c 2020 2020 3520 3130 2020 2020 2020  .|    5 10      
│ │ │ │  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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002b5b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b5c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b610: 2020 2020 207c 0a7c 2033 3878 2078 2020       |.| 38x x  
│ │ │ │ -0002b620: 2d20 3278 2078 2020 2b20 3339 7820 7820  - 2x x  + 39x x 
│ │ │ │ -0002b630: 202d 2031 3178 2078 2020 2d20 3238 7820   - 11x x  - 28x 
│ │ │ │ -0002b640: 7820 2020 2b20 3231 7820 7820 2020 2d20  x   + 21x x   - 
│ │ │ │ -0002b650: 3578 2078 2020 202d 2031 3078 2078 2020  5x x   - 10x x  
│ │ │ │ -0002b660: 202b 2078 207c 0a7c 2020 2020 3220 3920   + x |.|    2 9 
│ │ │ │ -0002b670: 2020 2020 3320 3920 2020 2020 2034 2039      3 9      4 9
│ │ │ │ -0002b680: 2020 2020 2020 3520 3920 2020 2020 2031        5 9      1
│ │ │ │ -0002b690: 2031 3020 2020 2020 2032 2031 3020 2020   10      2 10   
│ │ │ │ -0002b6a0: 2020 3320 3130 2020 2020 2020 3420 3130    3 10      4 10
│ │ │ │ -0002b6b0: 2020 2020 357c 0a7c 2020 2020 2020 2020      5|.|        
│ │ │ │ +0002b600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b610: 0a7c 2033 3878 2078 2020 2d20 3278 2078  .| 38x x  - 2x x
│ │ │ │ +0002b620: 2020 2b20 3339 7820 7820 202d 2031 3178    + 39x x  - 11x
│ │ │ │ +0002b630: 2078 2020 2d20 3238 7820 7820 2020 2b20   x  - 28x x   + 
│ │ │ │ +0002b640: 3231 7820 7820 2020 2d20 3578 2078 2020  21x x   - 5x x  
│ │ │ │ +0002b650: 202d 2031 3078 2078 2020 202b 2078 207c   - 10x x   + x |
│ │ │ │ +0002b660: 0a7c 2020 2020 3220 3920 2020 2020 3320  .|    2 9     3 
│ │ │ │ +0002b670: 3920 2020 2020 2034 2039 2020 2020 2020  9      4 9      
│ │ │ │ +0002b680: 3520 3920 2020 2020 2031 2031 3020 2020  5 9      1 10   
│ │ │ │ +0002b690: 2020 2032 2031 3020 2020 2020 3320 3130     2 10     3 10
│ │ │ │ +0002b6a0: 2020 2020 2020 3420 3130 2020 2020 357c        4 10    5|
│ │ │ │ +0002b6b0: 0a7c 2020 2020 2020 2020 2020 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 207c 0a7c 3137 7820 7820 202b       |.|17x x  +
│ │ │ │ -0002b710: 2031 3878 2078 2020 2b20 3435 7820 7820   18x x  + 45x x 
│ │ │ │ -0002b720: 202b 2031 3678 2078 2020 2b20 3578 2078   + 16x x  + 5x x
│ │ │ │ -0002b730: 2020 2b20 3336 7820 7820 202b 2033 3978    + 36x x  + 39x
│ │ │ │ -0002b740: 2078 2020 202d 2033 3678 2078 2020 202d   x   - 36x x   -
│ │ │ │ -0002b750: 2033 3378 207c 0a7c 2020 2031 2039 2020   33x |.|   1 9  
│ │ │ │ -0002b760: 2020 2020 3220 3920 2020 2020 2033 2039      2 9      3 9
│ │ │ │ -0002b770: 2020 2020 2020 3420 3920 2020 2020 3520        4 9     5 
│ │ │ │ -0002b780: 3920 2020 2020 2036 2039 2020 2020 2020  9      6 9      
│ │ │ │ -0002b790: 3120 3130 2020 2020 2020 3220 3130 2020  1 10      2 10  
│ │ │ │ -0002b7a0: 2020 2020 337c 0a7c 2020 2020 2020 2020      3|.|        
│ │ │ │ +0002b6f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b700: 0a7c 3137 7820 7820 202b 2031 3878 2078  .|17x x  + 18x x
│ │ │ │ +0002b710: 2020 2b20 3435 7820 7820 202b 2031 3678    + 45x x  + 16x
│ │ │ │ +0002b720: 2078 2020 2b20 3578 2078 2020 2b20 3336   x  + 5x x  + 36
│ │ │ │ +0002b730: 7820 7820 202b 2033 3978 2078 2020 202d  x x  + 39x x   -
│ │ │ │ +0002b740: 2033 3678 2078 2020 202d 2033 3378 207c   36x x   - 33x |
│ │ │ │ +0002b750: 0a7c 2020 2031 2039 2020 2020 2020 3220  .|   1 9      2 
│ │ │ │ +0002b760: 3920 2020 2020 2033 2039 2020 2020 2020  9      3 9      
│ │ │ │ +0002b770: 3420 3920 2020 2020 3520 3920 2020 2020  4 9     5 9     
│ │ │ │ +0002b780: 2036 2039 2020 2020 2020 3120 3130 2020   6 9      1 10  
│ │ │ │ +0002b790: 2020 2020 3220 3130 2020 2020 2020 337c      2 10      3|
│ │ │ │ +0002b7a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002b7e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b7f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002b800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b840: 2020 2020 207c 0a7c 3578 2078 2020 2d20       |.|5x x  - 
│ │ │ │ -0002b850: 3778 2078 2020 2b20 3435 7820 7820 202b  7x x  + 45x x  +
│ │ │ │ -0002b860: 2032 3578 2078 2020 2d20 3437 7820 7820   25x x  - 47x x 
│ │ │ │ -0002b870: 202b 2034 7820 7820 202d 2032 3978 2078   + 4x x  - 29x x
│ │ │ │ -0002b880: 2020 2b20 3332 7820 7820 202d 2031 3378    + 32x x  - 13x
│ │ │ │ -0002b890: 2078 2020 2d7c 0a7c 2020 3320 3820 2020   x  -|.|  3 8   
│ │ │ │ -0002b8a0: 2020 3420 3820 2020 2020 2035 2038 2020    4 8      5 8  
│ │ │ │ -0002b8b0: 2020 2020 3620 3820 2020 2020 2030 2039      6 8      0 9
│ │ │ │ -0002b8c0: 2020 2020 2031 2039 2020 2020 2020 3220       1 9      2 
│ │ │ │ -0002b8d0: 3920 2020 2020 2033 2039 2020 2020 2020  9      3 9      
│ │ │ │ -0002b8e0: 3420 3920 207c 0a7c 2020 2020 2020 2020  4 9  |.|        
│ │ │ │ +0002b830: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b840: 0a7c 3578 2078 2020 2d20 3778 2078 2020  .|5x x  - 7x x  
│ │ │ │ +0002b850: 2b20 3435 7820 7820 202b 2032 3578 2078  + 45x x  + 25x x
│ │ │ │ +0002b860: 2020 2d20 3437 7820 7820 202b 2034 7820    - 47x x  + 4x 
│ │ │ │ +0002b870: 7820 202d 2032 3978 2078 2020 2b20 3332  x  - 29x x  + 32
│ │ │ │ +0002b880: 7820 7820 202d 2031 3378 2078 2020 2d7c  x x  - 13x x  -|
│ │ │ │ +0002b890: 0a7c 2020 3320 3820 2020 2020 3420 3820  .|  3 8     4 8 
│ │ │ │ +0002b8a0: 2020 2020 2035 2038 2020 2020 2020 3620       5 8      6 
│ │ │ │ +0002b8b0: 3820 2020 2020 2030 2039 2020 2020 2031  8      0 9     1
│ │ │ │ +0002b8c0: 2039 2020 2020 2020 3220 3920 2020 2020   9      2 9     
│ │ │ │ +0002b8d0: 2033 2039 2020 2020 2020 3420 3920 207c   3 9      4 9  |
│ │ │ │ +0002b8e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b930: 2020 2020 207c 0a7c 2078 2020 2b20 3338       |.| x  + 38
│ │ │ │ -0002b940: 7820 7820 202b 2034 3678 2078 2020 2d20  x x  + 46x x  - 
│ │ │ │ -0002b950: 3678 2078 2020 2b20 3132 7820 7820 202d  6x x  + 12x x  -
│ │ │ │ -0002b960: 2034 3878 2078 2020 2b20 3137 7820 7820   48x x  + 17x x 
│ │ │ │ -0002b970: 202d 2033 3978 2078 2020 2d20 3434 7820   - 39x x  - 44x 
│ │ │ │ -0002b980: 7820 202d 207c 0a7c 3420 3820 2020 2020  x  - |.|4 8     
│ │ │ │ -0002b990: 2035 2038 2020 2020 2020 3620 3820 2020   5 8      6 8   
│ │ │ │ -0002b9a0: 2020 3020 3920 2020 2020 2031 2039 2020    0 9      1 9  
│ │ │ │ -0002b9b0: 2020 2020 3220 3920 2020 2020 2033 2039      2 9      3 9
│ │ │ │ -0002b9c0: 2020 2020 2020 3420 3920 2020 2020 2035        4 9      5
│ │ │ │ -0002b9d0: 2039 2020 207c 0a7c 2020 2020 2020 2020   9   |.|        
│ │ │ │ +0002b920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b930: 0a7c 2078 2020 2b20 3338 7820 7820 202b  .| x  + 38x x  +
│ │ │ │ +0002b940: 2034 3678 2078 2020 2d20 3678 2078 2020   46x x  - 6x x  
│ │ │ │ +0002b950: 2b20 3132 7820 7820 202d 2034 3878 2078  + 12x x  - 48x x
│ │ │ │ +0002b960: 2020 2b20 3137 7820 7820 202d 2033 3978    + 17x x  - 39x
│ │ │ │ +0002b970: 2078 2020 2d20 3434 7820 7820 202d 207c   x  - 44x x  - |
│ │ │ │ +0002b980: 0a7c 3420 3820 2020 2020 2035 2038 2020  .|4 8      5 8  
│ │ │ │ +0002b990: 2020 2020 3620 3820 2020 2020 3020 3920      6 8     0 9 
│ │ │ │ +0002b9a0: 2020 2020 2031 2039 2020 2020 2020 3220       1 9      2 
│ │ │ │ +0002b9b0: 3920 2020 2020 2033 2039 2020 2020 2020  9      3 9      
│ │ │ │ +0002b9c0: 3420 3920 2020 2020 2035 2039 2020 207c  4 9      5 9   |
│ │ │ │ +0002b9d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002ba10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002ba20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba70: 2020 2020 207c 0a7c 2034 7820 7820 202b       |.| 4x x  +
│ │ │ │ -0002ba80: 2031 3778 2078 2020 2d20 3434 7820 7820   17x x  - 44x x 
│ │ │ │ -0002ba90: 202b 2031 3878 2078 2020 2d20 3432 7820   + 18x x  - 42x 
│ │ │ │ -0002baa0: 7820 202b 2033 3978 2078 2020 2b20 3978  x  + 39x x  + 9x
│ │ │ │ -0002bab0: 2078 2020 2d20 3133 7820 7820 202b 2034   x  - 13x x  + 4
│ │ │ │ -0002bac0: 3878 2078 207c 0a7c 2020 2036 2037 2020  8x x |.|   6 7  
│ │ │ │ -0002bad0: 2020 2020 3020 3820 2020 2020 2031 2038      0 8      1 8
│ │ │ │ -0002bae0: 2020 2020 2020 3220 3820 2020 2020 2033        2 8      3
│ │ │ │ -0002baf0: 2038 2020 2020 2020 3420 3820 2020 2020   8      4 8     
│ │ │ │ -0002bb00: 3520 3820 2020 2020 2036 2038 2020 2020  5 8      6 8    
│ │ │ │ -0002bb10: 2020 3020 397c 0a7c 2020 2020 2020 2020    0 9|.|        
│ │ │ │ +0002ba60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002ba70: 0a7c 2034 7820 7820 202b 2031 3778 2078  .| 4x x  + 17x x
│ │ │ │ +0002ba80: 2020 2d20 3434 7820 7820 202b 2031 3878    - 44x x  + 18x
│ │ │ │ +0002ba90: 2078 2020 2d20 3432 7820 7820 202b 2033   x  - 42x x  + 3
│ │ │ │ +0002baa0: 3978 2078 2020 2b20 3978 2078 2020 2d20  9x x  + 9x x  - 
│ │ │ │ +0002bab0: 3133 7820 7820 202b 2034 3878 2078 207c  13x x  + 48x x |
│ │ │ │ +0002bac0: 0a7c 2020 2036 2037 2020 2020 2020 3020  .|   6 7      0 
│ │ │ │ +0002bad0: 3820 2020 2020 2031 2038 2020 2020 2020  8      1 8      
│ │ │ │ +0002bae0: 3220 3820 2020 2020 2033 2038 2020 2020  2 8      3 8    
│ │ │ │ +0002baf0: 2020 3420 3820 2020 2020 3520 3820 2020    4 8     5 8   
│ │ │ │ +0002bb00: 2020 2036 2038 2020 2020 2020 3020 397c     6 8      0 9|
│ │ │ │ +0002bb10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2033 3278 2078 2020       |.| 32x x  
│ │ │ │ -0002bb70: 2d20 3778 2078 2020 2d20 3239 7820 7820  - 7x x  - 29x x 
│ │ │ │ -0002bb80: 202d 2034 3778 2078 2020 2b20 3335 7820   - 47x x  + 35x 
│ │ │ │ -0002bb90: 7820 202b 2033 3878 2078 2020 2d20 3433  x  + 38x x  - 43
│ │ │ │ -0002bba0: 7820 7820 202d 2032 7820 7820 202d 2033  x x  - 2x x  - 3
│ │ │ │ -0002bbb0: 3278 2078 207c 0a7c 2020 2020 3320 3820  2x x |.|    3 8 
│ │ │ │ -0002bbc0: 2020 2020 3420 3820 2020 2020 2035 2038      4 8      5 8
│ │ │ │ -0002bbd0: 2020 2020 2020 3620 3820 2020 2020 2030        6 8      0
│ │ │ │ -0002bbe0: 2039 2020 2020 2020 3120 3920 2020 2020   9      1 9     
│ │ │ │ -0002bbf0: 2032 2039 2020 2020 2033 2039 2020 2020   2 9     3 9    
│ │ │ │ -0002bc00: 2020 3420 397c 0a7c 2d2d 2d2d 2d2d 2d2d    4 9|.|--------
│ │ │ │ +0002bb50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002bb60: 0a7c 2033 3278 2078 2020 2d20 3778 2078  .| 32x x  - 7x x
│ │ │ │ +0002bb70: 2020 2d20 3239 7820 7820 202d 2034 3778    - 29x x  - 47x
│ │ │ │ +0002bb80: 2078 2020 2b20 3335 7820 7820 202b 2033   x  + 35x x  + 3
│ │ │ │ +0002bb90: 3878 2078 2020 2d20 3433 7820 7820 202d  8x x  - 43x x  -
│ │ │ │ +0002bba0: 2032 7820 7820 202d 2033 3278 2078 207c   2x x  - 32x x |
│ │ │ │ +0002bbb0: 0a7c 2020 2020 3320 3820 2020 2020 3420  .|    3 8     4 
│ │ │ │ +0002bbc0: 3820 2020 2020 2035 2038 2020 2020 2020  8      5 8      
│ │ │ │ +0002bbd0: 3620 3820 2020 2020 2030 2039 2020 2020  6 8      0 9    
│ │ │ │ +0002bbe0: 2020 3120 3920 2020 2020 2032 2039 2020    1 9      2 9  
│ │ │ │ +0002bbf0: 2020 2033 2039 2020 2020 2020 3420 397c     3 9      4 9|
│ │ │ │ +0002bc00: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0002bc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bc50: 2d2d 2d2d 2d7c 0a7c 7820 202b 2032 3478  -----|.|x  + 24x
│ │ │ │ -0002bc60: 2078 2020 2d20 3237 7820 7820 202d 2035   x  - 27x x  - 5
│ │ │ │ -0002bc70: 7820 7820 2020 2b20 3238 7820 7820 2020  x x   + 28x x   
│ │ │ │ -0002bc80: 2b20 3337 7820 7820 2020 2b20 3978 2078  + 37x x   + 9x x
│ │ │ │ -0002bc90: 2020 202b 2032 3778 2078 2020 202d 2032     + 27x x   - 2
│ │ │ │ -0002bca0: 3578 2078 207c 0a7c 2039 2020 2020 2020  5x x |.| 9      
│ │ │ │ -0002bcb0: 3520 3920 2020 2020 2036 2039 2020 2020  5 9      6 9    
│ │ │ │ -0002bcc0: 2030 2031 3020 2020 2020 2031 2031 3020   0 10      1 10 
│ │ │ │ -0002bcd0: 2020 2020 2032 2031 3020 2020 2020 3420       2 10     4 
│ │ │ │ -0002bce0: 3130 2020 2020 2020 3620 3130 2020 2020  10      6 10    
│ │ │ │ -0002bcf0: 2020 3020 207c 0a7c 2020 2020 2020 2020    0  |.|        
│ │ │ │ +0002bc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0002bc50: 0a7c 7820 202b 2032 3478 2078 2020 2d20  .|x  + 24x x  - 
│ │ │ │ +0002bc60: 3237 7820 7820 202d 2035 7820 7820 2020  27x x  - 5x x   
│ │ │ │ +0002bc70: 2b20 3238 7820 7820 2020 2b20 3337 7820  + 28x x   + 37x 
│ │ │ │ +0002bc80: 7820 2020 2b20 3978 2078 2020 202b 2032  x   + 9x x   + 2
│ │ │ │ +0002bc90: 3778 2078 2020 202d 2032 3578 2078 207c  7x x   - 25x x |
│ │ │ │ +0002bca0: 0a7c 2039 2020 2020 2020 3520 3920 2020  .| 9      5 9   
│ │ │ │ +0002bcb0: 2020 2036 2039 2020 2020 2030 2031 3020     6 9     0 10 
│ │ │ │ +0002bcc0: 2020 2020 2031 2031 3020 2020 2020 2032       1 10      2
│ │ │ │ +0002bcd0: 2031 3020 2020 2020 3420 3130 2020 2020   10     4 10    
│ │ │ │ +0002bce0: 2020 3620 3130 2020 2020 2020 3020 207c    6 10      0  |
│ │ │ │ +0002bcf0: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002bd30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002bd40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd90: 2020 2020 207c 0a7c 2078 2020 2d20 3436       |.| x  - 46
│ │ │ │ -0002bda0: 7820 7820 202b 2032 3678 2078 2020 2b20  x x  + 26x x  + 
│ │ │ │ -0002bdb0: 3137 7820 7820 202b 2031 3578 2078 2020  17x x  + 15x x  
│ │ │ │ -0002bdc0: 202b 2033 3578 2078 2020 202b 2033 3478   + 35x x   + 34x
│ │ │ │ -0002bdd0: 2078 2020 202b 2032 3078 2078 2020 202b   x   + 20x x   +
│ │ │ │ -0002bde0: 2031 3478 207c 0a7c 3320 3920 2020 2020   14x |.|3 9     
│ │ │ │ -0002bdf0: 2034 2039 2020 2020 2020 3520 3920 2020   4 9      5 9   
│ │ │ │ -0002be00: 2020 2036 2039 2020 2020 2020 3020 3130     6 9      0 10
│ │ │ │ -0002be10: 2020 2020 2020 3120 3130 2020 2020 2020        1 10      
│ │ │ │ -0002be20: 3220 3130 2020 2020 2020 3420 3130 2020  2 10      4 10  
│ │ │ │ -0002be30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002bd80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002bd90: 0a7c 2078 2020 2d20 3436 7820 7820 202b  .| x  - 46x x  +
│ │ │ │ +0002bda0: 2032 3678 2078 2020 2b20 3137 7820 7820   26x x  + 17x x 
│ │ │ │ +0002bdb0: 202b 2031 3578 2078 2020 202b 2033 3578   + 15x x   + 35x
│ │ │ │ +0002bdc0: 2078 2020 202b 2033 3478 2078 2020 202b   x   + 34x x   +
│ │ │ │ +0002bdd0: 2032 3078 2078 2020 202b 2031 3478 207c   20x x   + 14x |
│ │ │ │ +0002bde0: 0a7c 3320 3920 2020 2020 2034 2039 2020  .|3 9      4 9  
│ │ │ │ +0002bdf0: 2020 2020 3520 3920 2020 2020 2036 2039      5 9      6 9
│ │ │ │ +0002be00: 2020 2020 2020 3020 3130 2020 2020 2020        0 10      
│ │ │ │ +0002be10: 3120 3130 2020 2020 2020 3220 3130 2020  1 10      2 10  
│ │ │ │ +0002be20: 2020 2020 3420 3130 2020 2020 2020 207c      4 10       |
│ │ │ │ +0002be30: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002be80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002be70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002be80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ -0002bed0: 2020 2020 207c 0a7c 2b20 3435 7820 7820       |.|+ 45x x 
│ │ │ │ -0002bee0: 202d 2031 3478 2078 2020 2d20 3235 7820   - 14x x  - 25x 
│ │ │ │ -0002bef0: 7820 2020 2b20 3435 7820 7820 2020 2d20  x   + 45x x   - 
│ │ │ │ -0002bf00: 3431 7820 7820 2020 2d20 3436 7820 7820  41x x   - 46x x 
│ │ │ │ -0002bf10: 2020 2b20 3878 2078 2020 202d 2032 3878    + 8x x   - 28x
│ │ │ │ -0002bf20: 2078 2020 207c 0a7c 2020 2020 2035 2039   x   |.|     5 9
│ │ │ │ -0002bf30: 2020 2020 2020 3620 3920 2020 2020 2030        6 9      0
│ │ │ │ -0002bf40: 2031 3020 2020 2020 2031 2031 3020 2020   10      1 10   
│ │ │ │ -0002bf50: 2020 2032 2031 3020 2020 2020 2034 2031     2 10      4 1
│ │ │ │ -0002bf60: 3020 2020 2020 3620 3130 2020 2020 2020  0     6 10      
│ │ │ │ -0002bf70: 3020 3131 207c 0a7c 2020 2020 2020 2020  0 11 |.|        
│ │ │ │ +0002bec0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002bed0: 0a7c 2b20 3435 7820 7820 202d 2031 3478  .|+ 45x x  - 14x
│ │ │ │ +0002bee0: 2078 2020 2d20 3235 7820 7820 2020 2b20   x  - 25x x   + 
│ │ │ │ +0002bef0: 3435 7820 7820 2020 2d20 3431 7820 7820  45x x   - 41x x 
│ │ │ │ +0002bf00: 2020 2d20 3436 7820 7820 2020 2b20 3878    - 46x x   + 8x
│ │ │ │ +0002bf10: 2078 2020 202d 2032 3878 2078 2020 207c   x   - 28x x   |
│ │ │ │ +0002bf20: 0a7c 2020 2020 2035 2039 2020 2020 2020  .|     5 9      
│ │ │ │ +0002bf30: 3620 3920 2020 2020 2030 2031 3020 2020  6 9      0 10   
│ │ │ │ +0002bf40: 2020 2031 2031 3020 2020 2020 2032 2031     1 10      2 1
│ │ │ │ +0002bf50: 3020 2020 2020 2034 2031 3020 2020 2020  0      4 10     
│ │ │ │ +0002bf60: 3620 3130 2020 2020 2020 3020 3131 207c  6 10      0 11 |
│ │ │ │ +0002bf70: 0a7c 2020 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002bfb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002bfc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020                  
│ │ │ │ -0002c010: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c000: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c010: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c050: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c060: 0a7c 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 2020 2020 2020 2020                  
│ │ │ │ -0002c0b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c0a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c0b0: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c100: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c0f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c100: 0a7c 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 2020                  
│ │ │ │ -0002c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c150: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c140: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c150: 0a7c 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 2020 2020 2020                  
│ │ │ │ -0002c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c1a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c190: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c1a0: 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c1f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c1e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c1f0: 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c240: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c230: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c240: 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c290: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c280: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c290: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c2e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c2d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c2e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002c2f0: 2020 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c330: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c320: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c330: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c380: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c380: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c3d0: 2020 2020 207c 0a7c 7820 2020 2d20 7820       |.|x   - x 
│ │ │ │ -0002c3e0: 7820 2020 2d20 3436 7820 7820 2020 2b20  x   - 46x x   + 
│ │ │ │ -0002c3f0: 3278 2078 2020 202b 2078 2078 2020 2c20  2x x   + x x  , 
│ │ │ │ +0002c3c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c3d0: 0a7c 7820 2020 2d20 7820 7820 2020 2d20  .|x   - x x   - 
│ │ │ │ +0002c3e0: 3436 7820 7820 2020 2b20 3278 2078 2020  46x x   + 2x x  
│ │ │ │ +0002c3f0: 202b 2078 2078 2020 2c20 2020 2020 2020   + x x  ,       
│ │ │ │  0002c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c420: 2020 2020 207c 0a7c 2031 3020 2020 2036       |.| 10    6
│ │ │ │ -0002c430: 2031 3020 2020 2020 2031 2031 3120 2020   10      1 11   
│ │ │ │ -0002c440: 2020 3220 3131 2020 2020 3520 3131 2020    2 11    5 11  
│ │ │ │ +0002c410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c420: 0a7c 2031 3020 2020 2036 2031 3020 2020  .| 10    6 10   
│ │ │ │ +0002c430: 2020 2031 2031 3120 2020 2020 3220 3131     1 11     2 11
│ │ │ │ +0002c440: 2020 2020 3520 3131 2020 2020 2020 2020      5 11        
│ │ │ │  0002c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c470: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c460: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c470: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c4c0: 2020 2020 207c 0a7c 7820 2020 2d20 3239       |.|x   - 29
│ │ │ │ -0002c4d0: 7820 7820 2020 2b20 3236 7820 7820 2020  x x   + 26x x   
│ │ │ │ -0002c4e0: 2d20 3236 7820 7820 2020 2d20 3431 7820  - 26x x   - 41x 
│ │ │ │ -0002c4f0: 7820 2020 2d20 3435 7820 7820 2020 2d20  x   - 45x x   - 
│ │ │ │ -0002c500: 3336 7820 7820 2020 2b20 3236 7820 7820  36x x   + 26x x 
│ │ │ │ -0002c510: 202c 2020 207c 0a7c 2031 3020 2020 2020   ,   |.| 10     
│ │ │ │ -0002c520: 2034 2031 3020 2020 2020 2035 2031 3020   4 10      5 10 
│ │ │ │ -0002c530: 2020 2020 2036 2031 3020 2020 2020 2031       6 10      1
│ │ │ │ -0002c540: 2031 3120 2020 2020 2032 2031 3120 2020   11      2 11   
│ │ │ │ -0002c550: 2020 2034 2031 3120 2020 2020 2035 2031     4 11      5 1
│ │ │ │ -0002c560: 3120 2020 207c 0a7c 2020 2020 2020 2020  1    |.|        
│ │ │ │ +0002c4b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c4c0: 0a7c 7820 2020 2d20 3239 7820 7820 2020  .|x   - 29x x   
│ │ │ │ +0002c4d0: 2b20 3236 7820 7820 2020 2d20 3236 7820  + 26x x   - 26x 
│ │ │ │ +0002c4e0: 7820 2020 2d20 3431 7820 7820 2020 2d20  x   - 41x x   - 
│ │ │ │ +0002c4f0: 3435 7820 7820 2020 2d20 3336 7820 7820  45x x   - 36x x 
│ │ │ │ +0002c500: 2020 2b20 3236 7820 7820 202c 2020 207c    + 26x x  ,   |
│ │ │ │ +0002c510: 0a7c 2031 3020 2020 2020 2034 2031 3020  .| 10      4 10 
│ │ │ │ +0002c520: 2020 2020 2035 2031 3020 2020 2020 2036       5 10      6
│ │ │ │ +0002c530: 2031 3020 2020 2020 2031 2031 3120 2020   10      1 11   
│ │ │ │ +0002c540: 2020 2032 2031 3120 2020 2020 2034 2031     2 11      4 1
│ │ │ │ +0002c550: 3120 2020 2020 2035 2031 3120 2020 207c  1      5 11    |
│ │ │ │ +0002c560: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c5b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c5a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c5b0: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c600: 2020 2020 207c 0a7c 2034 3778 2078 2020       |.| 47x x  
│ │ │ │ -0002c610: 2d20 3432 7820 7820 202d 2031 3878 2078  - 42x x  - 18x x
│ │ │ │ -0002c620: 2020 202d 2035 7820 7820 2020 2b20 3434     - 5x x   + 44
│ │ │ │ -0002c630: 7820 7820 2020 2d20 3339 7820 7820 2020  x x   - 39x x   
│ │ │ │ -0002c640: 2d20 3135 7820 7820 2020 2d20 3331 7820  - 15x x   - 31x 
│ │ │ │ -0002c650: 7820 2020 207c 0a7c 2020 2020 3520 3920  x    |.|    5 9 
│ │ │ │ -0002c660: 2020 2020 2036 2039 2020 2020 2020 3020       6 9      0 
│ │ │ │ -0002c670: 3130 2020 2020 2031 2031 3020 2020 2020  10     1 10     
│ │ │ │ -0002c680: 2032 2031 3020 2020 2020 2033 2031 3020   2 10      3 10 
│ │ │ │ -0002c690: 2020 2020 2034 2031 3020 2020 2020 2035       4 10      5
│ │ │ │ -0002c6a0: 2031 3020 207c 0a7c 2020 2020 2020 2020   10  |.|        
│ │ │ │ +0002c5f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c600: 0a7c 2034 3778 2078 2020 2d20 3432 7820  .| 47x x  - 42x 
│ │ │ │ +0002c610: 7820 202d 2031 3878 2078 2020 202d 2035  x  - 18x x   - 5
│ │ │ │ +0002c620: 7820 7820 2020 2b20 3434 7820 7820 2020  x x   + 44x x   
│ │ │ │ +0002c630: 2d20 3339 7820 7820 2020 2d20 3135 7820  - 39x x   - 15x 
│ │ │ │ +0002c640: 7820 2020 2d20 3331 7820 7820 2020 207c  x   - 31x x    |
│ │ │ │ +0002c650: 0a7c 2020 2020 3520 3920 2020 2020 2036  .|    5 9      6
│ │ │ │ +0002c660: 2039 2020 2020 2020 3020 3130 2020 2020   9      0 10    
│ │ │ │ +0002c670: 2031 2031 3020 2020 2020 2032 2031 3020   1 10      2 10 
│ │ │ │ +0002c680: 2020 2020 2033 2031 3020 2020 2020 2034       3 10      4
│ │ │ │ +0002c690: 2031 3020 2020 2020 2035 2031 3020 207c   10      5 10  |
│ │ │ │ +0002c6a0: 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c6f0: 2020 2020 207c 0a7c 3578 2078 2020 202d       |.|5x x   -
│ │ │ │ -0002c700: 2031 3578 2078 2020 202d 2031 3378 2078   15x x   - 13x x
│ │ │ │ -0002c710: 2020 202d 2032 3078 2078 2020 202d 2034     - 20x x   - 4
│ │ │ │ -0002c720: 3578 2078 2020 202b 2034 7820 7820 2020  5x x   + 4x x   
│ │ │ │ -0002c730: 2d20 3478 2078 2020 202b 2031 3078 2078  - 4x x   + 10x x
│ │ │ │ -0002c740: 2020 202b 207c 0a7c 2020 3020 3130 2020     + |.|  0 10  
│ │ │ │ -0002c750: 2020 2020 3120 3130 2020 2020 2020 3220      1 10      2 
│ │ │ │ -0002c760: 3130 2020 2020 2020 3320 3130 2020 2020  10      3 10    
│ │ │ │ -0002c770: 2020 3420 3130 2020 2020 2035 2031 3020    4 10     5 10 
│ │ │ │ -0002c780: 2020 2020 3620 3130 2020 2020 2020 3120      6 10      1 
│ │ │ │ -0002c790: 3131 2020 207c 0a7c 2020 2020 2020 2020  11   |.|        
│ │ │ │ +0002c6e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c6f0: 0a7c 3578 2078 2020 202d 2031 3578 2078  .|5x x   - 15x x
│ │ │ │ +0002c700: 2020 202d 2031 3378 2078 2020 202d 2032     - 13x x   - 2
│ │ │ │ +0002c710: 3078 2078 2020 202d 2034 3578 2078 2020  0x x   - 45x x  
│ │ │ │ +0002c720: 202b 2034 7820 7820 2020 2d20 3478 2078   + 4x x   - 4x x
│ │ │ │ +0002c730: 2020 202b 2031 3078 2078 2020 202b 207c     + 10x x   + |
│ │ │ │ +0002c740: 0a7c 2020 3020 3130 2020 2020 2020 3120  .|  0 10      1 
│ │ │ │ +0002c750: 3130 2020 2020 2020 3220 3130 2020 2020  10      2 10    
│ │ │ │ +0002c760: 2020 3320 3130 2020 2020 2020 3420 3130    3 10      4 10
│ │ │ │ +0002c770: 2020 2020 2035 2031 3020 2020 2020 3620       5 10     6 
│ │ │ │ +0002c780: 3130 2020 2020 2020 3120 3131 2020 207c  10      1 11   |
│ │ │ │ +0002c790: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c7d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c7e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c830: 2020 2020 207c 0a7c 202d 2033 3278 2078       |.| - 32x x
│ │ │ │ -0002c840: 2020 2d20 3337 7820 7820 202d 2034 7820    - 37x x  - 4x 
│ │ │ │ -0002c850: 7820 202d 2033 3878 2078 2020 2b20 3130  x  - 38x x  + 10
│ │ │ │ -0002c860: 7820 7820 202b 2031 3378 2078 2020 2b20  x x  + 13x x  + 
│ │ │ │ -0002c870: 3132 7820 7820 2020 2b20 3335 7820 7820  12x x   + 35x x 
│ │ │ │ -0002c880: 2020 2b20 207c 0a7c 2020 2020 2020 3120    +  |.|      1 
│ │ │ │ -0002c890: 3920 2020 2020 2032 2039 2020 2020 2033  9      2 9     3
│ │ │ │ -0002c8a0: 2039 2020 2020 2020 3420 3920 2020 2020   9      4 9     
│ │ │ │ -0002c8b0: 2035 2039 2020 2020 2020 3620 3920 2020   5 9      6 9   
│ │ │ │ -0002c8c0: 2020 2030 2031 3020 2020 2020 2031 2031     0 10      1 1
│ │ │ │ -0002c8d0: 3020 2020 207c 0a7c 2020 2020 2020 2020  0    |.|        
│ │ │ │ +0002c820: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c830: 0a7c 202d 2033 3278 2078 2020 2d20 3337  .| - 32x x  - 37
│ │ │ │ +0002c840: 7820 7820 202d 2034 7820 7820 202d 2033  x x  - 4x x  - 3
│ │ │ │ +0002c850: 3878 2078 2020 2b20 3130 7820 7820 202b  8x x  + 10x x  +
│ │ │ │ +0002c860: 2031 3378 2078 2020 2b20 3132 7820 7820   13x x  + 12x x 
│ │ │ │ +0002c870: 2020 2b20 3335 7820 7820 2020 2b20 207c    + 35x x   +  |
│ │ │ │ +0002c880: 0a7c 2020 2020 2020 3120 3920 2020 2020  .|      1 9     
│ │ │ │ +0002c890: 2032 2039 2020 2020 2033 2039 2020 2020   2 9     3 9    
│ │ │ │ +0002c8a0: 2020 3420 3920 2020 2020 2035 2039 2020    4 9      5 9  
│ │ │ │ +0002c8b0: 2020 2020 3620 3920 2020 2020 2030 2031      6 9      0 1
│ │ │ │ +0002c8c0: 3020 2020 2020 2031 2031 3020 2020 207c  0      1 10    |
│ │ │ │ +0002c8d0: 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c920: 2020 2020 207c 0a7c 202d 2034 3378 2078       |.| - 43x x
│ │ │ │ -0002c930: 2020 2b20 3132 7820 7820 202d 2031 3278    + 12x x  - 12x
│ │ │ │ -0002c940: 2078 2020 202b 2031 3478 2078 2020 202d   x   + 14x x   -
│ │ │ │ -0002c950: 2034 3778 2078 2020 202d 2034 3678 2078   47x x   - 46x x
│ │ │ │ -0002c960: 2020 202d 2031 3478 2078 2020 202b 2034     - 14x x   + 4
│ │ │ │ -0002c970: 3878 2078 207c 0a7c 2020 2020 2020 3520  8x x |.|      5 
│ │ │ │ -0002c980: 3920 2020 2020 2036 2039 2020 2020 2020  9      6 9      
│ │ │ │ -0002c990: 3020 3130 2020 2020 2020 3120 3130 2020  0 10      1 10  
│ │ │ │ -0002c9a0: 2020 2020 3220 3130 2020 2020 2020 3320      2 10      3 
│ │ │ │ -0002c9b0: 3130 2020 2020 2020 3420 3130 2020 2020  10      4 10    
│ │ │ │ -0002c9c0: 2020 3520 207c 0a7c 2d2d 2d2d 2d2d 2d2d    5  |.|--------
│ │ │ │ +0002c910: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c920: 0a7c 202d 2034 3378 2078 2020 2b20 3132  .| - 43x x  + 12
│ │ │ │ +0002c930: 7820 7820 202d 2031 3278 2078 2020 202b  x x  - 12x x   +
│ │ │ │ +0002c940: 2031 3478 2078 2020 202d 2034 3778 2078   14x x   - 47x x
│ │ │ │ +0002c950: 2020 202d 2034 3678 2078 2020 202d 2031     - 46x x   - 1
│ │ │ │ +0002c960: 3478 2078 2020 202b 2034 3878 2078 207c  4x x   + 48x x |
│ │ │ │ +0002c970: 0a7c 2020 2020 2020 3520 3920 2020 2020  .|      5 9     
│ │ │ │ +0002c980: 2036 2039 2020 2020 2020 3020 3130 2020   6 9      0 10  
│ │ │ │ +0002c990: 2020 2020 3120 3130 2020 2020 2020 3220      1 10      2 
│ │ │ │ +0002c9a0: 3130 2020 2020 2020 3320 3130 2020 2020  10      3 10    
│ │ │ │ +0002c9b0: 2020 3420 3130 2020 2020 2020 3520 207c    4 10      5  |
│ │ │ │ +0002c9c0: 0a7c 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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ca10: 2d2d 2d2d 2d7c 0a7c 2020 202b 2039 7820  -----|.|   + 9x 
│ │ │ │ -0002ca20: 7820 2020 2b20 3237 7820 7820 2020 2d20  x   + 27x x   - 
│ │ │ │ -0002ca30: 3237 7820 7820 202c 2020 2020 2020 2020  27x x  ,        
│ │ │ │ +0002ca00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0002ca10: 0a7c 2020 202b 2039 7820 7820 2020 2b20  .|   + 9x x   + 
│ │ │ │ +0002ca20: 3237 7820 7820 2020 2d20 3237 7820 7820  27x x   - 27x x 
│ │ │ │ +0002ca30: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  0002ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca60: 2020 2020 207c 0a7c 3131 2020 2020 2032       |.|11     2
│ │ │ │ -0002ca70: 2031 3120 2020 2020 2034 2031 3120 2020   11      4 11   
│ │ │ │ -0002ca80: 2020 2035 2031 3120 2020 2020 2020 2020     5 11         
│ │ │ │ +0002ca50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002ca60: 0a7c 3131 2020 2020 2032 2031 3120 2020  .|11     2 11   
│ │ │ │ +0002ca70: 2020 2034 2031 3120 2020 2020 2035 2031     4 11      5 1
│ │ │ │ +0002ca80: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  0002ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cab0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002caa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002cab0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002caf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002caf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002cb00: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb50: 2020 2020 207c 0a7c 2078 2020 202b 2033       |.| x   + 3
│ │ │ │ -0002cb60: 3678 2078 2020 202b 2033 3578 2078 2020  6x x   + 35x x  
│ │ │ │ -0002cb70: 202d 2031 3778 2078 2020 2c20 2020 2020   - 17x x  ,     
│ │ │ │ +0002cb40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002cb50: 0a7c 2078 2020 202b 2033 3678 2078 2020  .| x   + 36x x  
│ │ │ │ +0002cb60: 202b 2033 3578 2078 2020 202d 2031 3778   + 35x x   - 17x
│ │ │ │ +0002cb70: 2078 2020 2c20 2020 2020 2020 2020 2020   x  ,           
│ │ │ │  0002cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cba0: 2020 2020 207c 0a7c 3020 3131 2020 2020       |.|0 11    
│ │ │ │ -0002cbb0: 2020 3120 3131 2020 2020 2020 3220 3131    1 11      2 11
│ │ │ │ -0002cbc0: 2020 2020 2020 3420 3131 2020 2020 2020        4 11      
│ │ │ │ +0002cb90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002cba0: 0a7c 3020 3131 2020 2020 2020 3120 3131  .|0 11      1 11
│ │ │ │ +0002cbb0: 2020 2020 2020 3220 3131 2020 2020 2020        2 11      
│ │ │ │ +0002cbc0: 3420 3131 2020 2020 2020 2020 2020 2020  4 11            
│ │ │ │  0002cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cbf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002cbe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002cbf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002cc30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002cc40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 202b 2031 3178 2078       |.| + 11x x
│ │ │ │ -0002cca0: 2020 202b 2031 3478 2078 2020 202d 2038     + 14x x   - 8
│ │ │ │ -0002ccb0: 7820 7820 2020 2020 2020 2020 2020 2020  x x             
│ │ │ │ +0002cc80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002cc90: 0a7c 202b 2031 3178 2078 2020 202b 2031  .| + 11x x   + 1
│ │ │ │ +0002cca0: 3478 2078 2020 202d 2038 7820 7820 2020  4x x   - 8x x   
│ │ │ │ +0002ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ccd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cce0: 2020 2020 207c 0a7c 2020 2020 2020 3220       |.|      2 
│ │ │ │ -0002ccf0: 3131 2020 2020 2020 3420 3131 2020 2020  11      4 11    
│ │ │ │ -0002cd00: 2035 2031 3120 2020 2020 2020 2020 2020   5 11           
│ │ │ │ +0002ccd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002cce0: 0a7c 2020 2020 2020 3220 3131 2020 2020  .|      2 11    
│ │ │ │ +0002ccf0: 2020 3420 3131 2020 2020 2035 2031 3120    4 11     5 11 
│ │ │ │ +0002cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cd30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002cd20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002cd30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cd80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002cd70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002cd80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cdd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002cdc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002cdd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ce10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ce20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002ce10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002ce20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002ce30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020                  
│ │ │ │ -0002ce70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002ce60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002ce70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002ce80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ce90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ceb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cec0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002ceb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002cec0: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cf10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002cf00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002cf10: 0a7c 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 2020                  
│ │ │ │ -0002cf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cf60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002cf50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002cf60: 0a7c 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 2020 2020 2020                  
│ │ │ │ -0002cfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cfb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002cfa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002cfb0: 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d000: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002cff0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d000: 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d050: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d050: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d0a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d090: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d0a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d0f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d0e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d0f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d140: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d130: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d140: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d190: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d180: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d190: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d1a0: 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d1d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d1e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d220: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d230: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d280: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d270: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d280: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d2d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d2c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d2d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d320: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d310: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d320: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d370: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d360: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d370: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d3c0: 2020 2020 207c 0a7c 2b20 3135 7820 7820       |.|+ 15x x 
│ │ │ │ -0002d3d0: 2020 2d20 3137 7820 7820 2020 2d20 3239    - 17x x   - 29
│ │ │ │ -0002d3e0: 7820 7820 2020 2b20 3335 7820 7820 2020  x x   + 35x x   
│ │ │ │ -0002d3f0: 2d20 3137 7820 7820 2020 2d20 3331 7820  - 17x x   - 31x 
│ │ │ │ -0002d400: 7820 202c 2020 2020 2020 2020 2020 2020  x  ,            
│ │ │ │ -0002d410: 2020 2020 207c 0a7c 2020 2020 2036 2031       |.|     6 1
│ │ │ │ -0002d420: 3020 2020 2020 2030 2031 3120 2020 2020  0      0 11     
│ │ │ │ -0002d430: 2031 2031 3120 2020 2020 2032 2031 3120   1 11      2 11 
│ │ │ │ -0002d440: 2020 2020 2033 2031 3120 2020 2020 2035       3 11      5
│ │ │ │ -0002d450: 2031 3120 2020 2020 2020 2020 2020 2020   11             
│ │ │ │ -0002d460: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d3b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d3c0: 0a7c 2b20 3135 7820 7820 2020 2d20 3137  .|+ 15x x   - 17
│ │ │ │ +0002d3d0: 7820 7820 2020 2d20 3239 7820 7820 2020  x x   - 29x x   
│ │ │ │ +0002d3e0: 2b20 3335 7820 7820 2020 2d20 3137 7820  + 35x x   - 17x 
│ │ │ │ +0002d3f0: 7820 2020 2d20 3331 7820 7820 202c 2020  x   - 31x x  ,  
│ │ │ │ +0002d400: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d410: 0a7c 2020 2020 2036 2031 3020 2020 2020  .|     6 10     
│ │ │ │ +0002d420: 2030 2031 3120 2020 2020 2031 2031 3120   0 11      1 11 
│ │ │ │ +0002d430: 2020 2020 2032 2031 3120 2020 2020 2033       2 11      3
│ │ │ │ +0002d440: 2031 3120 2020 2020 2035 2031 3120 2020   11      5 11   
│ │ │ │ +0002d450: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d460: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d4b0: 2020 2020 207c 0a7c 2038 7820 7820 2020       |.| 8x x   
│ │ │ │ -0002d4c0: 2b20 3478 2078 2020 2c20 2020 2020 2020  + 4x x  ,       
│ │ │ │ +0002d4a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d4b0: 0a7c 2038 7820 7820 2020 2b20 3478 2078  .| 8x x   + 4x x
│ │ │ │ +0002d4c0: 2020 2c20 2020 2020 2020 2020 2020 2020    ,             
│ │ │ │  0002d4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d500: 2020 2020 207c 0a7c 2020 2032 2031 3120       |.|   2 11 
│ │ │ │ -0002d510: 2020 2020 3520 3131 2020 2020 2020 2020      5 11        
│ │ │ │ +0002d4f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d500: 0a7c 2020 2032 2031 3120 2020 2020 3520  .|   2 11     5 
│ │ │ │ +0002d510: 3131 2020 2020 2020 2020 2020 2020 2020  11              
│ │ │ │  0002d520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d550: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d540: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d550: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d5a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d590: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d5a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d5f0: 2020 2020 207c 0a7c 3133 7820 7820 2020       |.|13x x   
│ │ │ │ -0002d600: 2b20 3331 7820 7820 2020 2b20 3330 7820  + 31x x   + 30x 
│ │ │ │ -0002d610: 7820 2020 2d20 3435 7820 7820 2020 2d20  x   - 45x x   - 
│ │ │ │ -0002d620: 3135 7820 7820 2020 2b20 3337 7820 7820  15x x   + 37x x 
│ │ │ │ -0002d630: 2020 2b20 3434 7820 7820 2020 2b20 3778    + 44x x   + 7x
│ │ │ │ -0002d640: 2078 2020 207c 0a7c 2020 2032 2031 3020   x   |.|   2 10 
│ │ │ │ -0002d650: 2020 2020 2033 2031 3020 2020 2020 2034       3 10      4
│ │ │ │ -0002d660: 2031 3020 2020 2020 2035 2031 3020 2020   10      5 10   
│ │ │ │ -0002d670: 2020 2036 2031 3020 2020 2020 2030 2031     6 10      0 1
│ │ │ │ -0002d680: 3120 2020 2020 2031 2031 3120 2020 2020  1      1 11     
│ │ │ │ -0002d690: 3220 3131 207c 0a7c 2020 2020 2020 2020  2 11 |.|        
│ │ │ │ +0002d5e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d5f0: 0a7c 3133 7820 7820 2020 2b20 3331 7820  .|13x x   + 31x 
│ │ │ │ +0002d600: 7820 2020 2b20 3330 7820 7820 2020 2d20  x   + 30x x   - 
│ │ │ │ +0002d610: 3435 7820 7820 2020 2d20 3135 7820 7820  45x x   - 15x x 
│ │ │ │ +0002d620: 2020 2b20 3337 7820 7820 2020 2b20 3434    + 37x x   + 44
│ │ │ │ +0002d630: 7820 7820 2020 2b20 3778 2078 2020 207c  x x   + 7x x   |
│ │ │ │ +0002d640: 0a7c 2020 2032 2031 3020 2020 2020 2033  .|   2 10      3
│ │ │ │ +0002d650: 2031 3020 2020 2020 2034 2031 3020 2020   10      4 10   
│ │ │ │ +0002d660: 2020 2035 2031 3020 2020 2020 2036 2031     5 10      6 1
│ │ │ │ +0002d670: 3020 2020 2020 2030 2031 3120 2020 2020  0      0 11     
│ │ │ │ +0002d680: 2031 2031 3120 2020 2020 3220 3131 207c   1 11     2 11 |
│ │ │ │ +0002d690: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d6e0: 2020 2020 207c 0a7c 2020 202b 2031 3278       |.|   + 12x
│ │ │ │ -0002d6f0: 2078 2020 202d 2031 3278 2078 2020 202b   x   - 12x x   +
│ │ │ │ -0002d700: 2034 3878 2078 2020 202d 2078 2078 2020   48x x   - x x  
│ │ │ │ -0002d710: 202d 2039 7820 7820 2020 2b20 3438 7820   - 9x x   + 48x 
│ │ │ │ -0002d720: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ -0002d730: 2020 2020 207c 0a7c 3130 2020 2020 2020       |.|10      
│ │ │ │ -0002d740: 3620 3130 2020 2020 2020 3020 3131 2020  6 10      0 11  
│ │ │ │ -0002d750: 2020 2020 3120 3131 2020 2020 3220 3131      1 11    2 11
│ │ │ │ -0002d760: 2020 2020 2033 2031 3120 2020 2020 2035       3 11      5
│ │ │ │ -0002d770: 2031 3120 2020 2020 2020 2020 2020 2020   11             
│ │ │ │ -0002d780: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +0002d6d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d6e0: 0a7c 2020 202b 2031 3278 2078 2020 202d  .|   + 12x x   -
│ │ │ │ +0002d6f0: 2031 3278 2078 2020 202b 2034 3878 2078   12x x   + 48x x
│ │ │ │ +0002d700: 2020 202d 2078 2078 2020 202d 2039 7820     - x x   - 9x 
│ │ │ │ +0002d710: 7820 2020 2b20 3438 7820 7820 2020 2020  x   + 48x x     
│ │ │ │ +0002d720: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d730: 0a7c 3130 2020 2020 2020 3620 3130 2020  .|10      6 10  
│ │ │ │ +0002d740: 2020 2020 3020 3131 2020 2020 2020 3120      0 11      1 
│ │ │ │ +0002d750: 3131 2020 2020 3220 3131 2020 2020 2033  11    2 11     3
│ │ │ │ +0002d760: 2031 3120 2020 2020 2035 2031 3120 2020   11      5 11   
│ │ │ │ +0002d770: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d780: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0002d790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d7d0: 2d2d 2d2d 2d7c 0a7c 2b20 3978 2078 2020  -----|.|+ 9x x  
│ │ │ │ -0002d7e0: 202d 2032 3578 2078 2020 202d 2034 3578   - 25x x   - 45x
│ │ │ │ -0002d7f0: 2078 2020 2c20 2020 2020 2020 2020 2020   x  ,           
│ │ │ │ +0002d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0002d7d0: 0a7c 2b20 3978 2078 2020 202d 2032 3578  .|+ 9x x   - 25x
│ │ │ │ +0002d7e0: 2078 2020 202d 2034 3578 2078 2020 2c20   x   - 45x x  , 
│ │ │ │ +0002d7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d820: 2020 2020 207c 0a7c 2020 2020 3320 3131       |.|    3 11
│ │ │ │ -0002d830: 2020 2020 2020 3420 3131 2020 2020 2020        4 11      
│ │ │ │ -0002d840: 3520 3131 2020 2020 2020 2020 2020 2020  5 11            
│ │ │ │ +0002d810: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d820: 0a7c 2020 2020 3320 3131 2020 2020 2020  .|    3 11      
│ │ │ │ +0002d830: 3420 3131 2020 2020 2020 3520 3131 2020  4 11      5 11  
│ │ │ │ +0002d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d870: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002d860: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d870: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0002d880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d8c0: 2d2d 2d2d 2d2b 0a7c 6939 203a 202d 2d20  -----+.|i9 : -- 
│ │ │ │ -0002d8d0: 6275 742e 2e2e 2020 2020 2020 2020 2020  but...          
│ │ │ │ +0002d8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0002d8c0: 0a7c 6939 203a 202d 2d20 6275 742e 2e2e  .|i9 : -- but...
│ │ │ │ +0002d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d910: 2020 2020 207c 0a7c 2020 2020 2070 6869       |.|     phi
│ │ │ │ -0002d920: 202a 2070 7369 203d 3d20 3120 2020 2020   * psi == 1     
│ │ │ │ +0002d900: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d910: 0a7c 2020 2020 2070 6869 202a 2070 7369  .|     phi * psi
│ │ │ │ +0002d920: 203d 3d20 3120 2020 2020 2020 2020 2020   == 1           
│ │ │ │  0002d930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d960: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002d950: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d960: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d9b0: 2020 2020 207c 0a7c 6f39 203d 2066 616c       |.|o9 = fal
│ │ │ │ -0002d9c0: 7365 2020 2020 2020 2020 2020 2020 2020  se              
│ │ │ │ +0002d9a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d9b0: 0a7c 6f39 203d 2066 616c 7365 2020 2020  .|o9 = false    
│ │ │ │ +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 2020                  
│ │ │ │ -0002d9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002da00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002d9f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002da00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0002da10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2b 0a7c 6931 3020 3a20 2d2d  -----+.|i10 : --
│ │ │ │ -0002da60: 2069 6e20 7468 6973 2063 6173 6520 7765   in this case we
│ │ │ │ -0002da70: 2063 616e 2072 656d 6564 7920 656e 6162   can remedy enab
│ │ │ │ -0002da80: 6c69 6e67 2074 6865 206f 7074 696f 6e20  ling the option 
│ │ │ │ -0002da90: 4365 7274 6966 7920 2020 2020 2020 2020  Certify         
│ │ │ │ -0002daa0: 2020 2020 207c 0a7c 2020 2020 2020 7469       |.|      ti
│ │ │ │ -0002dab0: 6d65 2070 7369 203d 2061 7070 726f 7869  me psi = approxi
│ │ │ │ -0002dac0: 6d61 7465 496e 7665 7273 654d 6170 2870  mateInverseMap(p
│ │ │ │ -0002dad0: 6869 2c43 6f64 696d 4273 496e 763d 3e34  hi,CodimBsInv=>4
│ │ │ │ -0002dae0: 2c43 6572 7469 6679 3d3e 7472 7565 2920  ,Certify=>true) 
│ │ │ │ -0002daf0: 2020 2020 207c 0a7c 2d2d 2061 7070 726f       |.|-- appro
│ │ │ │ -0002db00: 7869 6d61 7465 496e 7665 7273 654d 6170  ximateInverseMap
│ │ │ │ -0002db10: 3a20 7374 6570 2031 206f 6620 3320 2020  : step 1 of 3   
│ │ │ │ +0002da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0002da50: 0a7c 6931 3020 3a20 2d2d 2069 6e20 7468  .|i10 : -- in th
│ │ │ │ +0002da60: 6973 2063 6173 6520 7765 2063 616e 2072  is case we can r
│ │ │ │ +0002da70: 656d 6564 7920 656e 6162 6c69 6e67 2074  emedy enabling t
│ │ │ │ +0002da80: 6865 206f 7074 696f 6e20 4365 7274 6966  he option Certif
│ │ │ │ +0002da90: 7920 2020 2020 2020 2020 2020 2020 207c  y              |
│ │ │ │ +0002daa0: 0a7c 2020 2020 2020 7469 6d65 2070 7369  .|      time psi
│ │ │ │ +0002dab0: 203d 2061 7070 726f 7869 6d61 7465 496e   = approximateIn
│ │ │ │ +0002dac0: 7665 7273 654d 6170 2870 6869 2c43 6f64  verseMap(phi,Cod
│ │ │ │ +0002dad0: 696d 4273 496e 763d 3e34 2c43 6572 7469  imBsInv=>4,Certi
│ │ │ │ +0002dae0: 6679 3d3e 7472 7565 2920 2020 2020 207c  fy=>true)      |
│ │ │ │ +0002daf0: 0a7c 2d2d 2061 7070 726f 7869 6d61 7465  .|-- approximate
│ │ │ │ +0002db00: 496e 7665 7273 654d 6170 3a20 7374 6570  InverseMap: step
│ │ │ │ +0002db10: 2031 206f 6620 3320 2020 2020 2020 2020   1 of 3         
│ │ │ │  0002db20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db40: 2020 2020 207c 0a7c 2d2d 2061 7070 726f       |.|-- appro
│ │ │ │ -0002db50: 7869 6d61 7465 496e 7665 7273 654d 6170  ximateInverseMap
│ │ │ │ -0002db60: 3a20 7374 6570 2032 206f 6620 3320 2020  : step 2 of 3   
│ │ │ │ +0002db30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002db40: 0a7c 2d2d 2061 7070 726f 7869 6d61 7465  .|-- approximate
│ │ │ │ +0002db50: 496e 7665 7273 654d 6170 3a20 7374 6570  InverseMap: step
│ │ │ │ +0002db60: 2032 206f 6620 3320 2020 2020 2020 2020   2 of 3         
│ │ │ │  0002db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db90: 2020 2020 207c 0a7c 2d2d 2061 7070 726f       |.|-- appro
│ │ │ │ -0002dba0: 7869 6d61 7465 496e 7665 7273 654d 6170  ximateInverseMap
│ │ │ │ -0002dbb0: 3a20 7374 6570 2033 206f 6620 3320 2020  : step 3 of 3   
│ │ │ │ +0002db80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002db90: 0a7c 2d2d 2061 7070 726f 7869 6d61 7465  .|-- approximate
│ │ │ │ +0002dba0: 496e 7665 7273 654d 6170 3a20 7374 6570  InverseMap: step
│ │ │ │ +0002dbb0: 2033 206f 6620 3320 2020 2020 2020 2020   3 of 3         
│ │ │ │  0002dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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!              
│ │ │ │ +0002dbd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002dbe0: 0a7c 4365 7274 6966 793a 206f 7574 7075  .|Certify: outpu
│ │ │ │ +0002dbf0: 7420 6365 7274 6966 6965 6421 2020 2020  t certified!    
│ │ │ │ +0002dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2e38 3833 3333 7320 2863 7075 293b   3.88333s (cpu);
│ │ │ │ -0002dc50: 2033 2e31 3136 3538 7320 2874 6872 6561   3.11658s (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       |.|        
│ │ │ │ +0002dc20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002dc30: 0a7c 202d 2d20 7573 6564 2033 2e32 3335  .| -- used 3.235
│ │ │ │ +0002dc40: 3235 7320 2863 7075 293b 2032 2e38 3031  25s (cpu); 2.801
│ │ │ │ +0002dc50: 3236 7320 2874 6872 6561 6429 3b20 3073  26s (thread); 0s
│ │ │ │ +0002dc60: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +0002dc70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002dc80: 0a7c 2020 2020 2020 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 = --
│ │ │ │ -0002dce0: 2072 6174 696f 6e61 6c20 6d61 7020 2d2d   rational map --
│ │ │ │ +0002dcc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002dcd0: 0a7c 6f31 3020 3d20 2d2d 2072 6174 696f  .|o10 = -- ratio
│ │ │ │ +0002dce0: 6e61 6c20 6d61 7020 2d2d 2020 2020 2020  nal map --      
│ │ │ │  0002dcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002dd10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002dd20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd40: 2020 2020 2020 2020 205a 5a20 2020 2020           ZZ     
│ │ │ │ +0002dd40: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │  0002dd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd70: 2020 2020 207c 0a7c 2020 2020 2020 736f       |.|      so
│ │ │ │ -0002dd80: 7572 6365 3a20 7375 6276 6172 6965 7479  urce: subvariety
│ │ │ │ -0002dd90: 206f 6620 5072 6f6a 282d 2d5b 7820 2c20   of Proj(--[x , 
│ │ │ │ -0002dda0: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -0002ddb0: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -0002ddc0: 7820 2c20 787c 0a7c 2020 2020 2020 2020  x , x|.|        
│ │ │ │ +0002dd60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002dd70: 0a7c 2020 2020 2020 736f 7572 6365 3a20  .|      source: 
│ │ │ │ +0002dd80: 7375 6276 6172 6965 7479 206f 6620 5072  subvariety of Pr
│ │ │ │ +0002dd90: 6f6a 282d 2d5b 7820 2c20 7820 2c20 7820  oj(--[x , x , x 
│ │ │ │ +0002dda0: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ +0002ddb0: 2c20 7820 2c20 7820 2c20 7820 2c20 787c  , x , x , x , x|
│ │ │ │ +0002ddc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dde0: 2020 2020 2020 2020 2039 3720 2030 2020           97  0  
│ │ │ │ -0002ddf0: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -0002de00: 2035 2020 2036 2020 2037 2020 2038 2020   5   6   7   8  
│ │ │ │ -0002de10: 2039 2020 207c 0a7c 2020 2020 2020 2020   9   |.|        
│ │ │ │ -0002de20: 2020 2020 2020 7b20 2020 2020 2020 2020        {         
│ │ │ │ +0002dde0: 2020 2039 3720 2030 2020 2031 2020 2032     97  0   1   2
│ │ │ │ +0002ddf0: 2020 2033 2020 2034 2020 2035 2020 2036     3   4   5   6
│ │ │ │ +0002de00: 2020 2037 2020 2038 2020 2039 2020 207c     7   8   9   |
│ │ │ │ +0002de10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002de20: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │  0002de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002de40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002de50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002de60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de80: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0002de80: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  0002de90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002deb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002dec0: 2020 2020 2020 2078 2078 2020 2d20 3878         x x  - 8x
│ │ │ │ -0002ded0: 2078 2020 2b20 3235 7820 202d 2032 3578   x  + 25x  - 25x
│ │ │ │ -0002dee0: 2078 2020 2d20 3232 7820 7820 202b 2078   x  - 22x x  + x
│ │ │ │ -0002def0: 2078 2020 2b20 3133 7820 7820 202b 2034   x  + 13x x  + 4
│ │ │ │ -0002df00: 3178 2078 207c 0a7c 2020 2020 2020 2020  1x x |.|        
│ │ │ │ -0002df10: 2020 2020 2020 2020 3120 3320 2020 2020          1 3     
│ │ │ │ -0002df20: 3220 3320 2020 2020 2033 2020 2020 2020  2 3      3      
│ │ │ │ -0002df30: 3220 3420 2020 2020 2033 2034 2020 2020  2 4      3 4    
│ │ │ │ -0002df40: 3020 3520 2020 2020 2032 2035 2020 2020  0 5      2 5    
│ │ │ │ -0002df50: 2020 3320 357c 0a7c 2020 2020 2020 2020    3 5|.|        
│ │ │ │ +0002dea0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002deb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002dec0: 2078 2078 2020 2d20 3878 2078 2020 2b20   x x  - 8x x  + 
│ │ │ │ +0002ded0: 3235 7820 202d 2032 3578 2078 2020 2d20  25x  - 25x x  - 
│ │ │ │ +0002dee0: 3232 7820 7820 202b 2078 2078 2020 2b20  22x x  + x x  + 
│ │ │ │ +0002def0: 3133 7820 7820 202b 2034 3178 2078 207c  13x x  + 41x x |
│ │ │ │ +0002df00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002df10: 2020 3120 3320 2020 2020 3220 3320 2020    1 3     2 3   
│ │ │ │ +0002df20: 2020 2033 2020 2020 2020 3220 3420 2020     3      2 4   
│ │ │ │ +0002df30: 2020 2033 2034 2020 2020 3020 3520 2020     3 4    0 5   
│ │ │ │ +0002df40: 2020 2032 2035 2020 2020 2020 3320 357c     2 5      3 5|
│ │ │ │ +0002df50: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dfa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002dfb0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -0002dfc0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0002df90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002dfa0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002dfb0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0002dfc0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0002dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dff0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002e000: 2020 2020 2020 2078 2020 2b20 3137 7820         x  + 17x 
│ │ │ │ -0002e010: 7820 202d 2031 3478 2020 2d20 3133 7820  x  - 14x  - 13x 
│ │ │ │ -0002e020: 7820 202b 2033 3478 2078 2020 2b20 3434  x  + 34x x  + 44
│ │ │ │ -0002e030: 7820 7820 202d 2033 3078 2078 2020 2b20  x x  - 30x x  + 
│ │ │ │ -0002e040: 3237 7820 787c 0a7c 2020 2020 2020 2020  27x x|.|        
│ │ │ │ -0002e050: 2020 2020 2020 2020 3220 2020 2020 2032          2      2
│ │ │ │ -0002e060: 2033 2020 2020 2020 3320 2020 2020 2032   3      3      2
│ │ │ │ -0002e070: 2034 2020 2020 2020 3320 3420 2020 2020   4      3 4     
│ │ │ │ -0002e080: 2030 2035 2020 2020 2020 3220 3520 2020   0 5      2 5   
│ │ │ │ -0002e090: 2020 2033 207c 0a7c 2020 2020 2020 2020     3 |.|        
│ │ │ │ +0002dfe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002dff0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e000: 2078 2020 2b20 3137 7820 7820 202d 2031   x  + 17x x  - 1
│ │ │ │ +0002e010: 3478 2020 2d20 3133 7820 7820 202b 2033  4x  - 13x x  + 3
│ │ │ │ +0002e020: 3478 2078 2020 2b20 3434 7820 7820 202d  4x x  + 44x x  -
│ │ │ │ +0002e030: 2033 3078 2078 2020 2b20 3237 7820 787c   30x x  + 27x x|
│ │ │ │ +0002e040: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e050: 2020 3220 2020 2020 2032 2033 2020 2020    2      2 3    
│ │ │ │ +0002e060: 2020 3320 2020 2020 2032 2034 2020 2020    3      2 4    
│ │ │ │ +0002e070: 2020 3320 3420 2020 2020 2030 2035 2020    3 4      0 5  
│ │ │ │ +0002e080: 2020 2020 3220 3520 2020 2020 2033 207c      2 5      3 |
│ │ │ │ +0002e090: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e0e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e0d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e0e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e100: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +0002e100: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  0002e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e130: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002e140: 2020 2020 2020 2078 2078 2020 2d20 3430         x x  - 40
│ │ │ │ -0002e150: 7820 7820 202b 2032 3878 2020 2d20 7820  x x  + 28x  - x 
│ │ │ │ -0002e160: 7820 202b 2035 7820 7820 202d 2031 3678  x  + 5x x  - 16x
│ │ │ │ -0002e170: 2078 2020 2b20 3578 2078 2020 2d20 3336   x  + 5x x  - 36
│ │ │ │ -0002e180: 7820 7820 207c 0a7c 2020 2020 2020 2020  x x  |.|        
│ │ │ │ -0002e190: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ -0002e1a0: 2032 2033 2020 2020 2020 3320 2020 2030   2 3      3    0
│ │ │ │ -0002e1b0: 2034 2020 2020 2032 2034 2020 2020 2020   4     2 4      
│ │ │ │ -0002e1c0: 3320 3420 2020 2020 3020 3520 2020 2020  3 4     0 5     
│ │ │ │ -0002e1d0: 2032 2035 207c 0a7c 2020 2020 2020 2020   2 5 |.|        
│ │ │ │ -0002e1e0: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │ +0002e120: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e130: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e140: 2078 2078 2020 2d20 3430 7820 7820 202b   x x  - 40x x  +
│ │ │ │ +0002e150: 2032 3878 2020 2d20 7820 7820 202b 2035   28x  - x x  + 5
│ │ │ │ +0002e160: 7820 7820 202d 2031 3678 2078 2020 2b20  x x  - 16x x  + 
│ │ │ │ +0002e170: 3578 2078 2020 2d20 3336 7820 7820 207c  5x x  - 36x x  |
│ │ │ │ +0002e180: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e190: 2020 3120 3220 2020 2020 2032 2033 2020    1 2      2 3  
│ │ │ │ +0002e1a0: 2020 2020 3320 2020 2030 2034 2020 2020      3    0 4    
│ │ │ │ +0002e1b0: 2032 2034 2020 2020 2020 3320 3420 2020   2 4      3 4   
│ │ │ │ +0002e1c0: 2020 3020 3520 2020 2020 2032 2035 207c    0 5      2 5 |
│ │ │ │ +0002e1d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e1e0: 7d20 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e220: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002e230: 2020 2020 2020 2020 2020 205a 5a20 2020             ZZ   
│ │ │ │ +0002e210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e220: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e230: 2020 2020 205a 5a20 2020 2020 2020 2020       ZZ         
│ │ │ │  0002e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e270: 2020 2020 207c 0a7c 2020 2020 2020 7461       |.|      ta
│ │ │ │ -0002e280: 7267 6574 3a20 5072 6f6a 282d 2d5b 7420  rget: Proj(--[t 
│ │ │ │ -0002e290: 2c20 7420 2c20 7420 2c20 7420 2c20 7420  , t , t , t , t 
│ │ │ │ -0002e2a0: 2c20 7420 2c20 7420 2c20 7420 2c20 7420  , t , t , t , t 
│ │ │ │ -0002e2b0: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ -0002e2c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002e2d0: 2020 2020 2020 2020 2020 2039 3720 2030             97  0
│ │ │ │ -0002e2e0: 2020 2031 2020 2032 2020 2033 2020 2034     1   2   3   4
│ │ │ │ -0002e2f0: 2020 2035 2020 2036 2020 2037 2020 2038     5   6   7   8
│ │ │ │ -0002e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e310: 2020 2020 207c 0a7c 2020 2020 2020 6465       |.|      de
│ │ │ │ -0002e320: 6669 6e69 6e67 2066 6f72 6d73 3a20 7b20  fining forms: { 
│ │ │ │ +0002e260: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e270: 0a7c 2020 2020 2020 7461 7267 6574 3a20  .|      target: 
│ │ │ │ +0002e280: 5072 6f6a 282d 2d5b 7420 2c20 7420 2c20  Proj(--[t , t , 
│ │ │ │ +0002e290: 7420 2c20 7420 2c20 7420 2c20 7420 2c20  t , t , t , t , 
│ │ │ │ +0002e2a0: 7420 2c20 7420 2c20 7420 5d29 2020 2020  t , t , t ])    
│ │ │ │ +0002e2b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e2c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e2d0: 2020 2020 2039 3720 2030 2020 2031 2020       97  0   1  
│ │ │ │ +0002e2e0: 2032 2020 2033 2020 2034 2020 2035 2020   2   3   4   5  
│ │ │ │ +0002e2f0: 2036 2020 2037 2020 2038 2020 2020 2020   6   7   8      
│ │ │ │ +0002e300: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e310: 0a7c 2020 2020 2020 6465 6669 6e69 6e67  .|      defining
│ │ │ │ +0002e320: 2066 6f72 6d73 3a20 7b20 2020 2020 2020   forms: {       
│ │ │ │  0002e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e360: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e360: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e380: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0002e380: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  0002e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e3b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002e3c0: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ -0002e3d0: 2032 7820 7820 202b 2036 7820 202d 2032   2x x  + 6x  - 2
│ │ │ │ -0002e3e0: 3478 2078 2020 2d20 3235 7820 7820 202b  4x x  - 25x x  +
│ │ │ │ -0002e3f0: 2033 3978 2078 2020 2b20 3378 2078 2020   39x x  + 3x x  
│ │ │ │ -0002e400: 2b20 3436 787c 0a7c 2020 2020 2020 2020  + 46x|.|        
│ │ │ │ -0002e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e420: 2020 2032 2033 2020 2020 2033 2020 2020     2 3     3    
│ │ │ │ -0002e430: 2020 3220 3420 2020 2020 2033 2034 2020    2 4      3 4  
│ │ │ │ -0002e440: 2020 2020 3020 3520 2020 2020 3220 3520      0 5     2 5 
│ │ │ │ -0002e450: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e3a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e3b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e3c0: 2020 2020 2020 2020 202d 2032 7820 7820           - 2x x 
│ │ │ │ +0002e3d0: 202b 2036 7820 202d 2032 3478 2078 2020   + 6x  - 24x x  
│ │ │ │ +0002e3e0: 2d20 3235 7820 7820 202b 2033 3978 2078  - 25x x  + 39x x
│ │ │ │ +0002e3f0: 2020 2b20 3378 2078 2020 2b20 3436 787c    + 3x x  + 46x|
│ │ │ │ +0002e400: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e410: 2020 2020 2020 2020 2020 2020 2032 2033               2 3
│ │ │ │ +0002e420: 2020 2020 2033 2020 2020 2020 3220 3420       3      2 4 
│ │ │ │ +0002e430: 2020 2020 2033 2034 2020 2020 2020 3020       3 4      0 
│ │ │ │ +0002e440: 3520 2020 2020 3220 3520 2020 2020 207c  5     2 5      |
│ │ │ │ +0002e450: 0a7c 2020 2020 2020 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e4a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002e4b0: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ -0002e4c0: 2031 3978 2078 2020 2d20 3430 7820 7820   19x x  - 40x x 
│ │ │ │ -0002e4d0: 202b 2031 3978 2078 2020 2b20 3139 7820   + 19x x  + 19x 
│ │ │ │ -0002e4e0: 7820 202d 2033 3878 2078 2020 2d20 3339  x  - 38x x  - 39
│ │ │ │ -0002e4f0: 7820 7820 207c 0a7c 2020 2020 2020 2020  x x  |.|        
│ │ │ │ -0002e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e510: 2020 2020 3220 3520 2020 2020 2033 2035      2 5      3 5
│ │ │ │ -0002e520: 2020 2020 2020 3120 3820 2020 2020 2032        1 8      2
│ │ │ │ -0002e530: 2038 2020 2020 2020 3320 3820 2020 2020   8      3 8     
│ │ │ │ -0002e540: 2035 2038 207c 0a7c 2020 2020 2020 2020   5 8 |.|        
│ │ │ │ +0002e490: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e4a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e4b0: 2020 2020 2020 2020 202d 2031 3978 2078           - 19x x
│ │ │ │ +0002e4c0: 2020 2d20 3430 7820 7820 202b 2031 3978    - 40x x  + 19x
│ │ │ │ +0002e4d0: 2078 2020 2b20 3139 7820 7820 202d 2033   x  + 19x x  - 3
│ │ │ │ +0002e4e0: 3878 2078 2020 2d20 3339 7820 7820 207c  8x x  - 39x x  |
│ │ │ │ +0002e4f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e500: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0002e510: 3520 2020 2020 2033 2035 2020 2020 2020  5      3 5      
│ │ │ │ +0002e520: 3120 3820 2020 2020 2032 2038 2020 2020  1 8      2 8    
│ │ │ │ +0002e530: 2020 3320 3820 2020 2020 2035 2038 207c    3 8      5 8 |
│ │ │ │ +0002e540: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e590: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002e5a0: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ -0002e5b0: 2033 3978 2078 2020 2d20 3339 7820 7820   39x x  - 39x x 
│ │ │ │ -0002e5c0: 202d 2033 3878 2078 2020 2d20 3339 7820   - 38x x  - 39x 
│ │ │ │ -0002e5d0: 7820 202b 2031 3978 2078 2020 2b20 3278  x  + 19x x  + 2x
│ │ │ │ -0002e5e0: 2078 2020 2b7c 0a7c 2020 2020 2020 2020   x  +|.|        
│ │ │ │ -0002e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e600: 2020 2020 3220 3320 2020 2020 2030 2034      2 3      0 4
│ │ │ │ -0002e610: 2020 2020 2020 3320 3420 2020 2020 2030        3 4      0
│ │ │ │ -0002e620: 2035 2020 2020 2020 3220 3520 2020 2020   5      2 5     
│ │ │ │ -0002e630: 3320 3520 207c 0a7c 2020 2020 2020 2020  3 5  |.|        
│ │ │ │ +0002e580: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e590: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e5a0: 2020 2020 2020 2020 202d 2033 3978 2078           - 39x x
│ │ │ │ +0002e5b0: 2020 2d20 3339 7820 7820 202d 2033 3878    - 39x x  - 38x
│ │ │ │ +0002e5c0: 2078 2020 2d20 3339 7820 7820 202b 2031   x  - 39x x  + 1
│ │ │ │ +0002e5d0: 3978 2078 2020 2b20 3278 2078 2020 2b7c  9x x  + 2x x  +|
│ │ │ │ +0002e5e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e5f0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0002e600: 3320 2020 2020 2030 2034 2020 2020 2020  3      0 4      
│ │ │ │ +0002e610: 3320 3420 2020 2020 2030 2035 2020 2020  3 4      0 5    
│ │ │ │ +0002e620: 2020 3220 3520 2020 2020 3320 3520 207c    2 5     3 5  |
│ │ │ │ +0002e630: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e680: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e6a0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0002e6b0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -0002e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e6d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002e6e0: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ -0002e6f0: 2033 3978 2020 2d20 3878 2078 2020 2b20   39x  - 8x x  + 
│ │ │ │ -0002e700: 3235 7820 202b 2031 3978 2078 2020 2b20  25x  + 19x x  + 
│ │ │ │ -0002e710: 3436 7820 7820 202d 2032 3278 2078 2020  46x x  - 22x x  
│ │ │ │ -0002e720: 2b20 7820 787c 0a7c 2020 2020 2020 2020  + x x|.|        
│ │ │ │ -0002e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e740: 2020 2020 3120 2020 2020 3220 3320 2020      1     2 3   
│ │ │ │ -0002e750: 2020 2033 2020 2020 2020 3020 3420 2020     3      0 4   
│ │ │ │ -0002e760: 2020 2032 2034 2020 2020 2020 3320 3420     2 4      3 4 
│ │ │ │ -0002e770: 2020 2030 207c 0a7c 2020 2020 2020 2020     0 |.|        
│ │ │ │ +0002e670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e680: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e690: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0002e6a0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +0002e6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e6c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e6d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e6e0: 2020 2020 2020 2020 202d 2033 3978 2020           - 39x  
│ │ │ │ +0002e6f0: 2d20 3878 2078 2020 2b20 3235 7820 202b  - 8x x  + 25x  +
│ │ │ │ +0002e700: 2031 3978 2078 2020 2b20 3436 7820 7820   19x x  + 46x x 
│ │ │ │ +0002e710: 202d 2032 3278 2078 2020 2b20 7820 787c   - 22x x  + x x|
│ │ │ │ +0002e720: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e730: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +0002e740: 2020 2020 3220 3320 2020 2020 2033 2020      2 3      3  
│ │ │ │ +0002e750: 2020 2020 3020 3420 2020 2020 2032 2034      0 4      2 4
│ │ │ │ +0002e760: 2020 2020 2020 3320 3420 2020 2030 207c        3 4    0 |
│ │ │ │ +0002e770: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e7c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e7b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e7c0: 0a7c 2020 2020 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 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0002e810: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002e820: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ -0002e830: 2033 3978 2078 2020 2b20 3139 7820 7820   39x x  + 19x x 
│ │ │ │ -0002e840: 202b 2078 2078 2020 2b20 3878 2078 2020   + x x  + 8x x  
│ │ │ │ -0002e850: 2d20 3235 7820 202d 2032 3378 2078 2020  - 25x  - 23x x  
│ │ │ │ -0002e860: 2d20 3330 787c 0a7c 2020 2020 2020 2020  - 30x|.|        
│ │ │ │ -0002e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e880: 2020 2020 3020 3120 2020 2020 2030 2032      0 1      0 2
│ │ │ │ -0002e890: 2020 2020 3020 3320 2020 2020 3220 3320      0 3     2 3 
│ │ │ │ -0002e8a0: 2020 2020 2033 2020 2020 2020 3220 3420       3      2 4 
│ │ │ │ -0002e8b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e7f0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +0002e800: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e810: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e820: 2020 2020 2020 2020 202d 2033 3978 2078           - 39x x
│ │ │ │ +0002e830: 2020 2b20 3139 7820 7820 202b 2078 2078    + 19x x  + x x
│ │ │ │ +0002e840: 2020 2b20 3878 2078 2020 2d20 3235 7820    + 8x x  - 25x 
│ │ │ │ +0002e850: 202d 2032 3378 2078 2020 2d20 3330 787c   - 23x x  - 30x|
│ │ │ │ +0002e860: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e870: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +0002e880: 3120 2020 2020 2030 2032 2020 2020 3020  1      0 2    0 
│ │ │ │ +0002e890: 3320 2020 2020 3220 3320 2020 2020 2033  3     2 3      3
│ │ │ │ +0002e8a0: 2020 2020 2020 3220 3420 2020 2020 207c        2 4      |
│ │ │ │ +0002e8b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e900: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002e910: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ -0002e920: 2032 3478 2078 2020 2d20 3235 7820 7820   24x x  - 25x x 
│ │ │ │ -0002e930: 202b 2032 3278 2078 2020 2b20 3438 7820   + 22x x  + 48x 
│ │ │ │ -0002e940: 7820 202d 2031 3778 2078 2020 2b20 3234  x  - 17x x  + 24
│ │ │ │ -0002e950: 7820 7820 207c 0a7c 2020 2020 2020 2020  x x  |.|        
│ │ │ │ -0002e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e970: 2020 2020 3220 3420 2020 2020 2033 2034      2 4      3 4
│ │ │ │ -0002e980: 2020 2020 2020 3220 3520 2020 2020 2033        2 5      3
│ │ │ │ -0002e990: 2035 2020 2020 2020 3220 3620 2020 2020   5      2 6     
│ │ │ │ -0002e9a0: 2031 2037 207c 0a7c 2020 2020 2020 2020   1 7 |.|        
│ │ │ │ +0002e8f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e900: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e910: 2020 2020 2020 2020 202d 2032 3478 2078           - 24x x
│ │ │ │ +0002e920: 2020 2d20 3235 7820 7820 202b 2032 3278    - 25x x  + 22x
│ │ │ │ +0002e930: 2078 2020 2b20 3438 7820 7820 202d 2031   x  + 48x x  - 1
│ │ │ │ +0002e940: 3778 2078 2020 2b20 3234 7820 7820 207c  7x x  + 24x x  |
│ │ │ │ +0002e950: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e960: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0002e970: 3420 2020 2020 2033 2034 2020 2020 2020  4      3 4      
│ │ │ │ +0002e980: 3220 3520 2020 2020 2033 2035 2020 2020  2 5      3 5    
│ │ │ │ +0002e990: 2020 3220 3620 2020 2020 2031 2037 207c    2 6      1 7 |
│ │ │ │ +0002e9a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e9f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002ea00: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -0002ea10: 3978 2078 2020 2b20 7820 7820 202b 2031  9x x  + x x  + 1
│ │ │ │ -0002ea20: 3978 2078 2020 2b20 3339 7820 7820 202d  9x x  + 39x x  -
│ │ │ │ -0002ea30: 2031 3978 2078 2020 2b20 3337 7820 7820   19x x  + 37x x 
│ │ │ │ -0002ea40: 202d 2033 367c 0a7c 2020 2020 2020 2020   - 36|.|        
│ │ │ │ -0002ea50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ea60: 2020 3220 3520 2020 2033 2035 2020 2020    2 5    3 5    
│ │ │ │ -0002ea70: 2020 3520 3720 2020 2020 2031 2038 2020    5 7      1 8  
│ │ │ │ -0002ea80: 2020 2020 3420 3820 2020 2020 2036 2038      4 8      6 8
│ │ │ │ -0002ea90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e9e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e9f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ea00: 2020 2020 2020 2020 2031 3978 2078 2020           19x x  
│ │ │ │ +0002ea10: 2b20 7820 7820 202b 2031 3978 2078 2020  + x x  + 19x x  
│ │ │ │ +0002ea20: 2b20 3339 7820 7820 202d 2031 3978 2078  + 39x x  - 19x x
│ │ │ │ +0002ea30: 2020 2b20 3337 7820 7820 202d 2033 367c    + 37x x  - 36|
│ │ │ │ +0002ea40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ea50: 2020 2020 2020 2020 2020 2020 3220 3520              2 5 
│ │ │ │ +0002ea60: 2020 2033 2035 2020 2020 2020 3520 3720     3 5      5 7 
│ │ │ │ +0002ea70: 2020 2020 2031 2038 2020 2020 2020 3420       1 8      4 
│ │ │ │ +0002ea80: 3820 2020 2020 2036 2038 2020 2020 207c  8      6 8     |
│ │ │ │ +0002ea90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002eaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eae0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002eaf0: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ -0002eb00: 3978 2078 2020 2d20 3139 7820 7820 202d  9x x  - 19x x  -
│ │ │ │ -0002eb10: 2032 7820 7820 202b 2033 3978 2078 2020   2x x  + 39x x  
│ │ │ │ -0002eb20: 2b20 3139 7820 7820 202d 2078 2078 2020  + 19x x  - x x  
│ │ │ │ -0002eb30: 2d20 3337 787c 0a7c 2020 2020 2020 2020  - 37x|.|        
│ │ │ │ -0002eb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eb50: 2020 3020 3520 2020 2020 2032 2035 2020    0 5      2 5  
│ │ │ │ -0002eb60: 2020 2033 2035 2020 2020 2020 3020 3820     3 5      0 8 
│ │ │ │ -0002eb70: 2020 2020 2031 2038 2020 2020 3320 3820       1 8    3 8 
│ │ │ │ -0002eb80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002ead0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002eae0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002eaf0: 2020 2020 2020 2020 2033 3978 2078 2020           39x x  
│ │ │ │ +0002eb00: 2d20 3139 7820 7820 202d 2032 7820 7820  - 19x x  - 2x x 
│ │ │ │ +0002eb10: 202b 2033 3978 2078 2020 2b20 3139 7820   + 39x x  + 19x 
│ │ │ │ +0002eb20: 7820 202d 2078 2078 2020 2d20 3337 787c  x  - x x  - 37x|
│ │ │ │ +0002eb30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002eb40: 2020 2020 2020 2020 2020 2020 3020 3520              0 5 
│ │ │ │ +0002eb50: 2020 2020 2032 2035 2020 2020 2033 2035       2 5     3 5
│ │ │ │ +0002eb60: 2020 2020 2020 3020 3820 2020 2020 2031        0 8      1
│ │ │ │ +0002eb70: 2038 2020 2020 3320 3820 2020 2020 207c   8    3 8      |
│ │ │ │ +0002eb80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ebd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002ebe0: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │ -0002ebf0: 3378 2078 2020 2d20 3330 7820 7820 202d  3x x  - 30x x  -
│ │ │ │ -0002ec00: 2031 3978 2078 2020 2d20 3430 7820 7820   19x x  - 40x x 
│ │ │ │ -0002ec10: 202b 2034 3478 2078 2020 2d20 3232 7820   + 44x x  - 22x 
│ │ │ │ -0002ec20: 7820 202d 207c 0a7c 2020 2020 2020 2020  x  - |.|        
│ │ │ │ -0002ec30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ec40: 2020 3220 3420 2020 2020 2033 2034 2020    2 4      3 4  
│ │ │ │ -0002ec50: 2020 2020 3020 3520 2020 2020 2032 2035      0 5      2 5
│ │ │ │ -0002ec60: 2020 2020 2020 3320 3520 2020 2020 2032        3 5      2
│ │ │ │ -0002ec70: 2036 2020 207c 0a7c 2020 2020 2020 2020   6   |.|        
│ │ │ │ -0002ec80: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
│ │ │ │ +0002ebc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002ebd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ebe0: 2020 2020 2020 2020 2034 3378 2078 2020           43x x  
│ │ │ │ +0002ebf0: 2d20 3330 7820 7820 202d 2031 3978 2078  - 30x x  - 19x x
│ │ │ │ +0002ec00: 2020 2d20 3430 7820 7820 202b 2034 3478    - 40x x  + 44x
│ │ │ │ +0002ec10: 2078 2020 2d20 3232 7820 7820 202d 207c   x  - 22x x  - |
│ │ │ │ +0002ec20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ec30: 2020 2020 2020 2020 2020 2020 3220 3420              2 4 
│ │ │ │ +0002ec40: 2020 2020 2033 2034 2020 2020 2020 3020       3 4      0 
│ │ │ │ +0002ec50: 3520 2020 2020 2032 2035 2020 2020 2020  5      2 5      
│ │ │ │ +0002ec60: 3320 3520 2020 2020 2032 2036 2020 207c  3 5      2 6   |
│ │ │ │ +0002ec70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ec80: 2020 2020 2020 2020 7d20 2020 2020 2020          }       
│ │ │ │  0002ec90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ecb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ecc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002ecb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002ecc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002ecd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ece0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ed00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ed10: 2020 2020 207c 0a7c 6f31 3020 3a20 5261       |.|o10 : Ra
│ │ │ │ -0002ed20: 7469 6f6e 616c 4d61 7020 2871 7561 6472  tionalMap (quadr
│ │ │ │ -0002ed30: 6174 6963 2062 6972 6174 696f 6e61 6c20  atic birational 
│ │ │ │ -0002ed40: 6d61 7020 6672 6f6d 2038 2d64 696d 656e  map from 8-dimen
│ │ │ │ -0002ed50: 7369 6f6e 616c 2073 7562 7661 7269 6574  sional subvariet
│ │ │ │ -0002ed60: 7920 6f66 207c 0a7c 2d2d 2d2d 2d2d 2d2d  y of |.|--------
│ │ │ │ +0002ed00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002ed10: 0a7c 6f31 3020 3a20 5261 7469 6f6e 616c  .|o10 : Rational
│ │ │ │ +0002ed20: 4d61 7020 2871 7561 6472 6174 6963 2062  Map (quadratic b
│ │ │ │ +0002ed30: 6972 6174 696f 6e61 6c20 6d61 7020 6672  irational map fr
│ │ │ │ +0002ed40: 6f6d 2038 2d64 696d 656e 7369 6f6e 616c  om 8-dimensional
│ │ │ │ +0002ed50: 2073 7562 7661 7269 6574 7920 6f66 207c   subvariety of |
│ │ │ │ +0002ed60: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0002ed70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ed80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ed90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002edb0: 2d2d 2d2d 2d7c 0a7c 2020 2c20 7820 205d  -----|.|  , x  ]
│ │ │ │ -0002edc0: 2920 6465 6669 6e65 6420 6279 2020 2020  ) defined by    
│ │ │ │ +0002eda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
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│ │ │ │ +0002edc0: 6e65 6420 6279 2020 2020 2020 2020 2020  ned by          
│ │ │ │  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 207c 0a7c 3130 2020 2031 3120       |.|10   11 
│ │ │ │ +0002edf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002ee00: 0a7c 3130 2020 2031 3120 2020 2020 2020  .|10   11       
│ │ │ │  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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ee50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002ee40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002ee50: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eea0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002ee90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002eea0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eef0: 2020 2020 207c 0a7c 202d 2078 2078 2020       |.| - x x  
│ │ │ │ -0002ef00: 2b20 3132 7820 7820 202b 2032 3578 2078  + 12x x  + 25x x
│ │ │ │ -0002ef10: 2020 2b20 3235 7820 7820 202b 2032 3378    + 25x x  + 23x
│ │ │ │ -0002ef20: 2078 2020 2d20 3378 2078 2020 2b20 3278   x  - 3x x  + 2x
│ │ │ │ -0002ef30: 2078 2020 2b20 3131 7820 7820 202d 2033   x  + 11x x  - 3
│ │ │ │ -0002ef40: 3778 2020 207c 0a7c 2020 2020 3020 3620  7x   |.|    0 6 
│ │ │ │ -0002ef50: 2020 2020 2032 2036 2020 2020 2020 3120       2 6      1 
│ │ │ │ -0002ef60: 3720 2020 2020 2033 2037 2020 2020 2020  7      3 7      
│ │ │ │ -0002ef70: 3520 3720 2020 2020 3620 3720 2020 2020  5 7     6 7     
│ │ │ │ -0002ef80: 3020 3820 2020 2020 2031 2038 2020 2020  0 8      1 8    
│ │ │ │ -0002ef90: 2020 3320 207c 0a7c 2020 2020 2020 2020    3  |.|        
│ │ │ │ +0002eee0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002eef0: 0a7c 202d 2078 2078 2020 2b20 3132 7820  .| - x x  + 12x 
│ │ │ │ +0002ef00: 7820 202b 2032 3578 2078 2020 2b20 3235  x  + 25x x  + 25
│ │ │ │ +0002ef10: 7820 7820 202b 2032 3378 2078 2020 2d20  x x  + 23x x  - 
│ │ │ │ +0002ef20: 3378 2078 2020 2b20 3278 2078 2020 2b20  3x x  + 2x x  + 
│ │ │ │ +0002ef30: 3131 7820 7820 202d 2033 3778 2020 207c  11x x  - 37x   |
│ │ │ │ +0002ef40: 0a7c 2020 2020 3020 3620 2020 2020 2032  .|    0 6      2
│ │ │ │ +0002ef50: 2036 2020 2020 2020 3120 3720 2020 2020   6      1 7     
│ │ │ │ +0002ef60: 2033 2037 2020 2020 2020 3520 3720 2020   3 7      5 7   
│ │ │ │ +0002ef70: 2020 3620 3720 2020 2020 3020 3820 2020    6 7     0 8   
│ │ │ │ +0002ef80: 2020 2031 2038 2020 2020 2020 3320 207c     1 8      3  |
│ │ │ │ +0002ef90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002efd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002efe0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f030: 2020 2020 207c 0a7c 2020 2b20 3331 7820       |.|  + 31x 
│ │ │ │ -0002f040: 7820 202d 2033 3678 2078 2020 2d20 7820  x  - 36x x  - x 
│ │ │ │ -0002f050: 7820 202b 2031 3378 2078 2020 2b20 3878  x  + 13x x  + 8x
│ │ │ │ -0002f060: 2078 2020 2b20 3978 2078 2020 2b20 3436   x  + 9x x  + 46
│ │ │ │ -0002f070: 7820 7820 202b 2034 3178 2078 2020 2d20  x x  + 41x x  - 
│ │ │ │ -0002f080: 3778 2020 207c 0a7c 3520 2020 2020 2032  7x   |.|5      2
│ │ │ │ -0002f090: 2036 2020 2020 2020 3320 3620 2020 2030   6      3 6    0
│ │ │ │ -0002f0a0: 2037 2020 2020 2020 3120 3720 2020 2020   7      1 7     
│ │ │ │ -0002f0b0: 3320 3720 2020 2020 3520 3720 2020 2020  3 7     5 7     
│ │ │ │ -0002f0c0: 2036 2037 2020 2020 2020 3020 3820 2020   6 7      0 8   
│ │ │ │ -0002f0d0: 2020 3120 207c 0a7c 2020 2020 2020 2020    1  |.|        
│ │ │ │ +0002f020: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f030: 0a7c 2020 2b20 3331 7820 7820 202d 2033  .|  + 31x x  - 3
│ │ │ │ +0002f040: 3678 2078 2020 2d20 7820 7820 202b 2031  6x x  - x x  + 1
│ │ │ │ +0002f050: 3378 2078 2020 2b20 3878 2078 2020 2b20  3x x  + 8x x  + 
│ │ │ │ +0002f060: 3978 2078 2020 2b20 3436 7820 7820 202b  9x x  + 46x x  +
│ │ │ │ +0002f070: 2034 3178 2078 2020 2d20 3778 2020 207c   41x x  - 7x   |
│ │ │ │ +0002f080: 0a7c 3520 2020 2020 2032 2036 2020 2020  .|5      2 6    
│ │ │ │ +0002f090: 2020 3320 3620 2020 2030 2037 2020 2020    3 6    0 7    
│ │ │ │ +0002f0a0: 2020 3120 3720 2020 2020 3320 3720 2020    1 7     3 7   
│ │ │ │ +0002f0b0: 2020 3520 3720 2020 2020 2036 2037 2020    5 7      6 7  
│ │ │ │ +0002f0c0: 2020 2020 3020 3820 2020 2020 3120 207c      0 8     1  |
│ │ │ │ +0002f0d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002f0e0: 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002f110: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f120: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2b20 3337 7820 7820       |.|+ 37x x 
│ │ │ │ -0002f180: 202b 2034 3878 2078 2020 2d20 3578 2078   + 48x x  - 5x x
│ │ │ │ -0002f190: 2020 2d20 3578 2078 2020 2b20 7820 7820    - 5x x  + x x 
│ │ │ │ -0002f1a0: 202b 2032 3078 2078 2020 2b20 3130 7820   + 20x x  + 10x 
│ │ │ │ -0002f1b0: 7820 202b 2033 3478 2078 2020 2b20 3431  x  + 34x x  + 41
│ │ │ │ -0002f1c0: 7820 7820 207c 0a7c 2020 2020 2033 2035  x x  |.|     3 5
│ │ │ │ -0002f1d0: 2020 2020 2020 3220 3620 2020 2020 3120        2 6     1 
│ │ │ │ -0002f1e0: 3720 2020 2020 3320 3720 2020 2035 2037  7     3 7    5 7
│ │ │ │ -0002f1f0: 2020 2020 2020 3620 3720 2020 2020 2030        6 7      0
│ │ │ │ -0002f200: 2038 2020 2020 2020 3120 3820 2020 2020   8      1 8     
│ │ │ │ -0002f210: 2033 2020 207c 0a7c 2020 2020 2020 2020   3   |.|        
│ │ │ │ +0002f160: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f170: 0a7c 2b20 3337 7820 7820 202b 2034 3878  .|+ 37x x  + 48x
│ │ │ │ +0002f180: 2078 2020 2d20 3578 2078 2020 2d20 3578   x  - 5x x  - 5x
│ │ │ │ +0002f190: 2078 2020 2b20 7820 7820 202b 2032 3078   x  + x x  + 20x
│ │ │ │ +0002f1a0: 2078 2020 2b20 3130 7820 7820 202b 2033   x  + 10x x  + 3
│ │ │ │ +0002f1b0: 3478 2078 2020 2b20 3431 7820 7820 207c  4x x  + 41x x  |
│ │ │ │ +0002f1c0: 0a7c 2020 2020 2033 2035 2020 2020 2020  .|     3 5      
│ │ │ │ +0002f1d0: 3220 3620 2020 2020 3120 3720 2020 2020  2 6     1 7     
│ │ │ │ +0002f1e0: 3320 3720 2020 2035 2037 2020 2020 2020  3 7    5 7      
│ │ │ │ +0002f1f0: 3620 3720 2020 2020 2030 2038 2020 2020  6 7      0 8    
│ │ │ │ +0002f200: 2020 3120 3820 2020 2020 2033 2020 207c    1 8      3   |
│ │ │ │ +0002f210: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002f220: 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002f250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f260: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f2b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002f2a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f2b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f300: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002f2f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f300: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002f310: 2020 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002f340: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f350: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002f390: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f3a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f3f0: 2020 2020 207c 0a7c 2078 2020 2d20 3339       |.| x  - 39
│ │ │ │ -0002f400: 7820 7820 202d 2031 3778 2078 2020 2d20  x x  - 17x x  - 
│ │ │ │ -0002f410: 3338 7820 7820 202b 2032 3478 2078 2020  38x x  + 24x x  
│ │ │ │ -0002f420: 2b20 3234 7820 7820 202b 2035 7820 7820  + 24x x  + 5x x 
│ │ │ │ -0002f430: 202b 2078 2078 2020 2d20 3139 7820 7820   + x x  - 19x x 
│ │ │ │ -0002f440: 202d 2020 207c 0a7c 3320 3520 2020 2020   -   |.|3 5     
│ │ │ │ -0002f450: 2030 2036 2020 2020 2020 3220 3620 2020   0 6      2 6   
│ │ │ │ -0002f460: 2020 2033 2036 2020 2020 2020 3120 3720     3 6      1 7 
│ │ │ │ -0002f470: 2020 2020 2033 2037 2020 2020 2035 2037       3 7     5 7
│ │ │ │ -0002f480: 2020 2020 3620 3720 2020 2020 2030 2038      6 7      0 8
│ │ │ │ -0002f490: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002f3e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f3f0: 0a7c 2078 2020 2d20 3339 7820 7820 202d  .| x  - 39x x  -
│ │ │ │ +0002f400: 2031 3778 2078 2020 2d20 3338 7820 7820   17x x  - 38x x 
│ │ │ │ +0002f410: 202b 2032 3478 2078 2020 2b20 3234 7820   + 24x x  + 24x 
│ │ │ │ +0002f420: 7820 202b 2035 7820 7820 202b 2078 2078  x  + 5x x  + x x
│ │ │ │ +0002f430: 2020 2d20 3139 7820 7820 202d 2020 207c    - 19x x  -   |
│ │ │ │ +0002f440: 0a7c 3320 3520 2020 2020 2030 2036 2020  .|3 5      0 6  
│ │ │ │ +0002f450: 2020 2020 3220 3620 2020 2020 2033 2036      2 6      3 6
│ │ │ │ +0002f460: 2020 2020 2020 3120 3720 2020 2020 2033        1 7      3
│ │ │ │ +0002f470: 2037 2020 2020 2035 2037 2020 2020 3620   7     5 7    6 
│ │ │ │ +0002f480: 3720 2020 2020 2030 2038 2020 2020 207c  7      0 8     |
│ │ │ │ +0002f490: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f4e0: 2020 2020 207c 0a7c 2d20 3337 7820 7820       |.|- 37x x 
│ │ │ │ -0002f4f0: 202d 2032 3178 2078 2020 2b20 7820 7820   - 21x x  + x x 
│ │ │ │ -0002f500: 2020 2b20 3278 2078 2020 202d 2035 7820    + 2x x   - 5x 
│ │ │ │ -0002f510: 7820 2020 2b20 7820 7820 2020 2d20 3578  x   + x x   - 5x
│ │ │ │ -0002f520: 2078 2020 202b 2035 7820 7820 202c 2020   x   + 5x x  ,  
│ │ │ │ -0002f530: 2020 2020 207c 0a7c 2020 2020 2036 2038       |.|     6 8
│ │ │ │ -0002f540: 2020 2020 2020 3520 3920 2020 2031 2031        5 9    1 1
│ │ │ │ -0002f550: 3020 2020 2020 3220 3130 2020 2020 2033  0     2 10     3
│ │ │ │ -0002f560: 2031 3020 2020 2035 2031 3020 2020 2020   10    5 10     
│ │ │ │ -0002f570: 3620 3130 2020 2020 2035 2031 3120 2020  6 10     5 11   
│ │ │ │ -0002f580: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002f4d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f4e0: 0a7c 2d20 3337 7820 7820 202d 2032 3178  .|- 37x x  - 21x
│ │ │ │ +0002f4f0: 2078 2020 2b20 7820 7820 2020 2b20 3278   x  + x x   + 2x
│ │ │ │ +0002f500: 2078 2020 202d 2035 7820 7820 2020 2b20   x   - 5x x   + 
│ │ │ │ +0002f510: 7820 7820 2020 2d20 3578 2078 2020 202b  x x   - 5x x   +
│ │ │ │ +0002f520: 2035 7820 7820 202c 2020 2020 2020 207c   5x x  ,       |
│ │ │ │ +0002f530: 0a7c 2020 2020 2036 2038 2020 2020 2020  .|     6 8      
│ │ │ │ +0002f540: 3520 3920 2020 2031 2031 3020 2020 2020  5 9    1 10     
│ │ │ │ +0002f550: 3220 3130 2020 2020 2033 2031 3020 2020  2 10     3 10   
│ │ │ │ +0002f560: 2035 2031 3020 2020 2020 3620 3130 2020   5 10     6 10  
│ │ │ │ +0002f570: 2020 2035 2031 3120 2020 2020 2020 207c     5 11        |
│ │ │ │ +0002f580: 0a7c 2020 2020 2020 2020 2020 2020 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f5d0: 2020 2020 207c 0a7c 2031 3978 2078 2020       |.| 19x x  
│ │ │ │ -0002f5e0: 2b20 3139 7820 7820 202b 2032 3178 2078  + 19x x  + 21x x
│ │ │ │ -0002f5f0: 2020 2b20 3139 7820 7820 202d 2031 3978    + 19x x  - 19x
│ │ │ │ -0002f600: 2078 2020 2b20 7820 7820 202d 2033 3978   x  + x x  - 39x
│ │ │ │ -0002f610: 2078 2020 2b20 3337 7820 7820 202b 2034   x  + 37x x  + 4
│ │ │ │ -0002f620: 3078 2020 207c 0a7c 2020 2020 3020 3720  0x   |.|    0 7 
│ │ │ │ -0002f630: 2020 2020 2032 2037 2020 2020 2020 3320       2 7      3 
│ │ │ │ -0002f640: 3720 2020 2020 2030 2038 2020 2020 2020  7      0 8      
│ │ │ │ -0002f650: 3120 3820 2020 2033 2038 2020 2020 2020  1 8    3 8      
│ │ │ │ -0002f660: 3420 3820 2020 2020 2036 2038 2020 2020  4 8      6 8    
│ │ │ │ -0002f670: 2020 3020 207c 0a7c 2020 2020 2020 2020    0  |.|        
│ │ │ │ +0002f5c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f5d0: 0a7c 2031 3978 2078 2020 2b20 3139 7820  .| 19x x  + 19x 
│ │ │ │ +0002f5e0: 7820 202b 2032 3178 2078 2020 2b20 3139  x  + 21x x  + 19
│ │ │ │ +0002f5f0: 7820 7820 202d 2031 3978 2078 2020 2b20  x x  - 19x x  + 
│ │ │ │ +0002f600: 7820 7820 202d 2033 3978 2078 2020 2b20  x x  - 39x x  + 
│ │ │ │ +0002f610: 3337 7820 7820 202b 2034 3078 2020 207c  37x x  + 40x   |
│ │ │ │ +0002f620: 0a7c 2020 2020 3020 3720 2020 2020 2032  .|    0 7      2
│ │ │ │ +0002f630: 2037 2020 2020 2020 3320 3720 2020 2020   7      3 7     
│ │ │ │ +0002f640: 2030 2038 2020 2020 2020 3120 3820 2020   0 8      1 8   
│ │ │ │ +0002f650: 2033 2038 2020 2020 2020 3420 3820 2020   3 8      4 8   
│ │ │ │ +0002f660: 2020 2036 2038 2020 2020 2020 3020 207c     6 8      0  |
│ │ │ │ +0002f670: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002f680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f6c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002f6b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f6c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f710: 2020 2020 207c 0a7c 2020 2d20 3878 2078       |.|  - 8x x
│ │ │ │ -0002f720: 2020 2b20 3332 7820 7820 202d 2033 3978    + 32x x  - 39x
│ │ │ │ -0002f730: 2078 2020 2b20 7820 7820 202d 2033 3878   x  + x x  - 38x
│ │ │ │ -0002f740: 2078 2020 2d20 3330 7820 7820 202d 2035   x  - 30x x  - 5
│ │ │ │ -0002f750: 7820 7820 202b 2033 3978 2078 2020 2b20  x x  + 39x x  + 
│ │ │ │ -0002f760: 7820 7820 207c 0a7c 3520 2020 2020 3220  x x  |.|5     2 
│ │ │ │ -0002f770: 3520 2020 2020 2033 2035 2020 2020 2020  5      3 5      
│ │ │ │ -0002f780: 3420 3520 2020 2030 2036 2020 2020 2020  4 5    0 6      
│ │ │ │ -0002f790: 3120 3620 2020 2020 2032 2036 2020 2020  1 6      2 6    
│ │ │ │ -0002f7a0: 2033 2036 2020 2020 2020 3420 3620 2020   3 6      4 6   
│ │ │ │ -0002f7b0: 2035 2020 207c 0a7c 2020 2020 2020 2020   5   |.|        
│ │ │ │ +0002f700: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f710: 0a7c 2020 2d20 3878 2078 2020 2b20 3332  .|  - 8x x  + 32
│ │ │ │ +0002f720: 7820 7820 202d 2033 3978 2078 2020 2b20  x x  - 39x x  + 
│ │ │ │ +0002f730: 7820 7820 202d 2033 3878 2078 2020 2d20  x x  - 38x x  - 
│ │ │ │ +0002f740: 3330 7820 7820 202d 2035 7820 7820 202b  30x x  - 5x x  +
│ │ │ │ +0002f750: 2033 3978 2078 2020 2b20 7820 7820 207c   39x x  + x x  |
│ │ │ │ +0002f760: 0a7c 3520 2020 2020 3220 3520 2020 2020  .|5     2 5     
│ │ │ │ +0002f770: 2033 2035 2020 2020 2020 3420 3520 2020   3 5      4 5   
│ │ │ │ +0002f780: 2030 2036 2020 2020 2020 3120 3620 2020   0 6      1 6   
│ │ │ │ +0002f790: 2020 2032 2036 2020 2020 2033 2036 2020     2 6     3 6  
│ │ │ │ +0002f7a0: 2020 2020 3420 3620 2020 2035 2020 207c      4 6    5   |
│ │ │ │ +0002f7b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f800: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002f7f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f800: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f850: 2020 2020 207c 0a7c 2078 2020 2d20 3231       |.| x  - 21
│ │ │ │ -0002f860: 7820 7820 202d 2031 3078 2078 2020 2d20  x x  - 10x x  - 
│ │ │ │ -0002f870: 3334 7820 7820 202d 2033 3878 2078 2020  34x x  - 38x x  
│ │ │ │ -0002f880: 2d20 3778 2078 2020 2b20 3578 2078 2020  - 7x x  + 5x x  
│ │ │ │ -0002f890: 2b20 3139 7820 7820 202b 2032 3378 2078  + 19x x  + 23x x
│ │ │ │ -0002f8a0: 2020 2b20 207c 0a7c 3320 3420 2020 2020    +  |.|3 4     
│ │ │ │ -0002f8b0: 2030 2035 2020 2020 2020 3220 3520 2020   0 5      2 5   
│ │ │ │ -0002f8c0: 2020 2033 2035 2020 2020 2020 3020 3620     3 5      0 6 
│ │ │ │ -0002f8d0: 2020 2020 3220 3620 2020 2020 3320 3620      2 6     3 6 
│ │ │ │ -0002f8e0: 2020 2020 2030 2037 2020 2020 2020 3120       0 7      1 
│ │ │ │ -0002f8f0: 3720 2020 207c 0a7c 2020 2020 2020 2020  7    |.|        
│ │ │ │ +0002f840: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f850: 0a7c 2078 2020 2d20 3231 7820 7820 202d  .| x  - 21x x  -
│ │ │ │ +0002f860: 2031 3078 2078 2020 2d20 3334 7820 7820   10x x  - 34x x 
│ │ │ │ +0002f870: 202d 2033 3878 2078 2020 2d20 3778 2078   - 38x x  - 7x x
│ │ │ │ +0002f880: 2020 2b20 3578 2078 2020 2b20 3139 7820    + 5x x  + 19x 
│ │ │ │ +0002f890: 7820 202b 2032 3378 2078 2020 2b20 207c  x  + 23x x  +  |
│ │ │ │ +0002f8a0: 0a7c 3320 3420 2020 2020 2030 2035 2020  .|3 4      0 5  
│ │ │ │ +0002f8b0: 2020 2020 3220 3520 2020 2020 2033 2035      2 5      3 5
│ │ │ │ +0002f8c0: 2020 2020 2020 3020 3620 2020 2020 3220        0 6     2 
│ │ │ │ +0002f8d0: 3620 2020 2020 3320 3620 2020 2020 2030  6     3 6      0
│ │ │ │ +0002f8e0: 2037 2020 2020 2020 3120 3720 2020 207c   7      1 7    |
│ │ │ │ +0002f8f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002f900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f940: 2020 2020 207c 0a7c 2b20 3234 7820 7820       |.|+ 24x x 
│ │ │ │ -0002f950: 202b 2035 7820 7820 202b 2078 2078 2020   + 5x x  + x x  
│ │ │ │ -0002f960: 2b20 3431 7820 7820 202b 2031 3278 2078  + 41x x  + 12x x
│ │ │ │ -0002f970: 2020 2d20 3578 2078 2020 2d20 3236 7820    - 5x x  - 26x 
│ │ │ │ -0002f980: 7820 202b 2034 3278 2078 2020 2d20 3234  x  + 42x x  - 24
│ │ │ │ -0002f990: 7820 7820 207c 0a7c 2020 2020 2033 2037  x x  |.|     3 7
│ │ │ │ -0002f9a0: 2020 2020 2035 2037 2020 2020 3620 3720       5 7    6 7 
│ │ │ │ -0002f9b0: 2020 2020 2031 2038 2020 2020 2020 3320       1 8      3 
│ │ │ │ -0002f9c0: 3820 2020 2020 3420 3820 2020 2020 2036  8     4 8      6
│ │ │ │ -0002f9d0: 2038 2020 2020 2020 3120 3920 2020 2020   8      1 9     
│ │ │ │ -0002f9e0: 2032 2020 207c 0a7c 2020 2020 2020 2020   2   |.|        
│ │ │ │ +0002f930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002f940: 0a7c 2b20 3234 7820 7820 202b 2035 7820  .|+ 24x x  + 5x 
│ │ │ │ +0002f950: 7820 202b 2078 2078 2020 2b20 3431 7820  x  + x x  + 41x 
│ │ │ │ +0002f960: 7820 202b 2031 3278 2078 2020 2d20 3578  x  + 12x x  - 5x
│ │ │ │ +0002f970: 2078 2020 2d20 3236 7820 7820 202b 2034   x  - 26x x  + 4
│ │ │ │ +0002f980: 3278 2078 2020 2d20 3234 7820 7820 207c  2x x  - 24x x  |
│ │ │ │ +0002f990: 0a7c 2020 2020 2033 2037 2020 2020 2035  .|     3 7     5
│ │ │ │ +0002f9a0: 2037 2020 2020 3620 3720 2020 2020 2031   7    6 7      1
│ │ │ │ +0002f9b0: 2038 2020 2020 2020 3320 3820 2020 2020   8      3 8     
│ │ │ │ +0002f9c0: 3420 3820 2020 2020 2036 2038 2020 2020  4 8      6 8    
│ │ │ │ +0002f9d0: 2020 3120 3920 2020 2020 2032 2020 207c    1 9      2   |
│ │ │ │ +0002f9e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002f9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fa30: 2020 2020 207c 0a7c 7820 7820 202d 2078       |.|x x  - x
│ │ │ │ -0002fa40: 2078 2020 202b 2035 7820 7820 2020 2d20   x   + 5x x   - 
│ │ │ │ -0002fa50: 3578 2078 2020 2c20 2020 2020 2020 2020  5x x  ,         
│ │ │ │ +0002fa20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002fa30: 0a7c 7820 7820 202d 2078 2078 2020 202b  .|x x  - x x   +
│ │ │ │ +0002fa40: 2035 7820 7820 2020 2d20 3578 2078 2020   5x x   - 5x x  
│ │ │ │ +0002fa50: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  0002fa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fa80: 2020 2020 207c 0a7c 2035 2039 2020 2020       |.| 5 9    
│ │ │ │ -0002fa90: 3420 3130 2020 2020 2036 2031 3020 2020  4 10     6 10   
│ │ │ │ -0002faa0: 2020 3520 3131 2020 2020 2020 2020 2020    5 11          
│ │ │ │ +0002fa70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002fa80: 0a7c 2035 2039 2020 2020 3420 3130 2020  .| 5 9    4 10  
│ │ │ │ +0002fa90: 2020 2036 2031 3020 2020 2020 3520 3131     6 10     5 11
│ │ │ │ +0002faa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fad0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002fac0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002fad0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002fae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002faf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fb20: 2020 2020 207c 0a7c 2078 2020 2b20 3337       |.| x  + 37
│ │ │ │ -0002fb30: 7820 7820 202b 2078 2078 2020 202b 2078  x x  + x x   + x
│ │ │ │ -0002fb40: 2078 2020 202d 2035 7820 7820 2020 2b20   x   - 5x x   + 
│ │ │ │ -0002fb50: 3578 2078 2020 2c20 2020 2020 2020 2020  5x x  ,         
│ │ │ │ -0002fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fb70: 2020 2020 207c 0a7c 3620 3820 2020 2020       |.|6 8     
│ │ │ │ -0002fb80: 2035 2039 2020 2020 3120 3130 2020 2020   5 9    1 10    
│ │ │ │ -0002fb90: 3220 3130 2020 2020 2036 2031 3020 2020  2 10     6 10   
│ │ │ │ -0002fba0: 2020 3520 3131 2020 2020 2020 2020 2020    5 11          
│ │ │ │ -0002fbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fbc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002fb10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002fb20: 0a7c 2078 2020 2b20 3337 7820 7820 202b  .| x  + 37x x  +
│ │ │ │ +0002fb30: 2078 2078 2020 202b 2078 2078 2020 202d   x x   + x x   -
│ │ │ │ +0002fb40: 2035 7820 7820 2020 2b20 3578 2078 2020   5x x   + 5x x  
│ │ │ │ +0002fb50: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +0002fb60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002fb70: 0a7c 3620 3820 2020 2020 2035 2039 2020  .|6 8      5 9  
│ │ │ │ +0002fb80: 2020 3120 3130 2020 2020 3220 3130 2020    1 10    2 10  
│ │ │ │ +0002fb90: 2020 2036 2031 3020 2020 2020 3520 3131     6 10     5 11
│ │ │ │ +0002fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002fbb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002fbc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002fbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fc10: 2020 2020 207c 0a7c 3578 2078 2020 2d20       |.|5x x  - 
│ │ │ │ -0002fc20: 3433 7820 7820 202d 2032 3278 2078 2020  43x x  - 22x x  
│ │ │ │ -0002fc30: 2d20 3433 7820 7820 202d 2033 7820 7820  - 43x x  - 3x x 
│ │ │ │ -0002fc40: 202d 2031 3978 2078 2020 2d20 3432 7820   - 19x x  - 42x 
│ │ │ │ -0002fc50: 7820 202d 2031 3378 2078 2020 2b20 3433  x  - 13x x  + 43
│ │ │ │ -0002fc60: 7820 7820 207c 0a7c 2020 3320 3620 2020  x x  |.|  3 6   
│ │ │ │ -0002fc70: 2020 2031 2037 2020 2020 2020 3320 3720     1 7      3 7 
│ │ │ │ -0002fc80: 2020 2020 2035 2037 2020 2020 2036 2037       5 7     6 7
│ │ │ │ -0002fc90: 2020 2020 2020 3020 3820 2020 2020 2031        0 8      1
│ │ │ │ -0002fca0: 2038 2020 2020 2020 3320 3820 2020 2020   8      3 8     
│ │ │ │ -0002fcb0: 2034 2020 207c 0a7c 2020 2020 2020 2020   4   |.|        
│ │ │ │ +0002fc00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002fc10: 0a7c 3578 2078 2020 2d20 3433 7820 7820  .|5x x  - 43x x 
│ │ │ │ +0002fc20: 202d 2032 3278 2078 2020 2d20 3433 7820   - 22x x  - 43x 
│ │ │ │ +0002fc30: 7820 202d 2033 7820 7820 202d 2031 3978  x  - 3x x  - 19x
│ │ │ │ +0002fc40: 2078 2020 2d20 3432 7820 7820 202d 2031   x  - 42x x  - 1
│ │ │ │ +0002fc50: 3378 2078 2020 2b20 3433 7820 7820 207c  3x x  + 43x x  |
│ │ │ │ +0002fc60: 0a7c 2020 3320 3620 2020 2020 2031 2037  .|  3 6      1 7
│ │ │ │ +0002fc70: 2020 2020 2020 3320 3720 2020 2020 2035        3 7      5
│ │ │ │ +0002fc80: 2037 2020 2020 2036 2037 2020 2020 2020   7     6 7      
│ │ │ │ +0002fc90: 3020 3820 2020 2020 2031 2038 2020 2020  0 8      1 8    
│ │ │ │ +0002fca0: 2020 3320 3820 2020 2020 2034 2020 207c    3 8      4   |
│ │ │ │ +0002fcb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002fcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fcd0: 2020 2020 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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002fcf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002fd00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002fd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fd50: 2020 2020 207c 0a7c 5050 5e31 3120 746f       |.|PP^11 to
│ │ │ │ -0002fd60: 2050 505e 3829 2020 2020 2020 2020 2020   PP^8)          
│ │ │ │ +0002fd40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002fd50: 0a7c 5050 5e31 3120 746f 2050 505e 3829  .|PP^11 to PP^8)
│ │ │ │ +0002fd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fda0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +0002fd90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002fda0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0002fdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002fdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002fdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002fde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002fdf0: 2d2d 2d2d 2d7c 0a7c 7820 202d 2032 3378  -----|.|x  - 23x
│ │ │ │ -0002fe00: 2078 2020 2d20 3333 7820 7820 202b 2038   x  - 33x x  + 8
│ │ │ │ -0002fe10: 7820 7820 202b 2031 3078 2078 2020 2d20  x x  + 10x x  - 
│ │ │ │ -0002fe20: 3235 7820 7820 202d 2039 7820 7820 202b  25x x  - 9x x  +
│ │ │ │ -0002fe30: 2033 7820 7820 202b 2032 3478 2078 2020   3x x  + 24x x  
│ │ │ │ -0002fe40: 2020 2020 207c 0a7c 2038 2020 2020 2020       |.| 8      
│ │ │ │ -0002fe50: 3420 3820 2020 2020 2036 2038 2020 2020  4 8      6 8    
│ │ │ │ -0002fe60: 2030 2039 2020 2020 2020 3120 3920 2020   0 9      1 9   
│ │ │ │ -0002fe70: 2020 2032 2039 2020 2020 2033 2039 2020     2 9     3 9  
│ │ │ │ -0002fe80: 2020 2034 2039 2020 2020 2020 3520 2020     4 9      5   
│ │ │ │ -0002fe90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002fde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0002fdf0: 0a7c 7820 202d 2032 3378 2078 2020 2d20  .|x  - 23x x  - 
│ │ │ │ +0002fe00: 3333 7820 7820 202b 2038 7820 7820 202b  33x x  + 8x x  +
│ │ │ │ +0002fe10: 2031 3078 2078 2020 2d20 3235 7820 7820   10x x  - 25x x 
│ │ │ │ +0002fe20: 202d 2039 7820 7820 202b 2033 7820 7820   - 9x x  + 3x x 
│ │ │ │ +0002fe30: 202b 2032 3478 2078 2020 2020 2020 207c   + 24x x       |
│ │ │ │ +0002fe40: 0a7c 2038 2020 2020 2020 3420 3820 2020  .| 8      4 8   
│ │ │ │ +0002fe50: 2020 2036 2038 2020 2020 2030 2039 2020     6 8     0 9  
│ │ │ │ +0002fe60: 2020 2020 3120 3920 2020 2020 2032 2039      1 9      2 9
│ │ │ │ +0002fe70: 2020 2020 2033 2039 2020 2020 2034 2039       3 9     4 9
│ │ │ │ +0002fe80: 2020 2020 2020 3520 2020 2020 2020 207c        5        |
│ │ │ │ +0002fe90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002fea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002feb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fee0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002fed0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002fee0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002fef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ff00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ff10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ff20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ff30: 2020 2020 207c 0a7c 7820 202d 2033 3478       |.|x  - 34x
│ │ │ │ -0002ff40: 2078 2020 2d20 3978 2078 2020 2d20 3436   x  - 9x x  - 46
│ │ │ │ -0002ff50: 7820 7820 202d 2031 3778 2078 2020 2b20  x x  - 17x x  + 
│ │ │ │ -0002ff60: 3332 7820 7820 202d 2038 7820 7820 202d  32x x  - 8x x  -
│ │ │ │ -0002ff70: 2033 3578 2078 2020 2d20 3436 7820 2020   35x x  - 46x   
│ │ │ │ -0002ff80: 2020 2020 207c 0a7c 2038 2020 2020 2020       |.| 8      
│ │ │ │ -0002ff90: 3320 3820 2020 2020 3420 3820 2020 2020  3 8     4 8     
│ │ │ │ -0002ffa0: 2036 2038 2020 2020 2020 3020 3920 2020   6 8      0 9   
│ │ │ │ -0002ffb0: 2020 2031 2039 2020 2020 2032 2039 2020     1 9     2 9  
│ │ │ │ -0002ffc0: 2020 2020 3320 3920 2020 2020 2034 2020      3 9      4  
│ │ │ │ -0002ffd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002ff20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002ff30: 0a7c 7820 202d 2033 3478 2078 2020 2d20  .|x  - 34x x  - 
│ │ │ │ +0002ff40: 3978 2078 2020 2d20 3436 7820 7820 202d  9x x  - 46x x  -
│ │ │ │ +0002ff50: 2031 3778 2078 2020 2b20 3332 7820 7820   17x x  + 32x x 
│ │ │ │ +0002ff60: 202d 2038 7820 7820 202d 2033 3578 2078   - 8x x  - 35x x
│ │ │ │ +0002ff70: 2020 2d20 3436 7820 2020 2020 2020 207c    - 46x        |
│ │ │ │ +0002ff80: 0a7c 2038 2020 2020 2020 3320 3820 2020  .| 8      3 8   
│ │ │ │ +0002ff90: 2020 3420 3820 2020 2020 2036 2038 2020    4 8      6 8  
│ │ │ │ +0002ffa0: 2020 2020 3020 3920 2020 2020 2031 2039      0 9      1 9
│ │ │ │ +0002ffb0: 2020 2020 2032 2039 2020 2020 2020 3320       2 9      3 
│ │ │ │ +0002ffc0: 3920 2020 2020 2034 2020 2020 2020 207c  9      4       |
│ │ │ │ +0002ffd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002ffe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030020: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030010: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030020: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00030030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030070: 2020 2020 207c 0a7c 2020 2d20 7820 7820       |.|  - x x 
│ │ │ │ -00030080: 202b 2078 2078 2020 2b20 3430 7820 7820   + x x  + 40x x 
│ │ │ │ -00030090: 202d 2033 3278 2078 2020 2b20 3578 2078   - 32x x  + 5x x
│ │ │ │ -000300a0: 2020 2d20 3131 7820 7820 202d 2032 3078    - 11x x  - 20x
│ │ │ │ -000300b0: 2078 2020 2b20 3435 7820 7820 202d 2020   x  + 45x x  -  
│ │ │ │ -000300c0: 2020 2020 207c 0a7c 3820 2020 2034 2038       |.|8    4 8
│ │ │ │ -000300d0: 2020 2020 3620 3820 2020 2020 2030 2039      6 8      0 9
│ │ │ │ -000300e0: 2020 2020 2020 3120 3920 2020 2020 3220        1 9     2 
│ │ │ │ -000300f0: 3920 2020 2020 2033 2039 2020 2020 2020  9      3 9      
│ │ │ │ -00030100: 3420 3920 2020 2020 2035 2039 2020 2020  4 9      5 9    
│ │ │ │ -00030110: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030060: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030070: 0a7c 2020 2d20 7820 7820 202b 2078 2078  .|  - x x  + x x
│ │ │ │ +00030080: 2020 2b20 3430 7820 7820 202d 2033 3278    + 40x x  - 32x
│ │ │ │ +00030090: 2078 2020 2b20 3578 2078 2020 2d20 3131   x  + 5x x  - 11
│ │ │ │ +000300a0: 7820 7820 202d 2032 3078 2078 2020 2b20  x x  - 20x x  + 
│ │ │ │ +000300b0: 3435 7820 7820 202d 2020 2020 2020 207c  45x x  -       |
│ │ │ │ +000300c0: 0a7c 3820 2020 2034 2038 2020 2020 3620  .|8    4 8    6 
│ │ │ │ +000300d0: 3820 2020 2020 2030 2039 2020 2020 2020  8      0 9      
│ │ │ │ +000300e0: 3120 3920 2020 2020 3220 3920 2020 2020  1 9     2 9     
│ │ │ │ +000300f0: 2033 2039 2020 2020 2020 3420 3920 2020   3 9      4 9   
│ │ │ │ +00030100: 2020 2035 2039 2020 2020 2020 2020 207c     5 9         |
│ │ │ │ +00030110: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00030160: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030150: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030160: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -000301b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000301a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000301b0: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00030200: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000301f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030200: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00030250: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030240: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030250: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -000302a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030290: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000302a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000302b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000302c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000302d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000302e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000302f0: 2020 2020 207c 0a7c 3337 7820 7820 202b       |.|37x x  +
│ │ │ │ -00030300: 2031 3278 2078 2020 2d20 3578 2078 2020   12x x  - 5x x  
│ │ │ │ -00030310: 2d20 3578 2078 2020 2b20 3231 7820 7820  - 5x x  + 21x x 
│ │ │ │ -00030320: 202b 2034 3278 2078 2020 2d20 3578 2078   + 42x x  - 5x x
│ │ │ │ -00030330: 2020 2b20 7820 7820 202d 2078 2078 2020    + x x  - x x  
│ │ │ │ -00030340: 2020 2020 207c 0a7c 2020 2031 2038 2020       |.|   1 8  
│ │ │ │ -00030350: 2020 2020 3320 3820 2020 2020 3420 3820      3 8     4 8 
│ │ │ │ -00030360: 2020 2020 3620 3820 2020 2020 2030 2039      6 8      0 9
│ │ │ │ -00030370: 2020 2020 2020 3120 3920 2020 2020 3220        1 9     2 
│ │ │ │ -00030380: 3920 2020 2033 2039 2020 2020 3420 2020  9    3 9    4   
│ │ │ │ -00030390: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000302e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000302f0: 0a7c 3337 7820 7820 202b 2031 3278 2078  .|37x x  + 12x x
│ │ │ │ +00030300: 2020 2d20 3578 2078 2020 2d20 3578 2078    - 5x x  - 5x x
│ │ │ │ +00030310: 2020 2b20 3231 7820 7820 202b 2034 3278    + 21x x  + 42x
│ │ │ │ +00030320: 2078 2020 2d20 3578 2078 2020 2b20 7820   x  - 5x x  + x 
│ │ │ │ +00030330: 7820 202d 2078 2078 2020 2020 2020 207c  x  - x x       |
│ │ │ │ +00030340: 0a7c 2020 2031 2038 2020 2020 2020 3320  .|   1 8      3 
│ │ │ │ +00030350: 3820 2020 2020 3420 3820 2020 2020 3620  8     4 8     6 
│ │ │ │ +00030360: 3820 2020 2020 2030 2039 2020 2020 2020  8      0 9      
│ │ │ │ +00030370: 3120 3920 2020 2020 3220 3920 2020 2033  1 9     2 9    3
│ │ │ │ +00030380: 2039 2020 2020 3420 2020 2020 2020 207c   9    4        |
│ │ │ │ +00030390: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -000303e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000303d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000303e0: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00030430: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030420: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030430: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -00030480: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030480: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00030490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000304a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000304b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000304c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000304d0: 2020 2020 207c 0a7c 7820 202b 2034 3078       |.|x  + 40x
│ │ │ │ -000304e0: 2078 2020 2d20 3578 2078 2020 2b20 3339   x  - 5x x  + 39
│ │ │ │ -000304f0: 7820 7820 202d 2033 3778 2078 2020 2b20  x x  - 37x x  + 
│ │ │ │ -00030500: 7820 7820 2020 2d20 7820 7820 2020 2d20  x x   - x x   - 
│ │ │ │ -00030510: 7820 7820 2020 2b20 7820 7820 2020 2020  x x   + x x     
│ │ │ │ -00030520: 2020 2020 207c 0a7c 2039 2020 2020 2020       |.| 9      
│ │ │ │ -00030530: 3220 3920 2020 2020 3320 3920 2020 2020  2 9     3 9     
│ │ │ │ -00030540: 2034 2039 2020 2020 2020 3520 3920 2020   4 9      5 9   
│ │ │ │ -00030550: 2030 2031 3020 2020 2031 2031 3020 2020   0 10    1 10   
│ │ │ │ -00030560: 2032 2031 3020 2020 2034 2031 3020 2020   2 10    4 10   
│ │ │ │ -00030570: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000304c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000304d0: 0a7c 7820 202b 2034 3078 2078 2020 2d20  .|x  + 40x x  - 
│ │ │ │ +000304e0: 3578 2078 2020 2b20 3339 7820 7820 202d  5x x  + 39x x  -
│ │ │ │ +000304f0: 2033 3778 2078 2020 2b20 7820 7820 2020   37x x  + x x   
│ │ │ │ +00030500: 2d20 7820 7820 2020 2d20 7820 7820 2020  - x x   - x x   
│ │ │ │ +00030510: 2b20 7820 7820 2020 2020 2020 2020 207c  + x x          |
│ │ │ │ +00030520: 0a7c 2039 2020 2020 2020 3220 3920 2020  .| 9      2 9   
│ │ │ │ +00030530: 2020 3320 3920 2020 2020 2034 2039 2020    3 9      4 9  
│ │ │ │ +00030540: 2020 2020 3520 3920 2020 2030 2031 3020      5 9    0 10 
│ │ │ │ +00030550: 2020 2031 2031 3020 2020 2032 2031 3020     1 10    2 10 
│ │ │ │ +00030560: 2020 2034 2031 3020 2020 2020 2020 207c     4 10        |
│ │ │ │ +00030570: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -000305c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000305b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000305c0: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00030610: 2020 2020 207c 0a7c 2020 2d20 3237 7820       |.|  - 27x 
│ │ │ │ -00030620: 7820 202d 2032 3778 2078 2020 2d20 3432  x  - 27x x  - 42
│ │ │ │ -00030630: 7820 7820 202b 2031 3178 2078 2020 2b20  x x  + 11x x  + 
│ │ │ │ -00030640: 3278 2078 2020 2b20 3232 7820 7820 202b  2x x  + 22x x  +
│ │ │ │ -00030650: 2033 3478 2078 2020 2b20 3432 7820 2020   34x x  + 42x   
│ │ │ │ -00030660: 2020 2020 207c 0a7c 3620 2020 2020 2031       |.|6      1
│ │ │ │ -00030670: 2037 2020 2020 2020 3320 3720 2020 2020   7      3 7     
│ │ │ │ -00030680: 2035 2037 2020 2020 2020 3620 3720 2020   5 7      6 7   
│ │ │ │ -00030690: 2020 3020 3820 2020 2020 2031 2038 2020    0 8      1 8  
│ │ │ │ -000306a0: 2020 2020 3320 3820 2020 2020 2034 2020      3 8      4  
│ │ │ │ -000306b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030610: 0a7c 2020 2d20 3237 7820 7820 202d 2032  .|  - 27x x  - 2
│ │ │ │ +00030620: 3778 2078 2020 2d20 3432 7820 7820 202b  7x x  - 42x x  +
│ │ │ │ +00030630: 2031 3178 2078 2020 2b20 3278 2078 2020   11x x  + 2x x  
│ │ │ │ +00030640: 2b20 3232 7820 7820 202b 2033 3478 2078  + 22x x  + 34x x
│ │ │ │ +00030650: 2020 2b20 3432 7820 2020 2020 2020 207c    + 42x        |
│ │ │ │ +00030660: 0a7c 3620 2020 2020 2031 2037 2020 2020  .|6      1 7    
│ │ │ │ +00030670: 2020 3320 3720 2020 2020 2035 2037 2020    3 7      5 7  
│ │ │ │ +00030680: 2020 2020 3620 3720 2020 2020 3020 3820      6 7     0 8 
│ │ │ │ +00030690: 2020 2020 2031 2038 2020 2020 2020 3320       1 8      3 
│ │ │ │ +000306a0: 3820 2020 2020 2034 2020 2020 2020 207c  8      4       |
│ │ │ │ +000306b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000306c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000306d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000306e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000306f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030700: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000306f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030700: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -00030750: 2020 2020 207c 0a7c 2032 3178 2078 2020       |.| 21x x  
│ │ │ │ -00030760: 2b20 3434 7820 7820 202d 2031 3478 2078  + 44x x  - 14x x
│ │ │ │ -00030770: 2020 2d20 3232 7820 7820 202d 2035 7820    - 22x x  - 5x 
│ │ │ │ -00030780: 7820 202d 2033 3478 2078 2020 2d20 3434  x  - 34x x  - 44
│ │ │ │ -00030790: 7820 7820 202b 2031 3078 2078 2020 2020  x x  + 10x x    
│ │ │ │ -000307a0: 2020 2020 207c 0a7c 2020 2020 3320 3720       |.|    3 7 
│ │ │ │ -000307b0: 2020 2020 2035 2037 2020 2020 2020 3620       5 7      6 
│ │ │ │ -000307c0: 3720 2020 2020 2030 2038 2020 2020 2031  7      0 8     1
│ │ │ │ -000307d0: 2038 2020 2020 2020 3320 3820 2020 2020   8      3 8     
│ │ │ │ -000307e0: 2034 2038 2020 2020 2020 3620 3820 2020   4 8      6 8   
│ │ │ │ -000307f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030740: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030750: 0a7c 2032 3178 2078 2020 2b20 3434 7820  .| 21x x  + 44x 
│ │ │ │ +00030760: 7820 202d 2031 3478 2078 2020 2d20 3232  x  - 14x x  - 22
│ │ │ │ +00030770: 7820 7820 202d 2035 7820 7820 202d 2033  x x  - 5x x  - 3
│ │ │ │ +00030780: 3478 2078 2020 2d20 3434 7820 7820 202b  4x x  - 44x x  +
│ │ │ │ +00030790: 2031 3078 2078 2020 2020 2020 2020 207c   10x x         |
│ │ │ │ +000307a0: 0a7c 2020 2020 3320 3720 2020 2020 2035  .|    3 7      5
│ │ │ │ +000307b0: 2037 2020 2020 2020 3620 3720 2020 2020   7      6 7     
│ │ │ │ +000307c0: 2030 2038 2020 2020 2031 2038 2020 2020   0 8     1 8    
│ │ │ │ +000307d0: 2020 3320 3820 2020 2020 2034 2038 2020    3 8      4 8  
│ │ │ │ +000307e0: 2020 2020 3620 3820 2020 2020 2020 207c      6 8        |
│ │ │ │ +000307f0: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00030840: 2020 2020 207c 0a7c 2020 2b20 3337 7820       |.|  + 37x 
│ │ │ │ -00030850: 7820 202d 2078 2078 2020 2d20 3335 7820  x  - x x  - 35x 
│ │ │ │ -00030860: 7820 202b 2039 7820 7820 202b 2032 3578  x  + 9x x  + 25x
│ │ │ │ -00030870: 2078 2020 202d 2031 3278 2078 2020 202d   x   - 12x x   -
│ │ │ │ -00030880: 2033 3678 2078 2020 202d 2031 3478 2020   36x x   - 14x  
│ │ │ │ -00030890: 2020 2020 207c 0a7c 3920 2020 2020 2033       |.|9      3
│ │ │ │ -000308a0: 2039 2020 2020 3420 3920 2020 2020 2035   9    4 9      5
│ │ │ │ -000308b0: 2039 2020 2020 2036 2039 2020 2020 2020   9     6 9      
│ │ │ │ -000308c0: 3120 3130 2020 2020 2020 3220 3130 2020  1 10      2 10  
│ │ │ │ -000308d0: 2020 2020 3420 3130 2020 2020 2020 2020      4 10        
│ │ │ │ -000308e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030830: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030840: 0a7c 2020 2b20 3337 7820 7820 202d 2078  .|  + 37x x  - x
│ │ │ │ +00030850: 2078 2020 2d20 3335 7820 7820 202b 2039   x  - 35x x  + 9
│ │ │ │ +00030860: 7820 7820 202b 2032 3578 2078 2020 202d  x x  + 25x x   -
│ │ │ │ +00030870: 2031 3278 2078 2020 202d 2033 3678 2078   12x x   - 36x x
│ │ │ │ +00030880: 2020 202d 2031 3478 2020 2020 2020 207c     - 14x       |
│ │ │ │ +00030890: 0a7c 3920 2020 2020 2033 2039 2020 2020  .|9      3 9    
│ │ │ │ +000308a0: 3420 3920 2020 2020 2035 2039 2020 2020  4 9      5 9    
│ │ │ │ +000308b0: 2036 2039 2020 2020 2020 3120 3130 2020   6 9      1 10  
│ │ │ │ +000308c0: 2020 2020 3220 3130 2020 2020 2020 3420      2 10      4 
│ │ │ │ +000308d0: 3130 2020 2020 2020 2020 2020 2020 207c  10             |
│ │ │ │ +000308e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -00030930: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030930: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00030980: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030970: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030980: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -000309d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000309c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000309d0: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030a10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030a20: 0a7c 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 2020 2020 2020                  
│ │ │ │ -00030a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030a60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030a70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ac0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030ab0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030ac0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00030ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030b10: 2020 2020 207c 0a7c 2020 2d20 3236 7820       |.|  - 26x 
│ │ │ │ -00030b20: 7820 202d 2034 3578 2078 2020 2b20 3232  x  - 45x x  + 22
│ │ │ │ -00030b30: 7820 7820 202d 2034 3278 2078 2020 2b20  x x  - 42x x  + 
│ │ │ │ -00030b40: 3378 2078 2020 2d20 3133 7820 7820 202d  3x x  - 13x x  -
│ │ │ │ -00030b50: 2034 7820 7820 202d 2032 3478 2078 2020   4x x  - 24x x  
│ │ │ │ -00030b60: 2020 2020 207c 0a7c 3820 2020 2020 2036       |.|8      6
│ │ │ │ -00030b70: 2038 2020 2020 2020 3120 3920 2020 2020   8      1 9     
│ │ │ │ -00030b80: 2032 2039 2020 2020 2020 3320 3920 2020   2 9      3 9   
│ │ │ │ -00030b90: 2020 3420 3920 2020 2020 2035 2039 2020    4 9      5 9  
│ │ │ │ -00030ba0: 2020 2036 2039 2020 2020 2020 3120 2020     6 9      1   
│ │ │ │ -00030bb0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00030b00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030b10: 0a7c 2020 2d20 3236 7820 7820 202d 2034  .|  - 26x x  - 4
│ │ │ │ +00030b20: 3578 2078 2020 2b20 3232 7820 7820 202d  5x x  + 22x x  -
│ │ │ │ +00030b30: 2034 3278 2078 2020 2b20 3378 2078 2020   42x x  + 3x x  
│ │ │ │ +00030b40: 2d20 3133 7820 7820 202d 2034 7820 7820  - 13x x  - 4x x 
│ │ │ │ +00030b50: 202d 2032 3478 2078 2020 2020 2020 207c   - 24x x       |
│ │ │ │ +00030b60: 0a7c 3820 2020 2020 2036 2038 2020 2020  .|8      6 8    
│ │ │ │ +00030b70: 2020 3120 3920 2020 2020 2032 2039 2020    1 9      2 9  
│ │ │ │ +00030b80: 2020 2020 3320 3920 2020 2020 3420 3920      3 9     4 9 
│ │ │ │ +00030b90: 2020 2020 2035 2039 2020 2020 2036 2039       5 9     6 9
│ │ │ │ +00030ba0: 2020 2020 2020 3120 2020 2020 2020 207c        1        |
│ │ │ │ +00030bb0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00030bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030c00: 2d2d 2d2d 2d7c 0a7c 2020 2d20 3237 7820  -----|.|  - 27x 
│ │ │ │ -00030c10: 7820 202d 2035 7820 7820 2020 2b20 3238  x  - 5x x   + 28
│ │ │ │ -00030c20: 7820 7820 2020 2b20 3337 7820 7820 2020  x x   + 37x x   
│ │ │ │ -00030c30: 2b20 3978 2078 2020 202b 2032 3778 2078  + 9x x   + 27x x
│ │ │ │ -00030c40: 2020 202d 2032 3578 2078 2020 202b 2039     - 25x x   + 9
│ │ │ │ -00030c50: 7820 7820 207c 0a7c 3920 2020 2020 2036  x x  |.|9      6
│ │ │ │ -00030c60: 2039 2020 2020 2030 2031 3020 2020 2020   9     0 10     
│ │ │ │ -00030c70: 2031 2031 3020 2020 2020 2032 2031 3020   1 10      2 10 
│ │ │ │ -00030c80: 2020 2020 3420 3130 2020 2020 2020 3620      4 10      6 
│ │ │ │ -00030c90: 3130 2020 2020 2020 3020 3131 2020 2020  10      0 11    
│ │ │ │ -00030ca0: 2032 2031 317c 0a7c 2020 2020 2020 2020   2 11|.|        
│ │ │ │ +00030bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00030c00: 0a7c 2020 2d20 3237 7820 7820 202d 2035  .|  - 27x x  - 5
│ │ │ │ +00030c10: 7820 7820 2020 2b20 3238 7820 7820 2020  x x   + 28x x   
│ │ │ │ +00030c20: 2b20 3337 7820 7820 2020 2b20 3978 2078  + 37x x   + 9x x
│ │ │ │ +00030c30: 2020 202b 2032 3778 2078 2020 202d 2032     + 27x x   - 2
│ │ │ │ +00030c40: 3578 2078 2020 202b 2039 7820 7820 207c  5x x   + 9x x  |
│ │ │ │ +00030c50: 0a7c 3920 2020 2020 2036 2039 2020 2020  .|9      6 9    
│ │ │ │ +00030c60: 2030 2031 3020 2020 2020 2031 2031 3020   0 10      1 10 
│ │ │ │ +00030c70: 2020 2020 2032 2031 3020 2020 2020 3420       2 10     4 
│ │ │ │ +00030c80: 3130 2020 2020 2020 3620 3130 2020 2020  10      6 10    
│ │ │ │ +00030c90: 2020 3020 3131 2020 2020 2032 2031 317c    0 11     2 11|
│ │ │ │ +00030ca0: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00030cf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030ce0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030cf0: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00030d40: 2020 2020 207c 0a7c 7820 202b 2032 3678       |.|x  + 26x
│ │ │ │ -00030d50: 2078 2020 2b20 3137 7820 7820 202b 2031   x  + 17x x  + 1
│ │ │ │ -00030d60: 3578 2078 2020 202b 2033 3578 2078 2020  5x x   + 35x x  
│ │ │ │ -00030d70: 202b 2033 3478 2078 2020 202b 2032 3078   + 34x x   + 20x
│ │ │ │ -00030d80: 2078 2020 202b 2031 3478 2078 2020 202b   x   + 14x x   +
│ │ │ │ -00030d90: 2033 3678 207c 0a7c 2039 2020 2020 2020   36x |.| 9      
│ │ │ │ -00030da0: 3520 3920 2020 2020 2036 2039 2020 2020  5 9      6 9    
│ │ │ │ -00030db0: 2020 3020 3130 2020 2020 2020 3120 3130    0 10      1 10
│ │ │ │ -00030dc0: 2020 2020 2020 3220 3130 2020 2020 2020        2 10      
│ │ │ │ -00030dd0: 3420 3130 2020 2020 2020 3020 3131 2020  4 10      0 11  
│ │ │ │ -00030de0: 2020 2020 317c 0a7c 2020 2020 2020 2020      1|.|        
│ │ │ │ +00030d30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030d40: 0a7c 7820 202b 2032 3678 2078 2020 2b20  .|x  + 26x x  + 
│ │ │ │ +00030d50: 3137 7820 7820 202b 2031 3578 2078 2020  17x x  + 15x x  
│ │ │ │ +00030d60: 202b 2033 3578 2078 2020 202b 2033 3478   + 35x x   + 34x
│ │ │ │ +00030d70: 2078 2020 202b 2032 3078 2078 2020 202b   x   + 20x x   +
│ │ │ │ +00030d80: 2031 3478 2078 2020 202b 2033 3678 207c   14x x   + 36x |
│ │ │ │ +00030d90: 0a7c 2039 2020 2020 2020 3520 3920 2020  .| 9      5 9   
│ │ │ │ +00030da0: 2020 2036 2039 2020 2020 2020 3020 3130     6 9      0 10
│ │ │ │ +00030db0: 2020 2020 2020 3120 3130 2020 2020 2020        1 10      
│ │ │ │ +00030dc0: 3220 3130 2020 2020 2020 3420 3130 2020  2 10      4 10  
│ │ │ │ +00030dd0: 2020 2020 3020 3131 2020 2020 2020 317c      0 11      1|
│ │ │ │ +00030de0: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00030e30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030e20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030e30: 0a7c 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 2020 2020 2020                  
│ │ │ │ -00030e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030e80: 2020 2020 207c 0a7c 2031 3478 2078 2020       |.| 14x x  
│ │ │ │ -00030e90: 2d20 3235 7820 7820 2020 2b20 3435 7820  - 25x x   + 45x 
│ │ │ │ -00030ea0: 7820 2020 2d20 3431 7820 7820 2020 2d20  x   - 41x x   - 
│ │ │ │ -00030eb0: 3436 7820 7820 2020 2b20 3878 2078 2020  46x x   + 8x x  
│ │ │ │ -00030ec0: 202d 2032 3878 2078 2020 202b 2031 3178   - 28x x   + 11x
│ │ │ │ -00030ed0: 2078 2020 207c 0a7c 2020 2020 3620 3920   x   |.|    6 9 
│ │ │ │ -00030ee0: 2020 2020 2030 2031 3020 2020 2020 2031       0 10      1
│ │ │ │ -00030ef0: 2031 3020 2020 2020 2032 2031 3020 2020   10      2 10   
│ │ │ │ -00030f00: 2020 2034 2031 3020 2020 2020 3620 3130     4 10     6 10
│ │ │ │ -00030f10: 2020 2020 2020 3020 3131 2020 2020 2020        0 11      
│ │ │ │ -00030f20: 3220 3131 207c 0a7c 2020 2020 2020 2020  2 11 |.|        
│ │ │ │ +00030e70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030e80: 0a7c 2031 3478 2078 2020 2d20 3235 7820  .| 14x x  - 25x 
│ │ │ │ +00030e90: 7820 2020 2b20 3435 7820 7820 2020 2d20  x   + 45x x   - 
│ │ │ │ +00030ea0: 3431 7820 7820 2020 2d20 3436 7820 7820  41x x   - 46x x 
│ │ │ │ +00030eb0: 2020 2b20 3878 2078 2020 202d 2032 3878    + 8x x   - 28x
│ │ │ │ +00030ec0: 2078 2020 202b 2031 3178 2078 2020 207c   x   + 11x x   |
│ │ │ │ +00030ed0: 0a7c 2020 2020 3620 3920 2020 2020 2030  .|    6 9      0
│ │ │ │ +00030ee0: 2031 3020 2020 2020 2031 2031 3020 2020   10      1 10   
│ │ │ │ +00030ef0: 2020 2032 2031 3020 2020 2020 2034 2031     2 10      4 1
│ │ │ │ +00030f00: 3020 2020 2020 3620 3130 2020 2020 2020  0     6 10      
│ │ │ │ +00030f10: 3020 3131 2020 2020 2020 3220 3131 207c  0 11      2 11 |
│ │ │ │ +00030f20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00030f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030f70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030f60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030f70: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00030fc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030fb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00030fc0: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00031010: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031000: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031010: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00031020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031060: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031050: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031060: 0a7c 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 2020 2020 2020                  
│ │ │ │ -000310a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000310b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000310a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000310b0: 0a7c 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 2020 2020 2020                  
│ │ │ │ -000310f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031100: 2020 2020 207c 0a7c 2020 2b20 3278 2078       |.|  + 2x x
│ │ │ │ -00031110: 2020 2b20 3438 7820 7820 202d 2078 2078    + 48x x  - x x
│ │ │ │ -00031120: 2020 202b 2032 3678 2078 2020 202d 2031     + 26x x   - 1
│ │ │ │ -00031130: 3278 2078 2020 202d 2033 3678 2078 2020  2x x   - 36x x  
│ │ │ │ -00031140: 202d 2031 3878 2078 2020 202d 2035 7820   - 18x x   - 5x 
│ │ │ │ -00031150: 7820 2020 2d7c 0a7c 3920 2020 2020 3520  x   -|.|9     5 
│ │ │ │ -00031160: 3920 2020 2020 2036 2039 2020 2020 3020  9      6 9    0 
│ │ │ │ -00031170: 3130 2020 2020 2020 3120 3130 2020 2020  10      1 10    
│ │ │ │ -00031180: 2020 3220 3130 2020 2020 2020 3420 3130    2 10      4 10
│ │ │ │ -00031190: 2020 2020 2020 3620 3130 2020 2020 2030        6 10     0
│ │ │ │ -000311a0: 2031 3120 207c 0a7c 2020 2020 2020 2020   11  |.|        
│ │ │ │ +000310f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031100: 0a7c 2020 2b20 3278 2078 2020 2b20 3438  .|  + 2x x  + 48
│ │ │ │ +00031110: 7820 7820 202d 2078 2078 2020 202b 2032  x x  - x x   + 2
│ │ │ │ +00031120: 3678 2078 2020 202d 2031 3278 2078 2020  6x x   - 12x x  
│ │ │ │ +00031130: 202d 2033 3678 2078 2020 202d 2031 3878   - 36x x   - 18x
│ │ │ │ +00031140: 2078 2020 202d 2035 7820 7820 2020 2d7c   x   - 5x x   -|
│ │ │ │ +00031150: 0a7c 3920 2020 2020 3520 3920 2020 2020  .|9     5 9     
│ │ │ │ +00031160: 2036 2039 2020 2020 3020 3130 2020 2020   6 9    0 10    
│ │ │ │ +00031170: 2020 3120 3130 2020 2020 2020 3220 3130    1 10      2 10
│ │ │ │ +00031180: 2020 2020 2020 3420 3130 2020 2020 2020        4 10      
│ │ │ │ +00031190: 3620 3130 2020 2020 2030 2031 3120 207c  6 10     0 11  |
│ │ │ │ +000311a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -000311f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000311e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000311f0: 0a7c 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 2020 2020 2020                  
│ │ │ │ -00031230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031240: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031230: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031240: 0a7c 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 2020 2020 2020                  
│ │ │ │ -00031280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031290: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031280: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031290: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000312a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000312b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000312c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000312d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000312e0: 2020 2020 207c 0a7c 2b20 3578 2078 2020       |.|+ 5x x  
│ │ │ │ -000312f0: 202d 2035 7820 7820 202c 2020 2020 2020   - 5x x  ,      
│ │ │ │ +000312d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000312e0: 0a7c 2b20 3578 2078 2020 202d 2035 7820  .|+ 5x x   - 5x 
│ │ │ │ +000312f0: 7820 202c 2020 2020 2020 2020 2020 2020  x  ,            
│ │ │ │  00031300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031330: 2020 2020 207c 0a7c 2020 2020 3620 3130       |.|    6 10
│ │ │ │ -00031340: 2020 2020 2035 2031 3120 2020 2020 2020       5 11       
│ │ │ │ +00031320: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031330: 0a7c 2020 2020 3620 3130 2020 2020 2035  .|    6 10     5
│ │ │ │ +00031340: 2031 3120 2020 2020 2020 2020 2020 2020   11             
│ │ │ │  00031350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031380: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031380: 0a7c 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 2020 2020 2020                  
│ │ │ │ -000313c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000313d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000313c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000313d0: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00031420: 2020 2020 207c 0a7c 7820 202b 2031 3478       |.|x  + 14x
│ │ │ │ -00031430: 2078 2020 2b20 3878 2078 2020 2b20 3136   x  + 8x x  + 16
│ │ │ │ -00031440: 7820 7820 202b 2032 3778 2078 2020 2b20  x x  + 27x x  + 
│ │ │ │ -00031450: 3234 7820 7820 202d 2031 3178 2078 2020  24x x  - 11x x  
│ │ │ │ -00031460: 2d20 3678 2078 2020 2b20 3278 2078 2020  - 6x x  + 2x x  
│ │ │ │ -00031470: 2d20 3578 207c 0a7c 2038 2020 2020 2020  - 5x |.| 8      
│ │ │ │ -00031480: 3620 3820 2020 2020 3020 3920 2020 2020  6 8     0 9     
│ │ │ │ -00031490: 2031 2039 2020 2020 2020 3220 3920 2020   1 9      2 9   
│ │ │ │ -000314a0: 2020 2033 2039 2020 2020 2020 3420 3920     3 9      4 9 
│ │ │ │ -000314b0: 2020 2020 3520 3920 2020 2020 3620 3920      5 9     6 9 
│ │ │ │ -000314c0: 2020 2020 307c 0a7c 2020 2020 2020 2020      0|.|        
│ │ │ │ +00031410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031420: 0a7c 7820 202b 2031 3478 2078 2020 2b20  .|x  + 14x x  + 
│ │ │ │ +00031430: 3878 2078 2020 2b20 3136 7820 7820 202b  8x x  + 16x x  +
│ │ │ │ +00031440: 2032 3778 2078 2020 2b20 3234 7820 7820   27x x  + 24x x 
│ │ │ │ +00031450: 202d 2031 3178 2078 2020 2d20 3678 2078   - 11x x  - 6x x
│ │ │ │ +00031460: 2020 2b20 3278 2078 2020 2d20 3578 207c    + 2x x  - 5x |
│ │ │ │ +00031470: 0a7c 2038 2020 2020 2020 3620 3820 2020  .| 8      6 8   
│ │ │ │ +00031480: 2020 3020 3920 2020 2020 2031 2039 2020    0 9      1 9  
│ │ │ │ +00031490: 2020 2020 3220 3920 2020 2020 2033 2039      2 9      3 9
│ │ │ │ +000314a0: 2020 2020 2020 3420 3920 2020 2020 3520        4 9     5 
│ │ │ │ +000314b0: 3920 2020 2020 3620 3920 2020 2020 307c  9     6 9     0|
│ │ │ │ +000314c0: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00031510: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031510: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00031560: 2020 2020 207c 0a7c 2d20 3434 7820 7820       |.|- 44x x 
│ │ │ │ -00031570: 202d 2032 3178 2078 2020 2d20 3231 7820   - 21x x  - 21x 
│ │ │ │ -00031580: 7820 202d 2039 7820 7820 202b 2031 3478  x  - 9x x  + 14x
│ │ │ │ -00031590: 2078 2020 2d20 3232 7820 7820 202b 2034   x  - 22x x  + 4
│ │ │ │ -000315a0: 3578 2078 2020 2b20 3578 2078 2020 202b  5x x  + 5x x   +
│ │ │ │ -000315b0: 2032 3078 207c 0a7c 2020 2020 2030 2039   20x |.|     0 9
│ │ │ │ -000315c0: 2020 2020 2020 3120 3920 2020 2020 2032        1 9      2
│ │ │ │ -000315d0: 2039 2020 2020 2033 2039 2020 2020 2020   9     3 9      
│ │ │ │ -000315e0: 3420 3920 2020 2020 2035 2039 2020 2020  4 9      5 9    
│ │ │ │ -000315f0: 2020 3620 3920 2020 2020 3020 3130 2020    6 9     0 10  
│ │ │ │ -00031600: 2020 2020 317c 0a7c 2020 2020 2020 2020      1|.|        
│ │ │ │ +00031550: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031560: 0a7c 2d20 3434 7820 7820 202d 2032 3178  .|- 44x x  - 21x
│ │ │ │ +00031570: 2078 2020 2d20 3231 7820 7820 202d 2039   x  - 21x x  - 9
│ │ │ │ +00031580: 7820 7820 202b 2031 3478 2078 2020 2d20  x x  + 14x x  - 
│ │ │ │ +00031590: 3232 7820 7820 202b 2034 3578 2078 2020  22x x  + 45x x  
│ │ │ │ +000315a0: 2b20 3578 2078 2020 202b 2032 3078 207c  + 5x x   + 20x |
│ │ │ │ +000315b0: 0a7c 2020 2020 2030 2039 2020 2020 2020  .|     0 9      
│ │ │ │ +000315c0: 3120 3920 2020 2020 2032 2039 2020 2020  1 9      2 9    
│ │ │ │ +000315d0: 2033 2039 2020 2020 2020 3420 3920 2020   3 9      4 9   
│ │ │ │ +000315e0: 2020 2035 2039 2020 2020 2020 3620 3920     5 9      6 9 
│ │ │ │ +000315f0: 2020 2020 3020 3130 2020 2020 2020 317c      0 10      1|
│ │ │ │ +00031600: 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -00031650: 2020 2020 207c 0a7c 2078 2020 202d 2033       |.| x   - 3
│ │ │ │ -00031660: 3778 2078 2020 202d 2039 7820 7820 2020  7x x   - 9x x   
│ │ │ │ -00031670: 2b20 3134 7820 7820 202c 2020 2020 2020  + 14x x  ,      
│ │ │ │ +00031640: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031650: 0a7c 2078 2020 202d 2033 3778 2078 2020  .| x   - 37x x  
│ │ │ │ +00031660: 202d 2039 7820 7820 2020 2b20 3134 7820   - 9x x   + 14x 
│ │ │ │ +00031670: 7820 202c 2020 2020 2020 2020 2020 2020  x  ,            
│ │ │ │  00031680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000316a0: 2020 2020 207c 0a7c 3620 3130 2020 2020       |.|6 10    
│ │ │ │ -000316b0: 2020 3220 3131 2020 2020 2034 2031 3120    2 11     4 11 
│ │ │ │ -000316c0: 2020 2020 2035 2031 3120 2020 2020 2020       5 11       
│ │ │ │ +00031690: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000316a0: 0a7c 3620 3130 2020 2020 2020 3220 3131  .|6 10      2 11
│ │ │ │ +000316b0: 2020 2020 2034 2031 3120 2020 2020 2035       4 11      5
│ │ │ │ +000316c0: 2031 3120 2020 2020 2020 2020 2020 2020   11             
│ │ │ │  000316d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000316e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000316f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000316e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000316f0: 0a7c 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 2020 2020 2020                  
│ │ │ │ -00031730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031740: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031730: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031740: 0a7c 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 2020 2020 2020                  
│ │ │ │ -00031780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031790: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031780: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031790: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000317a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000317b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000317c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000317d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000317e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000317d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000317e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000317f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031830: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031820: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031830: 0a7c 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 2020 2020 2020                  
│ │ │ │ -00031870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031880: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031870: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031880: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00031890: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000318b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000318c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000318d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000318c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000318d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000318e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000318f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031920: 2020 2020 207c 0a7c 2020 202b 2031 3378       |.|   + 13x
│ │ │ │ -00031930: 2078 2020 202b 2033 3978 2078 2020 202d   x   + 39x x   -
│ │ │ │ -00031940: 2036 7820 7820 2020 2b20 3578 2078 2020   6x x   + 5x x  
│ │ │ │ -00031950: 202b 2034 3278 2078 2020 202b 2034 7820   + 42x x   + 4x 
│ │ │ │ -00031960: 7820 2020 2b20 3678 2078 2020 2020 2020  x   + 6x x      
│ │ │ │ -00031970: 2020 2020 207c 0a7c 3130 2020 2020 2020       |.|10      
│ │ │ │ -00031980: 3220 3130 2020 2020 2020 3420 3130 2020  2 10      4 10  
│ │ │ │ -00031990: 2020 2036 2031 3020 2020 2020 3120 3131     6 10     1 11
│ │ │ │ -000319a0: 2020 2020 2020 3220 3131 2020 2020 2034        2 11     4
│ │ │ │ -000319b0: 2031 3120 2020 2020 3520 3131 2020 2020   11     5 11    
│ │ │ │ -000319c0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00031910: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031920: 0a7c 2020 202b 2031 3378 2078 2020 202b  .|   + 13x x   +
│ │ │ │ +00031930: 2033 3978 2078 2020 202d 2036 7820 7820   39x x   - 6x x 
│ │ │ │ +00031940: 2020 2b20 3578 2078 2020 202b 2034 3278    + 5x x   + 42x
│ │ │ │ +00031950: 2078 2020 202b 2034 7820 7820 2020 2b20   x   + 4x x   + 
│ │ │ │ +00031960: 3678 2078 2020 2020 2020 2020 2020 207c  6x x           |
│ │ │ │ +00031970: 0a7c 3130 2020 2020 2020 3220 3130 2020  .|10      2 10  
│ │ │ │ +00031980: 2020 2020 3420 3130 2020 2020 2036 2031      4 10     6 1
│ │ │ │ +00031990: 3020 2020 2020 3120 3131 2020 2020 2020  0     1 11      
│ │ │ │ +000319a0: 3220 3131 2020 2020 2034 2031 3120 2020  2 11     4 11   
│ │ │ │ +000319b0: 2020 3520 3131 2020 2020 2020 2020 207c    5 11         |
│ │ │ │ +000319c0: 0a7c 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 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031a10: 2d2d 2d2d 2d7c 0a7c 202b 2032 3778 2078  -----|.| + 27x x
│ │ │ │ -00031a20: 2020 202d 2032 3778 2078 2020 2c20 2020     - 27x x  ,   
│ │ │ │ +00031a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00031a10: 0a7c 202b 2032 3778 2078 2020 202d 2032  .| + 27x x   - 2
│ │ │ │ +00031a20: 3778 2078 2020 2c20 2020 2020 2020 2020  7x x  ,         
│ │ │ │  00031a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00031a50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031a60: 0a7c 2020 2020 2020 3420 3131 2020 2020  .|      4 11    
│ │ │ │ +00031a70: 2020 3520 3131 2020 2020 2020 2020 2020    5 11          
│ │ │ │  00031a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031ab0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031aa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031ab0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00031ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031b00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031af0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031b00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00031b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031b50: 2020 2020 207c 0a7c 7820 2020 2b20 3335       |.|x   + 35
│ │ │ │ -00031b60: 7820 7820 2020 2d20 3137 7820 7820 202c  x x   - 17x x  ,
│ │ │ │ +00031b40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +00031b60: 2d20 3137 7820 7820 202c 2020 2020 2020  - 17x x  ,      
│ │ │ │  00031b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00031ba0: 2020 2020 207c 0a7c 2031 3120 2020 2020       |.| 11     
│ │ │ │ -00031bb0: 2032 2031 3120 2020 2020 2034 2031 3120   2 11      4 11 
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│ │ │ │ +00031bb0: 2020 2020 2034 2031 3120 2020 2020 2020       4 11       
│ │ │ │  00031bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031bf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031be0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031bf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00031c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031c40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031c30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031c40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00031c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031c90: 2020 2020 207c 0a7c 2b20 3134 7820 7820       |.|+ 14x x 
│ │ │ │ -00031ca0: 2020 2d20 3878 2078 2020 2020 2020 2020    - 8x x        
│ │ │ │ +00031c80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031c90: 0a7c 2b20 3134 7820 7820 2020 2d20 3878  .|+ 14x x   - 8x
│ │ │ │ +00031ca0: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │  00031cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031ce0: 2020 2020 207c 0a7c 2020 2020 2034 2031       |.|     4 1
│ │ │ │ -00031cf0: 3120 2020 2020 3520 3131 2020 2020 2020  1     5 11      
│ │ │ │ +00031cd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031ce0: 0a7c 2020 2020 2034 2031 3120 2020 2020  .|     4 11     
│ │ │ │ +00031cf0: 3520 3131 2020 2020 2020 2020 2020 2020  5 11            
│ │ │ │  00031d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031d30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031d20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031d30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00031d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031d80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031d70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031d80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00031d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031dd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031dc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031dd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00031de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031e20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031e10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031e20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00031e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031e70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031e60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031e70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00031e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031ec0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ +00031ec0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00031ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031f10: 2020 2020 207c 0a7c 2033 3778 2078 2020       |.| 37x x  
│ │ │ │ -00031f20: 202d 2035 7820 7820 2020 2d20 3978 2078   - 5x x   - 9x x
│ │ │ │ -00031f30: 2020 202b 2031 3978 2078 2020 2c20 2020     + 19x x  ,   
│ │ │ │ +00031f00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031f10: 0a7c 2033 3778 2078 2020 202d 2035 7820  .| 37x x   - 5x 
│ │ │ │ +00031f20: 7820 2020 2d20 3978 2078 2020 202b 2031  x   - 9x x   + 1
│ │ │ │ +00031f30: 3978 2078 2020 2c20 2020 2020 2020 2020  9x x  ,         
│ │ │ │  00031f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00031f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00031fa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00031fb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │  00031fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00032010: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000320b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00032100: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00032150: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000321a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000321c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000321d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │  000321f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032210: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00032230: 2020 2020 207c 0a7c 7820 2020 2d20 3130       |.|x   - 10
│ │ │ │ -00032240: 7820 7820 2020 2d20 3334 7820 7820 2020  x x   - 34x x   
│ │ │ │ -00032250: 2d20 3878 2078 2020 202b 2031 3578 2078  - 8x x   + 15x x
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│ │ │ │ +00032220: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00032230: 0a7c 7820 2020 2d20 3130 7820 7820 2020  .|x   - 10x x   
│ │ │ │ +00032240: 2d20 3334 7820 7820 2020 2d20 3878 2078  - 34x x   - 8x x
│ │ │ │ +00032250: 2020 202b 2031 3578 2078 2020 202d 2032     + 15x x   - 2
│ │ │ │ +00032260: 3578 2078 2020 202d 2032 3478 2078 2020  5x x   - 24x x  
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│ │ │ │  000322e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00032300: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00032330: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00032370: 2020 2020 207c 0a7c 7820 2020 2b20 3334       |.|x   + 34
│ │ │ │ -00032380: 7820 7820 2020 2b20 3132 7820 7820 2020  x x   + 12x x   
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│ │ │ │ +00032370: 0a7c 7820 2020 2b20 3334 7820 7820 2020  .|x   + 34x x   
│ │ │ │ +00032380: 2b20 3132 7820 7820 2020 2d20 3235 7820  + 12x x   - 25x 
│ │ │ │ +00032390: 7820 2020 2b20 3230 7820 7820 2020 2d20  x   + 20x x   - 
│ │ │ │ +000323a0: 3578 2078 2020 202b 2039 7820 7820 2020  5x x   + 9x x   
│ │ │ │ +000323b0: 2d20 3435 7820 7820 2020 2b20 2020 207c  - 45x x   +    |
│ │ │ │ +000323c0: 0a7c 2031 3020 2020 2020 2032 2031 3020  .| 10      2 10 
│ │ │ │ +000323d0: 2020 2020 2034 2031 3020 2020 2020 2036       4 10      6
│ │ │ │ +000323e0: 2031 3020 2020 2020 2030 2031 3120 2020   10      0 11   
│ │ │ │ +000323f0: 2020 3120 3131 2020 2020 2032 2031 3120    1 11     2 11 
│ │ │ │ +00032400: 2020 2020 2034 2031 3120 2020 2020 207c       4 11      |
│ │ │ │ +00032410: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00032420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032460: 2d2d 2d2d 2d7c 0a7c 3135 7820 7820 202c  -----|.|15x x  ,
│ │ │ │ +00032450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00032460: 0a7c 3135 7820 7820 202c 2020 2020 2020  .|15x x  ,      
│ │ │ │  00032470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000324a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000324b0: 2020 2020 207c 0a7c 2020 2035 2031 3120       |.|   5 11 
│ │ │ │ +000324a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000324b0: 0a7c 2020 2035 2031 3120 2020 2020 2020  .|   5 11       
│ │ │ │  000324c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000324d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000324e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000324f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032500: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000324f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00032500: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00032510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032550: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00032540: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00032550: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00032560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000325a0: 2020 2020 207c 0a7c 3235 7820 7820 202c       |.|25x x  ,
│ │ │ │ +00032590: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000325a0: 0a7c 3235 7820 7820 202c 2020 2020 2020  .|25x x  ,      
│ │ │ │  000325b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000325c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000325d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000325e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000325f0: 2020 2020 207c 0a7c 2020 2035 2031 3120       |.|   5 11 
│ │ │ │ +000325e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000325f0: 0a7c 2020 2035 2031 3120 2020 2020 2020  .|   5 11       
│ │ │ │  00032600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032640: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00032630: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00032640: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00032650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032690: 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 6173  -----+.|i11 : as
│ │ │ │ -000326a0: 7365 7274 2870 6869 202a 2070 7369 203d  sert(phi * psi =
│ │ │ │ -000326b0: 3d20 3129 2020 2020 2020 2020 2020 2020  = 1)            
│ │ │ │ +00032680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00032690: 0a7c 6931 3120 3a20 6173 7365 7274 2870  .|i11 : assert(p
│ │ │ │ +000326a0: 6869 202a 2070 7369 203d 3d20 3129 2020  hi * psi == 1)  
│ │ │ │ +000326b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000326c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000326d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000326e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000326d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000326e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000326f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032730: 2d2d 2d2d 2d2b 0a0a 5468 6520 6d65 7468  -----+..The meth
│ │ │ │ -00032740: 6f64 2061 6c73 6f20 6163 6365 7074 7320  od also accepts 
│ │ │ │ -00032750: 6173 2069 6e70 7574 2061 202a 6e6f 7465  as input a *note
│ │ │ │ -00032760: 2072 696e 6720 6d61 703a 2028 4d61 6361   ring map: (Maca
│ │ │ │ -00032770: 756c 6179 3244 6f63 2952 696e 674d 6170  ulay2Doc)RingMap
│ │ │ │ -00032780: 2c0a 7265 7072 6573 656e 7469 6e67 2061  ,.representing a
│ │ │ │ -00032790: 2072 6174 696f 6e61 6c20 6d61 7020 6265   rational map be
│ │ │ │ -000327a0: 7477 6565 6e20 7072 6f6a 6563 7469 7665  tween projective
│ │ │ │ -000327b0: 2076 6172 6965 7469 6573 2e20 496e 2074   varieties. In t
│ │ │ │ -000327c0: 6869 7320 6361 7365 2c20 6120 2a6e 6f74  his case, a *not
│ │ │ │ -000327d0: 650a 7269 6e67 206d 6170 3a20 284d 6163  e.ring map: (Mac
│ │ │ │ -000327e0: 6175 6c61 7932 446f 6329 5269 6e67 4d61  aulay2Doc)RingMa
│ │ │ │ -000327f0: 702c 2069 7320 7265 7475 726e 6564 2061  p, is returned a
│ │ │ │ -00032800: 7320 7765 6c6c 2e0a 0a43 6176 6561 740a  s well...Caveat.
│ │ │ │ -00032810: 3d3d 3d3d 3d3d 0a0a 466f 7220 7468 6520  ======..For the 
│ │ │ │ -00032820: 7075 7270 6f73 6520 6f66 2074 6869 7320  purpose of this 
│ │ │ │ -00032830: 6d65 7468 6f64 2c20 7468 6520 6f70 7469  method, the opti
│ │ │ │ -00032840: 6f6e 202a 6e6f 7465 2043 6572 7469 6679  on *note Certify
│ │ │ │ -00032850: 3a20 4365 7274 6966 792c 3d3e 7472 7565  : Certify,=>true
│ │ │ │ -00032860: 2069 7320 746f 6f0a 7269 6769 642c 2065   is too.rigid, e
│ │ │ │ -00032870: 7370 6563 6961 6c6c 7920 7768 656e 2074  specially when t
│ │ │ │ -00032880: 6865 2073 6f75 7263 6520 6f66 2074 6865  he source of the
│ │ │ │ -00032890: 2070 6173 7365 6420 6d61 7020 6973 206e   passed map is n
│ │ │ │ -000328a0: 6f74 2061 2070 726f 6a65 6374 6976 6520  ot a projective 
│ │ │ │ -000328b0: 7370 6163 652e 0a0a 5365 6520 616c 736f  space...See also
│ │ │ │ -000328c0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ -000328d0: 6e6f 7465 2069 6e76 6572 7365 4d61 703a  note inverseMap:
│ │ │ │ -000328e0: 2069 6e76 6572 7365 4d61 702c 202d 2d20   inverseMap, -- 
│ │ │ │ -000328f0: 696e 7665 7273 6520 6f66 2061 2062 6972  inverse of a bir
│ │ │ │ -00032900: 6174 696f 6e61 6c20 6d61 700a 2020 2a20  ational map.  * 
│ │ │ │ -00032910: 2a6e 6f74 6520 696e 7665 7273 6528 5261  *note inverse(Ra
│ │ │ │ -00032920: 7469 6f6e 616c 4d61 7029 3a20 696e 7665  tionalMap): inve
│ │ │ │ -00032930: 7273 655f 6c70 5261 7469 6f6e 616c 4d61  rse_lpRationalMa
│ │ │ │ -00032940: 705f 7270 2c20 2d2d 2069 6e76 6572 7365  p_rp, -- inverse
│ │ │ │ -00032950: 206f 6620 610a 2020 2020 6269 7261 7469   of a.    birati
│ │ │ │ -00032960: 6f6e 616c 206d 6170 0a0a 5761 7973 2074  onal map..Ways t
│ │ │ │ -00032970: 6f20 7573 6520 6170 7072 6f78 696d 6174  o use approximat
│ │ │ │ -00032980: 6549 6e76 6572 7365 4d61 703a 0a3d 3d3d  eInverseMap:.===
│ │ │ │ +00032720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00032730: 0a0a 5468 6520 6d65 7468 6f64 2061 6c73  ..The method als
│ │ │ │ +00032740: 6f20 6163 6365 7074 7320 6173 2069 6e70  o accepts as inp
│ │ │ │ +00032750: 7574 2061 202a 6e6f 7465 2072 696e 6720  ut a *note ring 
│ │ │ │ +00032760: 6d61 703a 2028 4d61 6361 756c 6179 3244  map: (Macaulay2D
│ │ │ │ +00032770: 6f63 2952 696e 674d 6170 2c0a 7265 7072  oc)RingMap,.repr
│ │ │ │ +00032780: 6573 656e 7469 6e67 2061 2072 6174 696f  esenting a ratio
│ │ │ │ +00032790: 6e61 6c20 6d61 7020 6265 7477 6565 6e20  nal map between 
│ │ │ │ +000327a0: 7072 6f6a 6563 7469 7665 2076 6172 6965  projective varie
│ │ │ │ +000327b0: 7469 6573 2e20 496e 2074 6869 7320 6361  ties. In this ca
│ │ │ │ +000327c0: 7365 2c20 6120 2a6e 6f74 650a 7269 6e67  se, a *note.ring
│ │ │ │ +000327d0: 206d 6170 3a20 284d 6163 6175 6c61 7932   map: (Macaulay2
│ │ │ │ +000327e0: 446f 6329 5269 6e67 4d61 702c 2069 7320  Doc)RingMap, is 
│ │ │ │ +000327f0: 7265 7475 726e 6564 2061 7320 7765 6c6c  returned as well
│ │ │ │ +00032800: 2e0a 0a43 6176 6561 740a 3d3d 3d3d 3d3d  ...Caveat.======
│ │ │ │ +00032810: 0a0a 466f 7220 7468 6520 7075 7270 6f73  ..For the purpos
│ │ │ │ +00032820: 6520 6f66 2074 6869 7320 6d65 7468 6f64  e of this method
│ │ │ │ +00032830: 2c20 7468 6520 6f70 7469 6f6e 202a 6e6f  , the option *no
│ │ │ │ +00032840: 7465 2043 6572 7469 6679 3a20 4365 7274  te Certify: Cert
│ │ │ │ +00032850: 6966 792c 3d3e 7472 7565 2069 7320 746f  ify,=>true is to
│ │ │ │ +00032860: 6f0a 7269 6769 642c 2065 7370 6563 6961  o.rigid, especia
│ │ │ │ +00032870: 6c6c 7920 7768 656e 2074 6865 2073 6f75  lly when the sou
│ │ │ │ +00032880: 7263 6520 6f66 2074 6865 2070 6173 7365  rce of the passe
│ │ │ │ +00032890: 6420 6d61 7020 6973 206e 6f74 2061 2070  d map is not a p
│ │ │ │ +000328a0: 726f 6a65 6374 6976 6520 7370 6163 652e  rojective space.
│ │ │ │ +000328b0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ +000328c0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2069  ===..  * *note i
│ │ │ │ +000328d0: 6e76 6572 7365 4d61 703a 2069 6e76 6572  nverseMap: inver
│ │ │ │ +000328e0: 7365 4d61 702c 202d 2d20 696e 7665 7273  seMap, -- invers
│ │ │ │ +000328f0: 6520 6f66 2061 2062 6972 6174 696f 6e61  e of a birationa
│ │ │ │ +00032900: 6c20 6d61 700a 2020 2a20 2a6e 6f74 6520  l map.  * *note 
│ │ │ │ +00032910: 696e 7665 7273 6528 5261 7469 6f6e 616c  inverse(Rational
│ │ │ │ +00032920: 4d61 7029 3a20 696e 7665 7273 655f 6c70  Map): inverse_lp
│ │ │ │ +00032930: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ +00032940: 2d2d 2069 6e76 6572 7365 206f 6620 610a  -- inverse of a.
│ │ │ │ +00032950: 2020 2020 6269 7261 7469 6f6e 616c 206d      birational m
│ │ │ │ +00032960: 6170 0a0a 5761 7973 2074 6f20 7573 6520  ap..Ways to use 
│ │ │ │ +00032970: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ +00032980: 7365 4d61 703a 0a3d 3d3d 3d3d 3d3d 3d3d  seMap:.=========
│ │ │ │  00032990: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000329a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -000329b0: 0a20 202a 2022 6170 7072 6f78 696d 6174  .  * "approximat
│ │ │ │ -000329c0: 6549 6e76 6572 7365 4d61 7028 5261 7469  eInverseMap(Rati
│ │ │ │ -000329d0: 6f6e 616c 4d61 7029 220a 2020 2a20 2261  onalMap)".  * "a
│ │ │ │ -000329e0: 7070 726f 7869 6d61 7465 496e 7665 7273  pproximateInvers
│ │ │ │ -000329f0: 654d 6170 2852 6174 696f 6e61 6c4d 6170  eMap(RationalMap
│ │ │ │ -00032a00: 2c5a 5a29 220a 2020 2a20 2261 7070 726f  ,ZZ)".  * "appro
│ │ │ │ -00032a10: 7869 6d61 7465 496e 7665 7273 654d 6170  ximateInverseMap
│ │ │ │ -00032a20: 2852 696e 674d 6170 2922 0a20 202a 2022  (RingMap)".  * "
│ │ │ │ -00032a30: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ -00032a40: 7365 4d61 7028 5269 6e67 4d61 702c 5a5a  seMap(RingMap,ZZ
│ │ │ │ -00032a50: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ -00032a60: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ -00032a70: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ -00032a80: 626a 6563 7420 2a6e 6f74 6520 6170 7072  bject *note appr
│ │ │ │ -00032a90: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ -00032aa0: 703a 2061 7070 726f 7869 6d61 7465 496e  p: approximateIn
│ │ │ │ -00032ab0: 7665 7273 654d 6170 2c20 6973 2061 202a  verseMap, is a *
│ │ │ │ -00032ac0: 6e6f 7465 0a6d 6574 686f 6420 6675 6e63  note.method func
│ │ │ │ -00032ad0: 7469 6f6e 2077 6974 6820 6f70 7469 6f6e  tion with option
│ │ │ │ -00032ae0: 733a 2028 4d61 6361 756c 6179 3244 6f63  s: (Macaulay2Doc
│ │ │ │ -00032af0: 294d 6574 686f 6446 756e 6374 696f 6e57  )MethodFunctionW
│ │ │ │ -00032b00: 6974 684f 7074 696f 6e73 2c2e 0a0a 2d2d  ithOptions,...--
│ │ │ │ +000329a0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ +000329b0: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ +000329c0: 7365 4d61 7028 5261 7469 6f6e 616c 4d61  seMap(RationalMa
│ │ │ │ +000329d0: 7029 220a 2020 2a20 2261 7070 726f 7869  p)".  * "approxi
│ │ │ │ +000329e0: 6d61 7465 496e 7665 7273 654d 6170 2852  mateInverseMap(R
│ │ │ │ +000329f0: 6174 696f 6e61 6c4d 6170 2c5a 5a29 220a  ationalMap,ZZ)".
│ │ │ │ +00032a00: 2020 2a20 2261 7070 726f 7869 6d61 7465    * "approximate
│ │ │ │ +00032a10: 496e 7665 7273 654d 6170 2852 696e 674d  InverseMap(RingM
│ │ │ │ +00032a20: 6170 2922 0a20 202a 2022 6170 7072 6f78  ap)".  * "approx
│ │ │ │ +00032a30: 696d 6174 6549 6e76 6572 7365 4d61 7028  imateInverseMap(
│ │ │ │ +00032a40: 5269 6e67 4d61 702c 5a5a 2922 0a0a 466f  RingMap,ZZ)"..Fo
│ │ │ │ +00032a50: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +00032a60: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +00032a70: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ +00032a80: 2a6e 6f74 6520 6170 7072 6f78 696d 6174  *note approximat
│ │ │ │ +00032a90: 6549 6e76 6572 7365 4d61 703a 2061 7070  eInverseMap: app
│ │ │ │ +00032aa0: 726f 7869 6d61 7465 496e 7665 7273 654d  roximateInverseM
│ │ │ │ +00032ab0: 6170 2c20 6973 2061 202a 6e6f 7465 0a6d  ap, is a *note.m
│ │ │ │ +00032ac0: 6574 686f 6420 6675 6e63 7469 6f6e 2077  ethod function w
│ │ │ │ +00032ad0: 6974 6820 6f70 7469 6f6e 733a 2028 4d61  ith options: (Ma
│ │ │ │ +00032ae0: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ +00032af0: 6446 756e 6374 696f 6e57 6974 684f 7074  dFunctionWithOpt
│ │ │ │ +00032b00: 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d  ions,...--------
│ │ │ │  00032b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ -00032b60: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ -00032b70: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ -00032b80: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ -00032b90: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ -00032ba0: 6179 322d 312e 3235 2e31 312b 6473 2f4d  ay2-1.25.11+ds/M
│ │ │ │ -00032bb0: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ -00032bc0: 6167 6573 2f43 7265 6d6f 6e61 2f0a 646f  ages/Cremona/.do
│ │ │ │ -00032bd0: 6375 6d65 6e74 6174 696f 6e2e 6d32 3a32  cumentation.m2:2
│ │ │ │ -00032be0: 3237 3a30 2e0a 1f0a 4669 6c65 3a20 4372  27:0....File: Cr
│ │ │ │ -00032bf0: 656d 6f6e 612e 696e 666f 2c20 4e6f 6465  emona.info, Node
│ │ │ │ -00032c00: 3a20 426c 6f77 5570 5374 7261 7465 6779  : BlowUpStrategy
│ │ │ │ -00032c10: 2c20 4e65 7874 3a20 4365 7274 6966 792c  , Next: Certify,
│ │ │ │ -00032c20: 2050 7265 763a 2061 7070 726f 7869 6d61   Prev: approxima
│ │ │ │ -00032c30: 7465 496e 7665 7273 654d 6170 2c20 5570  teInverseMap, Up
│ │ │ │ -00032c40: 3a20 546f 700a 0a42 6c6f 7755 7053 7472  : Top..BlowUpStr
│ │ │ │ -00032c50: 6174 6567 790a 2a2a 2a2a 2a2a 2a2a 2a2a  ategy.**********
│ │ │ │ -00032c60: 2a2a 2a2a 0a0a 4465 7363 7269 7074 696f  ****..Descriptio
│ │ │ │ -00032c70: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  n.===========..T
│ │ │ │ -00032c80: 6869 7320 6973 2061 6e20 6f70 7469 6f6e  his is an option
│ │ │ │ -00032c90: 616c 2061 7267 756d 656e 7420 666f 7220  al argument for 
│ │ │ │ -00032ca0: 2a6e 6f74 6520 6772 6170 683a 2067 7261  *note graph: gra
│ │ │ │ -00032cb0: 7068 2c2c 2061 6e64 2066 6f72 2074 6865  ph,, and for the
│ │ │ │ -00032cc0: 206d 6574 686f 6473 2074 6861 740a 6576   methods that.ev
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│ │ │ │ -00032ce0: 2e20 4375 7272 656e 746c 792c 2074 6865  . Currently, the
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│ │ │ │ -00032d00: 2061 7265 2022 456c 696d 696e 6174 6522   are "Eliminate"
│ │ │ │ -00032d10: 2061 6e64 0a22 5361 7475 7261 7465 222c   and."Saturate",
│ │ │ │ -00032d20: 2077 6869 6368 2069 6e64 6963 6174 6520   which indicate 
│ │ │ │ -00032d30: 7477 6f20 6469 6666 6572 656e 7420 7761  two different wa
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│ │ │ │ -00032d50: 7468 6520 2863 6c6f 7375 7265 206f 6620  the (closure of 
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│ │ │ │ -00032d70: 7261 7469 6f6e 616c 206d 6170 2e20 5468  rational map. Th
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│ │ │ │ -00032d90: 2069 7320 2245 6c69 6d69 6e61 7465 2220   is "Eliminate" 
│ │ │ │ -00032da0: 616e 6420 7468 6973 2069 730a 6765 6e65  and this is.gene
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│ │ │ │ -00032dc0: 2e0a 0a46 756e 6374 696f 6e73 2077 6974  ...Functions wit
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│ │ │ │ -00032de0: 656e 7420 6e61 6d65 6420 426c 6f77 5570  ent named BlowUp
│ │ │ │ -00032df0: 5374 7261 7465 6779 3a0a 3d3d 3d3d 3d3d  Strategy:.======
│ │ │ │ +00032b50: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
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│ │ │ │ +00032b90: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ +00032ba0: 3235 2e31 312b 6473 2f4d 322f 4d61 6361  25.11+ds/M2/Maca
│ │ │ │ +00032bb0: 756c 6179 322f 7061 636b 6167 6573 2f43  ulay2/packages/C
│ │ │ │ +00032bc0: 7265 6d6f 6e61 2f0a 646f 6375 6d65 6e74  remona/.document
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│ │ │ │ +00032c00: 5570 5374 7261 7465 6779 2c20 4e65 7874  UpStrategy, Next
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│ │ │ │ +00032c20: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
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│ │ │ │ +00032c50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +00032c60: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
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│ │ │ │ +00032c90: 756d 656e 7420 666f 7220 2a6e 6f74 6520  ument for *note 
│ │ │ │ +00032ca0: 6772 6170 683a 2067 7261 7068 2c2c 2061  graph: graph,, a
│ │ │ │ +00032cb0: 6e64 2066 6f72 2074 6865 206d 6574 686f  nd for the metho
│ │ │ │ +00032cc0: 6473 2074 6861 740a 6576 656e 7475 616c  ds that.eventual
│ │ │ │ +00032cd0: 6c79 2063 616c 6c20 6974 2e20 4375 7272  ly call it. Curr
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│ │ │ │ +00032cf0: 626c 6520 7661 6c75 6573 2061 7265 2022  ble values are "
│ │ │ │ +00032d00: 456c 696d 696e 6174 6522 2061 6e64 0a22  Eliminate" and."
│ │ │ │ +00032d10: 5361 7475 7261 7465 222c 2077 6869 6368  Saturate", which
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│ │ │ │ +00032d40: 636f 6d70 7574 696e 6720 7468 6520 2863  computing the (c
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│ │ │ │ +00032d90: 6c69 6d69 6e61 7465 2220 616e 6420 7468  liminate" and th
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│ │ │ │ +00032db0: 7072 6566 6572 6162 6c65 2e0a 0a46 756e  preferable...Fun
│ │ │ │ +00032dc0: 6374 696f 6e73 2077 6974 6820 6f70 7469  ctions with opti
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│ │ │ │ +00032df0: 6779 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  gy:.============
│ │ │ │  00032e00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00032e10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00032e20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00032e30: 0a0a 2020 2a20 2243 6865 726e 5363 6877  ..  * "ChernSchw
│ │ │ │ -00032e40: 6172 747a 4d61 6350 6865 7273 6f6e 282e  artzMacPherson(.
│ │ │ │ -00032e50: 2e2e 2c42 6c6f 7755 7053 7472 6174 6567  ..,BlowUpStrateg
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│ │ │ │ -00032e80: 5570 5374 7261 7465 6779 3d3e 2e2e 2e29  UpStrategy=>...)
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│ │ │ │ -00032f40: 2020 2a20 2270 726f 6a65 6374 6976 6544    * "projectiveD
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│ │ │ │ -00032f70: 0a20 202a 2022 5365 6772 6543 6c61 7373  .  * "SegreClass
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│ │ │ │ -00032fb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00032e20: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +00032e30: 2243 6865 726e 5363 6877 6172 747a 4d61  "ChernSchwartzMa
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│ │ │ │ +00032e80: 7465 6779 3d3e 2e2e 2e29 220a 2020 2a20  tegy=>...)".  * 
│ │ │ │ +00032e90: 2245 756c 6572 4368 6172 6163 7465 7269  "EulerCharacteri
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│ │ │ │ +00032ec0: 202a 2022 6772 6170 6828 2e2e 2e2c 426c   * "graph(...,Bl
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│ │ │ │ +00032ef0: 4d61 7028 2e2e 2e2c 426c 6f77 5570 5374  Map(...,BlowUpSt
│ │ │ │ +00032f00: 7261 7465 6779 3d3e 2e2e 2e29 220a 2020  rategy=>...)".  
│ │ │ │ +00032f10: 2a20 2269 7342 6972 6174 696f 6e61 6c28  * "isBirational(
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│ │ │ │ +00032f90: 2e2e 2922 0a0a 466f 7220 7468 6520 7072  ..)"..For the pr
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│ │ │ │ -00033140: 2042 6c6f 7755 7053 7472 6174 6567 792c   BlowUpStrategy,
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│ │ │ │ +00033120: 6e53 6368 7761 7274 7a4d 6163 5068 6572  nSchwartzMacPher
│ │ │ │ +00033130: 736f 6e2c 2050 7265 763a 2042 6c6f 7755  son, Prev: BlowU
│ │ │ │ +00033140: 7053 7472 6174 6567 792c 2055 703a 2054  pStrategy, Up: T
│ │ │ │ +00033150: 6f70 0a0a 4365 7274 6966 7920 2d2d 2077  op..Certify -- w
│ │ │ │ +00033160: 6865 7468 6572 2074 6f20 656e 7375 7265  hether to ensure
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│ │ │ │  00033190: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000331a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00033230: 2e0a 0a49 6620 7475 726e 6564 206f 6e20  ...If turned on 
│ │ │ │ -00033240: 696e 2074 6865 206d 6574 686f 6473 202a  in the methods *
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│ │ │ │ +00033420: 202a 6e6f 7465 0a69 7344 6f6d 696e 616e   *note.isDominan
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│ │ │ │ +00033440: 2a6e 6f74 6520 5365 6772 6543 6c61 7373  *note SegreClass
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│ │ │ │ +00033480: 6861 7261 6374 6572 6973 7469 632c 2061  haracteristic, a
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│ │ │ │  00033540: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00033550: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00033640: 2020 2a20 2269 6e76 6572 7365 4d61 7028    * "inverseMap(
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│ │ │ │ -000336b0: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
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│ │ │ │ -000336d0: 2e2e 2922 0a20 202a 2022 5365 6772 6543  ..)".  * "SegreC
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│ │ │ │ -000336f0: 3d3e 2e2e 2e29 220a 0a46 6f72 2074 6865  =>...)"..For the
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│ │ │ │ -00033710: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -00033720: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ -00033730: 2043 6572 7469 6679 3a20 4365 7274 6966   Certify: Certif
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│ │ │ │ -00033760: 446f 6329 5379 6d62 6f6c 2c2e 0a0a 2d2d  Doc)Symbol,...--
│ │ │ │ +00033560: 3d3d 3d0a 0a20 202a 2022 6170 7072 6f78  ===..  * "approx
│ │ │ │ +00033570: 696d 6174 6549 6e76 6572 7365 4d61 7028  imateInverseMap(
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│ │ │ │ +00033590: 2922 0a20 202a 2022 4368 6572 6e53 6368  )".  * "ChernSch
│ │ │ │ +000335a0: 7761 7274 7a4d 6163 5068 6572 736f 6e28  wartzMacPherson(
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│ │ │ │ +000335c0: 2922 0a20 202a 2022 6465 6772 6565 4d61  )".  * "degreeMa
│ │ │ │ +000335d0: 7028 2e2e 2e2c 4365 7274 6966 793d 3e2e  p(...,Certify=>.
│ │ │ │ +000335e0: 2e2e 2922 0a20 202a 2022 4575 6c65 7243  ..)".  * "EulerC
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│ │ │ │ +00033610: 0a20 202a 2022 6578 6365 7074 696f 6e61  .  * "exceptiona
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│ │ │ │ +00033660: 2022 6973 4269 7261 7469 6f6e 616c 282e   "isBirational(.
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│ │ │ │ +00033680: 220a 2020 2a20 2269 7344 6f6d 696e 616e  ".  * "isDominan
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│ │ │ │ +000336a0: 2e2e 2922 0a20 202a 2022 7072 6f6a 6563  ..)".  * "projec
│ │ │ │ +000336b0: 7469 7665 4465 6772 6565 7328 2e2e 2e2c  tiveDegrees(...,
│ │ │ │ +000336c0: 4365 7274 6966 793d 3e2e 2e2e 2922 0a20  Certify=>...)". 
│ │ │ │ +000336d0: 202a 2022 5365 6772 6543 6c61 7373 282e   * "SegreClass(.
│ │ │ │ +000336e0: 2e2e 2c43 6572 7469 6679 3d3e 2e2e 2e29  ..,Certify=>...)
│ │ │ │ +000336f0: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ +00033700: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ +00033710: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
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│ │ │ │ +00033740: 6120 2a6e 6f74 6520 7379 6d62 6f6c 3a20  a *note symbol: 
│ │ │ │ +00033750: 284d 6163 6175 6c61 7932 446f 6329 5379  (Macaulay2Doc)Sy
│ │ │ │ +00033760: 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d  mbol,...--------
│ │ │ │  00033770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000337a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000337b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ -000337c0: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
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│ │ │ │ -000337f0: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ -00033800: 6179 322d 312e 3235 2e31 312b 6473 2f4d  ay2-1.25.11+ds/M
│ │ │ │ -00033810: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ -00033820: 6167 6573 2f43 7265 6d6f 6e61 2f0a 646f  ages/Cremona/.do
│ │ │ │ -00033830: 6375 6d65 6e74 6174 696f 6e2e 6d32 3a38  cumentation.m2:8
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│ │ │ │ -00033850: 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 653a  mona.info, Node:
│ │ │ │ -00033860: 2043 6865 726e 5363 6877 6172 747a 4d61   ChernSchwartzMa
│ │ │ │ -00033870: 6350 6865 7273 6f6e 2c20 4e65 7874 3a20  cPherson, Next: 
│ │ │ │ -00033880: 436f 6469 6d42 7349 6e76 2c20 5072 6576  CodimBsInv, Prev
│ │ │ │ -00033890: 3a20 4365 7274 6966 792c 2055 703a 2054  : Certify, Up: T
│ │ │ │ -000338a0: 6f70 0a0a 4368 6572 6e53 6368 7761 7274  op..ChernSchwart
│ │ │ │ -000338b0: 7a4d 6163 5068 6572 736f 6e20 2d2d 2043  zMacPherson -- C
│ │ │ │ -000338c0: 6865 726e 2d53 6368 7761 7274 7a2d 4d61  hern-Schwartz-Ma
│ │ │ │ -000338d0: 6350 6865 7273 6f6e 2063 6c61 7373 206f  cPherson class o
│ │ │ │ -000338e0: 6620 6120 7072 6f6a 6563 7469 7665 2073  f a projective s
│ │ │ │ -000338f0: 6368 656d 650a 2a2a 2a2a 2a2a 2a2a 2a2a  cheme.**********
│ │ │ │ +000337b0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ +000337c0: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ +000337d0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ +000337e0: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ +000337f0: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ +00033800: 3235 2e31 312b 6473 2f4d 322f 4d61 6361  25.11+ds/M2/Maca
│ │ │ │ +00033810: 756c 6179 322f 7061 636b 6167 6573 2f43  ulay2/packages/C
│ │ │ │ +00033820: 7265 6d6f 6e61 2f0a 646f 6375 6d65 6e74  remona/.document
│ │ │ │ +00033830: 6174 696f 6e2e 6d32 3a38 333a 302e 0a1f  ation.m2:83:0...
│ │ │ │ +00033840: 0a46 696c 653a 2043 7265 6d6f 6e61 2e69  .File: Cremona.i
│ │ │ │ +00033850: 6e66 6f2c 204e 6f64 653a 2043 6865 726e  nfo, Node: Chern
│ │ │ │ +00033860: 5363 6877 6172 747a 4d61 6350 6865 7273  SchwartzMacPhers
│ │ │ │ +00033870: 6f6e 2c20 4e65 7874 3a20 436f 6469 6d42  on, Next: CodimB
│ │ │ │ +00033880: 7349 6e76 2c20 5072 6576 3a20 4365 7274  sInv, Prev: Cert
│ │ │ │ +00033890: 6966 792c 2055 703a 2054 6f70 0a0a 4368  ify, Up: Top..Ch
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│ │ │ │  00033900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00033910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00033920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00033930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00033990: 202a 6e6f 7465 2069 6465 616c 3a20 284d   *note ideal: (M
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│ │ │ │ -000339e0: 6420 7375 6273 6368 656d 6520 2458 5c73  d subscheme $X\s
│ │ │ │ -000339f0: 7562 7365 745c 6d61 7468 6262 7b50 7d5e  ubset\mathbb{P}^
│ │ │ │ -00033a00: 6e24 0a20 202a 202a 6e6f 7465 204f 7074  n$.  * *note Opt
│ │ │ │ -00033a10: 696f 6e61 6c20 696e 7075 7473 3a20 284d  ional inputs: (M
│ │ │ │ -00033a20: 6163 6175 6c61 7932 446f 6329 7573 696e  acaulay2Doc)usin
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│ │ │ │ -00033a40: 206f 7074 696f 6e61 6c20 696e 7075 7473   optional inputs
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│ │ │ │ -00033a60: 2042 6c6f 7755 7053 7472 6174 6567 793a   BlowUpStrategy:
│ │ │ │ -00033a70: 2042 6c6f 7755 7053 7472 6174 6567 792c   BlowUpStrategy,
│ │ │ │ -00033a80: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ -00033a90: 2076 616c 7565 0a20 2020 2020 2020 2022   value.        "
│ │ │ │ -00033aa0: 456c 696d 696e 6174 6522 2c0a 2020 2020  Eliminate",.    
│ │ │ │ -00033ab0: 2020 2a20 2a6e 6f74 6520 4365 7274 6966    * *note Certif
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│ │ │ │ -00033ad0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
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│ │ │ │ -00033b00: 2020 2063 6f72 7265 6374 6e65 7373 206f     correctness o
│ │ │ │ -00033b10: 6620 6f75 7470 7574 0a20 2020 2020 202a  f output.      *
│ │ │ │ -00033b20: 202a 6e6f 7465 2056 6572 626f 7365 3a20   *note Verbose: 
│ │ │ │ -00033b30: 696e 7665 7273 654d 6170 5f6c 705f 7064  inverseMap_lp_pd
│ │ │ │ -00033b40: 5f70 645f 7064 5f63 6d56 6572 626f 7365  _pd_pd_cmVerbose
│ │ │ │ -00033b50: 3d3e 5f70 645f 7064 5f70 645f 7270 2c20  =>_pd_pd_pd_rp, 
│ │ │ │ -00033b60: 3d3e 202e 2e2e 2c0a 2020 2020 2020 2020  => ...,.        
│ │ │ │ -00033b70: 6465 6661 756c 7420 7661 6c75 6520 7472  default value tr
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│ │ │ │ -00033b90: 0a20 2020 2020 202a 2061 202a 6e6f 7465  .      * a *note
│ │ │ │ -00033ba0: 2072 696e 6720 656c 656d 656e 743a 2028   ring element: (
│ │ │ │ -00033bb0: 4d61 6361 756c 6179 3244 6f63 2952 696e  Macaulay2Doc)Rin
│ │ │ │ -00033bc0: 6745 6c65 6d65 6e74 2c2c 2074 6865 2070  gElement,, the p
│ │ │ │ -00033bd0: 7573 682d 666f 7277 6172 6420 746f 0a20  ush-forward to. 
│ │ │ │ -00033be0: 2020 2020 2020 2074 6865 2043 686f 7720         the Chow 
│ │ │ │ -00033bf0: 7269 6e67 206f 6620 245c 6d61 7468 6262  ring of $\mathbb
│ │ │ │ -00033c00: 7b50 7d5e 6e24 206f 6620 7468 6520 4368  {P}^n$ of the Ch
│ │ │ │ -00033c10: 6572 6e2d 5363 6877 6172 747a 2d4d 6163  ern-Schwartz-Mac
│ │ │ │ -00033c20: 5068 6572 736f 6e20 636c 6173 730a 2020  Pherson class.  
│ │ │ │ -00033c30: 2020 2020 2020 2463 5f7b 534d 7d28 5829        $c_{SM}(X)
│ │ │ │ -00033c40: 2420 6f66 2024 5824 2e20 496e 2070 6172  $ of $X$. In par
│ │ │ │ -00033c50: 7469 6375 6c61 722c 2074 6865 2063 6f65  ticular, the coe
│ │ │ │ -00033c60: 6666 6963 6965 6e74 206f 6620 2448 5e6e  fficient of $H^n
│ │ │ │ -00033c70: 2420 6769 7665 7320 7468 650a 2020 2020  $ gives the.    
│ │ │ │ -00033c80: 2020 2020 4575 6c65 7220 6368 6172 6163      Euler charac
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│ │ │ │ -00033ca0: 7375 7070 6f72 7420 6f66 2024 5824 2c20  support of $X$, 
│ │ │ │ -00033cb0: 7768 6572 6520 2448 2420 6465 6e6f 7465  where $H$ denote
│ │ │ │ -00033cc0: 7320 7468 650a 2020 2020 2020 2020 6879  s the.        hy
│ │ │ │ -00033cd0: 7065 7270 6c61 6e65 2063 6c61 7373 2e0a  perplane class..
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│ │ │ │ -00033d40: 733a 0a70 726f 6a65 6374 6976 6544 6567  s:.projectiveDeg
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│ │ │ │ -00033d70: 6f6d 7075 7469 6e67 2063 6861 7261 6374  omputing charact
│ │ │ │ -00033d80: 6572 6973 7469 6320 636c 6173 7365 7320  eristic classes 
│ │ │ │ -00033d90: 6f66 0a70 726f 6a65 6374 6976 6520 7363  of.projective sc
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│ │ │ │ -00033dc0: 6563 742e 636f 6d2f 7363 6965 6e63 652f  ect.com/science/
│ │ │ │ -00033dd0: 6172 7469 636c 652f 7069 692f 0a53 3037  article/pii/.S07
│ │ │ │ -00033de0: 3437 3731 3731 3032 3030 3038 3935 2029  47717102000895 )
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│ │ │ │ -00033e00: 5365 6520 616c 736f 2074 6865 2063 6f72  See also the cor
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│ │ │ │ -00033e40: 7470 3a2f 2f77 7777 2e6d 6174 682e 6673  tp://www.math.fs
│ │ │ │ -00033e50: 752e 6564 752f 7e61 6c75 6666 692f 4353  u.edu/~aluffi/CS
│ │ │ │ -00033e60: 4d2f 4353 4d2e 6874 6d6c 2029 2c20 6279  M/CSM.html ), by
│ │ │ │ -00033e70: 2050 2e0a 416c 7566 6669 2c20 616e 6420   P..Aluffi, and 
│ │ │ │ -00033e80: 2a6e 6f74 6520 4368 6172 6163 7465 7269  *note Characteri
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│ │ │ │ -00033ea0: 6172 6163 7465 7269 7374 6963 436c 6173  aracteristicClas
│ │ │ │ -00033eb0: 7365 7329 546f 702c 2c20 6279 204d 2e0a  ses)Top,, by M..
│ │ │ │ -00033ec0: 4865 6c6d 6572 2061 6e64 2043 2e20 4a6f  Helmer and C. Jo
│ │ │ │ -00033ed0: 7374 2e0a 0a49 6e20 7468 6520 6578 616d  st...In the exam
│ │ │ │ -00033ee0: 706c 6520 6265 6c6f 772c 2077 6520 636f  ple below, we co
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│ │ │ │ -00033f00: 6f72 7761 7264 2074 6f20 7468 6520 4368  orward to the Ch
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│ │ │ │ -00033f20: 6862 627b 507d 5e34 2420 6f66 2074 6865  hbb{P}^4$ of the
│ │ │ │ -00033f30: 2043 6865 726e 2d53 6368 7761 7274 7a2d   Chern-Schwartz-
│ │ │ │ -00033f40: 4d61 6350 6865 7273 6f6e 2063 6c61 7373  MacPherson class
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│ │ │ │ -00033f60: 7220 7468 650a 7477 6973 7465 6420 6375  r the.twisted cu
│ │ │ │ -00033f70: 6269 6320 6375 7276 652c 2075 7369 6e67  bic curve, using
│ │ │ │ -00033f80: 2062 6f74 6820 6120 7072 6f62 6162 696c   both a probabil
│ │ │ │ -00033f90: 6973 7469 6320 616e 6420 6120 6e6f 6e2d  istic and a non-
│ │ │ │ -00033fa0: 7072 6f62 6162 696c 6973 7469 630a 6170  probabilistic.ap
│ │ │ │ -00033fb0: 7072 6f61 6368 2e0a 0a2b 2d2d 2d2d 2d2d  proach...+------
│ │ │ │ +00033940: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20  *..  * Usage: . 
│ │ │ │ +00033950: 2020 2020 2020 2043 6865 726e 5363 6877         ChernSchw
│ │ │ │ +00033960: 6172 747a 4d61 6350 6865 7273 6f6e 2049  artzMacPherson I
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│ │ │ │ +00033980: 2020 202a 2049 2c20 616e 202a 6e6f 7465     * I, an *note
│ │ │ │ +00033990: 2069 6465 616c 3a20 284d 6163 6175 6c61   ideal: (Macaula
│ │ │ │ +000339a0: 7932 446f 6329 4964 6561 6c2c 2c20 6120  y2Doc)Ideal,, a 
│ │ │ │ +000339b0: 686f 6d6f 6765 6e65 6f75 7320 6964 6561  homogeneous idea
│ │ │ │ +000339c0: 6c20 6465 6669 6e69 6e67 2061 0a20 2020  l defining a.   
│ │ │ │ +000339d0: 2020 2020 2063 6c6f 7365 6420 7375 6273       closed subs
│ │ │ │ +000339e0: 6368 656d 6520 2458 5c73 7562 7365 745c  cheme $X\subset\
│ │ │ │ +000339f0: 6d61 7468 6262 7b50 7d5e 6e24 0a20 202a  mathbb{P}^n$.  *
│ │ │ │ +00033a00: 202a 6e6f 7465 204f 7074 696f 6e61 6c20   *note Optional 
│ │ │ │ +00033a10: 696e 7075 7473 3a20 284d 6163 6175 6c61  inputs: (Macaula
│ │ │ │ +00033a20: 7932 446f 6329 7573 696e 6720 6675 6e63  y2Doc)using func
│ │ │ │ +00033a30: 7469 6f6e 7320 7769 7468 206f 7074 696f  tions with optio
│ │ │ │ +00033a40: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020  nal inputs,:.   
│ │ │ │ +00033a50: 2020 202a 202a 6e6f 7465 2042 6c6f 7755     * *note BlowU
│ │ │ │ +00033a60: 7053 7472 6174 6567 793a 2042 6c6f 7755  pStrategy: BlowU
│ │ │ │ +00033a70: 7053 7472 6174 6567 792c 203d 3e20 2e2e  pStrategy, => ..
│ │ │ │ +00033a80: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ +00033a90: 0a20 2020 2020 2020 2022 456c 696d 696e  .        "Elimin
│ │ │ │ +00033aa0: 6174 6522 2c0a 2020 2020 2020 2a20 2a6e  ate",.      * *n
│ │ │ │ +00033ab0: 6f74 6520 4365 7274 6966 793a 2043 6572  ote Certify: Cer
│ │ │ │ +00033ac0: 7469 6679 2c20 3d3e 202e 2e2e 2c20 6465  tify, => ..., de
│ │ │ │ +00033ad0: 6661 756c 7420 7661 6c75 6520 6661 6c73  fault value fals
│ │ │ │ +00033ae0: 652c 2077 6865 7468 6572 2074 6f20 656e  e, whether to en
│ │ │ │ +00033af0: 7375 7265 0a20 2020 2020 2020 2063 6f72  sure.        cor
│ │ │ │ +00033b00: 7265 6374 6e65 7373 206f 6620 6f75 7470  rectness of outp
│ │ │ │ +00033b10: 7574 0a20 2020 2020 202a 202a 6e6f 7465  ut.      * *note
│ │ │ │ +00033b20: 2056 6572 626f 7365 3a20 696e 7665 7273   Verbose: invers
│ │ │ │ +00033b30: 654d 6170 5f6c 705f 7064 5f70 645f 7064  eMap_lp_pd_pd_pd
│ │ │ │ +00033b40: 5f63 6d56 6572 626f 7365 3d3e 5f70 645f  _cmVerbose=>_pd_
│ │ │ │ +00033b50: 7064 5f70 645f 7270 2c20 3d3e 202e 2e2e  pd_pd_rp, => ...
│ │ │ │ +00033b60: 2c0a 2020 2020 2020 2020 6465 6661 756c  ,.        defaul
│ │ │ │ +00033b70: 7420 7661 6c75 6520 7472 7565 2c0a 2020  t value true,.  
│ │ │ │ +00033b80: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +00033b90: 202a 2061 202a 6e6f 7465 2072 696e 6720   * a *note ring 
│ │ │ │ +00033ba0: 656c 656d 656e 743a 2028 4d61 6361 756c  element: (Macaul
│ │ │ │ +00033bb0: 6179 3244 6f63 2952 696e 6745 6c65 6d65  ay2Doc)RingEleme
│ │ │ │ +00033bc0: 6e74 2c2c 2074 6865 2070 7573 682d 666f  nt,, the push-fo
│ │ │ │ +00033bd0: 7277 6172 6420 746f 0a20 2020 2020 2020  rward to.       
│ │ │ │ +00033be0: 2074 6865 2043 686f 7720 7269 6e67 206f   the Chow ring o
│ │ │ │ +00033bf0: 6620 245c 6d61 7468 6262 7b50 7d5e 6e24  f $\mathbb{P}^n$
│ │ │ │ +00033c00: 206f 6620 7468 6520 4368 6572 6e2d 5363   of the Chern-Sc
│ │ │ │ +00033c10: 6877 6172 747a 2d4d 6163 5068 6572 736f  hwartz-MacPherso
│ │ │ │ +00033c20: 6e20 636c 6173 730a 2020 2020 2020 2020  n class.        
│ │ │ │ +00033c30: 2463 5f7b 534d 7d28 5829 2420 6f66 2024  $c_{SM}(X)$ of $
│ │ │ │ +00033c40: 5824 2e20 496e 2070 6172 7469 6375 6c61  X$. In particula
│ │ │ │ +00033c50: 722c 2074 6865 2063 6f65 6666 6963 6965  r, the coefficie
│ │ │ │ +00033c60: 6e74 206f 6620 2448 5e6e 2420 6769 7665  nt of $H^n$ give
│ │ │ │ +00033c70: 7320 7468 650a 2020 2020 2020 2020 4575  s the.        Eu
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│ │ │ │ +00033c90: 6963 206f 6620 7468 6520 7375 7070 6f72  ic of the suppor
│ │ │ │ +00033ca0: 7420 6f66 2024 5824 2c20 7768 6572 6520  t of $X$, where 
│ │ │ │ +00033cb0: 2448 2420 6465 6e6f 7465 7320 7468 650a  $H$ denotes the.
│ │ │ │ +00033cc0: 2020 2020 2020 2020 6879 7065 7270 6c61          hyperpla
│ │ │ │ +00033cd0: 6e65 2063 6c61 7373 2e0a 0a44 6573 6372  ne class...Descr
│ │ │ │ +00033ce0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +00033cf0: 3d3d 0a0a 5468 6973 2069 7320 616e 2065  ==..This is an e
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│ │ │ │ +00033d50: 2064 7565 2074 6f20 7265 7375 6c74 7320   due to results 
│ │ │ │ +00033d60: 7368 6f77 6e20 696e 2043 6f6d 7075 7469  shown in Computi
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│ │ │ │ +00033d90: 6a65 6374 6976 6520 7363 6865 6d65 7320  jective schemes 
│ │ │ │ +00033da0: 2873 6565 2068 7474 703a 2f2f 7777 772e  (see http://www.
│ │ │ │ +00033db0: 7363 6965 6e63 6564 6972 6563 742e 636f  sciencedirect.co
│ │ │ │ +00033dc0: 6d2f 7363 6965 6e63 652f 6172 7469 636c  m/science/articl
│ │ │ │ +00033dd0: 652f 7069 692f 0a53 3037 3437 3731 3731  e/pii/.S07477171
│ │ │ │ +00033de0: 3032 3030 3038 3935 2029 2c20 6279 2050  02000895 ), by P
│ │ │ │ +00033df0: 2e20 416c 7566 6669 2e20 5365 6520 616c  . Aluffi. See al
│ │ │ │ +00033e00: 736f 2074 6865 2063 6f72 7265 7370 6f6e  so the correspon
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│ │ │ │ +00033e20: 7468 650a 7061 636b 6167 6573 2043 534d  the.packages CSM
│ │ │ │ +00033e30: 2d41 2028 7365 6520 6874 7470 3a2f 2f77  -A (see http://w
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│ │ │ │ +00033e50: 7e61 6c75 6666 692f 4353 4d2f 4353 4d2e  ~aluffi/CSM/CSM.
│ │ │ │ +00033e60: 6874 6d6c 2029 2c20 6279 2050 2e0a 416c  html ), by P..Al
│ │ │ │ +00033e70: 7566 6669 2c20 616e 6420 2a6e 6f74 6520  uffi, and *note 
│ │ │ │ +00033e80: 4368 6172 6163 7465 7269 7374 6963 436c  CharacteristicCl
│ │ │ │ +00033e90: 6173 7365 733a 2028 4368 6172 6163 7465  asses: (Characte
│ │ │ │ +00033ea0: 7269 7374 6963 436c 6173 7365 7329 546f  risticClasses)To
│ │ │ │ +00033eb0: 702c 2c20 6279 204d 2e0a 4865 6c6d 6572  p,, by M..Helmer
│ │ │ │ +00033ec0: 2061 6e64 2043 2e20 4a6f 7374 2e0a 0a49   and C. Jost...I
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│ │ │ │ +00033ef0: 7468 6520 7075 7368 2d66 6f72 7761 7264  the push-forward
│ │ │ │ +00033f00: 2074 6f20 7468 6520 4368 6f77 2072 696e   to the Chow rin
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│ │ │ │ +00033f20: 5e34 2420 6f66 2074 6865 2043 6865 726e  ^4$ of the Chern
│ │ │ │ +00033f30: 2d53 6368 7761 7274 7a2d 4d61 6350 6865  -Schwartz-MacPhe
│ │ │ │ +00033f40: 7273 6f6e 2063 6c61 7373 206f 6620 7468  rson class of th
│ │ │ │ +00033f50: 6520 636f 6e65 206f 7665 7220 7468 650a  e cone over the.
│ │ │ │ +00033f60: 7477 6973 7465 6420 6375 6269 6320 6375  twisted cubic cu
│ │ │ │ +00033f70: 7276 652c 2075 7369 6e67 2062 6f74 6820  rve, using both 
│ │ │ │ +00033f80: 6120 7072 6f62 6162 696c 6973 7469 6320  a probabilistic 
│ │ │ │ +00033f90: 616e 6420 6120 6e6f 6e2d 7072 6f62 6162  and a non-probab
│ │ │ │ +00033fa0: 696c 6973 7469 630a 6170 7072 6f61 6368  ilistic.approach
│ │ │ │ +00033fb0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │  00033fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033ff0: 2b0a 7c69 3120 3a20 4746 2835 5e37 295b  +.|i1 : GF(5^7)[
│ │ │ │ -00034000: 785f 302e 2e78 5f34 5d20 2020 2020 2020  x_0..x_4]       
│ │ │ │ +00033fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ +00033ff0: 3a20 4746 2835 5e37 295b 785f 302e 2e78  : GF(5^7)[x_0..x
│ │ │ │ +00034000: 5f34 5d20 2020 2020 2020 2020 2020 2020  _4]             
│ │ │ │  00034010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034020: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00034020: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00034030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034060: 2020 7c0a 7c6f 3120 3d20 4746 2037 3831    |.|o1 = GF 781
│ │ │ │ -00034070: 3235 5b78 202e 2e78 205d 2020 2020 2020  25[x ..x ]      
│ │ │ │ +00034050: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00034060: 3120 3d20 4746 2037 3831 3235 5b78 202e  1 = GF 78125[x .
│ │ │ │ +00034070: 2e78 205d 2020 2020 2020 2020 2020 2020  .x ]            
│ │ │ │  00034080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034090: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000340a0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000340b0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -000340c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000340d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00034090: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000340a0: 2020 2020 2020 2030 2020 2034 2020 2020         0   4    
│ │ │ │ +000340b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000340c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000340d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000340e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000340f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034100: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00034110: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52  o1 : PolynomialR
│ │ │ │ -00034120: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +00034100: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ +00034110: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +00034120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034140: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00034140: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00034150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00034180: 0a7c 6932 203a 2043 203d 206d 696e 6f72  .|i2 : C = minor
│ │ │ │ -00034190: 7328 322c 6d61 7472 6978 7b7b 785f 302c  s(2,matrix{{x_0,
│ │ │ │ -000341a0: 785f 312c 785f 327d 2c7b 785f 312c 785f  x_1,x_2},{x_1,x_
│ │ │ │ -000341b0: 322c 785f 337d 7d29 7c0a 7c20 2020 2020  2,x_3}})|.|     
│ │ │ │ +00034170: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ +00034180: 2043 203d 206d 696e 6f72 7328 322c 6d61   C = minors(2,ma
│ │ │ │ +00034190: 7472 6978 7b7b 785f 302c 785f 312c 785f  trix{{x_0,x_1,x_
│ │ │ │ +000341a0: 327d 2c7b 785f 312c 785f 322c 785f 337d  2},{x_1,x_2,x_3}
│ │ │ │ +000341b0: 7d29 7c0a 7c20 2020 2020 2020 2020 2020  })|.|           
│ │ │ │  000341c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000341d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000341e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000341f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00034200: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00034210: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00034220: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -00034230: 3d20 6964 6561 6c20 282d 2078 2020 2b20  = ideal (- x  + 
│ │ │ │ -00034240: 7820 7820 2c20 2d20 7820 7820 202b 2078  x x , - x x  + x
│ │ │ │ -00034250: 2078 202c 202d 2078 2020 2b20 7820 7820   x , - x  + x x 
│ │ │ │ -00034260: 2920 207c 0a7c 2020 2020 2020 2020 2020  )  |.|          
│ │ │ │ -00034270: 2020 2020 2031 2020 2020 3020 3220 2020       1    0 2   
│ │ │ │ -00034280: 2020 3120 3220 2020 2030 2033 2020 2020    1 2    0 3    
│ │ │ │ -00034290: 2032 2020 2020 3120 3320 2020 7c0a 7c20   2    1 3   |.| 
│ │ │ │ +000341e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000341f0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00034200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034210: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00034220: 2020 2020 7c0a 7c6f 3220 3d20 6964 6561      |.|o2 = idea
│ │ │ │ +00034230: 6c20 282d 2078 2020 2b20 7820 7820 2c20  l (- x  + x x , 
│ │ │ │ +00034240: 2d20 7820 7820 202b 2078 2078 202c 202d  - x x  + x x , -
│ │ │ │ +00034250: 2078 2020 2b20 7820 7820 2920 207c 0a7c   x  + x x )  |.|
│ │ │ │ +00034260: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00034270: 2020 2020 3020 3220 2020 2020 3120 3220      0 2     1 2 
│ │ │ │ +00034280: 2020 2030 2033 2020 2020 2032 2020 2020     0 3     2    
│ │ │ │ +00034290: 3120 3320 2020 7c0a 7c20 2020 2020 2020  1 3   |.|       
│ │ │ │  000342a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000342b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000342c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000342d0: 2020 2020 207c 0a7c 6f32 203a 2049 6465       |.|o2 : Ide
│ │ │ │ -000342e0: 616c 206f 6620 4746 2037 3831 3235 5b78  al of GF 78125[x
│ │ │ │ -000342f0: 202e 2e78 205d 2020 2020 2020 2020 2020   ..x ]          
│ │ │ │ -00034300: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034310: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00034320: 2020 2020 2020 2020 2030 2020 2034 2020           0   4  
│ │ │ │ +000342c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000342d0: 0a7c 6f32 203a 2049 6465 616c 206f 6620  .|o2 : Ideal of 
│ │ │ │ +000342e0: 4746 2037 3831 3235 5b78 202e 2e78 205d  GF 78125[x ..x ]
│ │ │ │ +000342f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034300: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034320: 2020 2030 2020 2034 2020 2020 2020 2020     0   4        
│ │ │ │  00034330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034340: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00034340: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  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 2032 2e32 3538 3632 7320 2863  used 2.25862s (c
│ │ │ │ -000343d0: 7075 293b 2031 2e31 3833 3232 7320 2874  pu); 1.18322s (t
│ │ │ │ -000343e0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ -000343f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00034370: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ +00034380: 3a20 7469 6d65 2043 6865 726e 5363 6877  : time ChernSchw
│ │ │ │ +00034390: 6172 747a 4d61 6350 6865 7273 6f6e 2043  artzMacPherson C
│ │ │ │ +000343a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000343b0: 2020 207c 0a7c 202d 2d20 7573 6564 2032     |.| -- used 2
│ │ │ │ +000343c0: 2e35 3433 3433 7320 2863 7075 293b 2031  .54343s (cpu); 1
│ │ │ │ +000343d0: 2e33 3433 3034 7320 2874 6872 6561 6429  .34304s (thread)
│ │ │ │ +000343e0: 3b20 3073 2028 6763 2920 2020 7c0a 7c20  ; 0s (gc)   |.| 
│ │ │ │ +000343f0: 2020 2020 2020 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                  
│ │ │ │ -00034460: 2020 2020 7c0a 7c6f 3320 3d20 3348 2020      |.|o3 = 3H  
│ │ │ │ -00034470: 2b20 3548 2020 2b20 3348 2020 2020 2020  + 5H  + 3H      
│ │ │ │ +00034420: 2020 2020 207c 0a7c 2020 2020 2020 2034       |.|       4
│ │ │ │ +00034430: 2020 2020 2033 2020 2020 2032 2020 2020       3     2    
│ │ │ │ +00034440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034450: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00034460: 7c6f 3320 3d20 3348 2020 2b20 3548 2020  |o3 = 3H  + 5H  
│ │ │ │ +00034470: 2b20 3348 2020 2020 2020 2020 2020 2020  + 3H            
│ │ │ │  00034480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034490: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034490: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000344a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000344b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000344c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000344d0: 2020 2020 2020 7c0a 7c20 2020 2020 5a5a        |.|     ZZ
│ │ │ │ -000344e0: 5b48 5d20 2020 2020 2020 2020 2020 2020  [H]             
│ │ │ │ +000344d0: 7c0a 7c20 2020 2020 5a5a 5b48 5d20 2020  |.|     ZZ[H]   
│ │ │ │ +000344e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000344f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00034510: 0a7c 6f33 203a 202d 2d2d 2d2d 2020 2020  .|o3 : -----    
│ │ │ │ +00034500: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ +00034510: 202d 2d2d 2d2d 2020 2020 2020 2020 2020   -----          
│ │ │ │  00034520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034540: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00034550: 2020 2035 2020 2020 2020 2020 2020 2020     5            
│ │ │ │ +00034540: 2020 7c0a 7c20 2020 2020 2020 2035 2020    |.|        5  
│ │ │ │ +00034550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034580: 207c 0a7c 2020 2020 2020 2048 2020 2020   |.|       H    
│ │ │ │ +00034570: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00034580: 2020 2020 2048 2020 2020 2020 2020 2020       H          
│ │ │ │  00034590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000345a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000345b0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000345b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  000345c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000345d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000345e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000345f0: 2d2d 2d2b 0a7c 6934 203a 2074 696d 6520  ---+.|i4 : time 
│ │ │ │ -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 2e36 3137 3834 7320 2863 7075 293b   1.61784s (cpu);
│ │ │ │ -00034680: 2031 2e31 3430 3331 7320 2874 6872 6561   1.14031s (threa
│ │ │ │ -00034690: 6429 3b20 3073 2028 6763 2920 2020 7c0a  d); 0s (gc)   |.
│ │ │ │ -000346a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000345e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000345f0: 6934 203a 2074 696d 6520 4368 6572 6e53  i4 : time ChernS
│ │ │ │ +00034600: 6368 7761 7274 7a4d 6163 5068 6572 736f  chwartzMacPherso
│ │ │ │ +00034610: 6e28 432c 4365 7274 6966 793d 3e74 7275  n(C,Certify=>tru
│ │ │ │ +00034620: 6529 2020 2020 7c0a 7c43 6572 7469 6679  e)    |.|Certify
│ │ │ │ +00034630: 3a20 6f75 7470 7574 2063 6572 7469 6669  : output certifi
│ │ │ │ +00034640: 6564 2120 2020 2020 2020 2020 2020 2020  ed!             
│ │ │ │ +00034650: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00034660: 0a7c 202d 2d20 7573 6564 2031 2e35 3338  .| -- used 1.538
│ │ │ │ +00034670: 3132 7320 2863 7075 293b 2031 2e30 3533  12s (cpu); 1.053
│ │ │ │ +00034680: 3034 7320 2874 6872 6561 6429 3b20 3073  04s (thread); 0s
│ │ │ │ +00034690: 2028 6763 2920 2020 7c0a 7c20 2020 2020   (gc)   |.|     
│ │ │ │ +000346a0: 2020 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  
│ │ │ │ +000346d0: 207c 0a7c 2020 2020 2020 2034 2020 2020   |.|       4    
│ │ │ │ +000346e0: 2033 2020 2020 2032 2020 2020 2020 2020   3     2        
│ │ │ │  000346f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034710: 7c0a 7c6f 3420 3d20 3348 2020 2b20 3548  |.|o4 = 3H  + 5H
│ │ │ │ -00034720: 2020 2b20 3348 2020 2020 2020 2020 2020    + 3H          
│ │ │ │ +00034700: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ +00034710: 3d20 3348 2020 2b20 3548 2020 2b20 3348  = 3H  + 5H  + 3H
│ │ │ │ +00034720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034740: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00034740: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00034750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034780: 2020 7c0a 7c20 2020 2020 5a5a 5b48 5d20    |.|     ZZ[H] 
│ │ │ │ +00034770: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00034780: 2020 2020 5a5a 5b48 5d20 2020 2020 2020      ZZ[H]       
│ │ │ │  00034790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000347a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000347b0: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ -000347c0: 203a 202d 2d2d 2d2d 2020 2020 2020 2020   : -----        
│ │ │ │ +000347b0: 2020 2020 207c 0a7c 6f34 203a 202d 2d2d       |.|o4 : ---
│ │ │ │ +000347c0: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │  000347d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000347e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000347f0: 2020 2020 7c0a 7c20 2020 2020 2020 2035      |.|        5
│ │ │ │ +000347e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000347f0: 7c20 2020 2020 2020 2035 2020 2020 2020  |        5      
│ │ │ │  00034800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034820: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00034830: 2020 2020 2020 2048 2020 2020 2020 2020         H        
│ │ │ │ +00034820: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00034830: 2048 2020 2020 2020 2020 2020 2020 2020   H              
│ │ │ │  00034840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034860: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00034860: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00034870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000348a0: 0a7c 6935 203a 206f 6f20 3d3d 206f 6f6f  .|i5 : oo == ooo
│ │ │ │ +00034890: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ +000348a0: 206f 6f20 3d3d 206f 6f6f 2020 2020 2020   oo == ooo      
│ │ │ │  000348b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000348c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000348d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000348d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000348e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000348f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034910: 207c 0a7c 6f35 203d 2074 7275 6520 2020   |.|o5 = true   
│ │ │ │ +00034900: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +00034910: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │  00034920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034940: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00034940: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00034950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000349f0: 2420 6361 6e20 6265 2063 6f6d 7075 7465  $ can be compute
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│ │ │ │ -00034a80: 636c 6173 7320 6f66 2074 6865 2074 616e  class of the tan
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│ │ │ │ -00034aa0: 5824 2061 6e64 2063 616e 2062 6520 6f62  X$ and can be ob
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│ │ │ │ -00034ac0: 730a 2869 6e20 6765 6e65 7261 6c20 7468  s.(in general th
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│ │ │ │ -00034af0: 7277 6172 6420 6f66 2074 6865 2073 6f2d  rward of the so-
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│ │ │ │ +00034970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +00034980: 496e 2074 6865 2063 6173 6520 7768 656e  In the case when
│ │ │ │ +00034990: 2074 6865 2069 6e70 7574 2069 6465 616c   the input ideal
│ │ │ │ +000349a0: 2049 2064 6566 696e 6573 2061 2073 6d6f   I defines a smo
│ │ │ │ +000349b0: 6f74 6820 7072 6f6a 6563 7469 7665 2076  oth projective v
│ │ │ │ +000349c0: 6172 6965 7479 2024 5824 2c20 7468 650a  ariety $X$, the.
│ │ │ │ +000349d0: 7075 7368 2d66 6f72 7761 7264 206f 6620  push-forward of 
│ │ │ │ +000349e0: 2463 5f7b 534d 7d28 5829 2420 6361 6e20  $c_{SM}(X)$ can 
│ │ │ │ +000349f0: 6265 2063 6f6d 7075 7465 6420 6d75 6368  be computed much
│ │ │ │ +00034a00: 206d 6f72 6520 6566 6669 6369 656e 746c   more efficientl
│ │ │ │ +00034a10: 7920 7573 696e 6720 2a6e 6f74 650a 5365  y using *note.Se
│ │ │ │ +00034a20: 6772 6543 6c61 7373 3a20 5365 6772 6543  greClass: SegreC
│ │ │ │ +00034a30: 6c61 7373 2c2e 2049 6e64 6565 642c 2069  lass,. Indeed, i
│ │ │ │ +00034a40: 6e20 7468 6973 2063 6173 652c 2024 635f  n this case, $c_
│ │ │ │ +00034a50: 7b53 4d7d 2858 2924 2063 6f69 6e63 6964  {SM}(X)$ coincid
│ │ │ │ +00034a60: 6573 2077 6974 6820 7468 650a 2874 6f74  es with the.(tot
│ │ │ │ +00034a70: 616c 2920 4368 6572 6e20 636c 6173 7320  al) Chern class 
│ │ │ │ +00034a80: 6f66 2074 6865 2074 616e 6765 6e74 2062  of the tangent b
│ │ │ │ +00034a90: 756e 646c 6520 6f66 2024 5824 2061 6e64  undle of $X$ and
│ │ │ │ +00034aa0: 2063 616e 2062 6520 6f62 7461 696e 6564   can be obtained
│ │ │ │ +00034ab0: 2061 7320 666f 6c6c 6f77 730a 2869 6e20   as follows.(in 
│ │ │ │ +00034ac0: 6765 6e65 7261 6c20 7468 6520 6d65 7468  general the meth
│ │ │ │ +00034ad0: 6f64 2062 656c 6f77 2067 6976 6573 2074  od below gives t
│ │ │ │ +00034ae0: 6865 2070 7573 682d 666f 7277 6172 6420  he push-forward 
│ │ │ │ +00034af0: 6f66 2074 6865 2073 6f2d 6361 6c6c 6564  of the so-called
│ │ │ │ +00034b00: 0a43 6865 726e 2d46 756c 746f 6e20 636c  .Chern-Fulton cl
│ │ │ │ +00034b10: 6173 7329 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  ass)...+--------
│ │ │ │  00034b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ -00034b70: 203a 2043 6865 726e 436c 6173 7320 3d20   : ChernClass = 
│ │ │ │ -00034b80: 6d65 7468 6f64 284f 7074 696f 6e73 3d3e  method(Options=>
│ │ │ │ -00034b90: 7b43 6572 7469 6679 3d3e 6661 6c73 657d  {Certify=>false}
│ │ │ │ -00034ba0: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ -00034bb0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00034b60: 2d2d 2d2d 2d2b 0a7c 6936 203a 2043 6865  -----+.|i6 : Che
│ │ │ │ +00034b70: 726e 436c 6173 7320 3d20 6d65 7468 6f64  rnClass = method
│ │ │ │ +00034b80: 284f 7074 696f 6e73 3d3e 7b43 6572 7469  (Options=>{Certi
│ │ │ │ +00034b90: 6679 3d3e 6661 6c73 657d 293b 2020 2020  fy=>false});    
│ │ │ │ +00034ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034bb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00034bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ -00034c10: 203a 2043 6865 726e 436c 6173 7320 2849   : ChernClass (I
│ │ │ │ -00034c20: 6465 616c 2920 3a3d 206f 202d 3e20 2849  deal) := o -> (I
│ │ │ │ -00034c30: 2920 2d3e 2028 2020 2020 2020 2020 2020  ) -> (          
│ │ │ │ +00034c00: 2d2d 2d2d 2d2b 0a7c 6937 203a 2043 6865  -----+.|i7 : Che
│ │ │ │ +00034c10: 726e 436c 6173 7320 2849 6465 616c 2920  rnClass (Ideal) 
│ │ │ │ +00034c20: 3a3d 206f 202d 3e20 2849 2920 2d3e 2028  := o -> (I) -> (
│ │ │ │ +00034c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034c50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00034c60: 2020 2020 2020 7320 3a3d 2053 6567 7265        s := Segre
│ │ │ │ -00034c70: 436c 6173 7328 492c 4365 7274 6966 793d  Class(I,Certify=
│ │ │ │ -00034c80: 3e6f 2e43 6572 7469 6679 293b 2020 2020  >o.Certify);    
│ │ │ │ +00034c50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00034c60: 7320 3a3d 2053 6567 7265 436c 6173 7328  s := SegreClass(
│ │ │ │ +00034c70: 492c 4365 7274 6966 793d 3e6f 2e43 6572  I,Certify=>o.Cer
│ │ │ │ +00034c80: 7469 6679 293b 2020 2020 2020 2020 2020  tify);          
│ │ │ │  00034c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034ca0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00034cb0: 2020 2020 2020 732a 2831 2b66 6972 7374        s*(1+first
│ │ │ │ -00034cc0: 2067 656e 7320 7269 6e67 2073 295e 286e   gens ring s)^(n
│ │ │ │ -00034cd0: 756d 6765 6e73 2072 696e 6720 4929 293b  umgens ring I));
│ │ │ │ +00034ca0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00034cb0: 732a 2831 2b66 6972 7374 2067 656e 7320  s*(1+first gens 
│ │ │ │ +00034cc0: 7269 6e67 2073 295e 286e 756d 6765 6e73  ring s)^(numgens
│ │ │ │ +00034cd0: 2072 696e 6720 4929 293b 2020 2020 2020   ring I));      
│ │ │ │  00034ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034cf0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00034cf0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00034d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │ -00034d50: 203a 202d 2d20 6578 616d 706c 653a 2043   : -- example: C
│ │ │ │ -00034d60: 6865 726e 2063 6c61 7373 206f 6620 4728  hern class of G(
│ │ │ │ -00034d70: 312c 3429 2020 2020 2020 2020 2020 2020  1,4)            
│ │ │ │ +00034d40: 2d2d 2d2d 2d2b 0a7c 6938 203a 202d 2d20  -----+.|i8 : -- 
│ │ │ │ +00034d50: 6578 616d 706c 653a 2043 6865 726e 2063  example: Chern c
│ │ │ │ +00034d60: 6c61 7373 206f 6620 4728 312c 3429 2020  lass of G(1,4)  
│ │ │ │ +00034d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034d90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00034da0: 2020 2047 203d 2047 7261 7373 6d61 6e6e     G = Grassmann
│ │ │ │ -00034db0: 6961 6e28 312c 342c 436f 6566 6669 6369  ian(1,4,Coeffici
│ │ │ │ -00034dc0: 656e 7452 696e 673d 3e5a 5a2f 3139 3031  entRing=>ZZ/1901
│ │ │ │ -00034dd0: 3831 2920 2020 2020 2020 2020 2020 2020  81)             
│ │ │ │ -00034de0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00034d90: 2020 2020 207c 0a7c 2020 2020 2047 203d       |.|     G =
│ │ │ │ +00034da0: 2047 7261 7373 6d61 6e6e 6961 6e28 312c   Grassmannian(1,
│ │ │ │ +00034db0: 342c 436f 6566 6669 6369 656e 7452 696e  4,CoefficientRin
│ │ │ │ +00034dc0: 673d 3e5a 5a2f 3139 3031 3831 2920 2020  g=>ZZ/190181)   
│ │ │ │ +00034dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034de0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00034df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034e30: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ -00034e40: 203d 2069 6465 616c 2028 7020 2020 7020   = ideal (p   p 
│ │ │ │ -00034e50: 2020 202d 2070 2020 2070 2020 2020 2b20     - p   p    + 
│ │ │ │ -00034e60: 7020 2020 7020 2020 2c20 7020 2020 7020  p   p   , p   p 
│ │ │ │ -00034e70: 2020 202d 2070 2020 2070 2020 2020 2b20     - p   p    + 
│ │ │ │ -00034e80: 7020 2020 7020 2020 2c20 207c 0a7c 2020  p   p   ,  |.|  
│ │ │ │ -00034e90: 2020 2020 2020 2020 2020 2032 2c33 2031             2,3 1
│ │ │ │ -00034ea0: 2c34 2020 2020 312c 3320 322c 3420 2020  ,4    1,3 2,4   
│ │ │ │ -00034eb0: 2031 2c32 2033 2c34 2020 2032 2c33 2030   1,2 3,4   2,3 0
│ │ │ │ -00034ec0: 2c34 2020 2020 302c 3320 322c 3420 2020  ,4    0,3 2,4   
│ │ │ │ -00034ed0: 2030 2c32 2033 2c34 2020 207c 0a7c 2020   0,2 3,4   |.|  
│ │ │ │ -00034ee0: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │ +00034e30: 2020 2020 207c 0a7c 6f38 203d 2069 6465       |.|o8 = ide
│ │ │ │ +00034e40: 616c 2028 7020 2020 7020 2020 202d 2070  al (p   p    - p
│ │ │ │ +00034e50: 2020 2070 2020 2020 2b20 7020 2020 7020     p    + p   p 
│ │ │ │ +00034e60: 2020 2c20 7020 2020 7020 2020 202d 2070    , p   p    - p
│ │ │ │ +00034e70: 2020 2070 2020 2020 2b20 7020 2020 7020     p    + p   p 
│ │ │ │ +00034e80: 2020 2c20 207c 0a7c 2020 2020 2020 2020    ,  |.|        
│ │ │ │ +00034e90: 2020 2020 2032 2c33 2031 2c34 2020 2020       2,3 1,4    
│ │ │ │ +00034ea0: 312c 3320 322c 3420 2020 2031 2c32 2033  1,3 2,4    1,2 3
│ │ │ │ +00034eb0: 2c34 2020 2032 2c33 2030 2c34 2020 2020  ,4   2,3 0,4    
│ │ │ │ +00034ec0: 302c 3320 322c 3420 2020 2030 2c32 2033  0,3 2,4    0,2 3
│ │ │ │ +00034ed0: 2c34 2020 207c 0a7c 2020 2020 202d 2d2d  ,4   |.|     ---
│ │ │ │ +00034ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00034f30: 2020 2070 2020 2070 2020 2020 2d20 7020     p   p    - p 
│ │ │ │ -00034f40: 2020 7020 2020 202b 2070 2020 2070 2020    p    + p   p  
│ │ │ │ -00034f50: 202c 2070 2020 2070 2020 2020 2d20 7020   , p   p    - p 
│ │ │ │ -00034f60: 2020 7020 2020 202b 2070 2020 2070 2020    p    + p   p  
│ │ │ │ -00034f70: 202c 2070 2020 2070 2020 207c 0a7c 2020   , p   p   |.|  
│ │ │ │ -00034f80: 2020 2020 312c 3320 302c 3420 2020 2030      1,3 0,4    0
│ │ │ │ -00034f90: 2c33 2031 2c34 2020 2020 302c 3120 332c  ,3 1,4    0,1 3,
│ │ │ │ -00034fa0: 3420 2020 312c 3220 302c 3420 2020 2030  4   1,2 0,4    0
│ │ │ │ -00034fb0: 2c32 2031 2c34 2020 2020 302c 3120 322c  ,2 1,4    0,1 2,
│ │ │ │ -00034fc0: 3420 2020 312c 3220 302c 337c 0a7c 2020  4   1,2 0,3|.|  
│ │ │ │ -00034fd0: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │ +00034f20: 2d2d 2d2d 2d7c 0a7c 2020 2020 2070 2020  -----|.|     p  
│ │ │ │ +00034f30: 2070 2020 2020 2d20 7020 2020 7020 2020   p    - p   p   
│ │ │ │ +00034f40: 202b 2070 2020 2070 2020 202c 2070 2020   + p   p   , p  
│ │ │ │ +00034f50: 2070 2020 2020 2d20 7020 2020 7020 2020   p    - p   p   
│ │ │ │ +00034f60: 202b 2070 2020 2070 2020 202c 2070 2020   + p   p   , p  
│ │ │ │ +00034f70: 2070 2020 207c 0a7c 2020 2020 2020 312c   p   |.|      1,
│ │ │ │ +00034f80: 3320 302c 3420 2020 2030 2c33 2031 2c34  3 0,4    0,3 1,4
│ │ │ │ +00034f90: 2020 2020 302c 3120 332c 3420 2020 312c      0,1 3,4   1,
│ │ │ │ +00034fa0: 3220 302c 3420 2020 2030 2c32 2031 2c34  2 0,4    0,2 1,4
│ │ │ │ +00034fb0: 2020 2020 302c 3120 322c 3420 2020 312c      0,1 2,4   1,
│ │ │ │ +00034fc0: 3220 302c 337c 0a7c 2020 2020 202d 2d2d  2 0,3|.|     ---
│ │ │ │ +00034fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00035020: 2020 202d 2070 2020 2070 2020 2020 2b20     - p   p    + 
│ │ │ │ -00035030: 7020 2020 7020 2020 2920 2020 2020 2020  p   p   )       
│ │ │ │ +00035010: 2d2d 2d2d 2d7c 0a7c 2020 2020 202d 2070  -----|.|     - p
│ │ │ │ +00035020: 2020 2070 2020 2020 2b20 7020 2020 7020     p    + p   p 
│ │ │ │ +00035030: 2020 2920 2020 2020 2020 2020 2020 2020    )             
│ │ │ │  00035040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035060: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00035070: 2020 2020 2020 302c 3220 312c 3320 2020        0,2 1,3   
│ │ │ │ -00035080: 2030 2c31 2032 2c33 2020 2020 2020 2020   0,1 2,3        
│ │ │ │ +00035060: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00035070: 302c 3220 312c 3320 2020 2030 2c31 2032  0,2 1,3    0,1 2
│ │ │ │ +00035080: 2c33 2020 2020 2020 2020 2020 2020 2020  ,3              
│ │ │ │  00035090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000350a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000350b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000350b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000350c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000350d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000350e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000350f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035100: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00035110: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │ +00035100: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00035110: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │  00035120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035150: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ -00035160: 203a 2049 6465 616c 206f 6620 2d2d 2d2d   : Ideal of ----
│ │ │ │ -00035170: 2d2d 5b70 2020 202e 2e70 2020 202c 2070  --[p   ..p   , p
│ │ │ │ +00035150: 2020 2020 207c 0a7c 6f38 203a 2049 6465       |.|o8 : Ide
│ │ │ │ +00035160: 616c 206f 6620 2d2d 2d2d 2d2d 5b70 2020  al of ------[p  
│ │ │ │ +00035170: 202e 2e70 2020 202c 2070 2020 202c 2070   ..p   , p   , p
│ │ │ │  00035180: 2020 202c 2070 2020 202c 2070 2020 202c     , p   , p   ,
│ │ │ │  00035190: 2070 2020 202c 2070 2020 202c 2070 2020   p   , p   , p  
│ │ │ │ -000351a0: 202c 2070 2020 202c 2070 207c 0a7c 2020   , p   , p |.|  
│ │ │ │ -000351b0: 2020 2020 2020 2020 2020 2020 3139 3031              1901
│ │ │ │ -000351c0: 3831 2020 302c 3120 2020 302c 3220 2020  81  0,1   0,2   
│ │ │ │ -000351d0: 312c 3220 2020 302c 3320 2020 312c 3320  1,2   0,3   1,3 
│ │ │ │ -000351e0: 2020 322c 3320 2020 302c 3420 2020 312c    2,3   0,4   1,
│ │ │ │ -000351f0: 3420 2020 322c 3420 2020 337c 0a7c 2d2d  4   2,4   3|.|--
│ │ │ │ +000351a0: 202c 2070 207c 0a7c 2020 2020 2020 2020   , p |.|        
│ │ │ │ +000351b0: 2020 2020 2020 3139 3031 3831 2020 302c        190181  0,
│ │ │ │ +000351c0: 3120 2020 302c 3220 2020 312c 3220 2020  1   0,2   1,2   
│ │ │ │ +000351d0: 302c 3320 2020 312c 3320 2020 322c 3320  0,3   1,3   2,3 
│ │ │ │ +000351e0: 2020 302c 3420 2020 312c 3420 2020 322c    0,4   1,4   2,
│ │ │ │ +000351f0: 3420 2020 337c 0a7c 2d2d 2d2d 2d2d 2d2d  4   3|.|--------
│ │ │ │  00035200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00035250: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ +00035240: 2d2d 2d2d 2d7c 0a7c 2020 5d20 2020 2020  -----|.|  ]     
│ │ │ │ +00035250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035290: 2020 2020 2020 2020 2020 207c 0a7c 2c34             |.|,4
│ │ │ │ +00035290: 2020 2020 207c 0a7c 2c34 2020 2020 2020       |.|,4      
│ │ │ │  000352a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000352b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000352c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000352d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000352e0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000352e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  000352f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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            
│ │ │ │ +00035330: 2d2d 2d2d 2d2b 0a7c 6939 203a 2074 696d  -----+.|i9 : tim
│ │ │ │ +00035340: 6520 4368 6572 6e43 6c61 7373 2047 2020  e ChernClass G  
│ │ │ │ +00035350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2e34 3339 3935 3273  - used 0.439952s
│ │ │ │ -000353a0: 2028 6370 7529 3b20 302e 3235 3839 3332   (cpu); 0.258932
│ │ │ │ -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             |.|  
│ │ │ │ +00035380: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +00035390: 2030 2e33 3931 3635 7320 2863 7075 293b   0.39165s (cpu);
│ │ │ │ +000353a0: 2030 2e32 3134 3036 3973 2028 7468 7265   0.214069s (thre
│ │ │ │ +000353b0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +000353c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000353d0: 2020 2020 207c 0a7c 2020 2020 2020 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  
│ │ │ │ -00035440: 2020 2020 3720 2020 2020 2036 2020 2020      7      6    
│ │ │ │ -00035450: 2020 3520 2020 2020 2034 2020 2020 2033    5      4     3
│ │ │ │ +00035420: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00035430: 3920 2020 2020 2038 2020 2020 2020 3720  9      8      7 
│ │ │ │ +00035440: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ +00035450: 2020 2034 2020 2020 2033 2020 2020 2020     4     3      
│ │ │ │  00035460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035470: 2020 2020 2020 2020 2020 207c 0a7c 6f39             |.|o9
│ │ │ │ -00035480: 203d 2031 3048 2020 2b20 3330 4820 202b   = 10H  + 30H  +
│ │ │ │ -00035490: 2036 3048 2020 2b20 3735 4820 202b 2035   60H  + 75H  + 5
│ │ │ │ -000354a0: 3748 2020 2b20 3235 4820 202b 2035 4820  7H  + 25H  + 5H 
│ │ │ │ +00035470: 2020 2020 207c 0a7c 6f39 203d 2031 3048       |.|o9 = 10H
│ │ │ │ +00035480: 2020 2b20 3330 4820 202b 2036 3048 2020    + 30H  + 60H  
│ │ │ │ +00035490: 2b20 3735 4820 202b 2035 3748 2020 2b20  + 75H  + 57H  + 
│ │ │ │ +000354a0: 3235 4820 202b 2035 4820 2020 2020 2020  25H  + 5H       
│ │ │ │  000354b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000354c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000354c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000354d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000354e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000354f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035510: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00035520: 2020 205a 5a5b 485d 2020 2020 2020 2020     ZZ[H]        
│ │ │ │ +00035510: 2020 2020 207c 0a7c 2020 2020 205a 5a5b       |.|     ZZ[
│ │ │ │ +00035520: 485d 2020 2020 2020 2020 2020 2020 2020  H]              
│ │ │ │  00035530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035560: 2020 2020 2020 2020 2020 207c 0a7c 6f39             |.|o9
│ │ │ │ -00035570: 203a 202d 2d2d 2d2d 2020 2020 2020 2020   : -----        
│ │ │ │ +00035560: 2020 2020 207c 0a7c 6f39 203a 202d 2d2d       |.|o9 : ---
│ │ │ │ +00035570: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │  00035580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000355a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000355b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000355c0: 2020 2020 2031 3020 2020 2020 2020 2020       10         
│ │ │ │ +000355b0: 2020 2020 207c 0a7c 2020 2020 2020 2031       |.|       1
│ │ │ │ +000355c0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │  000355d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000355e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000355f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035600: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00035610: 2020 2020 4820 2020 2020 2020 2020 2020      H           
│ │ │ │ +00035600: 2020 2020 207c 0a7c 2020 2020 2020 4820       |.|      H 
│ │ │ │ +00035610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035650: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00035650: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00035660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000356a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -000356b0: 3020 3a20 7469 6d65 2043 6865 726e 436c  0 : time ChernCl
│ │ │ │ -000356c0: 6173 7328 472c 4365 7274 6966 793d 3e74  ass(G,Certify=>t
│ │ │ │ -000356d0: 7275 6529 2020 2020 2020 2020 2020 2020  rue)            
│ │ │ │ +000356a0: 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 7469  -----+.|i10 : ti
│ │ │ │ +000356b0: 6d65 2043 6865 726e 436c 6173 7328 472c  me ChernClass(G,
│ │ │ │ +000356c0: 4365 7274 6966 793d 3e74 7275 6529 2020  Certify=>true)  
│ │ │ │ +000356d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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!        
│ │ │ │ +000356f0: 2020 2020 207c 0a7c 4365 7274 6966 793a       |.|Certify:
│ │ │ │ +00035700: 206f 7574 7075 7420 6365 7274 6966 6965   output certifie
│ │ │ │ +00035710: 6421 2020 2020 2020 2020 2020 2020 2020  d!              
│ │ │ │  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 2e31 3332 3334 3773  - used 0.132347s
│ │ │ │ -00035760: 2028 6370 7529 3b20 302e 3034 3439 3738   (cpu); 0.044978
│ │ │ │ -00035770: 3973 2028 7468 7265 6164 293b 2030 7320  9s (thread); 0s 
│ │ │ │ -00035780: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ -00035790: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00035740: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +00035750: 2030 2e31 3534 3739 3373 2028 6370 7529   0.154793s (cpu)
│ │ │ │ +00035760: 3b20 302e 3034 3439 3231 3673 2028 7468  ; 0.0449216s (th
│ │ │ │ +00035770: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +00035780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035790: 2020 2020 207c 0a7c 2020 2020 2020 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             |.|  
│ │ │ │ -000357f0: 2020 2020 2020 2039 2020 2020 2020 3820         9      8 
│ │ │ │ -00035800: 2020 2020 2037 2020 2020 2020 3620 2020       7      6   
│ │ │ │ -00035810: 2020 2035 2020 2020 2020 3420 2020 2020     5      4     
│ │ │ │ -00035820: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -00035830: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00035840: 3020 3d20 3130 4820 202b 2033 3048 2020  0 = 10H  + 30H  
│ │ │ │ -00035850: 2b20 3630 4820 202b 2037 3548 2020 2b20  + 60H  + 75H  + 
│ │ │ │ -00035860: 3537 4820 202b 2032 3548 2020 2b20 3548  57H  + 25H  + 5H
│ │ │ │ +000357e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000357f0: 2039 2020 2020 2020 3820 2020 2020 2037   9      8      7
│ │ │ │ +00035800: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ +00035810: 2020 2020 3420 2020 2020 3320 2020 2020      4     3     
│ │ │ │ +00035820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035830: 2020 2020 207c 0a7c 6f31 3020 3d20 3130       |.|o10 = 10
│ │ │ │ +00035840: 4820 202b 2033 3048 2020 2b20 3630 4820  H  + 30H  + 60H 
│ │ │ │ +00035850: 202b 2037 3548 2020 2b20 3537 4820 202b   + 75H  + 57H  +
│ │ │ │ +00035860: 2032 3548 2020 2b20 3548 2020 2020 2020   25H  + 5H      
│ │ │ │  00035870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035880: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00035880: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00035890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000358a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000358b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000358c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000358d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000358e0: 2020 2020 5a5a 5b48 5d20 2020 2020 2020      ZZ[H]       
│ │ │ │ +000358d0: 2020 2020 207c 0a7c 2020 2020 2020 5a5a       |.|      ZZ
│ │ │ │ +000358e0: 5b48 5d20 2020 2020 2020 2020 2020 2020  [H]             
│ │ │ │  000358f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035920: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00035930: 3020 3a20 2d2d 2d2d 2d20 2020 2020 2020  0 : -----       
│ │ │ │ +00035920: 2020 2020 207c 0a7c 6f31 3020 3a20 2d2d       |.|o10 : --
│ │ │ │ +00035930: 2d2d 2d20 2020 2020 2020 2020 2020 2020  ---             
│ │ │ │  00035940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035970: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00035980: 2020 2020 2020 3130 2020 2020 2020 2020        10        
│ │ │ │ +00035970: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00035980: 3130 2020 2020 2020 2020 2020 2020 2020  10              
│ │ │ │  00035990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000359a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000359b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000359c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000359d0: 2020 2020 2048 2020 2020 2020 2020 2020       H          
│ │ │ │ +000359c0: 2020 2020 207c 0a7c 2020 2020 2020 2048       |.|       H
│ │ │ │ +000359d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000359e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000359f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035a10: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00035a10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00035a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365  -----------+..Se
│ │ │ │ -00035a70: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ -00035a80: 0a20 202a 202a 6e6f 7465 2053 6567 7265  .  * *note Segre
│ │ │ │ -00035a90: 436c 6173 733a 2053 6567 7265 436c 6173  Class: SegreClas
│ │ │ │ -00035aa0: 732c 202d 2d20 5365 6772 6520 636c 6173  s, -- Segre clas
│ │ │ │ -00035ab0: 7320 6f66 2061 2063 6c6f 7365 6420 7375  s of a closed su
│ │ │ │ -00035ac0: 6273 6368 656d 6520 6f66 2061 0a20 2020  bscheme of a.   
│ │ │ │ -00035ad0: 2070 726f 6a65 6374 6976 6520 7661 7269   projective vari
│ │ │ │ -00035ae0: 6574 790a 2020 2a20 2a6e 6f74 6520 4575  ety.  * *note Eu
│ │ │ │ -00035af0: 6c65 7243 6861 7261 6374 6572 6973 7469  lerCharacteristi
│ │ │ │ -00035b00: 633a 2045 756c 6572 4368 6172 6163 7465  c: EulerCharacte
│ │ │ │ -00035b10: 7269 7374 6963 2c20 2d2d 2074 6f70 6f6c  ristic, -- topol
│ │ │ │ -00035b20: 6f67 6963 616c 2045 756c 6572 0a20 2020  ogical Euler.   
│ │ │ │ -00035b30: 2063 6861 7261 6374 6572 6973 7469 6320   characteristic 
│ │ │ │ -00035b40: 6f66 2061 2028 736d 6f6f 7468 2920 7072  of a (smooth) pr
│ │ │ │ -00035b50: 6f6a 6563 7469 7665 2076 6172 6965 7479  ojective variety
│ │ │ │ -00035b60: 0a20 202a 202a 6e6f 7465 2065 756c 6572  .  * *note euler
│ │ │ │ -00035b70: 2850 726f 6a65 6374 6976 6556 6172 6965  (ProjectiveVarie
│ │ │ │ -00035b80: 7479 293a 2028 5661 7269 6574 6965 7329  ty): (Varieties)
│ │ │ │ -00035b90: 6575 6c65 725f 6c70 5072 6f6a 6563 7469  euler_lpProjecti
│ │ │ │ -00035ba0: 7665 5661 7269 6574 795f 7270 2c20 2d2d  veVariety_rp, --
│ │ │ │ -00035bb0: 0a20 2020 2074 6f70 6f6c 6f67 6963 616c  .    topological
│ │ │ │ -00035bc0: 2045 756c 6572 2063 6861 7261 6374 6572   Euler character
│ │ │ │ -00035bd0: 6973 7469 6320 6f66 2061 2028 736d 6f6f  istic of a (smoo
│ │ │ │ -00035be0: 7468 2920 7072 6f6a 6563 7469 7665 2076  th) projective v
│ │ │ │ -00035bf0: 6172 6965 7479 0a0a 5761 7973 2074 6f20  ariety..Ways to 
│ │ │ │ -00035c00: 7573 6520 4368 6572 6e53 6368 7761 7274  use ChernSchwart
│ │ │ │ -00035c10: 7a4d 6163 5068 6572 736f 6e3a 0a3d 3d3d  zMacPherson:.===
│ │ │ │ +00035a60: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f  -----+..See also
│ │ │ │ +00035a70: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ +00035a80: 6e6f 7465 2053 6567 7265 436c 6173 733a  note SegreClass:
│ │ │ │ +00035a90: 2053 6567 7265 436c 6173 732c 202d 2d20   SegreClass, -- 
│ │ │ │ +00035aa0: 5365 6772 6520 636c 6173 7320 6f66 2061  Segre class of a
│ │ │ │ +00035ab0: 2063 6c6f 7365 6420 7375 6273 6368 656d   closed subschem
│ │ │ │ +00035ac0: 6520 6f66 2061 0a20 2020 2070 726f 6a65  e of a.    proje
│ │ │ │ +00035ad0: 6374 6976 6520 7661 7269 6574 790a 2020  ctive variety.  
│ │ │ │ +00035ae0: 2a20 2a6e 6f74 6520 4575 6c65 7243 6861  * *note EulerCha
│ │ │ │ +00035af0: 7261 6374 6572 6973 7469 633a 2045 756c  racteristic: Eul
│ │ │ │ +00035b00: 6572 4368 6172 6163 7465 7269 7374 6963  erCharacteristic
│ │ │ │ +00035b10: 2c20 2d2d 2074 6f70 6f6c 6f67 6963 616c  , -- topological
│ │ │ │ +00035b20: 2045 756c 6572 0a20 2020 2063 6861 7261   Euler.    chara
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│ │ │ │  00035d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00035ee0: 726f 7869 6d61 7465 496e 7665 7273 654d  roximateInverseM
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│ │ │ │  00035fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00036000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00036050: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00036b50: 6c20 6d61 703a 2052 6174 696f 6e61 6c4d  l map: RationalM
│ │ │ │ -00036b60: 6170 2c0a 2020 2a20 4f75 7470 7574 733a  ap,.  * Outputs:
│ │ │ │ -00036b70: 0a20 2020 2020 202a 2061 202a 6e6f 7465  .      * a *note
│ │ │ │ -00036b80: 2072 696e 673a 2028 4d61 6361 756c 6179   ring: (Macaulay
│ │ │ │ -00036b90: 3244 6f63 2952 696e 672c 2c20 7468 6520  2Doc)Ring,, the 
│ │ │ │ -00036ba0: 636f 6566 6669 6369 656e 7420 7269 6e67  coefficient ring
│ │ │ │ -00036bb0: 206f 6620 7068 690a 0a53 6565 2061 6c73   of phi..See als
│ │ │ │ -00036bc0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ -00036bd0: 2a6e 6f74 6520 5261 7469 6f6e 616c 4d61  *note RationalMa
│ │ │ │ -00036be0: 7020 2a2a 2052 696e 673a 2052 6174 696f  p ** Ring: Ratio
│ │ │ │ -00036bf0: 6e61 6c4d 6170 205f 7374 5f73 7420 5269  nalMap _st_st Ri
│ │ │ │ -00036c00: 6e67 2c20 2d2d 2063 6861 6e67 6520 7468  ng, -- change th
│ │ │ │ -00036c10: 650a 2020 2020 636f 6566 6669 6369 656e  e.    coefficien
│ │ │ │ -00036c20: 7420 7269 6e67 206f 6620 6120 7261 7469  t ring of a rati
│ │ │ │ -00036c30: 6f6e 616c 206d 6170 0a0a 5761 7973 2074  onal map..Ways t
│ │ │ │ -00036c40: 6f20 7573 6520 7468 6973 206d 6574 686f  o use this metho
│ │ │ │ -00036c50: 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  d:.=============
│ │ │ │ -00036c60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -00036c70: 202a 6e6f 7465 2063 6f65 6666 6963 6965   *note coefficie
│ │ │ │ -00036c80: 6e74 5269 6e67 2852 6174 696f 6e61 6c4d  ntRing(RationalM
│ │ │ │ -00036c90: 6170 293a 2063 6f65 6666 6963 6965 6e74  ap): coefficient
│ │ │ │ -00036ca0: 5269 6e67 5f6c 7052 6174 696f 6e61 6c4d  Ring_lpRationalM
│ │ │ │ -00036cb0: 6170 5f72 702c 202d 2d0a 2020 2020 636f  ap_rp, --.    co
│ │ │ │ -00036cc0: 6566 6669 6369 656e 7420 7269 6e67 206f  efficient ring o
│ │ │ │ -00036cd0: 6620 6120 7261 7469 6f6e 616c 206d 6170  f a rational map
│ │ │ │ -00036ce0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │ +00036ab0: 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 7469  ****..  * Functi
│ │ │ │ +00036ac0: 6f6e 3a20 2a6e 6f74 6520 636f 6566 6669  on: *note coeffi
│ │ │ │ +00036ad0: 6369 656e 7452 696e 673a 2028 4d61 6361  cientRing: (Maca
│ │ │ │ +00036ae0: 756c 6179 3244 6f63 2963 6f65 6666 6963  ulay2Doc)coeffic
│ │ │ │ +00036af0: 6965 6e74 5269 6e67 2c0a 2020 2a20 5573  ientRing,.  * Us
│ │ │ │ +00036b00: 6167 653a 200a 2020 2020 2020 2020 636f  age: .        co
│ │ │ │ +00036b10: 6566 6669 6369 656e 7452 696e 6720 7068  efficientRing ph
│ │ │ │ +00036b20: 690a 2020 2a20 496e 7075 7473 3a0a 2020  i.  * Inputs:.  
│ │ │ │ +00036b30: 2020 2020 2a20 7068 692c 2061 202a 6e6f      * phi, a *no
│ │ │ │ +00036b40: 7465 2072 6174 696f 6e61 6c20 6d61 703a  te rational map:
│ │ │ │ +00036b50: 2052 6174 696f 6e61 6c4d 6170 2c0a 2020   RationalMap,.  
│ │ │ │ +00036b60: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +00036b70: 202a 2061 202a 6e6f 7465 2072 696e 673a   * a *note ring:
│ │ │ │ +00036b80: 2028 4d61 6361 756c 6179 3244 6f63 2952   (Macaulay2Doc)R
│ │ │ │ +00036b90: 696e 672c 2c20 7468 6520 636f 6566 6669  ing,, the coeffi
│ │ │ │ +00036ba0: 6369 656e 7420 7269 6e67 206f 6620 7068  cient ring of ph
│ │ │ │ +00036bb0: 690a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  i..See also.====
│ │ │ │ +00036bc0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ +00036bd0: 5261 7469 6f6e 616c 4d61 7020 2a2a 2052  RationalMap ** R
│ │ │ │ +00036be0: 696e 673a 2052 6174 696f 6e61 6c4d 6170  ing: RationalMap
│ │ │ │ +00036bf0: 205f 7374 5f73 7420 5269 6e67 2c20 2d2d   _st_st Ring, --
│ │ │ │ +00036c00: 2063 6861 6e67 6520 7468 650a 2020 2020   change the.    
│ │ │ │ +00036c10: 636f 6566 6669 6369 656e 7420 7269 6e67  coefficient ring
│ │ │ │ +00036c20: 206f 6620 6120 7261 7469 6f6e 616c 206d   of a rational m
│ │ │ │ +00036c30: 6170 0a0a 5761 7973 2074 6f20 7573 6520  ap..Ways to use 
│ │ │ │ +00036c40: 7468 6973 206d 6574 686f 643a 0a3d 3d3d  this method:.===
│ │ │ │ +00036c50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00036c60: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +00036c70: 2063 6f65 6666 6963 6965 6e74 5269 6e67   coefficientRing
│ │ │ │ +00036c80: 2852 6174 696f 6e61 6c4d 6170 293a 2063  (RationalMap): c
│ │ │ │ +00036c90: 6f65 6666 6963 6965 6e74 5269 6e67 5f6c  oefficientRing_l
│ │ │ │ +00036ca0: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ +00036cb0: 202d 2d0a 2020 2020 636f 6566 6669 6369   --.    coeffici
│ │ │ │ +00036cc0: 656e 7420 7269 6e67 206f 6620 6120 7261  ent ring of a ra
│ │ │ │ +00036cd0: 7469 6f6e 616c 206d 6170 0a2d 2d2d 2d2d  tional map.-----
│ │ │ │ +00036ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036d30: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
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│ │ │ │ -00036d60: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
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│ │ │ │ -00036d80: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ -00036d90: 6163 6b61 6765 732f 4372 656d 6f6e 612f  ackages/Cremona/
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│ │ │ │ -00036de0: 735f 6c70 5261 7469 6f6e 616c 4d61 705f  s_lpRationalMap_
│ │ │ │ -00036df0: 7270 2c20 4e65 7874 3a20 6465 6772 6565  rp, Next: degree
│ │ │ │ -00036e00: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ -00036e10: 702c 2050 7265 763a 2063 6f65 6666 6963  p, Prev: coeffic
│ │ │ │ -00036e20: 6965 6e74 5269 6e67 5f6c 7052 6174 696f  ientRing_lpRatio
│ │ │ │ -00036e30: 6e61 6c4d 6170 5f72 702c 2055 703a 2054  nalMap_rp, Up: T
│ │ │ │ -00036e40: 6f70 0a0a 636f 6566 6669 6369 656e 7473  op..coefficients
│ │ │ │ -00036e50: 2852 6174 696f 6e61 6c4d 6170 2920 2d2d  (RationalMap) --
│ │ │ │ -00036e60: 2063 6f65 6666 6963 6965 6e74 206d 6174   coefficient mat
│ │ │ │ -00036e70: 7269 7820 6f66 2061 2072 6174 696f 6e61  rix of a rationa
│ │ │ │ -00036e80: 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a  l map.**********
│ │ │ │ +00036d20: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
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│ │ │ │ +00036d60: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ +00036d70: 2d31 2e32 352e 3131 2b64 732f 4d32 2f4d  -1.25.11+ds/M2/M
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│ │ │ │ +00036da0: 656e 7461 7469 6f6e 2e6d 323a 3438 323a  entation.m2:482:
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│ │ │ │ +00036dc0: 6e61 2e69 6e66 6f2c 204e 6f64 653a 2063  na.info, Node: c
│ │ │ │ +00036dd0: 6f65 6666 6963 6965 6e74 735f 6c70 5261  oefficients_lpRa
│ │ │ │ +00036de0: 7469 6f6e 616c 4d61 705f 7270 2c20 4e65  tionalMap_rp, Ne
│ │ │ │ +00036df0: 7874 3a20 6465 6772 6565 5f6c 7052 6174  xt: degree_lpRat
│ │ │ │ +00036e00: 696f 6e61 6c4d 6170 5f72 702c 2050 7265  ionalMap_rp, Pre
│ │ │ │ +00036e10: 763a 2063 6f65 6666 6963 6965 6e74 5269  v: coefficientRi
│ │ │ │ +00036e20: 6e67 5f6c 7052 6174 696f 6e61 6c4d 6170  ng_lpRationalMap
│ │ │ │ +00036e30: 5f72 702c 2055 703a 2054 6f70 0a0a 636f  _rp, Up: Top..co
│ │ │ │ +00036e40: 6566 6669 6369 656e 7473 2852 6174 696f  efficients(Ratio
│ │ │ │ +00036e50: 6e61 6c4d 6170 2920 2d2d 2063 6f65 6666  nalMap) -- coeff
│ │ │ │ +00036e60: 6963 6965 6e74 206d 6174 7269 7820 6f66  icient matrix of
│ │ │ │ +00036e70: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ +00036e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00036e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00036ea0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00036eb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00036ec0: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 756e  *******..  * Fun
│ │ │ │ -00036ed0: 6374 696f 6e3a 202a 6e6f 7465 2063 6f65  ction: *note coe
│ │ │ │ -00036ee0: 6666 6963 6965 6e74 733a 2028 4d61 6361  fficients: (Maca
│ │ │ │ -00036ef0: 756c 6179 3244 6f63 2963 6f65 6666 6963  ulay2Doc)coeffic
│ │ │ │ -00036f00: 6965 6e74 732c 0a20 202a 2055 7361 6765  ients,.  * Usage
│ │ │ │ -00036f10: 3a20 0a20 2020 2020 2020 2063 6f65 6666  : .        coeff
│ │ │ │ -00036f20: 6963 6965 6e74 7320 7068 690a 2020 2a20  icients phi.  * 
│ │ │ │ -00036f30: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -00036f40: 7068 692c 2061 202a 6e6f 7465 2072 6174  phi, a *note rat
│ │ │ │ -00036f50: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ -00036f60: 6e61 6c4d 6170 2c0a 2020 2a20 2a6e 6f74  nalMap,.  * *not
│ │ │ │ -00036f70: 6520 4f70 7469 6f6e 616c 2069 6e70 7574  e Optional input
│ │ │ │ -00036f80: 733a 2028 4d61 6361 756c 6179 3244 6f63  s: (Macaulay2Doc
│ │ │ │ -00036f90: 2975 7369 6e67 2066 756e 6374 696f 6e73  )using functions
│ │ │ │ -00036fa0: 2077 6974 6820 6f70 7469 6f6e 616c 2069   with optional i
│ │ │ │ -00036fb0: 6e70 7574 732c 3a0a 2020 2020 2020 2a20  nputs,:.      * 
│ │ │ │ -00036fc0: 2a6e 6f74 6520 4d6f 6e6f 6d69 616c 733a  *note Monomials:
│ │ │ │ -00036fd0: 2028 4d61 6361 756c 6179 3244 6f63 2963   (Macaulay2Doc)c
│ │ │ │ -00036fe0: 6f65 6666 6963 6965 6e74 732c 203d 3e20  oefficients, => 
│ │ │ │ -00036ff0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ -00037000: 7565 0a20 2020 2020 2020 206e 756c 6c2c  ue.        null,
│ │ │ │ -00037010: 0a20 2020 2020 202a 202a 6e6f 7465 2056  .      * *note V
│ │ │ │ -00037020: 6172 6961 626c 6573 3a20 284d 6163 6175  ariables: (Macau
│ │ │ │ -00037030: 6c61 7932 446f 6329 636f 6566 6669 6369  lay2Doc)coeffici
│ │ │ │ -00037040: 656e 7473 2c20 3d3e 202e 2e2e 2c20 6465  ents, => ..., de
│ │ │ │ -00037050: 6661 756c 7420 7661 6c75 650a 2020 2020  fault value.    
│ │ │ │ -00037060: 2020 2020 6e75 6c6c 2c0a 2020 2a20 4f75      null,.  * Ou
│ │ │ │ -00037070: 7470 7574 733a 0a20 2020 2020 202a 2061  tputs:.      * a
│ │ │ │ -00037080: 202a 6e6f 7465 206d 6174 7269 783a 2028   *note matrix: (
│ │ │ │ -00037090: 4d61 6361 756c 6179 3244 6f63 294d 6174  Macaulay2Doc)Mat
│ │ │ │ -000370a0: 7269 782c 2c20 7468 6520 636f 6566 6669  rix,, the coeffi
│ │ │ │ -000370b0: 6369 656e 7420 6d61 7472 6978 206f 6620  cient matrix of 
│ │ │ │ -000370c0: 7468 650a 2020 2020 2020 2020 706f 6c79  the.        poly
│ │ │ │ -000370d0: 6e6f 6d69 616c 7320 6465 6669 6e69 6e67  nomials defining
│ │ │ │ -000370e0: 2070 6869 0a0a 4465 7363 7269 7074 696f   phi..Descriptio
│ │ │ │ -000370f0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b  n.===========..+
│ │ │ │ +00036ec0: 2a0a 0a20 202a 2046 756e 6374 696f 6e3a  *..  * Function:
│ │ │ │ +00036ed0: 202a 6e6f 7465 2063 6f65 6666 6963 6965   *note coefficie
│ │ │ │ +00036ee0: 6e74 733a 2028 4d61 6361 756c 6179 3244  nts: (Macaulay2D
│ │ │ │ +00036ef0: 6f63 2963 6f65 6666 6963 6965 6e74 732c  oc)coefficients,
│ │ │ │ +00036f00: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +00036f10: 2020 2020 2063 6f65 6666 6963 6965 6e74       coefficient
│ │ │ │ +00036f20: 7320 7068 690a 2020 2a20 496e 7075 7473  s phi.  * Inputs
│ │ │ │ +00036f30: 3a0a 2020 2020 2020 2a20 7068 692c 2061  :.      * phi, a
│ │ │ │ +00036f40: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ +00036f50: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ +00036f60: 2c0a 2020 2a20 2a6e 6f74 6520 4f70 7469  ,.  * *note Opti
│ │ │ │ +00036f70: 6f6e 616c 2069 6e70 7574 733a 2028 4d61  onal inputs: (Ma
│ │ │ │ +00036f80: 6361 756c 6179 3244 6f63 2975 7369 6e67  caulay2Doc)using
│ │ │ │ +00036f90: 2066 756e 6374 696f 6e73 2077 6974 6820   functions with 
│ │ │ │ +00036fa0: 6f70 7469 6f6e 616c 2069 6e70 7574 732c  optional inputs,
│ │ │ │ +00036fb0: 3a0a 2020 2020 2020 2a20 2a6e 6f74 6520  :.      * *note 
│ │ │ │ +00036fc0: 4d6f 6e6f 6d69 616c 733a 2028 4d61 6361  Monomials: (Maca
│ │ │ │ +00036fd0: 756c 6179 3244 6f63 2963 6f65 6666 6963  ulay2Doc)coeffic
│ │ │ │ +00036fe0: 6965 6e74 732c 203d 3e20 2e2e 2e2c 2064  ients, => ..., d
│ │ │ │ +00036ff0: 6566 6175 6c74 2076 616c 7565 0a20 2020  efault value.   
│ │ │ │ +00037000: 2020 2020 206e 756c 6c2c 0a20 2020 2020       null,.     
│ │ │ │ +00037010: 202a 202a 6e6f 7465 2056 6172 6961 626c   * *note Variabl
│ │ │ │ +00037020: 6573 3a20 284d 6163 6175 6c61 7932 446f  es: (Macaulay2Do
│ │ │ │ +00037030: 6329 636f 6566 6669 6369 656e 7473 2c20  c)coefficients, 
│ │ │ │ +00037040: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +00037050: 7661 6c75 650a 2020 2020 2020 2020 6e75  value.        nu
│ │ │ │ +00037060: 6c6c 2c0a 2020 2a20 4f75 7470 7574 733a  ll,.  * Outputs:
│ │ │ │ +00037070: 0a20 2020 2020 202a 2061 202a 6e6f 7465  .      * a *note
│ │ │ │ +00037080: 206d 6174 7269 783a 2028 4d61 6361 756c   matrix: (Macaul
│ │ │ │ +00037090: 6179 3244 6f63 294d 6174 7269 782c 2c20  ay2Doc)Matrix,, 
│ │ │ │ +000370a0: 7468 6520 636f 6566 6669 6369 656e 7420  the coefficient 
│ │ │ │ +000370b0: 6d61 7472 6978 206f 6620 7468 650a 2020  matrix of the.  
│ │ │ │ +000370c0: 2020 2020 2020 706f 6c79 6e6f 6d69 616c        polynomial
│ │ │ │ +000370d0: 7320 6465 6669 6e69 6e67 2070 6869 0a0a  s defining phi..
│ │ │ │ +000370e0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +000370f0: 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d  =======..+------
│ │ │ │  00037100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037130: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -00037140: 4b20 3d20 5151 3b20 7269 6e67 5039 203d  K = QQ; ringP9 =
│ │ │ │ -00037150: 204b 5b78 5f30 2e2e 785f 395d 3b20 2020   K[x_0..x_9];   
│ │ │ │ -00037160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037170: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00037130: 2d2d 2b0a 7c69 3120 3a20 4b20 3d20 5151  --+.|i1 : K = QQ
│ │ │ │ +00037140: 3b20 7269 6e67 5039 203d 204b 5b78 5f30  ; ringP9 = K[x_0
│ │ │ │ +00037150: 2e2e 785f 395d 3b20 2020 2020 2020 2020  ..x_9];         
│ │ │ │ +00037160: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00037170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000371a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000371b0: 7c69 3320 3a20 4d20 3d20 7261 6e64 6f6d  |i3 : M = random
│ │ │ │ -000371c0: 284b 5e31 302c 4b5e 3130 2920 2020 2020  (K^10,K^10)     
│ │ │ │ +000371a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +000371b0: 4d20 3d20 7261 6e64 6f6d 284b 5e31 302c  M = random(K^10,
│ │ │ │ +000371c0: 4b5e 3130 2920 2020 2020 2020 2020 2020  K^10)           
│ │ │ │  000371d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000371e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000371e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000371f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037220: 2020 2020 7c0a 7c6f 3320 3d20 7c20 392f      |.|o3 = | 9/
│ │ │ │ -00037230: 3220 2036 2f37 2020 372f 3820 352f 3320  2  6/7  7/8 5/3 
│ │ │ │ -00037240: 2035 2f34 2020 392f 3130 2031 3020 2020   5/4  9/10 10   
│ │ │ │ -00037250: 342f 3920 2035 2f36 2020 3320 2020 7c7c  4/9  5/6  3   ||
│ │ │ │ -00037260: 0a7c 2020 2020 207c 2039 2f34 2020 3620  .|     | 9/4  6 
│ │ │ │ -00037270: 2020 2035 2f36 2031 2f31 3020 322f 3920     5/6 1/10 2/9 
│ │ │ │ -00037280: 2035 2f34 2020 3130 2f39 2039 2f31 3020   5/4  10/9 9/10 
│ │ │ │ -00037290: 372f 3130 2037 2f35 207c 7c0a 7c20 2020  7/10 7/5 ||.|   
│ │ │ │ -000372a0: 2020 7c20 332f 3420 2035 2f34 2020 3520    | 3/4  5/4  5 
│ │ │ │ -000372b0: 2020 342f 3320 2038 2f35 2020 312f 3720    4/3  8/5  1/7 
│ │ │ │ -000372c0: 2038 2f33 2020 332f 3220 2031 2f38 2020   8/3  3/2  1/8  
│ │ │ │ -000372d0: 372f 3320 7c7c 0a7c 2020 2020 207c 2037  7/3 ||.|     | 7
│ │ │ │ -000372e0: 2f34 2020 322f 3920 2032 2f35 2033 2f37  /4  2/9  2/5 3/7
│ │ │ │ -000372f0: 2020 392f 3420 2037 2f35 2020 332f 3420    9/4  7/5  3/4 
│ │ │ │ -00037300: 2034 2f33 2020 382f 3720 2032 2f37 207c   4/3  8/7  2/7 |
│ │ │ │ -00037310: 7c0a 7c20 2020 2020 7c20 372f 3920 2033  |.|     | 7/9  3
│ │ │ │ -00037320: 2f31 3020 352f 3320 392f 3130 2032 2f39  /10 5/3 9/10 2/9
│ │ │ │ -00037330: 2020 312f 3520 2039 2f35 2020 312f 3820    1/5  9/5  1/8 
│ │ │ │ -00037340: 2037 2f33 2020 312f 3520 7c7c 0a7c 2020   7/3  1/5 ||.|  
│ │ │ │ -00037350: 2020 207c 2037 2f31 3020 332f 3720 2037     | 7/10 3/7  7
│ │ │ │ -00037360: 2f32 2034 2f37 2020 392f 3820 2035 2f37  /2 4/7  9/8  5/7
│ │ │ │ -00037370: 2020 332f 3520 2037 2f38 2020 3130 2f39    3/5  7/8  10/9
│ │ │ │ -00037380: 2035 2f37 207c 7c0a 7c20 2020 2020 7c20   5/7 ||.|     | 
│ │ │ │ -00037390: 372f 3130 2035 2020 2020 322f 3520 352f  7/10 5    2/5 5/
│ │ │ │ -000373a0: 3920 2031 2f38 2020 332f 3820 2034 2020  9  1/8  3/8  4  
│ │ │ │ -000373b0: 2020 3130 2f39 2038 2020 2020 372f 3320    10/9 8    7/3 
│ │ │ │ -000373c0: 7c7c 0a7c 2020 2020 207c 2037 2f33 2020  ||.|     | 7/3  
│ │ │ │ -000373d0: 3130 2f39 2036 2f35 2035 2f39 2020 3130  10/9 6/5 5/9  10
│ │ │ │ -000373e0: 2f33 2035 2f32 2020 372f 3520 2036 2f37  /3 5/2  7/5  6/7
│ │ │ │ -000373f0: 2020 3130 2f39 2032 2020 207c 7c0a 7c20    10/9 2   ||.| 
│ │ │ │ -00037400: 2020 2020 7c20 3720 2020 2031 3020 2020      | 7    10   
│ │ │ │ -00037410: 352f 3720 362f 3720 2034 2020 2020 312f  5/7 6/7  4    1/
│ │ │ │ -00037420: 3620 2035 2020 2020 372f 3620 2038 2020  6  5    7/6  8  
│ │ │ │ -00037430: 2020 3420 2020 7c7c 0a7c 2020 2020 207c    4   ||.|     |
│ │ │ │ -00037440: 2033 2f37 2020 332f 3220 2035 2f39 2036   3/7  3/2  5/9 6
│ │ │ │ -00037450: 2020 2020 312f 3320 2038 2f35 2020 372f      1/3  8/5  7/
│ │ │ │ -00037460: 3130 2035 2f36 2020 3220 2020 2038 2f33  10 5/6  2    8/3
│ │ │ │ -00037470: 207c 7c0a 7c20 2020 2020 2020 2020 2020   ||.|           
│ │ │ │ +00037210: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00037220: 7c6f 3320 3d20 7c20 392f 3220 2036 2f37  |o3 = | 9/2  6/7
│ │ │ │ +00037230: 2020 372f 3820 352f 3320 2035 2f34 2020    7/8 5/3  5/4  
│ │ │ │ +00037240: 392f 3130 2031 3020 2020 342f 3920 2035  9/10 10   4/9  5
│ │ │ │ +00037250: 2f36 2020 3320 2020 7c7c 0a7c 2020 2020  /6  3   ||.|    
│ │ │ │ +00037260: 207c 2039 2f34 2020 3620 2020 2035 2f36   | 9/4  6    5/6
│ │ │ │ +00037270: 2031 2f31 3020 322f 3920 2035 2f34 2020   1/10 2/9  5/4  
│ │ │ │ +00037280: 3130 2f39 2039 2f31 3020 372f 3130 2037  10/9 9/10 7/10 7
│ │ │ │ +00037290: 2f35 207c 7c0a 7c20 2020 2020 7c20 332f  /5 ||.|     | 3/
│ │ │ │ +000372a0: 3420 2035 2f34 2020 3520 2020 342f 3320  4  5/4  5   4/3 
│ │ │ │ +000372b0: 2038 2f35 2020 312f 3720 2038 2f33 2020   8/5  1/7  8/3  
│ │ │ │ +000372c0: 332f 3220 2031 2f38 2020 372f 3320 7c7c  3/2  1/8  7/3 ||
│ │ │ │ +000372d0: 0a7c 2020 2020 207c 2037 2f34 2020 322f  .|     | 7/4  2/
│ │ │ │ +000372e0: 3920 2032 2f35 2033 2f37 2020 392f 3420  9  2/5 3/7  9/4 
│ │ │ │ +000372f0: 2037 2f35 2020 332f 3420 2034 2f33 2020   7/5  3/4  4/3  
│ │ │ │ +00037300: 382f 3720 2032 2f37 207c 7c0a 7c20 2020  8/7  2/7 ||.|   
│ │ │ │ +00037310: 2020 7c20 372f 3920 2033 2f31 3020 352f    | 7/9  3/10 5/
│ │ │ │ +00037320: 3320 392f 3130 2032 2f39 2020 312f 3520  3 9/10 2/9  1/5 
│ │ │ │ +00037330: 2039 2f35 2020 312f 3820 2037 2f33 2020   9/5  1/8  7/3  
│ │ │ │ +00037340: 312f 3520 7c7c 0a7c 2020 2020 207c 2037  1/5 ||.|     | 7
│ │ │ │ +00037350: 2f31 3020 332f 3720 2037 2f32 2034 2f37  /10 3/7  7/2 4/7
│ │ │ │ +00037360: 2020 392f 3820 2035 2f37 2020 332f 3520    9/8  5/7  3/5 
│ │ │ │ +00037370: 2037 2f38 2020 3130 2f39 2035 2f37 207c   7/8  10/9 5/7 |
│ │ │ │ +00037380: 7c0a 7c20 2020 2020 7c20 372f 3130 2035  |.|     | 7/10 5
│ │ │ │ +00037390: 2020 2020 322f 3520 352f 3920 2031 2f38      2/5 5/9  1/8
│ │ │ │ +000373a0: 2020 332f 3820 2034 2020 2020 3130 2f39    3/8  4    10/9
│ │ │ │ +000373b0: 2038 2020 2020 372f 3320 7c7c 0a7c 2020   8    7/3 ||.|  
│ │ │ │ +000373c0: 2020 207c 2037 2f33 2020 3130 2f39 2036     | 7/3  10/9 6
│ │ │ │ +000373d0: 2f35 2035 2f39 2020 3130 2f33 2035 2f32  /5 5/9  10/3 5/2
│ │ │ │ +000373e0: 2020 372f 3520 2036 2f37 2020 3130 2f39    7/5  6/7  10/9
│ │ │ │ +000373f0: 2032 2020 207c 7c0a 7c20 2020 2020 7c20   2   ||.|     | 
│ │ │ │ +00037400: 3720 2020 2031 3020 2020 352f 3720 362f  7    10   5/7 6/
│ │ │ │ +00037410: 3720 2034 2020 2020 312f 3620 2035 2020  7  4    1/6  5  
│ │ │ │ +00037420: 2020 372f 3620 2038 2020 2020 3420 2020    7/6  8    4   
│ │ │ │ +00037430: 7c7c 0a7c 2020 2020 207c 2033 2f37 2020  ||.|     | 3/7  
│ │ │ │ +00037440: 332f 3220 2035 2f39 2036 2020 2020 312f  3/2  5/9 6    1/
│ │ │ │ +00037450: 3320 2038 2f35 2020 372f 3130 2035 2f36  3  8/5  7/10 5/6
│ │ │ │ +00037460: 2020 3220 2020 2038 2f33 207c 7c0a 7c20    2    8/3 ||.| 
│ │ │ │ +00037470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000374a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000374b0: 2020 2020 2020 2020 2020 2020 2020 3130                10
│ │ │ │ -000374c0: 2020 2020 2020 2031 3020 2020 2020 2020         10       
│ │ │ │ +000374a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000374b0: 2020 2020 2020 2020 3130 2020 2020 2020          10      
│ │ │ │ +000374c0: 2031 3020 2020 2020 2020 2020 2020 2020   10             
│ │ │ │  000374d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000374e0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ -000374f0: 4d61 7472 6978 2051 5120 2020 3c2d 2d20  Matrix QQ   <-- 
│ │ │ │ -00037500: 5151 2020 2020 2020 2020 2020 2020 2020  QQ              
│ │ │ │ -00037510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037520: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000374e0: 2020 7c0a 7c6f 3320 3a20 4d61 7472 6978    |.|o3 : Matrix
│ │ │ │ +000374f0: 2051 5120 2020 3c2d 2d20 5151 2020 2020   QQ   <-- QQ    
│ │ │ │ +00037500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037510: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00037520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00037560: 7c69 3420 3a20 7068 6920 3d20 7261 7469  |i4 : phi = rati
│ │ │ │ -00037570: 6f6e 616c 4d61 7020 2828 7661 7273 2072  onalMap ((vars r
│ │ │ │ -00037580: 696e 6750 3929 202a 2028 7472 616e 7370  ingP9) * (transp
│ │ │ │ -00037590: 6f73 6520 4d29 293b 207c 0a7c 2020 2020  ose M)); |.|    
│ │ │ │ +00037550: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ +00037560: 7068 6920 3d20 7261 7469 6f6e 616c 4d61  phi = rationalMa
│ │ │ │ +00037570: 7020 2828 7661 7273 2072 696e 6750 3929  p ((vars ringP9)
│ │ │ │ +00037580: 202a 2028 7472 616e 7370 6f73 6520 4d29   * (transpose M)
│ │ │ │ +00037590: 293b 207c 0a7c 2020 2020 2020 2020 2020  ); |.|          
│ │ │ │  000375a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000375b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000375c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000375d0: 2020 2020 7c0a 7c6f 3420 3a20 5261 7469      |.|o4 : Rati
│ │ │ │ -000375e0: 6f6e 616c 4d61 7020 286c 696e 6561 7220  onalMap (linear 
│ │ │ │ -000375f0: 7261 7469 6f6e 616c 206d 6170 2066 726f  rational map fro
│ │ │ │ -00037600: 6d20 5050 5e39 2074 6f20 5050 5e39 297c  m PP^9 to PP^9)|
│ │ │ │ -00037610: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000375c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000375d0: 7c6f 3420 3a20 5261 7469 6f6e 616c 4d61  |o4 : RationalMa
│ │ │ │ +000375e0: 7020 286c 696e 6561 7220 7261 7469 6f6e  p (linear ration
│ │ │ │ +000375f0: 616c 206d 6170 2066 726f 6d20 5050 5e39  al map from PP^9
│ │ │ │ +00037600: 2074 6f20 5050 5e39 297c 0a2b 2d2d 2d2d   to PP^9)|.+----
│ │ │ │ +00037610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037640: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ -00037650: 3a20 4d27 203d 2063 6f65 6666 6963 6965  : M' = coefficie
│ │ │ │ -00037660: 6e74 7320 7068 6920 2020 2020 2020 2020  nts phi         
│ │ │ │ -00037670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037680: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00037640: 2d2d 2d2d 2b0a 7c69 3520 3a20 4d27 203d  ----+.|i5 : M' =
│ │ │ │ +00037650: 2063 6f65 6666 6963 6965 6e74 7320 7068   coefficients ph
│ │ │ │ +00037660: 6920 2020 2020 2020 2020 2020 2020 2020  i               
│ │ │ │ +00037670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00037680: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00037690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000376a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000376b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000376c0: 7c0a 7c6f 3520 3d20 7c20 392f 3220 2036  |.|o5 = | 9/2  6
│ │ │ │ -000376d0: 2f37 2020 372f 3820 352f 3320 2035 2f34  /7  7/8 5/3  5/4
│ │ │ │ -000376e0: 2020 392f 3130 2031 3020 2020 342f 3920    9/10 10   4/9 
│ │ │ │ -000376f0: 2035 2f36 2020 3320 2020 7c7c 0a7c 2020   5/6  3   ||.|  
│ │ │ │ -00037700: 2020 207c 2039 2f34 2020 3620 2020 2035     | 9/4  6    5
│ │ │ │ -00037710: 2f36 2031 2f31 3020 322f 3920 2035 2f34  /6 1/10 2/9  5/4
│ │ │ │ -00037720: 2020 3130 2f39 2039 2f31 3020 372f 3130    10/9 9/10 7/10
│ │ │ │ -00037730: 2037 2f35 207c 7c0a 7c20 2020 2020 7c20   7/5 ||.|     | 
│ │ │ │ -00037740: 332f 3420 2035 2f34 2020 3520 2020 342f  3/4  5/4  5   4/
│ │ │ │ -00037750: 3320 2038 2f35 2020 312f 3720 2038 2f33  3  8/5  1/7  8/3
│ │ │ │ -00037760: 2020 332f 3220 2031 2f38 2020 372f 3320    3/2  1/8  7/3 
│ │ │ │ -00037770: 7c7c 0a7c 2020 2020 207c 2037 2f34 2020  ||.|     | 7/4  
│ │ │ │ -00037780: 322f 3920 2032 2f35 2033 2f37 2020 392f  2/9  2/5 3/7  9/
│ │ │ │ -00037790: 3420 2037 2f35 2020 332f 3420 2034 2f33  4  7/5  3/4  4/3
│ │ │ │ -000377a0: 2020 382f 3720 2032 2f37 207c 7c0a 7c20    8/7  2/7 ||.| 
│ │ │ │ -000377b0: 2020 2020 7c20 372f 3920 2033 2f31 3020      | 7/9  3/10 
│ │ │ │ -000377c0: 352f 3320 392f 3130 2032 2f39 2020 312f  5/3 9/10 2/9  1/
│ │ │ │ -000377d0: 3520 2039 2f35 2020 312f 3820 2037 2f33  5  9/5  1/8  7/3
│ │ │ │ -000377e0: 2020 312f 3520 7c7c 0a7c 2020 2020 207c    1/5 ||.|     |
│ │ │ │ -000377f0: 2037 2f31 3020 332f 3720 2037 2f32 2034   7/10 3/7  7/2 4
│ │ │ │ -00037800: 2f37 2020 392f 3820 2035 2f37 2020 332f  /7  9/8  5/7  3/
│ │ │ │ -00037810: 3520 2037 2f38 2020 3130 2f39 2035 2f37  5  7/8  10/9 5/7
│ │ │ │ -00037820: 207c 7c0a 7c20 2020 2020 7c20 372f 3130   ||.|     | 7/10
│ │ │ │ -00037830: 2035 2020 2020 322f 3520 352f 3920 2031   5    2/5 5/9  1
│ │ │ │ -00037840: 2f38 2020 332f 3820 2034 2020 2020 3130  /8  3/8  4    10
│ │ │ │ -00037850: 2f39 2038 2020 2020 372f 3320 7c7c 0a7c  /9 8    7/3 ||.|
│ │ │ │ -00037860: 2020 2020 207c 2037 2f33 2020 3130 2f39       | 7/3  10/9
│ │ │ │ -00037870: 2036 2f35 2035 2f39 2020 3130 2f33 2035   6/5 5/9  10/3 5
│ │ │ │ -00037880: 2f32 2020 372f 3520 2036 2f37 2020 3130  /2  7/5  6/7  10
│ │ │ │ -00037890: 2f39 2032 2020 207c 7c0a 7c20 2020 2020  /9 2   ||.|     
│ │ │ │ -000378a0: 7c20 3720 2020 2031 3020 2020 352f 3720  | 7    10   5/7 
│ │ │ │ -000378b0: 362f 3720 2034 2020 2020 312f 3620 2035  6/7  4    1/6  5
│ │ │ │ -000378c0: 2020 2020 372f 3620 2038 2020 2020 3420      7/6  8    4 
│ │ │ │ -000378d0: 2020 7c7c 0a7c 2020 2020 207c 2033 2f37    ||.|     | 3/7
│ │ │ │ -000378e0: 2020 332f 3220 2035 2f39 2036 2020 2020    3/2  5/9 6    
│ │ │ │ -000378f0: 312f 3320 2038 2f35 2020 372f 3130 2035  1/3  8/5  7/10 5
│ │ │ │ -00037900: 2f36 2020 3220 2020 2038 2f33 207c 7c0a  /6  2    8/3 ||.
│ │ │ │ -00037910: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000376b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ +000376c0: 3d20 7c20 392f 3220 2036 2f37 2020 372f  = | 9/2  6/7  7/
│ │ │ │ +000376d0: 3820 352f 3320 2035 2f34 2020 392f 3130  8 5/3  5/4  9/10
│ │ │ │ +000376e0: 2031 3020 2020 342f 3920 2035 2f36 2020   10   4/9  5/6  
│ │ │ │ +000376f0: 3320 2020 7c7c 0a7c 2020 2020 207c 2039  3   ||.|     | 9
│ │ │ │ +00037700: 2f34 2020 3620 2020 2035 2f36 2031 2f31  /4  6    5/6 1/1
│ │ │ │ +00037710: 3020 322f 3920 2035 2f34 2020 3130 2f39  0 2/9  5/4  10/9
│ │ │ │ +00037720: 2039 2f31 3020 372f 3130 2037 2f35 207c   9/10 7/10 7/5 |
│ │ │ │ +00037730: 7c0a 7c20 2020 2020 7c20 332f 3420 2035  |.|     | 3/4  5
│ │ │ │ +00037740: 2f34 2020 3520 2020 342f 3320 2038 2f35  /4  5   4/3  8/5
│ │ │ │ +00037750: 2020 312f 3720 2038 2f33 2020 332f 3220    1/7  8/3  3/2 
│ │ │ │ +00037760: 2031 2f38 2020 372f 3320 7c7c 0a7c 2020   1/8  7/3 ||.|  
│ │ │ │ +00037770: 2020 207c 2037 2f34 2020 322f 3920 2032     | 7/4  2/9  2
│ │ │ │ +00037780: 2f35 2033 2f37 2020 392f 3420 2037 2f35  /5 3/7  9/4  7/5
│ │ │ │ +00037790: 2020 332f 3420 2034 2f33 2020 382f 3720    3/4  4/3  8/7 
│ │ │ │ +000377a0: 2032 2f37 207c 7c0a 7c20 2020 2020 7c20   2/7 ||.|     | 
│ │ │ │ +000377b0: 372f 3920 2033 2f31 3020 352f 3320 392f  7/9  3/10 5/3 9/
│ │ │ │ +000377c0: 3130 2032 2f39 2020 312f 3520 2039 2f35  10 2/9  1/5  9/5
│ │ │ │ +000377d0: 2020 312f 3820 2037 2f33 2020 312f 3520    1/8  7/3  1/5 
│ │ │ │ +000377e0: 7c7c 0a7c 2020 2020 207c 2037 2f31 3020  ||.|     | 7/10 
│ │ │ │ +000377f0: 332f 3720 2037 2f32 2034 2f37 2020 392f  3/7  7/2 4/7  9/
│ │ │ │ +00037800: 3820 2035 2f37 2020 332f 3520 2037 2f38  8  5/7  3/5  7/8
│ │ │ │ +00037810: 2020 3130 2f39 2035 2f37 207c 7c0a 7c20    10/9 5/7 ||.| 
│ │ │ │ +00037820: 2020 2020 7c20 372f 3130 2035 2020 2020      | 7/10 5    
│ │ │ │ +00037830: 322f 3520 352f 3920 2031 2f38 2020 332f  2/5 5/9  1/8  3/
│ │ │ │ +00037840: 3820 2034 2020 2020 3130 2f39 2038 2020  8  4    10/9 8  
│ │ │ │ +00037850: 2020 372f 3320 7c7c 0a7c 2020 2020 207c    7/3 ||.|     |
│ │ │ │ +00037860: 2037 2f33 2020 3130 2f39 2036 2f35 2035   7/3  10/9 6/5 5
│ │ │ │ +00037870: 2f39 2020 3130 2f33 2035 2f32 2020 372f  /9  10/3 5/2  7/
│ │ │ │ +00037880: 3520 2036 2f37 2020 3130 2f39 2032 2020  5  6/7  10/9 2  
│ │ │ │ +00037890: 207c 7c0a 7c20 2020 2020 7c20 3720 2020   ||.|     | 7   
│ │ │ │ +000378a0: 2031 3020 2020 352f 3720 362f 3720 2034   10   5/7 6/7  4
│ │ │ │ +000378b0: 2020 2020 312f 3620 2035 2020 2020 372f      1/6  5    7/
│ │ │ │ +000378c0: 3620 2038 2020 2020 3420 2020 7c7c 0a7c  6  8    4   ||.|
│ │ │ │ +000378d0: 2020 2020 207c 2033 2f37 2020 332f 3220       | 3/7  3/2 
│ │ │ │ +000378e0: 2035 2f39 2036 2020 2020 312f 3320 2038   5/9 6    1/3  8
│ │ │ │ +000378f0: 2f35 2020 372f 3130 2035 2f36 2020 3220  /5  7/10 5/6  2 
│ │ │ │ +00037900: 2020 2038 2f33 207c 7c0a 7c20 2020 2020     8/3 ||.|     
│ │ │ │ +00037910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037940: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00037950: 2020 2020 2020 2020 2020 3130 2020 2020            10    
│ │ │ │ -00037960: 2020 2031 3020 2020 2020 2020 2020 2020     10           
│ │ │ │ -00037970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037980: 2020 2020 7c0a 7c6f 3520 3a20 4d61 7472      |.|o5 : Matr
│ │ │ │ -00037990: 6978 2051 5120 2020 3c2d 2d20 5151 2020  ix QQ   <-- QQ  
│ │ │ │ +00037940: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00037950: 2020 2020 3130 2020 2020 2020 2031 3020      10       10 
│ │ │ │ +00037960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037970: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00037980: 7c6f 3520 3a20 4d61 7472 6978 2051 5120  |o5 : Matrix QQ 
│ │ │ │ +00037990: 2020 3c2d 2d20 5151 2020 2020 2020 2020    <-- QQ        
│ │ │ │  000379a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000379b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000379c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000379b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000379c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000379d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000379e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000379f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ -00037a00: 3a20 4d20 3d3d 204d 2720 2020 2020 2020  : M == M'       
│ │ │ │ +000379f0: 2d2d 2d2d 2b0a 7c69 3620 3a20 4d20 3d3d  ----+.|i6 : M ==
│ │ │ │ +00037a00: 204d 2720 2020 2020 2020 2020 2020 2020   M'             
│ │ │ │  00037a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00037a20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00037a30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00037a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a70: 7c0a 7c6f 3620 3d20 7472 7565 2020 2020  |.|o6 = true    
│ │ │ │ +00037a60: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ +00037a70: 3d20 7472 7565 2020 2020 2020 2020 2020  = true          
│ │ │ │  00037a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037aa0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00037aa0: 2020 2020 207c 0a2b 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 2d2d 2b0a 0a57 6179 7320 746f  ------+..Ways to
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│ │ │ │ -00037b00: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ -00037b10: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00037b20: 2a6e 6f74 6520 636f 6566 6669 6369 656e  *note coefficien
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│ │ │ │ -00037b40: 2063 6f65 6666 6963 6965 6e74 735f 6c70   coefficients_lp
│ │ │ │ -00037b50: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ -00037b60: 2d2d 0a20 2020 2063 6f65 6666 6963 6965  --.    coefficie
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│ │ │ │ -00037b80: 6174 696f 6e61 6c20 6d61 700a 2d2d 2d2d  ational map.----
│ │ │ │ +00037ae0: 2b0a 0a57 6179 7320 746f 2075 7365 2074  +..Ways to use t
│ │ │ │ +00037af0: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d  his method:.====
│ │ │ │ +00037b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00037b10: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ +00037b20: 636f 6566 6669 6369 656e 7473 2852 6174  coefficients(Rat
│ │ │ │ +00037b30: 696f 6e61 6c4d 6170 293a 2063 6f65 6666  ionalMap): coeff
│ │ │ │ +00037b40: 6963 6965 6e74 735f 6c70 5261 7469 6f6e  icients_lpRation
│ │ │ │ +00037b50: 616c 4d61 705f 7270 2c20 2d2d 0a20 2020  alMap_rp, --.   
│ │ │ │ +00037b60: 2063 6f65 6666 6963 6965 6e74 206d 6174   coefficient mat
│ │ │ │ +00037b70: 7269 7820 6f66 2061 2072 6174 696f 6e61  rix of a rationa
│ │ │ │ +00037b80: 6c20 6d61 700a 2d2d 2d2d 2d2d 2d2d 2d2d  l map.----------
│ │ │ │  00037b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
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│ │ │ │ -00037c90: 6c4d 6170 5f72 702c 204e 6578 743a 2064  lMap_rp, Next: d
│ │ │ │ -00037ca0: 6567 7265 654d 6170 2c20 5072 6576 3a20  egreeMap, Prev: 
│ │ │ │ -00037cb0: 636f 6566 6669 6369 656e 7473 5f6c 7052  coefficients_lpR
│ │ │ │ -00037cc0: 6174 696f 6e61 6c4d 6170 5f72 702c 2055  ationalMap_rp, U
│ │ │ │ -00037cd0: 703a 2054 6f70 0a0a 6465 6772 6565 2852  p: Top..degree(R
│ │ │ │ -00037ce0: 6174 696f 6e61 6c4d 6170 2920 2d2d 2064  ationalMap) -- d
│ │ │ │ -00037cf0: 6567 7265 6520 6f66 2061 2072 6174 696f  egree of a ratio
│ │ │ │ -00037d00: 6e61 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a  nal map.********
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│ │ │ │ +00037c90: 702c 204e 6578 743a 2064 6567 7265 654d  p, Next: degreeM
│ │ │ │ +00037ca0: 6170 2c20 5072 6576 3a20 636f 6566 6669  ap, Prev: coeffi
│ │ │ │ +00037cb0: 6369 656e 7473 5f6c 7052 6174 696f 6e61  cients_lpRationa
│ │ │ │ +00037cc0: 6c4d 6170 5f72 702c 2055 703a 2054 6f70  lMap_rp, Up: Top
│ │ │ │ +00037cd0: 0a0a 6465 6772 6565 2852 6174 696f 6e61  ..degree(Rationa
│ │ │ │ +00037ce0: 6c4d 6170 2920 2d2d 2064 6567 7265 6520  lMap) -- degree 
│ │ │ │ +00037cf0: 6f66 2061 2072 6174 696f 6e61 6c20 6d61  of a rational ma
│ │ │ │ +00037d00: 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  p.**************
│ │ │ │  00037d10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00037d20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00037d30: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 756e  *******..  * Fun
│ │ │ │ -00037d40: 6374 696f 6e3a 202a 6e6f 7465 2064 6567  ction: *note deg
│ │ │ │ -00037d50: 7265 653a 2028 4d61 6361 756c 6179 3244  ree: (Macaulay2D
│ │ │ │ -00037d60: 6f63 2964 6567 7265 652c 0a20 202a 2055  oc)degree,.  * U
│ │ │ │ -00037d70: 7361 6765 3a20 0a20 2020 2020 2020 2064  sage: .        d
│ │ │ │ -00037d80: 6567 7265 6520 7068 690a 2020 2a20 496e  egree phi.  * In
│ │ │ │ -00037d90: 7075 7473 3a0a 2020 2020 2020 2a20 7068  puts:.      * ph
│ │ │ │ -00037da0: 692c 2061 202a 6e6f 7465 2072 6174 696f  i, a *note ratio
│ │ │ │ -00037db0: 6e61 6c20 6d61 703a 2052 6174 696f 6e61  nal map: Rationa
│ │ │ │ -00037dc0: 6c4d 6170 2c0a 2020 2a20 4f75 7470 7574  lMap,.  * Output
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│ │ │ │ -00037e70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
│ │ │ │ -00037e80: 7320 6973 2061 2073 686f 7274 6375 7420  s is a shortcut 
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│ │ │ │ -00037ea0: 692c 4365 7274 6966 793d 3e74 7275 652c  i,Certify=>true,
│ │ │ │ -00037eb0: 5665 7262 6f73 653d 3e66 616c 7365 292c  Verbose=>false),
│ │ │ │ -00037ec0: 2073 6565 202a 6e6f 7465 0a64 6567 7265   see *note.degre
│ │ │ │ -00037ed0: 654d 6170 2852 6174 696f 6e61 6c4d 6170  eMap(RationalMap
│ │ │ │ -00037ee0: 293a 2064 6567 7265 654d 6170 5f6c 7052  ): degreeMap_lpR
│ │ │ │ -00037ef0: 6174 696f 6e61 6c4d 6170 5f72 702c 2e0a  ationalMap_rp,..
│ │ │ │ -00037f00: 0a57 6179 7320 746f 2075 7365 2074 6869  .Ways to use thi
│ │ │ │ -00037f10: 7320 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d  s method:.======
│ │ │ │ -00037f20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00037f40: 6772 6565 2852 6174 696f 6e61 6c4d 6170  gree(RationalMap
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│ │ │ │ -00037f70: 6567 7265 6520 6f66 2061 2072 6174 696f  egree of a ratio
│ │ │ │ -00037f80: 6e61 6c0a 2020 2020 6d61 700a 2d2d 2d2d  nal.    map.----
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│ │ │ │ +00037d40: 202a 6e6f 7465 2064 6567 7265 653a 2028   *note degree: (
│ │ │ │ +00037d50: 4d61 6361 756c 6179 3244 6f63 2964 6567  Macaulay2Doc)deg
│ │ │ │ +00037d60: 7265 652c 0a20 202a 2055 7361 6765 3a20  ree,.  * Usage: 
│ │ │ │ +00037d70: 0a20 2020 2020 2020 2064 6567 7265 6520  .        degree 
│ │ │ │ +00037d80: 7068 690a 2020 2a20 496e 7075 7473 3a0a  phi.  * Inputs:.
│ │ │ │ +00037d90: 2020 2020 2020 2a20 7068 692c 2061 202a        * phi, a *
│ │ │ │ +00037da0: 6e6f 7465 2072 6174 696f 6e61 6c20 6d61  note rational ma
│ │ │ │ +00037db0: 703a 2052 6174 696f 6e61 6c4d 6170 2c0a  p: RationalMap,.
│ │ │ │ +00037dc0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +00037dd0: 2020 202a 2061 6e20 2a6e 6f74 6520 696e     * an *note in
│ │ │ │ +00037de0: 7465 6765 723a 2028 4d61 6361 756c 6179  teger: (Macaulay
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│ │ │ │ +00037e20: 2020 2069 7320 3120 6966 2061 6e64 206f     is 1 if and o
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│ │ │ │ +00037e70: 3d3d 3d3d 3d0a 0a54 6869 7320 6973 2061  =====..This is a
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│ │ │ │ +00037ea0: 6966 793d 3e74 7275 652c 5665 7262 6f73  ify=>true,Verbos
│ │ │ │ +00037eb0: 653d 3e66 616c 7365 292c 2073 6565 202a  e=>false), see *
│ │ │ │ +00037ec0: 6e6f 7465 0a64 6567 7265 654d 6170 2852  note.degreeMap(R
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│ │ │ │ +00037ee0: 7265 654d 6170 5f6c 7052 6174 696f 6e61  reeMap_lpRationa
│ │ │ │ +00037ef0: 6c4d 6170 5f72 702c 2e0a 0a57 6179 7320  lMap_rp,...Ways 
│ │ │ │ +00037f00: 746f 2075 7365 2074 6869 7320 6d65 7468  to use this meth
│ │ │ │ +00037f10: 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  od:.============
│ │ │ │ +00037f20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +00037f30: 2a20 2a6e 6f74 6520 6465 6772 6565 2852  * *note degree(R
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│ │ │ │ +00037f50: 7265 655f 6c70 5261 7469 6f6e 616c 4d61  ree_lpRationalMa
│ │ │ │ +00037f60: 705f 7270 2c20 2d2d 2064 6567 7265 6520  p_rp, -- degree 
│ │ │ │ +00037f70: 6f66 2061 2072 6174 696f 6e61 6c0a 2020  of a rational.  
│ │ │ │ +00037f80: 2020 6d61 700a 2d2d 2d2d 2d2d 2d2d 2d2d    map.----------
│ │ │ │  00037f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00038030: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ -00038040: 6573 2f43 7265 6d6f 6e61 2f0a 646f 6375  es/Cremona/.docu
│ │ │ │ -00038050: 6d65 6e74 6174 696f 6e2e 6d32 3a34 3932  mentation.m2:492
│ │ │ │ -00038060: 3a30 2e0a 1f0a 4669 6c65 3a20 4372 656d  :0....File: Crem
│ │ │ │ -00038070: 6f6e 612e 696e 666f 2c20 4e6f 6465 3a20  ona.info, Node: 
│ │ │ │ -00038080: 6465 6772 6565 4d61 702c 204e 6578 743a  degreeMap, Next:
│ │ │ │ -00038090: 2064 6567 7265 654d 6170 5f6c 7052 6174   degreeMap_lpRat
│ │ │ │ -000380a0: 696f 6e61 6c4d 6170 5f72 702c 2050 7265  ionalMap_rp, Pre
│ │ │ │ -000380b0: 763a 2064 6567 7265 655f 6c70 5261 7469  v: degree_lpRati
│ │ │ │ -000380c0: 6f6e 616c 4d61 705f 7270 2c20 5570 3a20  onalMap_rp, Up: 
│ │ │ │ -000380d0: 546f 700a 0a64 6567 7265 654d 6170 202d  Top..degreeMap -
│ │ │ │ -000380e0: 2d20 6465 6772 6565 206f 6620 6120 7261  - degree of a ra
│ │ │ │ -000380f0: 7469 6f6e 616c 206d 6170 2062 6574 7765  tional map betwe
│ │ │ │ -00038100: 656e 2070 726f 6a65 6374 6976 6520 7661  en projective va
│ │ │ │ -00038110: 7269 6574 6965 730a 2a2a 2a2a 2a2a 2a2a  rieties.********
│ │ │ │ +00037fd0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ +00037fe0: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ +00037ff0: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ +00038000: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ +00038010: 682f 6d61 6361 756c 6179 322d 312e 3235  h/macaulay2-1.25
│ │ │ │ +00038020: 2e31 312b 6473 2f4d 322f 4d61 6361 756c  .11+ds/M2/Macaul
│ │ │ │ +00038030: 6179 322f 7061 636b 6167 6573 2f43 7265  ay2/packages/Cre
│ │ │ │ +00038040: 6d6f 6e61 2f0a 646f 6375 6d65 6e74 6174  mona/.documentat
│ │ │ │ +00038050: 696f 6e2e 6d32 3a34 3932 3a30 2e0a 1f0a  ion.m2:492:0....
│ │ │ │ +00038060: 4669 6c65 3a20 4372 656d 6f6e 612e 696e  File: Cremona.in
│ │ │ │ +00038070: 666f 2c20 4e6f 6465 3a20 6465 6772 6565  fo, Node: degree
│ │ │ │ +00038080: 4d61 702c 204e 6578 743a 2064 6567 7265  Map, Next: degre
│ │ │ │ +00038090: 654d 6170 5f6c 7052 6174 696f 6e61 6c4d  eMap_lpRationalM
│ │ │ │ +000380a0: 6170 5f72 702c 2050 7265 763a 2064 6567  ap_rp, Prev: deg
│ │ │ │ +000380b0: 7265 655f 6c70 5261 7469 6f6e 616c 4d61  ree_lpRationalMa
│ │ │ │ +000380c0: 705f 7270 2c20 5570 3a20 546f 700a 0a64  p_rp, Up: Top..d
│ │ │ │ +000380d0: 6567 7265 654d 6170 202d 2d20 6465 6772  egreeMap -- degr
│ │ │ │ +000380e0: 6565 206f 6620 6120 7261 7469 6f6e 616c  ee of a rational
│ │ │ │ +000380f0: 206d 6170 2062 6574 7765 656e 2070 726f   map between pro
│ │ │ │ +00038100: 6a65 6374 6976 6520 7661 7269 6574 6965  jective varietie
│ │ │ │ +00038110: 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  s.**************
│ │ │ │  00038120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00038130: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00038140: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00038150: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -00038160: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -00038170: 6465 6772 6565 4d61 7020 7068 690a 2020  degreeMap phi.  
│ │ │ │ -00038180: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -00038190: 2a20 7068 692c 2061 202a 6e6f 7465 2072  * phi, a *note r
│ │ │ │ -000381a0: 696e 6720 6d61 703a 2028 4d61 6361 756c  ing map: (Macaul
│ │ │ │ -000381b0: 6179 3244 6f63 2952 696e 674d 6170 2c2c  ay2Doc)RingMap,,
│ │ │ │ -000381c0: 2077 6869 6368 2072 6570 7265 7365 6e74   which represent
│ │ │ │ -000381d0: 7320 610a 2020 2020 2020 2020 7261 7469  s a.        rati
│ │ │ │ -000381e0: 6f6e 616c 206d 6170 2024 5c50 6869 2420  onal map $\Phi$ 
│ │ │ │ -000381f0: 6265 7477 6565 6e20 7072 6f6a 6563 7469  between projecti
│ │ │ │ -00038200: 7665 2076 6172 6965 7469 6573 0a20 202a  ve varieties.  *
│ │ │ │ -00038210: 202a 6e6f 7465 204f 7074 696f 6e61 6c20   *note Optional 
│ │ │ │ -00038220: 696e 7075 7473 3a20 284d 6163 6175 6c61  inputs: (Macaula
│ │ │ │ -00038230: 7932 446f 6329 7573 696e 6720 6675 6e63  y2Doc)using func
│ │ │ │ -00038240: 7469 6f6e 7320 7769 7468 206f 7074 696f  tions with optio
│ │ │ │ -00038250: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020  nal inputs,:.   
│ │ │ │ -00038260: 2020 202a 202a 6e6f 7465 2042 6c6f 7755     * *note BlowU
│ │ │ │ -00038270: 7053 7472 6174 6567 793a 2042 6c6f 7755  pStrategy: BlowU
│ │ │ │ -00038280: 7053 7472 6174 6567 792c 203d 3e20 2e2e  pStrategy, => ..
│ │ │ │ -00038290: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -000382a0: 0a20 2020 2020 2020 2022 456c 696d 696e  .        "Elimin
│ │ │ │ -000382b0: 6174 6522 2c0a 2020 2020 2020 2a20 2a6e  ate",.      * *n
│ │ │ │ -000382c0: 6f74 6520 4365 7274 6966 793a 2043 6572  ote Certify: Cer
│ │ │ │ -000382d0: 7469 6679 2c20 3d3e 202e 2e2e 2c20 6465  tify, => ..., de
│ │ │ │ -000382e0: 6661 756c 7420 7661 6c75 6520 6661 6c73  fault value fals
│ │ │ │ -000382f0: 652c 2077 6865 7468 6572 2074 6f20 656e  e, whether to en
│ │ │ │ -00038300: 7375 7265 0a20 2020 2020 2020 2063 6f72  sure.        cor
│ │ │ │ -00038310: 7265 6374 6e65 7373 206f 6620 6f75 7470  rectness of outp
│ │ │ │ -00038320: 7574 0a20 2020 2020 202a 202a 6e6f 7465  ut.      * *note
│ │ │ │ -00038330: 2056 6572 626f 7365 3a20 696e 7665 7273   Verbose: invers
│ │ │ │ -00038340: 654d 6170 5f6c 705f 7064 5f70 645f 7064  eMap_lp_pd_pd_pd
│ │ │ │ -00038350: 5f63 6d56 6572 626f 7365 3d3e 5f70 645f  _cmVerbose=>_pd_
│ │ │ │ -00038360: 7064 5f70 645f 7270 2c20 3d3e 202e 2e2e  pd_pd_rp, => ...
│ │ │ │ -00038370: 2c0a 2020 2020 2020 2020 6465 6661 756c  ,.        defaul
│ │ │ │ -00038380: 7420 7661 6c75 6520 7472 7565 2c0a 2020  t value true,.  
│ │ │ │ -00038390: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -000383a0: 202a 2061 6e20 2a6e 6f74 6520 696e 7465   * an *note inte
│ │ │ │ -000383b0: 6765 723a 2028 4d61 6361 756c 6179 3244  ger: (Macaulay2D
│ │ │ │ -000383c0: 6f63 295a 5a2c 2c20 7468 6520 6465 6772  oc)ZZ,, the degr
│ │ │ │ -000383d0: 6565 206f 6620 245c 5068 6924 2e20 536f  ee of $\Phi$. So
│ │ │ │ -000383e0: 2074 6869 730a 2020 2020 2020 2020 7661   this.        va
│ │ │ │ -000383f0: 6c75 6520 6973 2031 2069 6620 616e 6420  lue is 1 if and 
│ │ │ │ -00038400: 6f6e 6c79 2069 6620 7468 6520 6d61 7020  only if the map 
│ │ │ │ -00038410: 6973 2062 6972 6174 696f 6e61 6c20 6f6e  is birational on
│ │ │ │ -00038420: 746f 2069 7473 2069 6d61 6765 2e0a 0a44  to its image...D
│ │ │ │ -00038430: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -00038440: 3d3d 3d3d 3d3d 0a0a 4f6e 6520 696d 706f  ======..One impo
│ │ │ │ -00038450: 7274 616e 7420 6361 7365 2069 7320 7768  rtant case is wh
│ │ │ │ -00038460: 656e 2024 5c50 6869 3a5c 6d61 7468 6262  en $\Phi:\mathbb
│ │ │ │ -00038470: 7b50 7d5e 6e3d 5072 6f6a 284b 5b78 5f30  {P}^n=Proj(K[x_0
│ │ │ │ -00038480: 2c5c 6c64 6f74 732c 785f 6e5d 290a 5c64  ,\ldots,x_n]).\d
│ │ │ │ -00038490: 6173 6872 6967 6874 6172 726f 7720 5c6d  ashrightarrow \m
│ │ │ │ -000384a0: 6174 6862 627b 507d 5e6d 3d50 726f 6a28  athbb{P}^m=Proj(
│ │ │ │ -000384b0: 4b5b 795f 302c 5c6c 646f 7473 2c79 5f6d  K[y_0,\ldots,y_m
│ │ │ │ -000384c0: 5d29 2420 6973 2061 2072 6174 696f 6e61  ])$ is a rationa
│ │ │ │ -000384d0: 6c20 6d61 7020 6265 7477 6565 6e0a 7072  l map between.pr
│ │ │ │ -000384e0: 6f6a 6563 7469 7665 2073 7061 6365 732c  ojective spaces,
│ │ │ │ -000384f0: 2063 6f72 7265 7370 6f6e 6469 6e67 2074   corresponding t
│ │ │ │ -00038500: 6f20 6120 7269 6e67 206d 6170 2024 5c70  o a ring map $\p
│ │ │ │ -00038510: 6869 242e 2049 6620 2470 2420 6973 2061  hi$. If $p$ is a
│ │ │ │ -00038520: 2067 656e 6572 616c 0a70 6f69 6e74 206f   general.point o
│ │ │ │ -00038530: 6620 245c 6d61 7468 6262 7b50 7d5e 6e24  f $\mathbb{P}^n$
│ │ │ │ -00038540: 2c20 6465 6e6f 7465 2062 7920 2446 5f70  , denote by $F_p
│ │ │ │ -00038550: 285c 5068 6929 2420 7468 6520 636c 6f73  (\Phi)$ the clos
│ │ │ │ -00038560: 7572 6520 6f66 0a24 5c50 6869 5e7b 2d31  ure of.$\Phi^{-1
│ │ │ │ -00038570: 7d28 5c50 6869 2870 2929 5c73 7562 7365  }(\Phi(p))\subse
│ │ │ │ -00038580: 7465 7120 5c6d 6174 6862 627b 507d 5e6e  teq \mathbb{P}^n
│ │ │ │ -00038590: 242e 2054 6865 2064 6567 7265 6520 6f66  $. The degree of
│ │ │ │ -000385a0: 2024 5c50 6869 2420 6973 2064 6566 696e   $\Phi$ is defin
│ │ │ │ -000385b0: 6564 2061 730a 7468 6520 6465 6772 6565  ed as.the degree
│ │ │ │ -000385c0: 206f 6620 2446 5f70 285c 5068 6929 2420   of $F_p(\Phi)$ 
│ │ │ │ -000385d0: 6966 2024 6469 6d20 465f 7028 5c50 6869  if $dim F_p(\Phi
│ │ │ │ -000385e0: 2920 3d20 3024 2061 6e64 2024 3024 206f  ) = 0$ and $0$ o
│ │ │ │ -000385f0: 7468 6572 7769 7365 2e20 4966 2024 5c50  therwise. If $\P
│ │ │ │ -00038600: 6869 240a 6973 2064 6566 696e 6564 2062  hi$.is defined b
│ │ │ │ -00038610: 7920 666f 726d 7320 2446 5f30 2878 5f30  y forms $F_0(x_0
│ │ │ │ -00038620: 2c5c 6c64 6f74 732c 785f 6e29 2c5c 6c64  ,\ldots,x_n),\ld
│ │ │ │ -00038630: 6f74 732c 465f 6d28 785f 302c 5c6c 646f  ots,F_m(x_0,\ldo
│ │ │ │ -00038640: 7473 2c78 5f6e 2924 2061 6e64 2024 495f  ts,x_n)$ and $I_
│ │ │ │ -00038650: 7024 0a69 7320 7468 6520 6964 6561 6c20  p$.is the ideal 
│ │ │ │ -00038660: 6f66 2074 6865 2070 6f69 6e74 2024 7024  of the point $p$
│ │ │ │ -00038670: 2c20 7468 656e 2074 6865 2069 6465 616c  , then the ideal
│ │ │ │ -00038680: 206f 6620 2446 5f70 285c 5068 6929 2420   of $F_p(\Phi)$ 
│ │ │ │ -00038690: 6973 206e 6f74 6869 6e67 2062 7574 2074  is nothing but t
│ │ │ │ -000386a0: 6865 0a73 6174 7572 6174 696f 6e20 247b  he.saturation ${
│ │ │ │ -000386b0: 285c 7068 6928 5c70 6869 5e7b 2d31 7d28  (\phi(\phi^{-1}(
│ │ │ │ -000386c0: 495f 7029 2929 3a28 465f 302c 2e2e 2e2e  I_p))):(F_0,....
│ │ │ │ -000386d0: 2c46 5f6d 297d 5e7b 5c69 6e66 7479 7d24  ,F_m)}^{\infty}$
│ │ │ │ -000386e0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │ +00038150: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ +00038160: 200a 2020 2020 2020 2020 6465 6772 6565   .        degree
│ │ │ │ +00038170: 4d61 7020 7068 690a 2020 2a20 496e 7075  Map phi.  * Inpu
│ │ │ │ +00038180: 7473 3a0a 2020 2020 2020 2a20 7068 692c  ts:.      * phi,
│ │ │ │ +00038190: 2061 202a 6e6f 7465 2072 696e 6720 6d61   a *note ring ma
│ │ │ │ +000381a0: 703a 2028 4d61 6361 756c 6179 3244 6f63  p: (Macaulay2Doc
│ │ │ │ +000381b0: 2952 696e 674d 6170 2c2c 2077 6869 6368  )RingMap,, which
│ │ │ │ +000381c0: 2072 6570 7265 7365 6e74 7320 610a 2020   represents a.  
│ │ │ │ +000381d0: 2020 2020 2020 7261 7469 6f6e 616c 206d        rational m
│ │ │ │ +000381e0: 6170 2024 5c50 6869 2420 6265 7477 6565  ap $\Phi$ betwee
│ │ │ │ +000381f0: 6e20 7072 6f6a 6563 7469 7665 2076 6172  n projective var
│ │ │ │ +00038200: 6965 7469 6573 0a20 202a 202a 6e6f 7465  ieties.  * *note
│ │ │ │ +00038210: 204f 7074 696f 6e61 6c20 696e 7075 7473   Optional inputs
│ │ │ │ +00038220: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00038230: 7573 696e 6720 6675 6e63 7469 6f6e 7320  using functions 
│ │ │ │ +00038240: 7769 7468 206f 7074 696f 6e61 6c20 696e  with optional in
│ │ │ │ +00038250: 7075 7473 2c3a 0a20 2020 2020 202a 202a  puts,:.      * *
│ │ │ │ +00038260: 6e6f 7465 2042 6c6f 7755 7053 7472 6174  note BlowUpStrat
│ │ │ │ +00038270: 6567 793a 2042 6c6f 7755 7053 7472 6174  egy: BlowUpStrat
│ │ │ │ +00038280: 6567 792c 203d 3e20 2e2e 2e2c 2064 6566  egy, => ..., def
│ │ │ │ +00038290: 6175 6c74 2076 616c 7565 0a20 2020 2020  ault value.     
│ │ │ │ +000382a0: 2020 2022 456c 696d 696e 6174 6522 2c0a     "Eliminate",.
│ │ │ │ +000382b0: 2020 2020 2020 2a20 2a6e 6f74 6520 4365        * *note Ce
│ │ │ │ +000382c0: 7274 6966 793a 2043 6572 7469 6679 2c20  rtify: Certify, 
│ │ │ │ +000382d0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +000382e0: 7661 6c75 6520 6661 6c73 652c 2077 6865  value false, whe
│ │ │ │ +000382f0: 7468 6572 2074 6f20 656e 7375 7265 0a20  ther to ensure. 
│ │ │ │ +00038300: 2020 2020 2020 2063 6f72 7265 6374 6e65         correctne
│ │ │ │ +00038310: 7373 206f 6620 6f75 7470 7574 0a20 2020  ss of output.   
│ │ │ │ +00038320: 2020 202a 202a 6e6f 7465 2056 6572 626f     * *note Verbo
│ │ │ │ +00038330: 7365 3a20 696e 7665 7273 654d 6170 5f6c  se: inverseMap_l
│ │ │ │ +00038340: 705f 7064 5f70 645f 7064 5f63 6d56 6572  p_pd_pd_pd_cmVer
│ │ │ │ +00038350: 626f 7365 3d3e 5f70 645f 7064 5f70 645f  bose=>_pd_pd_pd_
│ │ │ │ +00038360: 7270 2c20 3d3e 202e 2e2e 2c0a 2020 2020  rp, => ...,.    
│ │ │ │ +00038370: 2020 2020 6465 6661 756c 7420 7661 6c75      default valu
│ │ │ │ +00038380: 6520 7472 7565 2c0a 2020 2a20 4f75 7470  e true,.  * Outp
│ │ │ │ +00038390: 7574 733a 0a20 2020 2020 202a 2061 6e20  uts:.      * an 
│ │ │ │ +000383a0: 2a6e 6f74 6520 696e 7465 6765 723a 2028  *note integer: (
│ │ │ │ +000383b0: 4d61 6361 756c 6179 3244 6f63 295a 5a2c  Macaulay2Doc)ZZ,
│ │ │ │ +000383c0: 2c20 7468 6520 6465 6772 6565 206f 6620  , the degree of 
│ │ │ │ +000383d0: 245c 5068 6924 2e20 536f 2074 6869 730a  $\Phi$. So this.
│ │ │ │ +000383e0: 2020 2020 2020 2020 7661 6c75 6520 6973          value is
│ │ │ │ +000383f0: 2031 2069 6620 616e 6420 6f6e 6c79 2069   1 if and only i
│ │ │ │ +00038400: 6620 7468 6520 6d61 7020 6973 2062 6972  f the map is bir
│ │ │ │ +00038410: 6174 696f 6e61 6c20 6f6e 746f 2069 7473  ational onto its
│ │ │ │ +00038420: 2069 6d61 6765 2e0a 0a44 6573 6372 6970   image...Descrip
│ │ │ │ +00038430: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ +00038440: 0a0a 4f6e 6520 696d 706f 7274 616e 7420  ..One important 
│ │ │ │ +00038450: 6361 7365 2069 7320 7768 656e 2024 5c50  case is when $\P
│ │ │ │ +00038460: 6869 3a5c 6d61 7468 6262 7b50 7d5e 6e3d  hi:\mathbb{P}^n=
│ │ │ │ +00038470: 5072 6f6a 284b 5b78 5f30 2c5c 6c64 6f74  Proj(K[x_0,\ldot
│ │ │ │ +00038480: 732c 785f 6e5d 290a 5c64 6173 6872 6967  s,x_n]).\dashrig
│ │ │ │ +00038490: 6874 6172 726f 7720 5c6d 6174 6862 627b  htarrow \mathbb{
│ │ │ │ +000384a0: 507d 5e6d 3d50 726f 6a28 4b5b 795f 302c  P}^m=Proj(K[y_0,
│ │ │ │ +000384b0: 5c6c 646f 7473 2c79 5f6d 5d29 2420 6973  \ldots,y_m])$ is
│ │ │ │ +000384c0: 2061 2072 6174 696f 6e61 6c20 6d61 7020   a rational map 
│ │ │ │ +000384d0: 6265 7477 6565 6e0a 7072 6f6a 6563 7469  between.projecti
│ │ │ │ +000384e0: 7665 2073 7061 6365 732c 2063 6f72 7265  ve spaces, corre
│ │ │ │ +000384f0: 7370 6f6e 6469 6e67 2074 6f20 6120 7269  sponding to a ri
│ │ │ │ +00038500: 6e67 206d 6170 2024 5c70 6869 242e 2049  ng map $\phi$. I
│ │ │ │ +00038510: 6620 2470 2420 6973 2061 2067 656e 6572  f $p$ is a gener
│ │ │ │ +00038520: 616c 0a70 6f69 6e74 206f 6620 245c 6d61  al.point of $\ma
│ │ │ │ +00038530: 7468 6262 7b50 7d5e 6e24 2c20 6465 6e6f  thbb{P}^n$, deno
│ │ │ │ +00038540: 7465 2062 7920 2446 5f70 285c 5068 6929  te by $F_p(\Phi)
│ │ │ │ +00038550: 2420 7468 6520 636c 6f73 7572 6520 6f66  $ the closure of
│ │ │ │ +00038560: 0a24 5c50 6869 5e7b 2d31 7d28 5c50 6869  .$\Phi^{-1}(\Phi
│ │ │ │ +00038570: 2870 2929 5c73 7562 7365 7465 7120 5c6d  (p))\subseteq \m
│ │ │ │ +00038580: 6174 6862 627b 507d 5e6e 242e 2054 6865  athbb{P}^n$. The
│ │ │ │ +00038590: 2064 6567 7265 6520 6f66 2024 5c50 6869   degree of $\Phi
│ │ │ │ +000385a0: 2420 6973 2064 6566 696e 6564 2061 730a  $ is defined as.
│ │ │ │ +000385b0: 7468 6520 6465 6772 6565 206f 6620 2446  the degree of $F
│ │ │ │ +000385c0: 5f70 285c 5068 6929 2420 6966 2024 6469  _p(\Phi)$ if $di
│ │ │ │ +000385d0: 6d20 465f 7028 5c50 6869 2920 3d20 3024  m F_p(\Phi) = 0$
│ │ │ │ +000385e0: 2061 6e64 2024 3024 206f 7468 6572 7769   and $0$ otherwi
│ │ │ │ +000385f0: 7365 2e20 4966 2024 5c50 6869 240a 6973  se. If $\Phi$.is
│ │ │ │ +00038600: 2064 6566 696e 6564 2062 7920 666f 726d   defined by form
│ │ │ │ +00038610: 7320 2446 5f30 2878 5f30 2c5c 6c64 6f74  s $F_0(x_0,\ldot
│ │ │ │ +00038620: 732c 785f 6e29 2c5c 6c64 6f74 732c 465f  s,x_n),\ldots,F_
│ │ │ │ +00038630: 6d28 785f 302c 5c6c 646f 7473 2c78 5f6e  m(x_0,\ldots,x_n
│ │ │ │ +00038640: 2924 2061 6e64 2024 495f 7024 0a69 7320  )$ and $I_p$.is 
│ │ │ │ +00038650: 7468 6520 6964 6561 6c20 6f66 2074 6865  the ideal of the
│ │ │ │ +00038660: 2070 6f69 6e74 2024 7024 2c20 7468 656e   point $p$, then
│ │ │ │ +00038670: 2074 6865 2069 6465 616c 206f 6620 2446   the ideal of $F
│ │ │ │ +00038680: 5f70 285c 5068 6929 2420 6973 206e 6f74  _p(\Phi)$ is not
│ │ │ │ +00038690: 6869 6e67 2062 7574 2074 6865 0a73 6174  hing but the.sat
│ │ │ │ +000386a0: 7572 6174 696f 6e20 247b 285c 7068 6928  uration ${(\phi(
│ │ │ │ +000386b0: 5c70 6869 5e7b 2d31 7d28 495f 7029 2929  \phi^{-1}(I_p)))
│ │ │ │ +000386c0: 3a28 465f 302c 2e2e 2e2e 2c46 5f6d 297d  :(F_0,....,F_m)}
│ │ │ │ +000386d0: 5e7b 5c69 6e66 7479 7d24 2e0a 0a2b 2d2d  ^{\infty}$...+--
│ │ │ │ +000386e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000386f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038730: 2d2b 0a7c 6931 203a 202d 2d20 5461 6b65  -+.|i1 : -- Take
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│ │ │ │ -00038750: 7068 693a 505e 382d 2d2d 3e47 2831 2c35  phi:P^8--->G(1,5
│ │ │ │ -00038760: 2920 7375 6273 6574 2050 5e31 3420 6465  ) subset P^14 de
│ │ │ │ -00038770: 6669 6e65 6420 6279 2020 2020 2020 2020  fined by        
│ │ │ │ -00038780: 207c 0a7c 2020 2020 202d 2d20 6f66 2061   |.|     -- of a
│ │ │ │ -00038790: 2067 656e 6572 6963 2032 2078 2036 206d   generic 2 x 6 m
│ │ │ │ -000387a0: 6174 7269 7820 6f66 206c 696e 6561 7220  atrix of linear 
│ │ │ │ -000387b0: 666f 726d 7320 6f6e 2050 5e38 2028 7468  forms on P^8 (th
│ │ │ │ -000387c0: 7573 2070 6869 2069 7320 2020 2020 2020  us phi is       
│ │ │ │ -000387d0: 207c 0a7c 2020 2020 204b 3d5a 5a2f 3333   |.|     K=ZZ/33
│ │ │ │ -000387e0: 3331 3b20 7269 6e67 5038 3d4b 5b78 5f30  31; ringP8=K[x_0
│ │ │ │ -000387f0: 2e2e 785f 385d 3b20 7269 6e67 5031 343d  ..x_8]; ringP14=
│ │ │ │ -00038800: 4b5b 745f 302e 2e74 5f31 345d 3b20 2020  K[t_0..t_14];   
│ │ │ │ -00038810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038820: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ +00038720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00038730: 203a 202d 2d20 5461 6b65 2061 2072 6174   : -- Take a rat
│ │ │ │ +00038740: 696f 6e61 6c20 6d61 7020 7068 693a 505e  ional map phi:P^
│ │ │ │ +00038750: 382d 2d2d 3e47 2831 2c35 2920 7375 6273  8--->G(1,5) subs
│ │ │ │ +00038760: 6574 2050 5e31 3420 6465 6669 6e65 6420  et P^14 defined 
│ │ │ │ +00038770: 6279 2020 2020 2020 2020 207c 0a7c 2020  by         |.|  
│ │ │ │ +00038780: 2020 202d 2d20 6f66 2061 2067 656e 6572     -- of a gener
│ │ │ │ +00038790: 6963 2032 2078 2036 206d 6174 7269 7820  ic 2 x 6 matrix 
│ │ │ │ +000387a0: 6f66 206c 696e 6561 7220 666f 726d 7320  of linear forms 
│ │ │ │ +000387b0: 6f6e 2050 5e38 2028 7468 7573 2070 6869  on P^8 (thus phi
│ │ │ │ +000387c0: 2069 7320 2020 2020 2020 207c 0a7c 2020   is        |.|  
│ │ │ │ +000387d0: 2020 204b 3d5a 5a2f 3333 3331 3b20 7269     K=ZZ/3331; ri
│ │ │ │ +000387e0: 6e67 5038 3d4b 5b78 5f30 2e2e 785f 385d  ngP8=K[x_0..x_8]
│ │ │ │ +000387f0: 3b20 7269 6e67 5031 343d 4b5b 745f 302e  ; ringP14=K[t_0.
│ │ │ │ +00038800: 2e74 5f31 345d 3b20 2020 2020 2020 2020  .t_14];         
│ │ │ │ +00038810: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ +00038820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038830: 2d2d 2d2d 2d2d 2d2d 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: 2d7c 0a7c 7468 6520 6d61 7869 6d61 6c20  -|.|the maximal 
│ │ │ │ -00038880: 6d69 6e6f 7273 2020 2020 2020 2020 2020  minors          
│ │ │ │ +00038860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7468  -----------|.|th
│ │ │ │ +00038870: 6520 6d61 7869 6d61 6c20 6d69 6e6f 7273  e maximal minors
│ │ │ │ +00038880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000388a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000388b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000388c0: 207c 0a7c 2062 6972 6174 696f 6e61 6c20   |.| birational 
│ │ │ │ -000388d0: 6f6e 746f 2069 7473 2069 6d61 6765 2920  onto its image) 
│ │ │ │ +000388b0: 2020 2020 2020 2020 2020 207c 0a7c 2062             |.| b
│ │ │ │ +000388c0: 6972 6174 696f 6e61 6c20 6f6e 746f 2069  irational onto i
│ │ │ │ +000388d0: 7473 2069 6d61 6765 2920 2020 2020 2020  ts image)       
│ │ │ │  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: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00038900: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00038910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038960: 2d2b 0a7c 6934 203a 2070 6869 3d6d 6170  -+.|i4 : phi=map
│ │ │ │ -00038970: 2872 696e 6750 382c 7269 6e67 5031 342c  (ringP8,ringP14,
│ │ │ │ -00038980: 6765 6e73 206d 696e 6f72 7328 322c 6d61  gens minors(2,ma
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│ │ │ │ -000389a0: 6920 746f 2020 2020 2020 2020 2020 2020  i to            
│ │ │ │ -000389b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00038950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +00038960: 203a 2070 6869 3d6d 6170 2872 696e 6750   : phi=map(ringP
│ │ │ │ +00038970: 382c 7269 6e67 5031 342c 6765 6e73 206d  8,ringP14,gens m
│ │ │ │ +00038980: 696e 6f72 7328 322c 6d61 7472 6978 2070  inors(2,matrix p
│ │ │ │ +00038990: 6163 6b28 362c 666f 7220 6920 746f 2020  ack(6,for i to  
│ │ │ │ +000389a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000389b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000389c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000389d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000389e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000389f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00038a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038a20: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -00038a30: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -00038a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038a50: 207c 0a7c 6f34 203d 206d 6170 2028 7269   |.|o4 = map (ri
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│ │ │ │ -00038a90: 7820 7820 2020 2020 2020 2020 2020 2020  x x             
│ │ │ │ -00038aa0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00038ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038ac0: 2020 2020 2030 2020 2020 2020 2030 2031       0       0 1
│ │ │ │ -00038ad0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -00038ae0: 2030 2032 2020 2020 2020 2020 2020 2020   0 2            
│ │ │ │ -00038af0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000389f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038a10: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00038a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038a30: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00038a40: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
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│ │ │ │ +00038a60: 7269 6e67 5031 342c 207b 2d20 3935 7820  ringP14, {- 95x 
│ │ │ │ +00038a70: 202b 2031 3831 7820 7820 202b 2031 3032   + 181x x  + 102
│ │ │ │ +00038a80: 3878 2020 2d20 3133 3834 7820 7820 2020  8x  - 1384x x   
│ │ │ │ +00038a90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038ab0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00038ac0: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ +00038ad0: 2020 3120 2020 2020 2020 2030 2032 2020    1        0 2  
│ │ │ │ +00038ae0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00038af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038b40: 207c 0a7c 6f34 203a 2052 696e 674d 6170   |.|o4 : RingMap
│ │ │ │ -00038b50: 2072 696e 6750 3820 3c2d 2d20 7269 6e67   ringP8 <-- ring
│ │ │ │ -00038b60: 5031 3420 2020 2020 2020 2020 2020 2020  P14             
│ │ │ │ +00038b30: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +00038b40: 203a 2052 696e 674d 6170 2072 696e 6750   : RingMap ringP
│ │ │ │ +00038b50: 3820 3c2d 2d20 7269 6e67 5031 3420 2020  8 <-- ringP14   
│ │ │ │ +00038b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00038b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00038c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00038c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00038c90: 2020 2020 2032 2020 2020 2020 2020 2020       2          
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│ │ │ │ -00038d30: 2020 2020 2032 2020 2020 2020 2030 2033       2       0 3
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│ │ │ │  00038d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00038e20: 3638 7820 7820 202d 2036 3736 7820 7820  68x x  - 676x x 
│ │ │ │ -00038e30: 202b 2035 3031 7820 202b 2037 3378 2078   + 501x  + 73x x
│ │ │ │ -00038e40: 2020 2b20 3132 3633 7820 7820 202b 2031    + 1263x x  + 1
│ │ │ │ -00038e50: 3033 3578 2078 2020 2b20 3834 3478 2078  035x x  + 844x x
│ │ │ │ -00038e60: 207c 0a7c 2020 2020 3120 3420 2020 2020   |.|    1 4     
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│ │ │ │ -00038e80: 2020 2020 2020 2034 2020 2020 2020 3020         4      0 
│ │ │ │ -00038e90: 3520 2020 2020 2020 2031 2035 2020 2020  5        1 5    
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│ │ │ │ +00038e50: 2020 2b20 3834 3478 2078 207c 0a7c 2020    + 844x x |.|  
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│ │ │ │ +00038e90: 2020 2031 2035 2020 2020 2020 2020 3220     1 5        2 
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│ │ │ │ +00038eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00038ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00038f00: 2020 2020 2020 2020 2020 2020 2020 2032                 2
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│ │ │ │ -0003a8b0: 2020 2020 2020 3620 3720 2020 2020 2020        6 7       
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│ │ │ │ -0003a8e0: 2020 2020 2020 2020 3320 3820 2020 2020          3 8     
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│ │ │ │ +0003a8c0: 2020 3020 3820 2020 2020 2020 2031 2038    0 8        1 8
│ │ │ │ +0003a8d0: 2020 2020 2020 2032 2038 2020 2020 2020         2 8      
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│ │ │ │ -0003a9e0: 207c 0a7c 2020 2020 2034 2038 2020 2020   |.|     4 8    
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│ │ │ │ -0003aa10: 2038 2020 2020 2020 2020 2030 2020 2020   8         0    
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│ │ │ │ -0003b3f0: 2020 3220 3420 2020 2020 2020 3320 3420    2 4       3 4 
│ │ │ │ -0003b400: 2020 2020 2020 3420 2020 2020 2020 2030        4        0
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│ │ │ │ +0003b3b0: 2020 2d20 3132 3131 7820 7820 202b 2031    - 1211x x  + 1
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│ │ │ │  0003b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003b670: 2020 2036 2020 2020 2020 2030 2037 2020     6       0 7  
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│ │ │ │  0003fcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003fce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  0003fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0003fd40: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
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│ │ │ │ +0003fd80: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +0003fd90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ -0003fde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0003fdf0: 203d 2031 2020 2020 2020 2020 2020 2020   = 1            
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│ │ │ │  0003fe60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003fe70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003fe80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0003fef0: 696d 656e 7369 6f6e 2035 2028 736f 2074  imension 5 (so t
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│ │ │ │ -0003ff20: 505e 3820 6d75 7374 2068 6176 6520 2020  P^8 must have   
│ │ │ │ -0003ff30: 207c 0a7c 2020 2020 2070 6869 273d 7068   |.|     phi'=ph
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│ │ │ │ -0003ff50: 6e67 5038 2c66 6f72 2069 2074 6f20 3820  ngP8,for i to 8 
│ │ │ │ -0003ff60: 6c69 7374 2072 616e 646f 6d28 312c 7269  list random(1,ri
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│ │ │ │ -0003ff80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003fe80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ +0003fe90: 203a 202d 2d20 436f 6d70 6f73 6520 7068   : -- Compose ph
│ │ │ │ +0003fea0: 693a 505e 382d 2d2d 3e50 5e31 3420 7769  i:P^8--->P^14 wi
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│ │ │ │ +0003ff20: 7374 2068 6176 6520 2020 207c 0a7c 2020  st have    |.|  
│ │ │ │ +0003ff30: 2020 2070 6869 273d 7068 692a 6d61 7028     phi'=phi*map(
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│ │ │ │ +0003ff50: 6f72 2069 2074 6f20 3820 6c69 7374 2072  or i to 8 list r
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│ │ │ │ -00040060: 2078 2020 2d20 3932 3878 2078 2020 2020   x  - 928x x    
│ │ │ │ -00040070: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00040080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040090: 2020 2020 2030 2020 2020 2020 2030 2031       0       0 1
│ │ │ │ -000400a0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -000400b0: 3020 3220 2020 2020 2020 3120 3220 2020  0 2       1 2   
│ │ │ │ -000400c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ +0003ffd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0003fff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00040050: 3778 2020 2d20 3133 3278 2078 2020 2d20  7x  - 132x x  - 
│ │ │ │ +00040060: 3932 3878 2078 2020 2020 207c 0a7c 2020  928x x     |.|  
│ │ │ │ +00040070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040080: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00040090: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
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│ │ │ │ -00040130: 5038 2020 2020 2020 2020 2020 2020 2020  P8              
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│ │ │ │ +00040160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  000401e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000401f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00040290: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ +00040320: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00040370: 3035 7820 7820 202d 2038 3330 7820 7820  05x x  - 830x x 
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│ │ │ │ +00040440: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000404c0: 202b 2034 3632 7820 7820 202d 2020 2020   + 462x x  -    
│ │ │ │ -000404d0: 207c 0a7c 2020 2020 2020 3320 3420 2020   |.|      3 4   
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│ │ │ │ -00040500: 2020 3220 3520 2020 2020 2020 2033 2035    2 5        3 5
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│ │ │ │  00040540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00044870: 202a 202a 6e6f 7465 2070 726f 6a65 6374   * *note project
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│ │ │ │ -00044990: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ -000449a0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ -000449b0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ -000449c0: 7420 2a6e 6f74 6520 6465 6772 6565 4d61  t *note degreeMa
│ │ │ │ -000449d0: 703a 2064 6567 7265 654d 6170 2c20 6973  p: degreeMap, is
│ │ │ │ -000449e0: 2061 202a 6e6f 7465 206d 6574 686f 6420   a *note method 
│ │ │ │ -000449f0: 6675 6e63 7469 6f6e 2077 6974 6820 6f70  function with op
│ │ │ │ -00044a00: 7469 6f6e 733a 0a28 4d61 6361 756c 6179  tions:.(Macaulay
│ │ │ │ -00044a10: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ -00044a20: 696f 6e57 6974 684f 7074 696f 6e73 2c2e  ionWithOptions,.
│ │ │ │ -00044a30: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..--------------
│ │ │ │ +00044910: 0a0a 2020 2a20 2264 6567 7265 654d 6170  ..  * "degreeMap
│ │ │ │ +00044920: 2852 696e 674d 6170 2922 0a20 202a 202a  (RingMap)".  * *
│ │ │ │ +00044930: 6e6f 7465 2064 6567 7265 654d 6170 2852  note degreeMap(R
│ │ │ │ +00044940: 6174 696f 6e61 6c4d 6170 293a 2064 6567  ationalMap): deg
│ │ │ │ +00044950: 7265 654d 6170 5f6c 7052 6174 696f 6e61  reeMap_lpRationa
│ │ │ │ +00044960: 6c4d 6170 5f72 702c 202d 2d20 6465 6772  lMap_rp, -- degr
│ │ │ │ +00044970: 6565 206f 6620 610a 2020 2020 7261 7469  ee of a.    rati
│ │ │ │ +00044980: 6f6e 616c 206d 6170 0a0a 466f 7220 7468  onal map..For th
│ │ │ │ +00044990: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ +000449a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +000449b0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ +000449c0: 6520 6465 6772 6565 4d61 703a 2064 6567  e degreeMap: deg
│ │ │ │ +000449d0: 7265 654d 6170 2c20 6973 2061 202a 6e6f  reeMap, is a *no
│ │ │ │ +000449e0: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
│ │ │ │ +000449f0: 6f6e 2077 6974 6820 6f70 7469 6f6e 733a  on with options:
│ │ │ │ +00044a00: 0a28 4d61 6361 756c 6179 3244 6f63 294d  .(Macaulay2Doc)M
│ │ │ │ +00044a10: 6574 686f 6446 756e 6374 696f 6e57 6974  ethodFunctionWit
│ │ │ │ +00044a20: 684f 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d  hOptions,...----
│ │ │ │ +00044a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00044a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00044a80: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66  -..The source of
│ │ │ │ -00044a90: 2074 6869 7320 646f 6375 6d65 6e74 2069   this document i
│ │ │ │ -00044aa0: 7320 696e 0a2f 6275 696c 642f 7265 7072  s in./build/repr
│ │ │ │ -00044ab0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61  oducible-path/ma
│ │ │ │ -00044ac0: 6361 756c 6179 322d 312e 3235 2e31 312b  caulay2-1.25.11+
│ │ │ │ -00044ad0: 6473 2f4d 322f 4d61 6361 756c 6179 322f  ds/M2/Macaulay2/
│ │ │ │ -00044ae0: 7061 636b 6167 6573 2f43 7265 6d6f 6e61  packages/Cremona
│ │ │ │ -00044af0: 2f0a 646f 6375 6d65 6e74 6174 696f 6e2e  /.documentation.
│ │ │ │ -00044b00: 6d32 3a35 343a 302e 0a1f 0a46 696c 653a  m2:54:0....File:
│ │ │ │ -00044b10: 2043 7265 6d6f 6e61 2e69 6e66 6f2c 204e   Cremona.info, N
│ │ │ │ -00044b20: 6f64 653a 2064 6567 7265 654d 6170 5f6c  ode: degreeMap_l
│ │ │ │ -00044b30: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ -00044b40: 204e 6578 743a 2064 6567 7265 6573 5f6c   Next: degrees_l
│ │ │ │ -00044b50: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ -00044b60: 2050 7265 763a 2064 6567 7265 654d 6170   Prev: degreeMap
│ │ │ │ -00044b70: 2c20 5570 3a20 546f 700a 0a64 6567 7265  , Up: Top..degre
│ │ │ │ -00044b80: 654d 6170 2852 6174 696f 6e61 6c4d 6170  eMap(RationalMap
│ │ │ │ -00044b90: 2920 2d2d 2064 6567 7265 6520 6f66 2061  ) -- degree of a
│ │ │ │ -00044ba0: 2072 6174 696f 6e61 6c20 6d61 700a 2a2a   rational map.**
│ │ │ │ +00044a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ +00044a80: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ +00044a90: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ +00044aa0: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ +00044ab0: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ +00044ac0: 322d 312e 3235 2e31 312b 6473 2f4d 322f  2-1.25.11+ds/M2/
│ │ │ │ +00044ad0: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ +00044ae0: 6573 2f43 7265 6d6f 6e61 2f0a 646f 6375  es/Cremona/.docu
│ │ │ │ +00044af0: 6d65 6e74 6174 696f 6e2e 6d32 3a35 343a  mentation.m2:54:
│ │ │ │ +00044b00: 302e 0a1f 0a46 696c 653a 2043 7265 6d6f  0....File: Cremo
│ │ │ │ +00044b10: 6e61 2e69 6e66 6f2c 204e 6f64 653a 2064  na.info, Node: d
│ │ │ │ +00044b20: 6567 7265 654d 6170 5f6c 7052 6174 696f  egreeMap_lpRatio
│ │ │ │ +00044b30: 6e61 6c4d 6170 5f72 702c 204e 6578 743a  nalMap_rp, Next:
│ │ │ │ +00044b40: 2064 6567 7265 6573 5f6c 7052 6174 696f   degrees_lpRatio
│ │ │ │ +00044b50: 6e61 6c4d 6170 5f72 702c 2050 7265 763a  nalMap_rp, Prev:
│ │ │ │ +00044b60: 2064 6567 7265 654d 6170 2c20 5570 3a20   degreeMap, Up: 
│ │ │ │ +00044b70: 546f 700a 0a64 6567 7265 654d 6170 2852  Top..degreeMap(R
│ │ │ │ +00044b80: 6174 696f 6e61 6c4d 6170 2920 2d2d 2064  ationalMap) -- d
│ │ │ │ +00044b90: 6567 7265 6520 6f66 2061 2072 6174 696f  egree of a ratio
│ │ │ │ +00044ba0: 6e61 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a  nal map.********
│ │ │ │  00044bb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00044bc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00044bd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00044be0: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20  ..  * Function: 
│ │ │ │ -00044bf0: 2a6e 6f74 6520 6465 6772 6565 4d61 703a  *note degreeMap:
│ │ │ │ -00044c00: 2064 6567 7265 654d 6170 2c0a 2020 2a20   degreeMap,.  * 
│ │ │ │ -00044c10: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -00044c20: 6465 6772 6565 4d61 7020 5068 690a 2020  degreeMap Phi.  
│ │ │ │ -00044c30: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -00044c40: 2a20 5068 692c 2061 202a 6e6f 7465 2072  * Phi, a *note r
│ │ │ │ -00044c50: 6174 696f 6e61 6c20 6d61 703a 2052 6174  ational map: Rat
│ │ │ │ -00044c60: 696f 6e61 6c4d 6170 2c0a 2020 2a20 2a6e  ionalMap,.  * *n
│ │ │ │ -00044c70: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70  ote Optional inp
│ │ │ │ -00044c80: 7574 733a 2028 4d61 6361 756c 6179 3244  uts: (Macaulay2D
│ │ │ │ -00044c90: 6f63 2975 7369 6e67 2066 756e 6374 696f  oc)using functio
│ │ │ │ -00044ca0: 6e73 2077 6974 6820 6f70 7469 6f6e 616c  ns with optional
│ │ │ │ -00044cb0: 2069 6e70 7574 732c 3a0a 2020 2020 2020   inputs,:.      
│ │ │ │ -00044cc0: 2a20 2a6e 6f74 6520 426c 6f77 5570 5374  * *note BlowUpSt
│ │ │ │ -00044cd0: 7261 7465 6779 3a20 426c 6f77 5570 5374  rategy: BlowUpSt
│ │ │ │ -00044ce0: 7261 7465 6779 2c20 3d3e 202e 2e2e 2c20  rategy, => ..., 
│ │ │ │ -00044cf0: 6465 6661 756c 7420 7661 6c75 650a 2020  default value.  
│ │ │ │ -00044d00: 2020 2020 2020 2245 6c69 6d69 6e61 7465        "Eliminate
│ │ │ │ -00044d10: 222c 0a20 2020 2020 202a 202a 6e6f 7465  ",.      * *note
│ │ │ │ -00044d20: 2043 6572 7469 6679 3a20 4365 7274 6966   Certify: Certif
│ │ │ │ -00044d30: 792c 203d 3e20 2e2e 2e2c 2064 6566 6175  y, => ..., defau
│ │ │ │ -00044d40: 6c74 2076 616c 7565 2066 616c 7365 2c20  lt value false, 
│ │ │ │ -00044d50: 7768 6574 6865 7220 746f 2065 6e73 7572  whether to ensur
│ │ │ │ -00044d60: 650a 2020 2020 2020 2020 636f 7272 6563  e.        correc
│ │ │ │ -00044d70: 746e 6573 7320 6f66 206f 7574 7075 740a  tness of output.
│ │ │ │ -00044d80: 2020 2020 2020 2a20 2a6e 6f74 6520 5665        * *note Ve
│ │ │ │ -00044d90: 7262 6f73 653a 2069 6e76 6572 7365 4d61  rbose: inverseMa
│ │ │ │ -00044da0: 705f 6c70 5f70 645f 7064 5f70 645f 636d  p_lp_pd_pd_pd_cm
│ │ │ │ -00044db0: 5665 7262 6f73 653d 3e5f 7064 5f70 645f  Verbose=>_pd_pd_
│ │ │ │ -00044dc0: 7064 5f72 702c 203d 3e20 2e2e 2e2c 0a20  pd_rp, => ...,. 
│ │ │ │ -00044dd0: 2020 2020 2020 2064 6566 6175 6c74 2076         default v
│ │ │ │ -00044de0: 616c 7565 2074 7275 652c 0a20 202a 204f  alue true,.  * O
│ │ │ │ -00044df0: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ -00044e00: 616e 202a 6e6f 7465 2069 6e74 6567 6572  an *note integer
│ │ │ │ -00044e10: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00044e20: 5a5a 2c2c 2074 6865 2064 6567 7265 6520  ZZ,, the degree 
│ │ │ │ -00044e30: 6f66 2050 6869 2e20 536f 2074 6869 7320  of Phi. So this 
│ │ │ │ -00044e40: 7661 6c75 650a 2020 2020 2020 2020 6973  value.        is
│ │ │ │ -00044e50: 2031 2069 6620 616e 6420 6f6e 6c79 2069   1 if and only i
│ │ │ │ -00044e60: 6620 7468 6520 6d61 7020 6973 2062 6972  f the map is bir
│ │ │ │ -00044e70: 6174 696f 6e61 6c20 6f6e 746f 2069 7473  ational onto its
│ │ │ │ -00044e80: 2069 6d61 6765 2e0a 0a44 6573 6372 6970   image...Descrip
│ │ │ │ -00044e90: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -00044ea0: 0a0a 5468 6973 2063 6f6d 7075 7461 7469  ..This computati
│ │ │ │ -00044eb0: 6f6e 2069 7320 646f 6e65 2074 6872 6f75  on is done throu
│ │ │ │ -00044ec0: 6768 2074 6865 2063 6f72 7265 7370 6f6e  gh the correspon
│ │ │ │ -00044ed0: 6469 6e67 206d 6574 686f 6420 666f 7220  ding method for 
│ │ │ │ -00044ee0: 7269 6e67 206d 6170 732e 2053 6565 0a2a  ring maps. See.*
│ │ │ │ -00044ef0: 6e6f 7465 2064 6567 7265 654d 6170 2852  note degreeMap(R
│ │ │ │ -00044f00: 696e 674d 6170 293a 2064 6567 7265 654d  ingMap): degreeM
│ │ │ │ -00044f10: 6170 2c20 666f 7220 6d6f 7265 2064 6574  ap, for more det
│ │ │ │ -00044f20: 6169 6c73 2061 6e64 2065 7861 6d70 6c65  ails and example
│ │ │ │ -00044f30: 732e 0a0a 5365 6520 616c 736f 0a3d 3d3d  s...See also.===
│ │ │ │ -00044f40: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -00044f50: 2064 6567 7265 654d 6170 2852 696e 674d   degreeMap(RingM
│ │ │ │ -00044f60: 6170 293a 2064 6567 7265 654d 6170 2c20  ap): degreeMap, 
│ │ │ │ -00044f70: 2d2d 2064 6567 7265 6520 6f66 2061 2072  -- degree of a r
│ │ │ │ -00044f80: 6174 696f 6e61 6c20 6d61 7020 6265 7477  ational map betw
│ │ │ │ -00044f90: 6565 6e0a 2020 2020 7072 6f6a 6563 7469  een.    projecti
│ │ │ │ -00044fa0: 7665 2076 6172 6965 7469 6573 0a20 202a  ve varieties.  *
│ │ │ │ -00044fb0: 202a 6e6f 7465 2070 726f 6a65 6374 6976   *note projectiv
│ │ │ │ -00044fc0: 6544 6567 7265 6573 3a20 7072 6f6a 6563  eDegrees: projec
│ │ │ │ -00044fd0: 7469 7665 4465 6772 6565 732c 202d 2d20  tiveDegrees, -- 
│ │ │ │ -00044fe0: 7072 6f6a 6563 7469 7665 2064 6567 7265  projective degre
│ │ │ │ -00044ff0: 6573 206f 6620 610a 2020 2020 7261 7469  es of a.    rati
│ │ │ │ -00045000: 6f6e 616c 206d 6170 2062 6574 7765 656e  onal map between
│ │ │ │ -00045010: 2070 726f 6a65 6374 6976 6520 7661 7269   projective vari
│ │ │ │ -00045020: 6574 6965 730a 2020 2a20 2a6e 6f74 6520  eties.  * *note 
│ │ │ │ -00045030: 6465 6772 6565 2852 6174 696f 6e61 6c4d  degree(RationalM
│ │ │ │ -00045040: 6170 293a 2064 6567 7265 655f 6c70 5261  ap): degree_lpRa
│ │ │ │ -00045050: 7469 6f6e 616c 4d61 705f 7270 2c20 2d2d  tionalMap_rp, --
│ │ │ │ -00045060: 2064 6567 7265 6520 6f66 2061 2072 6174   degree of a rat
│ │ │ │ -00045070: 696f 6e61 6c0a 2020 2020 6d61 700a 0a57  ional.    map..W
│ │ │ │ -00045080: 6179 7320 746f 2075 7365 2074 6869 7320  ays to use this 
│ │ │ │ -00045090: 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d  method:.========
│ │ │ │ -000450a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000450b0: 0a0a 2020 2a20 2a6e 6f74 6520 6465 6772  ..  * *note degr
│ │ │ │ -000450c0: 6565 4d61 7028 5261 7469 6f6e 616c 4d61  eeMap(RationalMa
│ │ │ │ -000450d0: 7029 3a20 6465 6772 6565 4d61 705f 6c70  p): degreeMap_lp
│ │ │ │ -000450e0: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ -000450f0: 2d2d 2064 6567 7265 6520 6f66 2061 0a20  -- degree of a. 
│ │ │ │ -00045100: 2020 2072 6174 696f 6e61 6c20 6d61 700a     rational map.
│ │ │ │ +00044bd0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +00044be0: 4675 6e63 7469 6f6e 3a20 2a6e 6f74 6520  Function: *note 
│ │ │ │ +00044bf0: 6465 6772 6565 4d61 703a 2064 6567 7265  degreeMap: degre
│ │ │ │ +00044c00: 654d 6170 2c0a 2020 2a20 5573 6167 653a  eMap,.  * Usage:
│ │ │ │ +00044c10: 200a 2020 2020 2020 2020 6465 6772 6565   .        degree
│ │ │ │ +00044c20: 4d61 7020 5068 690a 2020 2a20 496e 7075  Map Phi.  * Inpu
│ │ │ │ +00044c30: 7473 3a0a 2020 2020 2020 2a20 5068 692c  ts:.      * Phi,
│ │ │ │ +00044c40: 2061 202a 6e6f 7465 2072 6174 696f 6e61   a *note rationa
│ │ │ │ +00044c50: 6c20 6d61 703a 2052 6174 696f 6e61 6c4d  l map: RationalM
│ │ │ │ +00044c60: 6170 2c0a 2020 2a20 2a6e 6f74 6520 4f70  ap,.  * *note Op
│ │ │ │ +00044c70: 7469 6f6e 616c 2069 6e70 7574 733a 2028  tional inputs: (
│ │ │ │ +00044c80: 4d61 6361 756c 6179 3244 6f63 2975 7369  Macaulay2Doc)usi
│ │ │ │ +00044c90: 6e67 2066 756e 6374 696f 6e73 2077 6974  ng functions wit
│ │ │ │ +00044ca0: 6820 6f70 7469 6f6e 616c 2069 6e70 7574  h optional input
│ │ │ │ +00044cb0: 732c 3a0a 2020 2020 2020 2a20 2a6e 6f74  s,:.      * *not
│ │ │ │ +00044cc0: 6520 426c 6f77 5570 5374 7261 7465 6779  e BlowUpStrategy
│ │ │ │ +00044cd0: 3a20 426c 6f77 5570 5374 7261 7465 6779  : BlowUpStrategy
│ │ │ │ +00044ce0: 2c20 3d3e 202e 2e2e 2c20 6465 6661 756c  , => ..., defaul
│ │ │ │ +00044cf0: 7420 7661 6c75 650a 2020 2020 2020 2020  t value.        
│ │ │ │ +00044d00: 2245 6c69 6d69 6e61 7465 222c 0a20 2020  "Eliminate",.   
│ │ │ │ +00044d10: 2020 202a 202a 6e6f 7465 2043 6572 7469     * *note Certi
│ │ │ │ +00044d20: 6679 3a20 4365 7274 6966 792c 203d 3e20  fy: Certify, => 
│ │ │ │ +00044d30: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ +00044d40: 7565 2066 616c 7365 2c20 7768 6574 6865  ue false, whethe
│ │ │ │ +00044d50: 7220 746f 2065 6e73 7572 650a 2020 2020  r to ensure.    
│ │ │ │ +00044d60: 2020 2020 636f 7272 6563 746e 6573 7320      correctness 
│ │ │ │ +00044d70: 6f66 206f 7574 7075 740a 2020 2020 2020  of output.      
│ │ │ │ +00044d80: 2a20 2a6e 6f74 6520 5665 7262 6f73 653a  * *note Verbose:
│ │ │ │ +00044d90: 2069 6e76 6572 7365 4d61 705f 6c70 5f70   inverseMap_lp_p
│ │ │ │ +00044da0: 645f 7064 5f70 645f 636d 5665 7262 6f73  d_pd_pd_cmVerbos
│ │ │ │ +00044db0: 653d 3e5f 7064 5f70 645f 7064 5f72 702c  e=>_pd_pd_pd_rp,
│ │ │ │ +00044dc0: 203d 3e20 2e2e 2e2c 0a20 2020 2020 2020   => ...,.       
│ │ │ │ +00044dd0: 2064 6566 6175 6c74 2076 616c 7565 2074   default value t
│ │ │ │ +00044de0: 7275 652c 0a20 202a 204f 7574 7075 7473  rue,.  * Outputs
│ │ │ │ +00044df0: 3a0a 2020 2020 2020 2a20 616e 202a 6e6f  :.      * an *no
│ │ │ │ +00044e00: 7465 2069 6e74 6567 6572 3a20 284d 6163  te integer: (Mac
│ │ │ │ +00044e10: 6175 6c61 7932 446f 6329 5a5a 2c2c 2074  aulay2Doc)ZZ,, t
│ │ │ │ +00044e20: 6865 2064 6567 7265 6520 6f66 2050 6869  he degree of Phi
│ │ │ │ +00044e30: 2e20 536f 2074 6869 7320 7661 6c75 650a  . So this value.
│ │ │ │ +00044e40: 2020 2020 2020 2020 6973 2031 2069 6620          is 1 if 
│ │ │ │ +00044e50: 616e 6420 6f6e 6c79 2069 6620 7468 6520  and only if the 
│ │ │ │ +00044e60: 6d61 7020 6973 2062 6972 6174 696f 6e61  map is birationa
│ │ │ │ +00044e70: 6c20 6f6e 746f 2069 7473 2069 6d61 6765  l onto its image
│ │ │ │ +00044e80: 2e0a 0a44 6573 6372 6970 7469 6f6e 0a3d  ...Description.=
│ │ │ │ +00044e90: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973  ==========..This
│ │ │ │ +00044ea0: 2063 6f6d 7075 7461 7469 6f6e 2069 7320   computation is 
│ │ │ │ +00044eb0: 646f 6e65 2074 6872 6f75 6768 2074 6865  done through the
│ │ │ │ +00044ec0: 2063 6f72 7265 7370 6f6e 6469 6e67 206d   corresponding m
│ │ │ │ +00044ed0: 6574 686f 6420 666f 7220 7269 6e67 206d  ethod for ring m
│ │ │ │ +00044ee0: 6170 732e 2053 6565 0a2a 6e6f 7465 2064  aps. See.*note d
│ │ │ │ +00044ef0: 6567 7265 654d 6170 2852 696e 674d 6170  egreeMap(RingMap
│ │ │ │ +00044f00: 293a 2064 6567 7265 654d 6170 2c20 666f  ): degreeMap, fo
│ │ │ │ +00044f10: 7220 6d6f 7265 2064 6574 6169 6c73 2061  r more details a
│ │ │ │ +00044f20: 6e64 2065 7861 6d70 6c65 732e 0a0a 5365  nd examples...Se
│ │ │ │ +00044f30: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ +00044f40: 0a20 202a 202a 6e6f 7465 2064 6567 7265  .  * *note degre
│ │ │ │ +00044f50: 654d 6170 2852 696e 674d 6170 293a 2064  eMap(RingMap): d
│ │ │ │ +00044f60: 6567 7265 654d 6170 2c20 2d2d 2064 6567  egreeMap, -- deg
│ │ │ │ +00044f70: 7265 6520 6f66 2061 2072 6174 696f 6e61  ree of a rationa
│ │ │ │ +00044f80: 6c20 6d61 7020 6265 7477 6565 6e0a 2020  l map between.  
│ │ │ │ +00044f90: 2020 7072 6f6a 6563 7469 7665 2076 6172    projective var
│ │ │ │ +00044fa0: 6965 7469 6573 0a20 202a 202a 6e6f 7465  ieties.  * *note
│ │ │ │ +00044fb0: 2070 726f 6a65 6374 6976 6544 6567 7265   projectiveDegre
│ │ │ │ +00044fc0: 6573 3a20 7072 6f6a 6563 7469 7665 4465  es: projectiveDe
│ │ │ │ +00044fd0: 6772 6565 732c 202d 2d20 7072 6f6a 6563  grees, -- projec
│ │ │ │ +00044fe0: 7469 7665 2064 6567 7265 6573 206f 6620  tive degrees of 
│ │ │ │ +00044ff0: 610a 2020 2020 7261 7469 6f6e 616c 206d  a.    rational m
│ │ │ │ +00045000: 6170 2062 6574 7765 656e 2070 726f 6a65  ap between proje
│ │ │ │ +00045010: 6374 6976 6520 7661 7269 6574 6965 730a  ctive varieties.
│ │ │ │ +00045020: 2020 2a20 2a6e 6f74 6520 6465 6772 6565    * *note degree
│ │ │ │ +00045030: 2852 6174 696f 6e61 6c4d 6170 293a 2064  (RationalMap): d
│ │ │ │ +00045040: 6567 7265 655f 6c70 5261 7469 6f6e 616c  egree_lpRational
│ │ │ │ +00045050: 4d61 705f 7270 2c20 2d2d 2064 6567 7265  Map_rp, -- degre
│ │ │ │ +00045060: 6520 6f66 2061 2072 6174 696f 6e61 6c0a  e of a rational.
│ │ │ │ +00045070: 2020 2020 6d61 700a 0a57 6179 7320 746f      map..Ways to
│ │ │ │ +00045080: 2075 7365 2074 6869 7320 6d65 7468 6f64   use this method
│ │ │ │ +00045090: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +000450a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +000450b0: 2a6e 6f74 6520 6465 6772 6565 4d61 7028  *note degreeMap(
│ │ │ │ +000450c0: 5261 7469 6f6e 616c 4d61 7029 3a20 6465  RationalMap): de
│ │ │ │ +000450d0: 6772 6565 4d61 705f 6c70 5261 7469 6f6e  greeMap_lpRation
│ │ │ │ +000450e0: 616c 4d61 705f 7270 2c20 2d2d 2064 6567  alMap_rp, -- deg
│ │ │ │ +000450f0: 7265 6520 6f66 2061 0a20 2020 2072 6174  ree of a.    rat
│ │ │ │ +00045100: 696f 6e61 6c20 6d61 700a 2d2d 2d2d 2d2d  ional map.------
│ │ │ │  00045110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00045120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00045130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00045140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00045150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ -00045160: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
│ │ │ │ -00045170: 6869 7320 646f 6375 6d65 6e74 2069 7320  his document is 
│ │ │ │ -00045180: 696e 0a2f 6275 696c 642f 7265 7072 6f64  in./build/reprod
│ │ │ │ -00045190: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ -000451a0: 756c 6179 322d 312e 3235 2e31 312b 6473  ulay2-1.25.11+ds
│ │ │ │ -000451b0: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ -000451c0: 636b 6167 6573 2f43 7265 6d6f 6e61 2f0a  ckages/Cremona/.
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│ │ │ │ -000451e0: 3a36 3132 3a30 2e0a 1f0a 4669 6c65 3a20  :612:0....File: 
│ │ │ │ -000451f0: 4372 656d 6f6e 612e 696e 666f 2c20 4e6f  Cremona.info, No
│ │ │ │ -00045200: 6465 3a20 6465 6772 6565 735f 6c70 5261  de: degrees_lpRa
│ │ │ │ -00045210: 7469 6f6e 616c 4d61 705f 7270 2c20 4e65  tionalMap_rp, Ne
│ │ │ │ -00045220: 7874 3a20 6465 7363 7269 6265 5f6c 7052  xt: describe_lpR
│ │ │ │ -00045230: 6174 696f 6e61 6c4d 6170 5f72 702c 2050  ationalMap_rp, P
│ │ │ │ -00045240: 7265 763a 2064 6567 7265 654d 6170 5f6c  rev: degreeMap_l
│ │ │ │ -00045250: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ -00045260: 2055 703a 2054 6f70 0a0a 6465 6772 6565   Up: Top..degree
│ │ │ │ -00045270: 7328 5261 7469 6f6e 616c 4d61 7029 202d  s(RationalMap) -
│ │ │ │ -00045280: 2d20 7072 6f6a 6563 7469 7665 2064 6567  - projective deg
│ │ │ │ -00045290: 7265 6573 206f 6620 6120 7261 7469 6f6e  rees of a ration
│ │ │ │ -000452a0: 616c 206d 6170 0a2a 2a2a 2a2a 2a2a 2a2a  al map.*********
│ │ │ │ +00045150: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ +00045160: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ +00045170: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +00045180: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +00045190: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +000451a0: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ +000451b0: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ +000451c0: 2f43 7265 6d6f 6e61 2f0a 646f 6375 6d65  /Cremona/.docume
│ │ │ │ +000451d0: 6e74 6174 696f 6e2e 6d32 3a36 3132 3a30  ntation.m2:612:0
│ │ │ │ +000451e0: 2e0a 1f0a 4669 6c65 3a20 4372 656d 6f6e  ....File: Cremon
│ │ │ │ +000451f0: 612e 696e 666f 2c20 4e6f 6465 3a20 6465  a.info, Node: de
│ │ │ │ +00045200: 6772 6565 735f 6c70 5261 7469 6f6e 616c  grees_lpRational
│ │ │ │ +00045210: 4d61 705f 7270 2c20 4e65 7874 3a20 6465  Map_rp, Next: de
│ │ │ │ +00045220: 7363 7269 6265 5f6c 7052 6174 696f 6e61  scribe_lpRationa
│ │ │ │ +00045230: 6c4d 6170 5f72 702c 2050 7265 763a 2064  lMap_rp, Prev: d
│ │ │ │ +00045240: 6567 7265 654d 6170 5f6c 7052 6174 696f  egreeMap_lpRatio
│ │ │ │ +00045250: 6e61 6c4d 6170 5f72 702c 2055 703a 2054  nalMap_rp, Up: T
│ │ │ │ +00045260: 6f70 0a0a 6465 6772 6565 7328 5261 7469  op..degrees(Rati
│ │ │ │ +00045270: 6f6e 616c 4d61 7029 202d 2d20 7072 6f6a  onalMap) -- proj
│ │ │ │ +00045280: 6563 7469 7665 2064 6567 7265 6573 206f  ective degrees o
│ │ │ │ +00045290: 6620 6120 7261 7469 6f6e 616c 206d 6170  f a rational map
│ │ │ │ +000452a0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │  000452b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000452c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000452d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000452e0: 2a2a 2a0a 0a20 202a 2046 756e 6374 696f  ***..  * Functio
│ │ │ │ -000452f0: 6e3a 202a 6e6f 7465 2064 6567 7265 6573  n: *note degrees
│ │ │ │ -00045300: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00045310: 6465 6772 6565 732c 0a20 202a 2055 7361  degrees,.  * Usa
│ │ │ │ -00045320: 6765 3a20 0a20 2020 2020 2020 2064 6567  ge: .        deg
│ │ │ │ -00045330: 7265 6573 2070 6869 200a 2020 2020 2020  rees phi .      
│ │ │ │ -00045340: 2020 6d75 6c74 6964 6567 7265 6520 7068    multidegree ph
│ │ │ │ -00045350: 690a 2020 2a20 496e 7075 7473 3a0a 2020  i.  * Inputs:.  
│ │ │ │ -00045360: 2020 2020 2a20 7068 692c 2061 202a 6e6f      * phi, a *no
│ │ │ │ -00045370: 7465 2072 6174 696f 6e61 6c20 6d61 703a  te rational map:
│ │ │ │ -00045380: 2052 6174 696f 6e61 6c4d 6170 2c0a 2020   RationalMap,.  
│ │ │ │ -00045390: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -000453a0: 202a 2061 202a 6e6f 7465 206c 6973 743a   * a *note list:
│ │ │ │ -000453b0: 2028 4d61 6361 756c 6179 3244 6f63 294c   (Macaulay2Doc)L
│ │ │ │ -000453c0: 6973 742c 2c20 7468 6520 6c69 7374 206f  ist,, the list o
│ │ │ │ -000453d0: 6620 7072 6f6a 6563 7469 7665 2064 6567  f projective deg
│ │ │ │ -000453e0: 7265 6573 206f 660a 2020 2020 2020 2020  rees of.        
│ │ │ │ -000453f0: 7068 690a 0a44 6573 6372 6970 7469 6f6e  phi..Description
│ │ │ │ -00045400: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  .===========..Th
│ │ │ │ -00045410: 6973 2069 7320 6120 7368 6f72 7463 7574  is is a shortcut
│ │ │ │ -00045420: 2066 6f72 2070 726f 6a65 6374 6976 6544   for projectiveD
│ │ │ │ -00045430: 6567 7265 6573 2870 6869 2c43 6572 7469  egrees(phi,Certi
│ │ │ │ -00045440: 6679 3d3e 7472 7565 2c56 6572 626f 7365  fy=>true,Verbose
│ │ │ │ -00045450: 3d3e 6661 6c73 6529 2c20 7365 650a 2a6e  =>false), see.*n
│ │ │ │ -00045460: 6f74 6520 7072 6f6a 6563 7469 7665 4465  ote projectiveDe
│ │ │ │ -00045470: 6772 6565 7328 5261 7469 6f6e 616c 4d61  grees(RationalMa
│ │ │ │ -00045480: 7029 3a20 7072 6f6a 6563 7469 7665 4465  p): projectiveDe
│ │ │ │ -00045490: 6772 6565 735f 6c70 5261 7469 6f6e 616c  grees_lpRational
│ │ │ │ -000454a0: 4d61 705f 7270 2c2e 0a0a 5761 7973 2074  Map_rp,...Ways t
│ │ │ │ -000454b0: 6f20 7573 6520 7468 6973 206d 6574 686f  o use this metho
│ │ │ │ -000454c0: 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  d:.=============
│ │ │ │ -000454d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -000454e0: 202a 6e6f 7465 2064 6567 7265 6573 2852   *note degrees(R
│ │ │ │ -000454f0: 6174 696f 6e61 6c4d 6170 293a 2064 6567  ationalMap): deg
│ │ │ │ -00045500: 7265 6573 5f6c 7052 6174 696f 6e61 6c4d  rees_lpRationalM
│ │ │ │ -00045510: 6170 5f72 702c 202d 2d20 7072 6f6a 6563  ap_rp, -- projec
│ │ │ │ -00045520: 7469 7665 2064 6567 7265 6573 0a20 2020  tive degrees.   
│ │ │ │ -00045530: 206f 6620 6120 7261 7469 6f6e 616c 206d   of a rational m
│ │ │ │ -00045540: 6170 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ap.-------------
│ │ │ │ +000452d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20  *************.. 
│ │ │ │ +000452e0: 202a 2046 756e 6374 696f 6e3a 202a 6e6f   * Function: *no
│ │ │ │ +000452f0: 7465 2064 6567 7265 6573 3a20 284d 6163  te degrees: (Mac
│ │ │ │ +00045300: 6175 6c61 7932 446f 6329 6465 6772 6565  aulay2Doc)degree
│ │ │ │ +00045310: 732c 0a20 202a 2055 7361 6765 3a20 0a20  s,.  * Usage: . 
│ │ │ │ +00045320: 2020 2020 2020 2064 6567 7265 6573 2070         degrees p
│ │ │ │ +00045330: 6869 200a 2020 2020 2020 2020 6d75 6c74  hi .        mult
│ │ │ │ +00045340: 6964 6567 7265 6520 7068 690a 2020 2a20  idegree phi.  * 
│ │ │ │ +00045350: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +00045360: 7068 692c 2061 202a 6e6f 7465 2072 6174  phi, a *note rat
│ │ │ │ +00045370: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ +00045380: 6e61 6c4d 6170 2c0a 2020 2a20 4f75 7470  nalMap,.  * Outp
│ │ │ │ +00045390: 7574 733a 0a20 2020 2020 202a 2061 202a  uts:.      * a *
│ │ │ │ +000453a0: 6e6f 7465 206c 6973 743a 2028 4d61 6361  note list: (Maca
│ │ │ │ +000453b0: 756c 6179 3244 6f63 294c 6973 742c 2c20  ulay2Doc)List,, 
│ │ │ │ +000453c0: 7468 6520 6c69 7374 206f 6620 7072 6f6a  the list of proj
│ │ │ │ +000453d0: 6563 7469 7665 2064 6567 7265 6573 206f  ective degrees o
│ │ │ │ +000453e0: 660a 2020 2020 2020 2020 7068 690a 0a44  f.        phi..D
│ │ │ │ +000453f0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00045400: 3d3d 3d3d 3d3d 0a0a 5468 6973 2069 7320  ======..This is 
│ │ │ │ +00045410: 6120 7368 6f72 7463 7574 2066 6f72 2070  a shortcut for p
│ │ │ │ +00045420: 726f 6a65 6374 6976 6544 6567 7265 6573  rojectiveDegrees
│ │ │ │ +00045430: 2870 6869 2c43 6572 7469 6679 3d3e 7472  (phi,Certify=>tr
│ │ │ │ +00045440: 7565 2c56 6572 626f 7365 3d3e 6661 6c73  ue,Verbose=>fals
│ │ │ │ +00045450: 6529 2c20 7365 650a 2a6e 6f74 6520 7072  e), see.*note pr
│ │ │ │ +00045460: 6f6a 6563 7469 7665 4465 6772 6565 7328  ojectiveDegrees(
│ │ │ │ +00045470: 5261 7469 6f6e 616c 4d61 7029 3a20 7072  RationalMap): pr
│ │ │ │ +00045480: 6f6a 6563 7469 7665 4465 6772 6565 735f  ojectiveDegrees_
│ │ │ │ +00045490: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ +000454a0: 2c2e 0a0a 5761 7973 2074 6f20 7573 6520  ,...Ways to use 
│ │ │ │ +000454b0: 7468 6973 206d 6574 686f 643a 0a3d 3d3d  this method:.===
│ │ │ │ +000454c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000454d0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +000454e0: 2064 6567 7265 6573 2852 6174 696f 6e61   degrees(Rationa
│ │ │ │ +000454f0: 6c4d 6170 293a 2064 6567 7265 6573 5f6c  lMap): degrees_l
│ │ │ │ +00045500: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ +00045510: 202d 2d20 7072 6f6a 6563 7469 7665 2064   -- projective d
│ │ │ │ +00045520: 6567 7265 6573 0a20 2020 206f 6620 6120  egrees.    of a 
│ │ │ │ +00045530: 7261 7469 6f6e 616c 206d 6170 0a2d 2d2d  rational map.---
│ │ │ │ +00045540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00045550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00045560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00045570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00045580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00045590: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
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│ │ │ │ -000455f0: 2f70 6163 6b61 6765 732f 4372 656d 6f6e  /packages/Cremon
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│ │ │ │ -00045610: 2e6d 323a 3532 333a 302e 0a1f 0a46 696c  .m2:523:0....Fil
│ │ │ │ -00045620: 653a 2043 7265 6d6f 6e61 2e69 6e66 6f2c  e: Cremona.info,
│ │ │ │ -00045630: 204e 6f64 653a 2064 6573 6372 6962 655f   Node: describe_
│ │ │ │ -00045640: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ -00045650: 2c20 4e65 7874 3a20 446f 6d69 6e61 6e74  , Next: Dominant
│ │ │ │ -00045660: 2c20 5072 6576 3a20 6465 6772 6565 735f  , Prev: degrees_
│ │ │ │ -00045670: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ -00045680: 2c20 5570 3a20 546f 700a 0a64 6573 6372  , Up: Top..descr
│ │ │ │ -00045690: 6962 6528 5261 7469 6f6e 616c 4d61 7029  ibe(RationalMap)
│ │ │ │ -000456a0: 202d 2d20 6465 7363 7269 6265 2061 2072   -- describe a r
│ │ │ │ -000456b0: 6174 696f 6e61 6c20 6d61 700a 2a2a 2a2a  ational map.****
│ │ │ │ +00045580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
│ │ │ │ +00045590: 6520 736f 7572 6365 206f 6620 7468 6973  e source of this
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│ │ │ │ +000455f0: 6765 732f 4372 656d 6f6e 612f 0a64 6f63  ges/Cremona/.doc
│ │ │ │ +00045600: 756d 656e 7461 7469 6f6e 2e6d 323a 3532  umentation.m2:52
│ │ │ │ +00045610: 333a 302e 0a1f 0a46 696c 653a 2043 7265  3:0....File: Cre
│ │ │ │ +00045620: 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 653a  mona.info, Node:
│ │ │ │ +00045630: 2064 6573 6372 6962 655f 6c70 5261 7469   describe_lpRati
│ │ │ │ +00045640: 6f6e 616c 4d61 705f 7270 2c20 4e65 7874  onalMap_rp, Next
│ │ │ │ +00045650: 3a20 446f 6d69 6e61 6e74 2c20 5072 6576  : Dominant, Prev
│ │ │ │ +00045660: 3a20 6465 6772 6565 735f 6c70 5261 7469  : degrees_lpRati
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│ │ │ │  000456c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000456d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000456e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020  ************..  
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│ │ │ │ -00045730: 2020 2020 2020 2020 6465 7363 7269 6265          describe
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│ │ │ │ -000457e0: 2020 2020 6361 6c63 756c 6174 6564 2e0a      calculated..
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│ │ │ │ +00045720: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
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│ │ │ │ +00045760: 7261 7469 6f6e 616c 206d 6170 3a20 5261  rational map: Ra
│ │ │ │ +00045770: 7469 6f6e 616c 4d61 702c 0a20 202a 204f  tionalMap,.  * O
│ │ │ │ +00045780: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ +00045790: 6120 6465 7363 7269 7074 696f 6e20 6f66  a description of
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│ │ │ │ +000457f0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
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│ │ │ │  00045820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00045830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00045840: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
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│ │ │ │ -00045860: 745f 345d 3b20 2020 2020 2020 2020 2020  t_4];           
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│ │ │ │ +00045860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045880: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00045880: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  00045890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000458a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000458b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000458c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
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│ │ │ │ -00045910: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000458c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
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│ │ │ │ -00045940: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00045940: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
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│ │ │ │ +00045990: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000459a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000459b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000459c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  000459f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045a10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00045a10: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00045a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00045ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045ad0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ +00045b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00045b60: 2d2d 2d2d 2d2d 2d2d 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  ----------------
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│ │ │ │ +00045bb0: 7068 693b 2020 2020 2020 2020 2020 2020  phi;            
│ │ │ │  00045bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045be0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00045be0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
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│ │ │ │ -00045c20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00045c30: 2020 2020 2020 2020 2020 205a 5a20 2020             ZZ   
│ │ │ │ +00045c20: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00045c30: 2020 2020 205a 5a20 2020 2020 2020 2020       ZZ         
│ │ │ │  00045c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00045cb0: 2020 2020 2020 2033 3333 3331 2020 3020         33331  0 
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│ │ │ │ -00045e10: 5e34 2020 2020 2020 2020 2020 2020 2020  ^4              
│ │ │ │ +00045da0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ +00045ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00045ef0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │ +00045f10: 616c 7365 2020 2020 2020 2020 2020 2020  alse            
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│ │ │ │ -00045f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00045f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045f80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00045f30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │  00045f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00045fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00045fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00045fc0: 2d2d 2d2d 2b0a 7c69 3620 3a20 3f20 4920  ----+.|i6 : ? I 
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│ │ │ │  00045fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00046000: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
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│ │ │ │  00046020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046030: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000460a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000460b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000460c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
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│ │ │ │ +000460d0: 6921 3b20 2020 2020 2020 2020 2020 2020  i!;             
│ │ │ │  000460e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000460f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046100: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00046110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00046100: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │  000461a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000461b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000461c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00046210: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
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│ │ │ │ -000462d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  00046300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046310: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ +00046350: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │ -00046390: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000463d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000463e0: 2020 2020 7c0a 7c20 2020 2020 6465 6772      |.|     degr
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│ │ │ │ +000463d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ -00046460: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00046470: 6e75 6d62 6572 206f 6620 6d69 6e69 6d61  number of minima
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│ │ │ │ +00046490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000464a0: 2020 2020 7c0a 7c20 2020 2020 6469 6d65      |.|     dime
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│ │ │ │  00046510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046520: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ -00046560: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00046580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000465a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000465f0: 2d2d 2063 616c 6375 6c61 7465 7320 6576  -- calculates ev
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│ │ │ │ -00046610: 6e67 0a0a 5761 7973 2074 6f20 7573 6520  ng..Ways to use 
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│ │ │ │ -00046630: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00046640: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
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│ │ │ │ -00046690: 0a20 2020 2072 6174 696f 6e61 6c20 6d61  .    rational ma
│ │ │ │ -000466a0: 700a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  p.--------------
│ │ │ │ +000465a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ +000465b0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ +000465c0: 0a0a 2020 2a20 2a6e 6f74 6520 5261 7469  ..  * *note Rati
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│ │ │ │ +00046630: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
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│ │ │ │ +00046650: 6962 6528 5261 7469 6f6e 616c 4d61 7029  ibe(RationalMap)
│ │ │ │ +00046660: 3a20 6465 7363 7269 6265 5f6c 7052 6174  : describe_lpRat
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│ │ │ │ +000466a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000466b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000466c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000466d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000466e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000467a0: 4e65 7874 3a20 656e 7472 6965 735f 6c70  Next: entries_lp
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│ │ │ │ -000467e0: 2055 703a 2054 6f70 0a0a 446f 6d69 6e61   Up: Top..Domina
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│ │ │ │  00046ba0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00046bf0: 7469 6f6e 3a20 2a6e 6f74 6520 656e 7472  tion: *note entr
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│ │ │ │ -00046c20: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
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│ │ │ │ -00046c40: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -00046c50: 5068 692c 2061 202a 6e6f 7465 2072 6174  Phi, a *note rat
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│ │ │ │ +00046e00: 2020 2a20 2a6e 6f74 6520 656e 7472 6965    * *note entrie
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│ │ │ │ +00046e20: 656e 7472 6965 735f 6c70 5261 7469 6f6e  entries_lpRation
│ │ │ │ +00046e30: 616c 4d61 705f 7270 2c20 2d2d 2074 6865  alMap_rp, -- the
│ │ │ │ +00046e40: 2065 6e74 7269 6573 206f 6620 7468 650a   entries of the.
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│ │ │ │  00046e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00046ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
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│ │ │ │ -00046f90: 6570 7469 6f6e 616c 4c6f 6375 732c 2050  eptionalLocus, P
│ │ │ │ -00046fa0: 7265 763a 2065 6e74 7269 6573 5f6c 7052  rev: entries_lpR
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│ │ │ │ -00046fc0: 703a 2054 6f70 0a0a 4575 6c65 7243 6861  p: Top..EulerCha
│ │ │ │ -00046fd0: 7261 6374 6572 6973 7469 6320 2d2d 2074  racteristic -- t
│ │ │ │ -00046fe0: 6f70 6f6c 6f67 6963 616c 2045 756c 6572  opological Euler
│ │ │ │ -00046ff0: 2063 6861 7261 6374 6572 6973 7469 6320   characteristic 
│ │ │ │ -00047000: 6f66 2061 2028 736d 6f6f 7468 2920 7072  of a (smooth) pr
│ │ │ │ -00047010: 6f6a 6563 7469 7665 2076 6172 6965 7479  ojective variety
│ │ │ │ -00047020: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +00046ec0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
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│ │ │ │ +00046f20: 756c 6179 322f 7061 636b 6167 6573 2f43  ulay2/packages/C
│ │ │ │ +00046f30: 7265 6d6f 6e61 2f0a 646f 6375 6d65 6e74  remona/.document
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│ │ │ │ +00046f60: 696e 666f 2c20 4e6f 6465 3a20 4575 6c65  info, Node: Eule
│ │ │ │ +00046f70: 7243 6861 7261 6374 6572 6973 7469 632c  rCharacteristic,
│ │ │ │ +00046f80: 204e 6578 743a 2065 7863 6570 7469 6f6e   Next: exception
│ │ │ │ +00046f90: 616c 4c6f 6375 732c 2050 7265 763a 2065  alLocus, Prev: e
│ │ │ │ +00046fa0: 6e74 7269 6573 5f6c 7052 6174 696f 6e61  ntries_lpRationa
│ │ │ │ +00046fb0: 6c4d 6170 5f72 702c 2055 703a 2054 6f70  lMap_rp, Up: Top
│ │ │ │ +00046fc0: 0a0a 4575 6c65 7243 6861 7261 6374 6572  ..EulerCharacter
│ │ │ │ +00046fd0: 6973 7469 6320 2d2d 2074 6f70 6f6c 6f67  istic -- topolog
│ │ │ │ +00046fe0: 6963 616c 2045 756c 6572 2063 6861 7261  ical Euler chara
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│ │ │ │ +00047000: 736d 6f6f 7468 2920 7072 6f6a 6563 7469  smooth) projecti
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│ │ │ │ +00047020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00047030: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00047040: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00047050: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00047060: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00047070: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ -00047080: 7361 6765 3a20 0a20 2020 2020 2020 2045  sage: .        E
│ │ │ │ -00047090: 756c 6572 4368 6172 6163 7465 7269 7374  ulerCharacterist
│ │ │ │ -000470a0: 6963 2049 0a20 202a 2049 6e70 7574 733a  ic I.  * Inputs:
│ │ │ │ -000470b0: 0a20 2020 2020 202a 2049 2c20 616e 202a  .      * I, an *
│ │ │ │ -000470c0: 6e6f 7465 2069 6465 616c 3a20 284d 6163  note ideal: (Mac
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│ │ │ │ -00047100: 0a20 2020 2020 2020 2073 6d6f 6f74 6820  .        smooth 
│ │ │ │ -00047110: 7072 6f6a 6563 7469 7665 2076 6172 6965  projective varie
│ │ │ │ -00047120: 7479 2024 585c 7375 6273 6574 5c6d 6174  ty $X\subset\mat
│ │ │ │ -00047130: 6862 627b 507d 5e6e 240a 2020 2a20 2a6e  hbb{P}^n$.  * *n
│ │ │ │ -00047140: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70  ote Optional inp
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│ │ │ │ -00047160: 6f63 2975 7369 6e67 2066 756e 6374 696f  oc)using functio
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│ │ │ │ -00047190: 2a20 2a6e 6f74 6520 426c 6f77 5570 5374  * *note BlowUpSt
│ │ │ │ -000471a0: 7261 7465 6779 3a20 426c 6f77 5570 5374  rategy: BlowUpSt
│ │ │ │ -000471b0: 7261 7465 6779 2c20 3d3e 202e 2e2e 2c20  rategy, => ..., 
│ │ │ │ -000471c0: 6465 6661 756c 7420 7661 6c75 650a 2020  default value.  
│ │ │ │ -000471d0: 2020 2020 2020 2245 6c69 6d69 6e61 7465        "Eliminate
│ │ │ │ -000471e0: 222c 0a20 2020 2020 202a 202a 6e6f 7465  ",.      * *note
│ │ │ │ -000471f0: 2043 6572 7469 6679 3a20 4365 7274 6966   Certify: Certif
│ │ │ │ -00047200: 792c 203d 3e20 2e2e 2e2c 2064 6566 6175  y, => ..., defau
│ │ │ │ -00047210: 6c74 2076 616c 7565 2066 616c 7365 2c20  lt value false, 
│ │ │ │ -00047220: 7768 6574 6865 7220 746f 2065 6e73 7572  whether to ensur
│ │ │ │ -00047230: 650a 2020 2020 2020 2020 636f 7272 6563  e.        correc
│ │ │ │ -00047240: 746e 6573 7320 6f66 206f 7574 7075 740a  tness of output.
│ │ │ │ -00047250: 2020 2020 2020 2a20 2a6e 6f74 6520 5665        * *note Ve
│ │ │ │ -00047260: 7262 6f73 653a 2069 6e76 6572 7365 4d61  rbose: inverseMa
│ │ │ │ -00047270: 705f 6c70 5f70 645f 7064 5f70 645f 636d  p_lp_pd_pd_pd_cm
│ │ │ │ -00047280: 5665 7262 6f73 653d 3e5f 7064 5f70 645f  Verbose=>_pd_pd_
│ │ │ │ -00047290: 7064 5f72 702c 203d 3e20 2e2e 2e2c 0a20  pd_rp, => ...,. 
│ │ │ │ -000472a0: 2020 2020 2020 2064 6566 6175 6c74 2076         default v
│ │ │ │ -000472b0: 616c 7565 2074 7275 652c 0a20 202a 204f  alue true,.  * O
│ │ │ │ -000472c0: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ -000472d0: 616e 202a 6e6f 7465 2069 6e74 6567 6572  an *note integer
│ │ │ │ -000472e0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -000472f0: 5a5a 2c2c 2074 6865 2074 6f70 6f6c 6f67  ZZ,, the topolog
│ │ │ │ -00047300: 6963 616c 2045 756c 6572 0a20 2020 2020  ical Euler.     
│ │ │ │ -00047310: 2020 2063 6861 7261 6374 6572 6973 7469     characteristi
│ │ │ │ -00047320: 6373 206f 6620 2458 242e 0a0a 4465 7363  cs of $X$...Desc
│ │ │ │ -00047330: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ -00047340: 3d3d 3d0a 0a54 6869 7320 6973 2061 6e20  ===..This is an 
│ │ │ │ -00047350: 6170 706c 6963 6174 696f 6e20 6f66 2074  application of t
│ │ │ │ -00047360: 6865 206d 6574 686f 6420 2a6e 6f74 6520  he method *note 
│ │ │ │ -00047370: 5365 6772 6543 6c61 7373 3a20 5365 6772  SegreClass: Segr
│ │ │ │ -00047380: 6543 6c61 7373 2c2e 2053 6565 2061 6c73  eClass,. See als
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│ │ │ │ -000473c0: 4120 2873 6565 0a68 7474 703a 2f2f 7777  A (see.http://ww
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│ │ │ │ -000473e0: 616c 7566 6669 2f43 534d 2f43 534d 2e68  aluffi/CSM/CSM.h
│ │ │ │ -000473f0: 746d 6c20 292c 2062 7920 502e 2041 6c75  tml ), by P. Alu
│ │ │ │ -00047400: 6666 692c 2061 6e64 202a 6e6f 7465 0a43  ffi, and *note.C
│ │ │ │ -00047410: 6861 7261 6374 6572 6973 7469 6343 6c61  haracteristicCla
│ │ │ │ -00047420: 7373 6573 3a20 2843 6861 7261 6374 6572  sses: (Character
│ │ │ │ -00047430: 6973 7469 6343 6c61 7373 6573 2954 6f70  isticClasses)Top
│ │ │ │ -00047440: 2c2c 2062 7920 4d2e 2048 656c 6d65 7220  ,, by M. Helmer 
│ │ │ │ -00047450: 616e 6420 432e 204a 6f73 742e 0a0a 496e  and C. Jost...In
│ │ │ │ -00047460: 2067 656e 6572 616c 2c20 6576 656e 2069   general, even i
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│ │ │ │ -00047480: 6c20 6465 6669 6e65 7320 6120 7369 6e67  l defines a sing
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│ │ │ │ -000474a0: 2c20 7468 650a 7265 7475 726e 6564 2076  , the.returned v
│ │ │ │ -000474b0: 616c 7565 2065 7175 616c 7320 7468 6520  alue equals the 
│ │ │ │ -000474c0: 6465 6772 6565 206f 6620 7468 6520 636f  degree of the co
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│ │ │ │ -00047500: 206f 6620 2458 242e 2054 6865 2045 756c   of $X$. The Eul
│ │ │ │ -00047510: 6572 2063 6861 7261 6374 6572 6973 7469  er characteristi
│ │ │ │ -00047520: 6320 6f66 2061 2073 696e 6775 6c61 7220  c of a singular 
│ │ │ │ -00047530: 7661 7269 6574 7920 6361 6e0a 6265 2063  variety can.be c
│ │ │ │ -00047540: 6f6d 7075 7465 6420 7669 6120 7468 6520  omputed via the 
│ │ │ │ -00047550: 6d65 7468 6f64 202a 6e6f 7465 2043 6865  method *note Che
│ │ │ │ -00047560: 726e 5363 6877 6172 747a 4d61 6350 6865  rnSchwartzMacPhe
│ │ │ │ -00047570: 7273 6f6e 3a0a 4368 6572 6e53 6368 7761  rson:.ChernSchwa
│ │ │ │ -00047580: 7274 7a4d 6163 5068 6572 736f 6e2c 2e0a  rtzMacPherson,..
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│ │ │ │ -000475a0: 6265 6c6f 772c 2077 6520 636f 6d70 7574  below, we comput
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│ │ │ │ -000475e0: 7562 7365 745c 6d61 7468 6262 7b50 7d5e  ubset\mathbb{P}^
│ │ │ │ -000475f0: 7b39 7d24 2c20 7573 696e 6720 626f 7468  {9}$, using both
│ │ │ │ -00047600: 2061 2070 726f 6261 6269 6c69 7374 6963   a probabilistic
│ │ │ │ -00047610: 2061 6e64 2061 0a6e 6f6e 2d70 726f 6261   and a.non-proba
│ │ │ │ -00047620: 6269 6c69 7374 6963 2061 7070 726f 6163  bilistic approac
│ │ │ │ -00047630: 682e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  h...+-----------
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│ │ │ │ +00047090: 6172 6163 7465 7269 7374 6963 2049 0a20  aracteristic I. 
│ │ │ │ +000470a0: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +000470b0: 202a 2049 2c20 616e 202a 6e6f 7465 2069   * I, an *note i
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│ │ │ │ +00047120: 7375 6273 6574 5c6d 6174 6862 627b 507d  subset\mathbb{P}
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│ │ │ │ +00047150: 4d61 6361 756c 6179 3244 6f63 2975 7369  Macaulay2Doc)usi
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│ │ │ │ +00047190: 6520 426c 6f77 5570 5374 7261 7465 6779  e BlowUpStrategy
│ │ │ │ +000471a0: 3a20 426c 6f77 5570 5374 7261 7465 6779  : BlowUpStrategy
│ │ │ │ +000471b0: 2c20 3d3e 202e 2e2e 2c20 6465 6661 756c  , => ..., defaul
│ │ │ │ +000471c0: 7420 7661 6c75 650a 2020 2020 2020 2020  t value.        
│ │ │ │ +000471d0: 2245 6c69 6d69 6e61 7465 222c 0a20 2020  "Eliminate",.   
│ │ │ │ +000471e0: 2020 202a 202a 6e6f 7465 2043 6572 7469     * *note Certi
│ │ │ │ +000471f0: 6679 3a20 4365 7274 6966 792c 203d 3e20  fy: Certify, => 
│ │ │ │ +00047200: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ +00047210: 7565 2066 616c 7365 2c20 7768 6574 6865  ue false, whethe
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│ │ │ │ +00047230: 2020 2020 636f 7272 6563 746e 6573 7320      correctness 
│ │ │ │ +00047240: 6f66 206f 7574 7075 740a 2020 2020 2020  of output.      
│ │ │ │ +00047250: 2a20 2a6e 6f74 6520 5665 7262 6f73 653a  * *note Verbose:
│ │ │ │ +00047260: 2069 6e76 6572 7365 4d61 705f 6c70 5f70   inverseMap_lp_p
│ │ │ │ +00047270: 645f 7064 5f70 645f 636d 5665 7262 6f73  d_pd_pd_cmVerbos
│ │ │ │ +00047280: 653d 3e5f 7064 5f70 645f 7064 5f72 702c  e=>_pd_pd_pd_rp,
│ │ │ │ +00047290: 203d 3e20 2e2e 2e2c 0a20 2020 2020 2020   => ...,.       
│ │ │ │ +000472a0: 2064 6566 6175 6c74 2076 616c 7565 2074   default value t
│ │ │ │ +000472b0: 7275 652c 0a20 202a 204f 7574 7075 7473  rue,.  * Outputs
│ │ │ │ +000472c0: 3a0a 2020 2020 2020 2a20 616e 202a 6e6f  :.      * an *no
│ │ │ │ +000472d0: 7465 2069 6e74 6567 6572 3a20 284d 6163  te integer: (Mac
│ │ │ │ +000472e0: 6175 6c61 7932 446f 6329 5a5a 2c2c 2074  aulay2Doc)ZZ,, t
│ │ │ │ +000472f0: 6865 2074 6f70 6f6c 6f67 6963 616c 2045  he topological E
│ │ │ │ +00047300: 756c 6572 0a20 2020 2020 2020 2063 6861  uler.        cha
│ │ │ │ +00047310: 7261 6374 6572 6973 7469 6373 206f 6620  racteristics of 
│ │ │ │ +00047320: 2458 242e 0a0a 4465 7363 7269 7074 696f  $X$...Descriptio
│ │ │ │ +00047330: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  n.===========..T
│ │ │ │ +00047340: 6869 7320 6973 2061 6e20 6170 706c 6963  his is an applic
│ │ │ │ +00047350: 6174 696f 6e20 6f66 2074 6865 206d 6574  ation of the met
│ │ │ │ +00047360: 686f 6420 2a6e 6f74 6520 5365 6772 6543  hod *note SegreC
│ │ │ │ +00047370: 6c61 7373 3a20 5365 6772 6543 6c61 7373  lass: SegreClass
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│ │ │ │ +00047390: 636f 7272 6573 706f 6e64 696e 6720 6d65  corresponding me
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│ │ │ │ +000473d0: 2e66 7375 2e65 6475 2f7e 616c 7566 6669  .fsu.edu/~aluffi
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│ │ │ │ +00047730: 5a20 2020 2020 2020 2020 2020 2020 2020  Z               
│ │ │ │  00047740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047770: 2020 7c0a 7c6f 3120 3a20 4964 6561 6c20    |.|o1 : Ideal 
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│ │ │ │ -000477d0: 2020 2031 3930 3138 3120 2030 2c31 2020     190181  0,1  
│ │ │ │ -000477e0: 2030 2c32 2020 2031 2c32 2020 2030 2c33   0,2   1,2   0,3
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│ │ │ │ +000477c0: 2020 2020 2020 2020 2020 2020 2031 3930               190
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│ │ │ │  00047830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00047870: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00047890: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000478b0: 2020 7c0a 7c20 332c 3420 2020 2020 2020    |.| 3,4       
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│ │ │ │ +000478b0: 332c 3420 2020 2020 2020 2020 2020 2020  3,4             
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│ │ │ │  000478d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00047920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00047970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047990: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000479e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000479f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00047990: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │ +000479d0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
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│ │ │ │ +000479f0: 2020 2020 2020 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                  
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│ │ │ │ +00047a30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00047a40: 3220 3d20 3130 2020 2020 2020 2020 2020  2 = 10          
│ │ │ │  00047a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047a90: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
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│ │ │ │ +00047a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00047ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00047b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047b30: 2020 7c0a 7c43 6572 7469 6679 3a20 6f75    |.|Certify: ou
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│ │ │ │ +00047b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  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                  
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│ │ │ │ -00047bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047bd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00047b70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00047b80: 2d2d 2075 7365 6420 302e 3033 3033 3731  -- used 0.030371
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│ │ │ │ +00047bb0: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +00047bc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00047bd0: 2020 2020 2020 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                  
│ │ │ │ -00047c20: 2020 7c0a 7c6f 3320 3d20 3130 2020 2020    |.|o3 = 10    
│ │ │ │ +00047c10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00047c20: 3320 3d20 3130 2020 2020 2020 2020 2020  3 = 10          
│ │ │ │  00047c30: 2020 2020 2020 2020 2020 2020 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 2020                  
│ │ │ │ -00047c70: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00047c60: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00047c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00047cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00047cc0: 2d2d 2b0a 0a43 6176 6561 740a 3d3d 3d3d  --+..Caveat.====
│ │ │ │ -00047cd0: 3d3d 0a0a 4e6f 2074 6573 7420 6973 206d  ==..No test is m
│ │ │ │ -00047ce0: 6164 6520 746f 2073 6565 2069 6620 7468  ade to see if th
│ │ │ │ -00047cf0: 6520 7072 6f6a 6563 7469 7665 2076 6172  e projective var
│ │ │ │ -00047d00: 6965 7479 2069 7320 736d 6f6f 7468 2e0a  iety is smooth..
│ │ │ │ -00047d10: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -00047d20: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6575  ==..  * *note eu
│ │ │ │ -00047d30: 6c65 7228 5072 6f6a 6563 7469 7665 5661  ler(ProjectiveVa
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│ │ │ │ -00047d60: 6374 6976 6556 6172 6965 7479 5f72 702c  ctiveVariety_rp,
│ │ │ │ -00047d70: 202d 2d0a 2020 2020 746f 706f 6c6f 6769   --.    topologi
│ │ │ │ -00047d80: 6361 6c20 4575 6c65 7220 6368 6172 6163  cal Euler charac
│ │ │ │ -00047d90: 7465 7269 7374 6963 206f 6620 6120 2873  teristic of a (s
│ │ │ │ -00047da0: 6d6f 6f74 6829 2070 726f 6a65 6374 6976  mooth) projectiv
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│ │ │ │ -00047dc0: 6f74 6520 4368 6572 6e53 6368 7761 7274  ote ChernSchwart
│ │ │ │ -00047dd0: 7a4d 6163 5068 6572 736f 6e3a 2043 6865  zMacPherson: Che
│ │ │ │ -00047de0: 726e 5363 6877 6172 747a 4d61 6350 6865  rnSchwartzMacPhe
│ │ │ │ -00047df0: 7273 6f6e 2c20 2d2d 0a20 2020 2043 6865  rson, --.    Che
│ │ │ │ -00047e00: 726e 2d53 6368 7761 7274 7a2d 4d61 6350  rn-Schwartz-MacP
│ │ │ │ -00047e10: 6865 7273 6f6e 2063 6c61 7373 206f 6620  herson class of 
│ │ │ │ -00047e20: 6120 7072 6f6a 6563 7469 7665 2073 6368  a projective sch
│ │ │ │ -00047e30: 656d 650a 2020 2a20 2a6e 6f74 6520 5365  eme.  * *note Se
│ │ │ │ -00047e40: 6772 6543 6c61 7373 3a20 5365 6772 6543  greClass: SegreC
│ │ │ │ -00047e50: 6c61 7373 2c20 2d2d 2053 6567 7265 2063  lass, -- Segre c
│ │ │ │ -00047e60: 6c61 7373 206f 6620 6120 636c 6f73 6564  lass of a closed
│ │ │ │ -00047e70: 2073 7562 7363 6865 6d65 206f 6620 610a   subscheme of a.
│ │ │ │ -00047e80: 2020 2020 7072 6f6a 6563 7469 7665 2076      projective v
│ │ │ │ -00047e90: 6172 6965 7479 0a0a 5761 7973 2074 6f20  ariety..Ways to 
│ │ │ │ -00047ea0: 7573 6520 4575 6c65 7243 6861 7261 6374  use EulerCharact
│ │ │ │ -00047eb0: 6572 6973 7469 633a 0a3d 3d3d 3d3d 3d3d  eristic:.=======
│ │ │ │ +00047cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a43  ------------+..C
│ │ │ │ +00047cc0: 6176 6561 740a 3d3d 3d3d 3d3d 0a0a 4e6f  aveat.======..No
│ │ │ │ +00047cd0: 2074 6573 7420 6973 206d 6164 6520 746f   test is made to
│ │ │ │ +00047ce0: 2073 6565 2069 6620 7468 6520 7072 6f6a   see if the proj
│ │ │ │ +00047cf0: 6563 7469 7665 2076 6172 6965 7479 2069  ective variety i
│ │ │ │ +00047d00: 7320 736d 6f6f 7468 2e0a 0a53 6565 2061  s smooth...See a
│ │ │ │ +00047d10: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ +00047d20: 2a20 2a6e 6f74 6520 6575 6c65 7228 5072  * *note euler(Pr
│ │ │ │ +00047d30: 6f6a 6563 7469 7665 5661 7269 6574 7929  ojectiveVariety)
│ │ │ │ +00047d40: 3a20 2856 6172 6965 7469 6573 2965 756c  : (Varieties)eul
│ │ │ │ +00047d50: 6572 5f6c 7050 726f 6a65 6374 6976 6556  er_lpProjectiveV
│ │ │ │ +00047d60: 6172 6965 7479 5f72 702c 202d 2d0a 2020  ariety_rp, --.  
│ │ │ │ +00047d70: 2020 746f 706f 6c6f 6769 6361 6c20 4575    topological Eu
│ │ │ │ +00047d80: 6c65 7220 6368 6172 6163 7465 7269 7374  ler characterist
│ │ │ │ +00047d90: 6963 206f 6620 6120 2873 6d6f 6f74 6829  ic of a (smooth)
│ │ │ │ +00047da0: 2070 726f 6a65 6374 6976 6520 7661 7269   projective vari
│ │ │ │ +00047db0: 6574 790a 2020 2a20 2a6e 6f74 6520 4368  ety.  * *note Ch
│ │ │ │ +00047dc0: 6572 6e53 6368 7761 7274 7a4d 6163 5068  ernSchwartzMacPh
│ │ │ │ +00047dd0: 6572 736f 6e3a 2043 6865 726e 5363 6877  erson: ChernSchw
│ │ │ │ +00047de0: 6172 747a 4d61 6350 6865 7273 6f6e 2c20  artzMacPherson, 
│ │ │ │ +00047df0: 2d2d 0a20 2020 2043 6865 726e 2d53 6368  --.    Chern-Sch
│ │ │ │ +00047e00: 7761 7274 7a2d 4d61 6350 6865 7273 6f6e  wartz-MacPherson
│ │ │ │ +00047e10: 2063 6c61 7373 206f 6620 6120 7072 6f6a   class of a proj
│ │ │ │ +00047e20: 6563 7469 7665 2073 6368 656d 650a 2020  ective scheme.  
│ │ │ │ +00047e30: 2a20 2a6e 6f74 6520 5365 6772 6543 6c61  * *note SegreCla
│ │ │ │ +00047e40: 7373 3a20 5365 6772 6543 6c61 7373 2c20  ss: SegreClass, 
│ │ │ │ +00047e50: 2d2d 2053 6567 7265 2063 6c61 7373 206f  -- Segre class o
│ │ │ │ +00047e60: 6620 6120 636c 6f73 6564 2073 7562 7363  f a closed subsc
│ │ │ │ +00047e70: 6865 6d65 206f 6620 610a 2020 2020 7072  heme of a.    pr
│ │ │ │ +00047e80: 6f6a 6563 7469 7665 2076 6172 6965 7479  ojective variety
│ │ │ │ +00047e90: 0a0a 5761 7973 2074 6f20 7573 6520 4575  ..Ways to use Eu
│ │ │ │ +00047ea0: 6c65 7243 6861 7261 6374 6572 6973 7469  lerCharacteristi
│ │ │ │ +00047eb0: 633a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  c:.=============
│ │ │ │  00047ec0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00047ed0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ -00047ee0: 4575 6c65 7243 6861 7261 6374 6572 6973  EulerCharacteris
│ │ │ │ -00047ef0: 7469 6328 4964 6561 6c29 220a 0a46 6f72  tic(Ideal)"..For
│ │ │ │ -00047f00: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ -00047f10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00047f20: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ -00047f30: 6e6f 7465 2045 756c 6572 4368 6172 6163  note EulerCharac
│ │ │ │ -00047f40: 7465 7269 7374 6963 3a20 4575 6c65 7243  teristic: EulerC
│ │ │ │ -00047f50: 6861 7261 6374 6572 6973 7469 632c 2069  haracteristic, i
│ │ │ │ -00047f60: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64  s a *note method
│ │ │ │ -00047f70: 0a66 756e 6374 696f 6e20 7769 7468 206f  .function with o
│ │ │ │ -00047f80: 7074 696f 6e73 3a20 284d 6163 6175 6c61  ptions: (Macaula
│ │ │ │ -00047f90: 7932 446f 6329 4d65 7468 6f64 4675 6e63  y2Doc)MethodFunc
│ │ │ │ -00047fa0: 7469 6f6e 5769 7468 4f70 7469 6f6e 732c  tionWithOptions,
│ │ │ │ -00047fb0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │ +00047ed0: 3d3d 3d0a 0a20 202a 2022 4575 6c65 7243  ===..  * "EulerC
│ │ │ │ +00047ee0: 6861 7261 6374 6572 6973 7469 6328 4964  haracteristic(Id
│ │ │ │ +00047ef0: 6561 6c29 220a 0a46 6f72 2074 6865 2070  eal)"..For the p
│ │ │ │ +00047f00: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +00047f10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +00047f20: 6520 6f62 6a65 6374 202a 6e6f 7465 2045  e object *note E
│ │ │ │ +00047f30: 756c 6572 4368 6172 6163 7465 7269 7374  ulerCharacterist
│ │ │ │ +00047f40: 6963 3a20 4575 6c65 7243 6861 7261 6374  ic: EulerCharact
│ │ │ │ +00047f50: 6572 6973 7469 632c 2069 7320 6120 2a6e  eristic, is a *n
│ │ │ │ +00047f60: 6f74 6520 6d65 7468 6f64 0a66 756e 6374  ote method.funct
│ │ │ │ +00047f70: 696f 6e20 7769 7468 206f 7074 696f 6e73  ion with options
│ │ │ │ +00047f80: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00047f90: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769  MethodFunctionWi
│ │ │ │ +00047fa0: 7468 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d  thOptions,...---
│ │ │ │ +00047fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00047ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048000: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ -00048010: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ -00048020: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ -00048030: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ -00048040: 6163 6175 6c61 7932 2d31 2e32 352e 3131  acaulay2-1.25.11
│ │ │ │ -00048050: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ -00048060: 2f70 6163 6b61 6765 732f 4372 656d 6f6e  /packages/Cremon
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│ │ │ │ -00048080: 2e6d 323a 3334 343a 302e 0a1f 0a46 696c  .m2:344:0....Fil
│ │ │ │ -00048090: 653a 2043 7265 6d6f 6e61 2e69 6e66 6f2c  e: Cremona.info,
│ │ │ │ -000480a0: 204e 6f64 653a 2065 7863 6570 7469 6f6e   Node: exception
│ │ │ │ -000480b0: 616c 4c6f 6375 732c 204e 6578 743a 2066  alLocus, Next: f
│ │ │ │ -000480c0: 6c61 7474 656e 5f6c 7052 6174 696f 6e61  latten_lpRationa
│ │ │ │ -000480d0: 6c4d 6170 5f72 702c 2050 7265 763a 2045  lMap_rp, Prev: E
│ │ │ │ -000480e0: 756c 6572 4368 6172 6163 7465 7269 7374  ulerCharacterist
│ │ │ │ -000480f0: 6963 2c20 5570 3a20 546f 700a 0a65 7863  ic, Up: Top..exc
│ │ │ │ -00048100: 6570 7469 6f6e 616c 4c6f 6375 7320 2d2d  eptionalLocus --
│ │ │ │ -00048110: 2065 7863 6570 7469 6f6e 616c 206c 6f63   exceptional loc
│ │ │ │ -00048120: 7573 206f 6620 6120 6269 7261 7469 6f6e  us of a biration
│ │ │ │ -00048130: 616c 206d 6170 0a2a 2a2a 2a2a 2a2a 2a2a  al map.*********
│ │ │ │ +00047ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
│ │ │ │ +00048000: 6520 736f 7572 6365 206f 6620 7468 6973  e source of this
│ │ │ │ +00048010: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
│ │ │ │ +00048020: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ +00048030: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ +00048040: 7932 2d31 2e32 352e 3131 2b64 732f 4d32  y2-1.25.11+ds/M2
│ │ │ │ +00048050: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ +00048060: 6765 732f 4372 656d 6f6e 612f 0a64 6f63  ges/Cremona/.doc
│ │ │ │ +00048070: 756d 656e 7461 7469 6f6e 2e6d 323a 3334  umentation.m2:34
│ │ │ │ +00048080: 343a 302e 0a1f 0a46 696c 653a 2043 7265  4:0....File: Cre
│ │ │ │ +00048090: 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 653a  mona.info, Node:
│ │ │ │ +000480a0: 2065 7863 6570 7469 6f6e 616c 4c6f 6375   exceptionalLocu
│ │ │ │ +000480b0: 732c 204e 6578 743a 2066 6c61 7474 656e  s, Next: flatten
│ │ │ │ +000480c0: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ +000480d0: 702c 2050 7265 763a 2045 756c 6572 4368  p, Prev: EulerCh
│ │ │ │ +000480e0: 6172 6163 7465 7269 7374 6963 2c20 5570  aracteristic, Up
│ │ │ │ +000480f0: 3a20 546f 700a 0a65 7863 6570 7469 6f6e  : Top..exception
│ │ │ │ +00048100: 616c 4c6f 6375 7320 2d2d 2065 7863 6570  alLocus -- excep
│ │ │ │ +00048110: 7469 6f6e 616c 206c 6f63 7573 206f 6620  tional locus of 
│ │ │ │ +00048120: 6120 6269 7261 7469 6f6e 616c 206d 6170  a birational map
│ │ │ │ +00048130: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │  00048140: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00048150: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00048160: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00048170: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ -00048180: 2020 2020 2020 6578 6365 7074 696f 6e61        exceptiona
│ │ │ │ -00048190: 6c4c 6f63 7573 2070 6869 0a20 202a 2049  lLocus phi.  * I
│ │ │ │ -000481a0: 6e70 7574 733a 0a20 2020 2020 202a 2070  nputs:.      * p
│ │ │ │ -000481b0: 6869 2c20 6120 2a6e 6f74 6520 7261 7469  hi, a *note rati
│ │ │ │ -000481c0: 6f6e 616c 206d 6170 3a20 5261 7469 6f6e  onal map: Ration
│ │ │ │ -000481d0: 616c 4d61 702c 2c20 6120 6269 7261 7469  alMap,, a birati
│ │ │ │ -000481e0: 6f6e 616c 206d 6170 0a20 2020 2020 2020  onal map.       
│ │ │ │ -000481f0: 2024 585c 6461 7368 7269 6768 7461 7272   $X\dashrightarr
│ │ │ │ -00048200: 6f77 2059 240a 2020 2a20 2a6e 6f74 6520  ow Y$.  * *note 
│ │ │ │ -00048210: 4f70 7469 6f6e 616c 2069 6e70 7574 733a  Optional inputs:
│ │ │ │ -00048220: 2028 4d61 6361 756c 6179 3244 6f63 2975   (Macaulay2Doc)u
│ │ │ │ -00048230: 7369 6e67 2066 756e 6374 696f 6e73 2077  sing functions w
│ │ │ │ -00048240: 6974 6820 6f70 7469 6f6e 616c 2069 6e70  ith optional inp
│ │ │ │ -00048250: 7574 732c 3a0a 2020 2020 2020 2a20 2a6e  uts,:.      * *n
│ │ │ │ -00048260: 6f74 6520 4365 7274 6966 793a 2043 6572  ote Certify: Cer
│ │ │ │ -00048270: 7469 6679 2c20 3d3e 202e 2e2e 2c20 6465  tify, => ..., de
│ │ │ │ -00048280: 6661 756c 7420 7661 6c75 6520 6661 6c73  fault value fals
│ │ │ │ -00048290: 652c 2077 6865 7468 6572 2074 6f20 656e  e, whether to en
│ │ │ │ -000482a0: 7375 7265 0a20 2020 2020 2020 2063 6f72  sure.        cor
│ │ │ │ -000482b0: 7265 6374 6e65 7373 206f 6620 6f75 7470  rectness of outp
│ │ │ │ -000482c0: 7574 0a20 202a 204f 7574 7075 7473 3a0a  ut.  * Outputs:.
│ │ │ │ -000482d0: 2020 2020 2020 2a20 616e 202a 6e6f 7465        * an *note
│ │ │ │ -000482e0: 2069 6465 616c 3a20 284d 6163 6175 6c61   ideal: (Macaula
│ │ │ │ -000482f0: 7932 446f 6329 4964 6561 6c2c 2c20 7468  y2Doc)Ideal,, th
│ │ │ │ -00048300: 6520 6964 6561 6c20 6465 6669 6e69 6e67  e ideal defining
│ │ │ │ -00048310: 2074 6865 2063 6c6f 7375 7265 2069 6e0a   the closure in.
│ │ │ │ -00048320: 2020 2020 2020 2020 5820 6f66 2074 6865          X of the
│ │ │ │ -00048330: 206c 6f63 7573 2077 6865 7265 2070 6869   locus where phi
│ │ │ │ -00048340: 2069 7320 6e6f 7420 6120 6c6f 6361 6c20   is not a local 
│ │ │ │ -00048350: 6973 6f6d 6f72 7068 6973 6d0a 0a44 6573  isomorphism..Des
│ │ │ │ -00048360: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -00048370: 3d3d 3d3d 0a0a 5468 6973 206d 6574 686f  ====..This metho
│ │ │ │ -00048380: 6420 7369 6d70 6c79 2063 616c 6375 6c61  d simply calcula
│ │ │ │ -00048390: 7465 7320 7468 6520 2a6e 6f74 6520 696e  tes the *note in
│ │ │ │ -000483a0: 7665 7273 6520 696d 6167 653a 2052 6174  verse image: Rat
│ │ │ │ -000483b0: 696f 6e61 6c4d 6170 205e 5f73 745f 7374  ionalMap ^_st_st
│ │ │ │ -000483c0: 0a49 6465 616c 2c20 6f66 2074 6865 202a  .Ideal, of the *
│ │ │ │ -000483d0: 6e6f 7465 2062 6173 6520 6c6f 6375 733a  note base locus:
│ │ │ │ -000483e0: 2069 6465 616c 5f6c 7052 6174 696f 6e61   ideal_lpRationa
│ │ │ │ -000483f0: 6c4d 6170 5f72 702c 206f 6620 7468 6520  lMap_rp, of the 
│ │ │ │ -00048400: 696e 7665 7273 6520 6d61 702c 0a77 6869  inverse map,.whi
│ │ │ │ -00048410: 6368 2069 6e20 7475 726e 2069 7320 6465  ch in turn is de
│ │ │ │ -00048420: 7465 726d 696e 6564 2074 6872 6f75 6768  termined through
│ │ │ │ -00048430: 2074 6865 206d 6574 686f 6420 2a6e 6f74   the method *not
│ │ │ │ -00048440: 6520 696e 7665 7273 653a 0a69 6e76 6572  e inverse:.inver
│ │ │ │ -00048450: 7365 5f6c 7052 6174 696f 6e61 6c4d 6170  se_lpRationalMap
│ │ │ │ -00048460: 5f72 702c 2e0a 0a42 656c 6f77 2c20 7765  _rp,...Below, we
│ │ │ │ -00048470: 2063 6f6d 7075 7465 2074 6865 2065 7863   compute the exc
│ │ │ │ -00048480: 6570 7469 6f6e 616c 206c 6f63 7573 206f  eptional locus o
│ │ │ │ -00048490: 6620 7468 6520 6d61 7020 6465 6669 6e65  f the map define
│ │ │ │ -000484a0: 6420 6279 2074 6865 206c 696e 6561 7220  d by the linear 
│ │ │ │ -000484b0: 7379 7374 656d 0a6f 6620 7175 6164 7269  system.of quadri
│ │ │ │ -000484c0: 6373 2074 6872 6f75 6768 2074 6865 2071  cs through the q
│ │ │ │ -000484d0: 7569 6e74 6963 2072 6174 696f 6e61 6c20  uintic rational 
│ │ │ │ -000484e0: 6e6f 726d 616c 2063 7572 7665 2069 6e20  normal curve in 
│ │ │ │ -000484f0: 245c 6d61 7468 6262 7b50 7d5e 3524 2e0a  $\mathbb{P}^5$..
│ │ │ │ -00048500: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00048160: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +00048170: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +00048180: 6578 6365 7074 696f 6e61 6c4c 6f63 7573  exceptionalLocus
│ │ │ │ +00048190: 2070 6869 0a20 202a 2049 6e70 7574 733a   phi.  * Inputs:
│ │ │ │ +000481a0: 0a20 2020 2020 202a 2070 6869 2c20 6120  .      * phi, a 
│ │ │ │ +000481b0: 2a6e 6f74 6520 7261 7469 6f6e 616c 206d  *note rational m
│ │ │ │ +000481c0: 6170 3a20 5261 7469 6f6e 616c 4d61 702c  ap: RationalMap,
│ │ │ │ +000481d0: 2c20 6120 6269 7261 7469 6f6e 616c 206d  , a birational m
│ │ │ │ +000481e0: 6170 0a20 2020 2020 2020 2024 585c 6461  ap.        $X\da
│ │ │ │ +000481f0: 7368 7269 6768 7461 7272 6f77 2059 240a  shrightarrow Y$.
│ │ │ │ +00048200: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e    * *note Option
│ │ │ │ +00048210: 616c 2069 6e70 7574 733a 2028 4d61 6361  al inputs: (Maca
│ │ │ │ +00048220: 756c 6179 3244 6f63 2975 7369 6e67 2066  ulay2Doc)using f
│ │ │ │ +00048230: 756e 6374 696f 6e73 2077 6974 6820 6f70  unctions with op
│ │ │ │ +00048240: 7469 6f6e 616c 2069 6e70 7574 732c 3a0a  tional inputs,:.
│ │ │ │ +00048250: 2020 2020 2020 2a20 2a6e 6f74 6520 4365        * *note Ce
│ │ │ │ +00048260: 7274 6966 793a 2043 6572 7469 6679 2c20  rtify: Certify, 
│ │ │ │ +00048270: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +00048280: 7661 6c75 6520 6661 6c73 652c 2077 6865  value false, whe
│ │ │ │ +00048290: 7468 6572 2074 6f20 656e 7375 7265 0a20  ther to ensure. 
│ │ │ │ +000482a0: 2020 2020 2020 2063 6f72 7265 6374 6e65         correctne
│ │ │ │ +000482b0: 7373 206f 6620 6f75 7470 7574 0a20 202a  ss of output.  *
│ │ │ │ +000482c0: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +000482d0: 2a20 616e 202a 6e6f 7465 2069 6465 616c  * an *note ideal
│ │ │ │ +000482e0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +000482f0: 4964 6561 6c2c 2c20 7468 6520 6964 6561  Ideal,, the idea
│ │ │ │ +00048300: 6c20 6465 6669 6e69 6e67 2074 6865 2063  l defining the c
│ │ │ │ +00048310: 6c6f 7375 7265 2069 6e0a 2020 2020 2020  losure in.      
│ │ │ │ +00048320: 2020 5820 6f66 2074 6865 206c 6f63 7573    X of the locus
│ │ │ │ +00048330: 2077 6865 7265 2070 6869 2069 7320 6e6f   where phi is no
│ │ │ │ +00048340: 7420 6120 6c6f 6361 6c20 6973 6f6d 6f72  t a local isomor
│ │ │ │ +00048350: 7068 6973 6d0a 0a44 6573 6372 6970 7469  phism..Descripti
│ │ │ │ +00048360: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ +00048370: 5468 6973 206d 6574 686f 6420 7369 6d70  This method simp
│ │ │ │ +00048380: 6c79 2063 616c 6375 6c61 7465 7320 7468  ly calculates th
│ │ │ │ +00048390: 6520 2a6e 6f74 6520 696e 7665 7273 6520  e *note inverse 
│ │ │ │ +000483a0: 696d 6167 653a 2052 6174 696f 6e61 6c4d  image: RationalM
│ │ │ │ +000483b0: 6170 205e 5f73 745f 7374 0a49 6465 616c  ap ^_st_st.Ideal
│ │ │ │ +000483c0: 2c20 6f66 2074 6865 202a 6e6f 7465 2062  , of the *note b
│ │ │ │ +000483d0: 6173 6520 6c6f 6375 733a 2069 6465 616c  ase locus: ideal
│ │ │ │ +000483e0: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ +000483f0: 702c 206f 6620 7468 6520 696e 7665 7273  p, of the invers
│ │ │ │ +00048400: 6520 6d61 702c 0a77 6869 6368 2069 6e20  e map,.which in 
│ │ │ │ +00048410: 7475 726e 2069 7320 6465 7465 726d 696e  turn is determin
│ │ │ │ +00048420: 6564 2074 6872 6f75 6768 2074 6865 206d  ed through the m
│ │ │ │ +00048430: 6574 686f 6420 2a6e 6f74 6520 696e 7665  ethod *note inve
│ │ │ │ +00048440: 7273 653a 0a69 6e76 6572 7365 5f6c 7052  rse:.inverse_lpR
│ │ │ │ +00048450: 6174 696f 6e61 6c4d 6170 5f72 702c 2e0a  ationalMap_rp,..
│ │ │ │ +00048460: 0a42 656c 6f77 2c20 7765 2063 6f6d 7075  .Below, we compu
│ │ │ │ +00048470: 7465 2074 6865 2065 7863 6570 7469 6f6e  te the exception
│ │ │ │ +00048480: 616c 206c 6f63 7573 206f 6620 7468 6520  al locus of the 
│ │ │ │ +00048490: 6d61 7020 6465 6669 6e65 6420 6279 2074  map defined by t
│ │ │ │ +000484a0: 6865 206c 696e 6561 7220 7379 7374 656d  he linear system
│ │ │ │ +000484b0: 0a6f 6620 7175 6164 7269 6373 2074 6872  .of quadrics thr
│ │ │ │ +000484c0: 6f75 6768 2074 6865 2071 7569 6e74 6963  ough the quintic
│ │ │ │ +000484d0: 2072 6174 696f 6e61 6c20 6e6f 726d 616c   rational normal
│ │ │ │ +000484e0: 2063 7572 7665 2069 6e20 245c 6d61 7468   curve in $\math
│ │ │ │ +000484f0: 6262 7b50 7d5e 3524 2e0a 0a2b 2d2d 2d2d  bb{P}^5$...+----
│ │ │ │ +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 2d2d  ----------------
│ │ │ │  00048530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00048550: 0a7c 6931 203a 2050 3520 3a3d 205a 5a2f  .|i1 : P5 := ZZ/
│ │ │ │ -00048560: 3130 3030 3033 5b78 5f30 2e2e 785f 355d  100003[x_0..x_5]
│ │ │ │ -00048570: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00048540: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ +00048550: 2050 3520 3a3d 205a 5a2f 3130 3030 3033   P5 := ZZ/100003
│ │ │ │ +00048560: 5b78 5f30 2e2e 785f 355d 3b20 2020 2020  [x_0..x_5];     
│ │ │ │ +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 207c                 |
│ │ │ │ -000485a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00048590: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000485a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000485b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000485c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000485d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000485e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000485f0: 0a7c 6932 203a 2070 6869 203d 2072 6174  .|i2 : phi = rat
│ │ │ │ -00048600: 696f 6e61 6c4d 6170 286d 696e 6f72 7328  ionalMap(minors(
│ │ │ │ -00048610: 322c 6d61 7472 6978 7b7b 785f 302c 785f  2,matrix{{x_0,x_
│ │ │ │ -00048620: 312c 785f 322c 785f 332c 785f 347d 2c7b  1,x_2,x_3,x_4},{
│ │ │ │ -00048630: 785f 312c 785f 2020 2020 2020 2020 207c  x_1,x_         |
│ │ │ │ -00048640: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000485e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ +000485f0: 2070 6869 203d 2072 6174 696f 6e61 6c4d   phi = rationalM
│ │ │ │ +00048600: 6170 286d 696e 6f72 7328 322c 6d61 7472  ap(minors(2,matr
│ │ │ │ +00048610: 6978 7b7b 785f 302c 785f 312c 785f 322c  ix{{x_0,x_1,x_2,
│ │ │ │ +00048620: 785f 332c 785f 347d 2c7b 785f 312c 785f  x_3,x_4},{x_1,x_
│ │ │ │ +00048630: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00048640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048680: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00048690: 0a7c 6f32 203a 2052 6174 696f 6e61 6c4d  .|o2 : RationalM
│ │ │ │ -000486a0: 6170 2028 7175 6164 7261 7469 6320 7261  ap (quadratic ra
│ │ │ │ -000486b0: 7469 6f6e 616c 206d 6170 2066 726f 6d20  tional map from 
│ │ │ │ -000486c0: 5050 5e35 2074 6f20 352d 6469 6d65 6e73  PP^5 to 5-dimens
│ │ │ │ -000486d0: 696f 6e61 6c20 2020 2020 2020 2020 207c  ional          |
│ │ │ │ -000486e0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00048680: 2020 2020 2020 2020 207c 0a7c 6f32 203a           |.|o2 :
│ │ │ │ +00048690: 2052 6174 696f 6e61 6c4d 6170 2028 7175   RationalMap (qu
│ │ │ │ +000486a0: 6164 7261 7469 6320 7261 7469 6f6e 616c  adratic rational
│ │ │ │ +000486b0: 206d 6170 2066 726f 6d20 5050 5e35 2074   map from PP^5 t
│ │ │ │ +000486c0: 6f20 352d 6469 6d65 6e73 696f 6e61 6c20  o 5-dimensional 
│ │ │ │ +000486d0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │ +000486e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000486f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00048730: 0a7c 322c 785f 332c 785f 342c 785f 357d  .|2,x_3,x_4,x_5}
│ │ │ │ -00048740: 7d29 2c44 6f6d 696e 616e 743d 3e32 293b  }),Dominant=>2);
│ │ │ │ +00048720: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 322c 785f  ---------|.|2,x_
│ │ │ │ +00048730: 332c 785f 342c 785f 357d 7d29 2c44 6f6d  3,x_4,x_5}}),Dom
│ │ │ │ +00048740: 696e 616e 743d 3e32 293b 2020 2020 2020  inant=>2);      
│ │ │ │  00048750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048770: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00048780: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00048770: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00048780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000487a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000487b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000487c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000487d0: 0a7c 7375 6276 6172 6965 7479 206f 6620  .|subvariety of 
│ │ │ │ -000487e0: 5050 5e39 2920 2020 2020 2020 2020 2020  PP^9)           
│ │ │ │ +000487c0: 2020 2020 2020 2020 207c 0a7c 7375 6276           |.|subv
│ │ │ │ +000487d0: 6172 6965 7479 206f 6620 5050 5e39 2920  ariety of PP^9) 
│ │ │ │ +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 2020 2020 2020 2020 207c                 |
│ │ │ │ -00048820: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00048810: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00048820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00048870: 0a7c 6933 203a 2045 203d 2065 7863 6570  .|i3 : E = excep
│ │ │ │ -00048880: 7469 6f6e 616c 4c6f 6375 7320 7068 693b  tionalLocus phi;
│ │ │ │ +00048860: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ +00048870: 2045 203d 2065 7863 6570 7469 6f6e 616c   E = exceptional
│ │ │ │ +00048880: 4c6f 6375 7320 7068 693b 2020 2020 2020  Locus phi;      
│ │ │ │  00048890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000488a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000488b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000488c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000488b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000488c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000488d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000488e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000488f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048900: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00048910: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00048920: 2020 5a5a 2020 2020 2020 2020 2020 2020    ZZ            
│ │ │ │ +00048900: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00048910: 2020 2020 2020 2020 2020 2020 5a5a 2020              ZZ  
│ │ │ │ +00048920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048950: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00048960: 0a7c 6f33 203a 2049 6465 616c 206f 6620  .|o3 : Ideal of 
│ │ │ │ -00048970: 2d2d 2d2d 2d2d 5b78 202e 2e78 205d 2020  ------[x ..x ]  
│ │ │ │ +00048950: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ +00048960: 2049 6465 616c 206f 6620 2d2d 2d2d 2d2d   Ideal of ------
│ │ │ │ +00048970: 5b78 202e 2e78 205d 2020 2020 2020 2020  [x ..x ]        
│ │ │ │  00048980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000489a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000489b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000489c0: 3130 3030 3033 2020 3020 2020 3520 2020  100003  0   5   
│ │ │ │ +000489a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000489b0: 2020 2020 2020 2020 2020 3130 3030 3033            100003
│ │ │ │ +000489c0: 2020 3020 2020 3520 2020 2020 2020 2020    0   5         
│ │ │ │  000489d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000489e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000489f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00048a00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000489f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00048a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00048a50: 0a7c 6934 203a 2061 7373 6572 7428 4520  .|i4 : assert(E 
│ │ │ │ -00048a60: 3d3d 2070 6869 5e2a 2069 6465 616c 2070  == phi^* ideal p
│ │ │ │ -00048a70: 6869 5e2d 3129 2020 2020 2020 2020 2020  hi^-1)          
│ │ │ │ +00048a40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ +00048a50: 2061 7373 6572 7428 4520 3d3d 2070 6869   assert(E == phi
│ │ │ │ +00048a60: 5e2a 2069 6465 616c 2070 6869 5e2d 3129  ^* ideal phi^-1)
│ │ │ │ +00048a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048a90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00048aa0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00048a90: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00048aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00048af0: 0a7c 6935 203a 2061 7373 6572 7428 4520  .|i5 : assert(E 
│ │ │ │ -00048b00: 3d3d 206d 696e 6f72 7328 332c 6d61 7472  == minors(3,matr
│ │ │ │ -00048b10: 6978 7b7b 785f 302c 785f 312c 785f 322c  ix{{x_0,x_1,x_2,
│ │ │ │ -00048b20: 785f 337d 2c7b 785f 312c 785f 322c 785f  x_3},{x_1,x_2,x_
│ │ │ │ -00048b30: 332c 785f 347d 2c7b 785f 322c 785f 337c  3,x_4},{x_2,x_3|
│ │ │ │ -00048b40: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00048ae0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ +00048af0: 2061 7373 6572 7428 4520 3d3d 206d 696e   assert(E == min
│ │ │ │ +00048b00: 6f72 7328 332c 6d61 7472 6978 7b7b 785f  ors(3,matrix{{x_
│ │ │ │ +00048b10: 302c 785f 312c 785f 322c 785f 337d 2c7b  0,x_1,x_2,x_3},{
│ │ │ │ +00048b20: 785f 312c 785f 322c 785f 332c 785f 347d  x_1,x_2,x_3,x_4}
│ │ │ │ +00048b30: 2c7b 785f 322c 785f 337c 0a7c 2d2d 2d2d  ,{x_2,x_3|.|----
│ │ │ │ +00048b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00048b90: 0a7c 2c78 5f34 2c78 5f35 7d7d 2929 2020  .|,x_4,x_5}}))  
│ │ │ │ +00048b80: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2c78 5f34  ---------|.|,x_4
│ │ │ │ +00048b90: 2c78 5f35 7d7d 2929 2020 2020 2020 2020  ,x_5}}))        
│ │ │ │  00048ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048bd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00048be0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00048bd0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00048be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00048c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00048c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00048c30: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -00048c40: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2069  ===..  * *note i
│ │ │ │ -00048c50: 6465 616c 2852 6174 696f 6e61 6c4d 6170  deal(RationalMap
│ │ │ │ -00048c60: 293a 2069 6465 616c 5f6c 7052 6174 696f  ): ideal_lpRatio
│ │ │ │ -00048c70: 6e61 6c4d 6170 5f72 702c 202d 2d20 6261  nalMap_rp, -- ba
│ │ │ │ -00048c80: 7365 206c 6f63 7573 206f 6620 610a 2020  se locus of a.  
│ │ │ │ -00048c90: 2020 7261 7469 6f6e 616c 206d 6170 0a20    rational map. 
│ │ │ │ -00048ca0: 202a 202a 6e6f 7465 2069 6e76 6572 7365   * *note inverse
│ │ │ │ -00048cb0: 4d61 7028 5261 7469 6f6e 616c 4d61 7029  Map(RationalMap)
│ │ │ │ -00048cc0: 3a20 696e 7665 7273 654d 6170 2c20 2d2d  : inverseMap, --
│ │ │ │ -00048cd0: 2069 6e76 6572 7365 206f 6620 6120 6269   inverse of a bi
│ │ │ │ -00048ce0: 7261 7469 6f6e 616c 206d 6170 0a20 202a  rational map.  *
│ │ │ │ -00048cf0: 202a 6e6f 7465 2052 6174 696f 6e61 6c4d   *note RationalM
│ │ │ │ -00048d00: 6170 205e 2a2a 2049 6465 616c 3a20 5261  ap ^** Ideal: Ra
│ │ │ │ -00048d10: 7469 6f6e 616c 4d61 7020 5e5f 7374 5f73  tionalMap ^_st_s
│ │ │ │ -00048d20: 7420 4964 6561 6c2c 202d 2d20 696e 7665  t Ideal, -- inve
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│ │ │ │ -00048d60: 6d6f 7270 6869 736d 2852 6174 696f 6e61  morphism(Rationa
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│ │ │ │ -00048dd0: 7465 2066 6f72 6365 496e 7665 7273 654d  te forceInverseM
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│ │ │ │ +00048cf0: 2052 6174 696f 6e61 6c4d 6170 205e 2a2a   RationalMap ^**
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│ │ │ │ -00049080: 4c6f 6375 732c 2055 703a 2054 6f70 0a0a  Locus, Up: Top..
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│ │ │ │ +00049050: 6170 5f72 702c 204e 6578 743a 2066 6f72  ap_rp, Next: for
│ │ │ │ +00049060: 6365 496d 6167 652c 2050 7265 763a 2065  ceImage, Prev: e
│ │ │ │ +00049070: 7863 6570 7469 6f6e 616c 4c6f 6375 732c  xceptionalLocus,
│ │ │ │ +00049080: 2055 703a 2054 6f70 0a0a 666c 6174 7465   Up: Top..flatte
│ │ │ │ +00049090: 6e28 5261 7469 6f6e 616c 4d61 7029 202d  n(RationalMap) -
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│ │ │ │  000490e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00049110: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00049140: 656e 3a20 284d 6163 6175 6c61 7932 446f  en: (Macaulay2Do
│ │ │ │ -00049150: 6329 666c 6174 7465 6e2c 0a20 202a 2055  c)flatten,.  * U
│ │ │ │ -00049160: 7361 6765 3a20 0a20 2020 2020 2020 2066  sage: .        f
│ │ │ │ -00049170: 6c61 7474 656e 2070 6869 0a20 202a 2049  latten phi.  * I
│ │ │ │ -00049180: 6e70 7574 733a 0a20 2020 2020 202a 2070  nputs:.      * p
│ │ │ │ -00049190: 6869 2c20 6120 2a6e 6f74 6520 7261 7469  hi, a *note rati
│ │ │ │ -000491a0: 6f6e 616c 206d 6170 3a20 5261 7469 6f6e  onal map: Ration
│ │ │ │ -000491b0: 616c 4d61 702c 0a20 202a 204f 7574 7075  alMap,.  * Outpu
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│ │ │ │ -00049290: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d  ===========..+--
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│ │ │ │ +00049160: 0a20 2020 2020 2020 2066 6c61 7474 656e  .        flatten
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│ │ │ │ +00049190: 2a6e 6f74 6520 7261 7469 6f6e 616c 206d  *note rational m
│ │ │ │ +000491a0: 6170 3a20 5261 7469 6f6e 616c 4d61 702c  ap: RationalMap,
│ │ │ │ +000491b0: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ +000491c0: 2020 2020 2a20 6120 2a6e 6f74 6520 7261      * a *note ra
│ │ │ │ +000491d0: 7469 6f6e 616c 206d 6170 3a20 5261 7469  tional map: Rati
│ │ │ │ +000491e0: 6f6e 616c 4d61 702c 2c20 6120 7261 7469  onalMap,, a rati
│ │ │ │ +000491f0: 6f6e 616c 206d 6170 2069 736f 6d6f 7270  onal map isomorp
│ │ │ │ +00049200: 6869 6320 746f 2074 6865 0a20 2020 2020  hic to the.     
│ │ │ │ +00049210: 2020 206f 7269 6769 6e61 6c20 6d61 702c     original map,
│ │ │ │ +00049220: 2066 6c61 7474 656e 6564 2069 6e20 7468   flattened in th
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│ │ │ │ +00049240: 2069 6465 616c 7320 6f66 2073 6f75 7263   ideals of sourc
│ │ │ │ +00049250: 6520 616e 640a 2020 2020 2020 2020 7461  e and.        ta
│ │ │ │ +00049260: 7267 6574 2063 6f6e 7461 696e 206e 6f20  rget contain no 
│ │ │ │ +00049270: 6c69 6e65 6172 2066 6f72 6d73 0a0a 4465  linear forms..De
│ │ │ │ +00049280: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +00049290: 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d 2d2d  =====..+--------
│ │ │ │  000492a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000492b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000492c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000492d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000492e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -000492f0: 203a 2050 3520 3d20 5151 5b74 5f30 2e2e   : P5 = QQ[t_0..
│ │ │ │ -00049300: 745f 355d 3b20 7068 6920 3d20 7261 7469  t_5]; phi = rati
│ │ │ │ -00049310: 6f6e 616c 4d61 7028 5035 2f28 3335 2a74  onalMap(P5/(35*t
│ │ │ │ -00049320: 5f31 2b34 352a 745f 322b 3231 2a74 2020  _1+45*t_2+21*t  
│ │ │ │ -00049330: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000492e0: 2d2d 2d2d 2d2b 0a7c 6931 203a 2050 3520  -----+.|i1 : P5 
│ │ │ │ +000492f0: 3d20 5151 5b74 5f30 2e2e 745f 355d 3b20  = QQ[t_0..t_5]; 
│ │ │ │ +00049300: 7068 6920 3d20 7261 7469 6f6e 616c 4d61  phi = rationalMa
│ │ │ │ +00049310: 7028 5035 2f28 3335 2a74 5f31 2b34 352a  p(P5/(35*t_1+45*
│ │ │ │ +00049320: 745f 322b 3231 2a74 2020 2020 2020 2020  t_2+21*t        
│ │ │ │ +00049330: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00049340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049380: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00049390: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ -000493a0: 6c69 6e65 6172 2072 6174 696f 6e61 6c20  linear rational 
│ │ │ │ -000493b0: 6d61 7020 6672 6f6d 2074 6872 6565 666f  map from threefo
│ │ │ │ -000493c0: 6c64 2069 6e20 5050 5e35 2074 6f20 2020  ld in PP^5 to   
│ │ │ │ -000493d0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ +00049380: 2020 2020 207c 0a7c 6f32 203a 2052 6174       |.|o2 : Rat
│ │ │ │ +00049390: 696f 6e61 6c4d 6170 2028 6c69 6e65 6172  ionalMap (linear
│ │ │ │ +000493a0: 2072 6174 696f 6e61 6c20 6d61 7020 6672   rational map fr
│ │ │ │ +000493b0: 6f6d 2074 6872 6565 666f 6c64 2069 6e20  om threefold in 
│ │ │ │ +000493c0: 5050 5e35 2074 6f20 2020 2020 2020 2020  PP^5 to         
│ │ │ │ +000493d0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  000493e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000493f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00049420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 5f33  -----------|.|_3
│ │ │ │ -00049430: 2b35 3235 2a74 5f34 2b31 3336 352a 745f  +525*t_4+1365*t_
│ │ │ │ -00049440: 352c 3135 3735 2a74 5f30 2a74 5f32 2d33  5,1575*t_0*t_2-3
│ │ │ │ -00049450: 3235 302a 745f 325e 322b 3733 352a 745f  250*t_2^2+735*t_
│ │ │ │ -00049460: 302a 745f 332d 3138 3930 2a74 5f32 2a74  0*t_3-1890*t_2*t
│ │ │ │ -00049470: 5f33 2d31 3636 362a 745f 337c 0a7c 2020  _3-1666*t_3|.|  
│ │ │ │ +00049420: 2d2d 2d2d 2d7c 0a7c 5f33 2b35 3235 2a74  -----|.|_3+525*t
│ │ │ │ +00049430: 5f34 2b31 3336 352a 745f 352c 3135 3735  _4+1365*t_5,1575
│ │ │ │ +00049440: 2a74 5f30 2a74 5f32 2d33 3235 302a 745f  *t_0*t_2-3250*t_
│ │ │ │ +00049450: 325e 322b 3733 352a 745f 302a 745f 332d  2^2+735*t_0*t_3-
│ │ │ │ +00049460: 3138 3930 2a74 5f32 2a74 5f33 2d31 3636  1890*t_2*t_3-166
│ │ │ │ +00049470: 362a 745f 337c 0a7c 2020 2020 2020 2020  6*t_3|.|        
│ │ │ │  00049480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049490: 2020 2020 2020 2020 2020 2020 2020 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 207c 0a7c 6879             |.|hy
│ │ │ │ -000494d0: 7065 7273 7572 6661 6365 2069 6e20 5050  persurface in PP
│ │ │ │ -000494e0: 5e35 2920 2020 2020 2020 2020 2020 2020  ^5)             
│ │ │ │ +000494c0: 2020 2020 207c 0a7c 6879 7065 7273 7572       |.|hypersur
│ │ │ │ +000494d0: 6661 6365 2069 6e20 5050 5e35 2920 2020  face in PP^5)   
│ │ │ │ +000494e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000494f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049510: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ +00049510: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  00049520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d7c 0a7c 5e32  -----------|.|^2
│ │ │ │ -00049570: 2b31 3731 3530 2a74 5f30 2a74 5f34 2d34  +17150*t_0*t_4-4
│ │ │ │ -00049580: 3732 3530 2a74 5f32 2a74 5f34 2d32 3230  7250*t_2*t_4-220
│ │ │ │ -00049590: 3530 2a74 5f33 2a74 5f34 2d32 3736 3835  50*t_3*t_4-27685
│ │ │ │ -000495a0: 302a 745f 345e 322b 3436 3535 302a 745f  0*t_4^2+46550*t_
│ │ │ │ -000495b0: 302a 745f 352d 2020 2020 207c 0a7c 2d2d  0*t_5-     |.|--
│ │ │ │ +00049560: 2d2d 2d2d 2d7c 0a7c 5e32 2b31 3731 3530  -----|.|^2+17150
│ │ │ │ +00049570: 2a74 5f30 2a74 5f34 2d34 3732 3530 2a74  *t_0*t_4-47250*t
│ │ │ │ +00049580: 5f32 2a74 5f34 2d32 3230 3530 2a74 5f33  _2*t_4-22050*t_3
│ │ │ │ +00049590: 2a74 5f34 2d32 3736 3835 302a 745f 345e  *t_4-276850*t_4^
│ │ │ │ +000495a0: 322b 3436 3535 302a 745f 302a 745f 352d  2+46550*t_0*t_5-
│ │ │ │ +000495b0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  000495c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000495d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000495e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000495f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00049600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3132  -----------|.|12
│ │ │ │ -00049610: 3238 3530 2a74 5f32 2a74 5f35 2d35 3733  2850*t_2*t_5-573
│ │ │ │ -00049620: 3330 2a74 5f33 2a74 5f35 2d31 3433 3332  30*t_3*t_5-14332
│ │ │ │ -00049630: 3530 2a74 5f34 2a74 5f35 2d31 3836 3434  50*t_4*t_5-18644
│ │ │ │ -00049640: 3530 2a74 5f35 5e32 292c 5035 2f28 3331  50*t_5^2),P5/(31
│ │ │ │ -00049650: 352a 745f 302b 3238 302a 747c 0a7c 2d2d  5*t_0+280*t|.|--
│ │ │ │ +00049600: 2d2d 2d2d 2d7c 0a7c 3132 3238 3530 2a74  -----|.|122850*t
│ │ │ │ +00049610: 5f32 2a74 5f35 2d35 3733 3330 2a74 5f33  _2*t_5-57330*t_3
│ │ │ │ +00049620: 2a74 5f35 2d31 3433 3332 3530 2a74 5f34  *t_5-1433250*t_4
│ │ │ │ +00049630: 2a74 5f35 2d31 3836 3434 3530 2a74 5f35  *t_5-1864450*t_5
│ │ │ │ +00049640: 5e32 292c 5035 2f28 3331 352a 745f 302b  ^2),P5/(315*t_0+
│ │ │ │ +00049650: 3238 302a 747c 0a7c 2d2d 2d2d 2d2d 2d2d  280*t|.|--------
│ │ │ │  00049660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000496a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 5f31  -----------|.|_1
│ │ │ │ -000496b0: 2b34 352a 745f 322b 3231 2a74 5f33 2b32  +45*t_2+21*t_3+2
│ │ │ │ -000496c0: 3130 2a74 5f34 2b31 3035 302a 745f 3529  10*t_4+1050*t_5)
│ │ │ │ -000496d0: 2c7b 2d34 352a 745f 322d 3231 2a74 5f33  ,{-45*t_2-21*t_3
│ │ │ │ -000496e0: 2d34 3930 2a74 5f34 2d31 3333 302a 745f  -490*t_4-1330*t_
│ │ │ │ -000496f0: 352c 2020 2020 2020 2020 207c 0a7c 2d2d  5,         |.|--
│ │ │ │ +000496a0: 2d2d 2d2d 2d7c 0a7c 5f31 2b34 352a 745f  -----|.|_1+45*t_
│ │ │ │ +000496b0: 322b 3231 2a74 5f33 2b32 3130 2a74 5f34  2+21*t_3+210*t_4
│ │ │ │ +000496c0: 2b31 3035 302a 745f 3529 2c7b 2d34 352a  +1050*t_5),{-45*
│ │ │ │ +000496d0: 745f 322d 3231 2a74 5f33 2d34 3930 2a74  t_2-21*t_3-490*t
│ │ │ │ +000496e0: 5f34 2d31 3333 302a 745f 352c 2020 2020  _4-1330*t_5,    
│ │ │ │ +000496f0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  00049700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00049740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3435  -----------|.|45
│ │ │ │ -00049750: 2a74 5f32 2b32 312a 745f 332b 3532 352a  *t_2+21*t_3+525*
│ │ │ │ -00049760: 745f 342b 3133 3635 2a74 5f35 2c20 3335  t_4+1365*t_5, 35
│ │ │ │ -00049770: 2a74 5f32 2c20 3335 2a74 5f33 2c20 3335  *t_2, 35*t_3, 35
│ │ │ │ -00049780: 2a74 5f34 2c20 3335 2a74 5f35 7d29 3b20  *t_4, 35*t_5}); 
│ │ │ │ -00049790: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00049740: 2d2d 2d2d 2d7c 0a7c 3435 2a74 5f32 2b32  -----|.|45*t_2+2
│ │ │ │ +00049750: 312a 745f 332b 3532 352a 745f 342b 3133  1*t_3+525*t_4+13
│ │ │ │ +00049760: 3635 2a74 5f35 2c20 3335 2a74 5f32 2c20  65*t_5, 35*t_2, 
│ │ │ │ +00049770: 3335 2a74 5f33 2c20 3335 2a74 5f34 2c20  35*t_3, 35*t_4, 
│ │ │ │ +00049780: 3335 2a74 5f35 7d29 3b20 2020 2020 2020  35*t_5});       
│ │ │ │ +00049790: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  000497a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000497b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000497c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000497d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000497e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ -000497f0: 203a 2064 6573 6372 6962 6520 7068 6920   : describe phi 
│ │ │ │ +000497e0: 2d2d 2d2d 2d2b 0a7c 6933 203a 2064 6573  -----+.|i3 : des
│ │ │ │ +000497f0: 6372 6962 6520 7068 6920 2020 2020 2020  cribe phi       
│ │ │ │  00049800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049830: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00049830: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00049840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049880: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ -00049890: 203d 2072 6174 696f 6e61 6c20 6d61 7020   = rational map 
│ │ │ │ -000498a0: 6465 6669 6e65 6420 6279 2066 6f72 6d73  defined by forms
│ │ │ │ -000498b0: 206f 6620 6465 6772 6565 2031 2020 2020   of degree 1    
│ │ │ │ +00049880: 2020 2020 207c 0a7c 6f33 203d 2072 6174       |.|o3 = rat
│ │ │ │ +00049890: 696f 6e61 6c20 6d61 7020 6465 6669 6e65  ional map define
│ │ │ │ +000498a0: 6420 6279 2066 6f72 6d73 206f 6620 6465  d by forms of de
│ │ │ │ +000498b0: 6772 6565 2031 2020 2020 2020 2020 2020  gree 1          
│ │ │ │  000498c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000498d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000498e0: 2020 2073 6f75 7263 6520 7661 7269 6574     source variet
│ │ │ │ -000498f0: 793a 2073 6d6f 6f74 6820 636f 6d70 6c65  y: smooth comple
│ │ │ │ -00049900: 7465 2069 6e74 6572 7365 6374 696f 6e20  te intersection 
│ │ │ │ -00049910: 6f66 2074 7970 6520 2831 2c32 2920 696e  of type (1,2) in
│ │ │ │ -00049920: 2050 505e 3520 2020 2020 207c 0a7c 2020   PP^5      |.|  
│ │ │ │ -00049930: 2020 2074 6172 6765 7420 7661 7269 6574     target variet
│ │ │ │ -00049940: 793a 2068 7970 6572 706c 616e 6520 696e  y: hyperplane in
│ │ │ │ -00049950: 2050 505e 3520 2020 2020 2020 2020 2020   PP^5           
│ │ │ │ +000498d0: 2020 2020 207c 0a7c 2020 2020 2073 6f75       |.|     sou
│ │ │ │ +000498e0: 7263 6520 7661 7269 6574 793a 2073 6d6f  rce variety: smo
│ │ │ │ +000498f0: 6f74 6820 636f 6d70 6c65 7465 2069 6e74  oth complete int
│ │ │ │ +00049900: 6572 7365 6374 696f 6e20 6f66 2074 7970  ersection of typ
│ │ │ │ +00049910: 6520 2831 2c32 2920 696e 2050 505e 3520  e (1,2) in PP^5 
│ │ │ │ +00049920: 2020 2020 207c 0a7c 2020 2020 2074 6172       |.|     tar
│ │ │ │ +00049930: 6765 7420 7661 7269 6574 793a 2068 7970  get variety: hyp
│ │ │ │ +00049940: 6572 706c 616e 6520 696e 2050 505e 3520  erplane in PP^5 
│ │ │ │ +00049950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049970: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00049980: 2020 2063 6f65 6666 6963 6965 6e74 2072     coefficient r
│ │ │ │ -00049990: 696e 673a 2051 5120 2020 2020 2020 2020  ing: QQ         
│ │ │ │ +00049970: 2020 2020 207c 0a7c 2020 2020 2063 6f65       |.|     coe
│ │ │ │ +00049980: 6666 6963 6965 6e74 2072 696e 673a 2051  fficient ring: Q
│ │ │ │ +00049990: 5120 2020 2020 2020 2020 2020 2020 2020  Q               
│ │ │ │  000499a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000499b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000499c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000499c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  000499d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000499e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000499f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00049a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ -00049a20: 203a 2070 7369 203d 2066 6c61 7474 656e   : psi = flatten
│ │ │ │ -00049a30: 2070 6869 3b20 2020 2020 2020 2020 2020   phi;           
│ │ │ │ +00049a10: 2d2d 2d2d 2d2b 0a7c 6934 203a 2070 7369  -----+.|i4 : psi
│ │ │ │ +00049a20: 203d 2066 6c61 7474 656e 2070 6869 3b20   = flatten phi; 
│ │ │ │ +00049a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049a60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00049a60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00049a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049ab0: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ -00049ac0: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ -00049ad0: 6c69 6e65 6172 2072 6174 696f 6e61 6c20  linear rational 
│ │ │ │ -00049ae0: 6d61 7020 6672 6f6d 2068 7970 6572 7375  map from hypersu
│ │ │ │ -00049af0: 7266 6163 6520 696e 2050 505e 3420 746f  rface in PP^4 to
│ │ │ │ -00049b00: 2050 505e 3429 2020 2020 207c 0a2b 2d2d   PP^4)     |.+--
│ │ │ │ +00049ab0: 2020 2020 207c 0a7c 6f34 203a 2052 6174       |.|o4 : Rat
│ │ │ │ +00049ac0: 696f 6e61 6c4d 6170 2028 6c69 6e65 6172  ionalMap (linear
│ │ │ │ +00049ad0: 2072 6174 696f 6e61 6c20 6d61 7020 6672   rational map fr
│ │ │ │ +00049ae0: 6f6d 2068 7970 6572 7375 7266 6163 6520  om hypersurface 
│ │ │ │ +00049af0: 696e 2050 505e 3420 746f 2050 505e 3429  in PP^4 to PP^4)
│ │ │ │ +00049b00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00049b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00049b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ -00049b60: 203a 2064 6573 6372 6962 6520 7073 6920   : describe psi 
│ │ │ │ +00049b50: 2d2d 2d2d 2d2b 0a7c 6935 203a 2064 6573  -----+.|i5 : des
│ │ │ │ +00049b60: 6372 6962 6520 7073 6920 2020 2020 2020  cribe psi       
│ │ │ │  00049b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049ba0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00049ba0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00049bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049bf0: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ -00049c00: 203d 2072 6174 696f 6e61 6c20 6d61 7020   = rational map 
│ │ │ │ -00049c10: 6465 6669 6e65 6420 6279 2066 6f72 6d73  defined by forms
│ │ │ │ -00049c20: 206f 6620 6465 6772 6565 2031 2020 2020   of degree 1    
│ │ │ │ +00049bf0: 2020 2020 207c 0a7c 6f35 203d 2072 6174       |.|o5 = rat
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│ │ │ │ -00049c90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │ -00049cf0: 2020 2063 6f65 6666 6963 6965 6e74 2072     coefficient r
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│ │ │ │ +00049d00: 5120 2020 2020 2020 2020 2020 2020 2020  Q               
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│ │ │ │  00049d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00049d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00049db0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -00049dc0: 0a20 202a 202a 6e6f 7465 2066 6c61 7474  .  * *note flatt
│ │ │ │ -00049dd0: 656e 2852 6174 696f 6e61 6c4d 6170 293a  en(RationalMap):
│ │ │ │ -00049de0: 2066 6c61 7474 656e 5f6c 7052 6174 696f   flatten_lpRatio
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│ │ │ │ +00049da0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +00049db0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  =========..  * *
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│ │ │ │ +00049e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00049e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00049e80: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
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│ │ │ │ -00049eb0: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
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│ │ │ │ -00049f10: 696c 653a 2043 7265 6d6f 6e61 2e69 6e66  ile: Cremona.inf
│ │ │ │ -00049f20: 6f2c 204e 6f64 653a 2066 6f72 6365 496d  o, Node: forceIm
│ │ │ │ -00049f30: 6167 652c 204e 6578 743a 2066 6f72 6365  age, Next: force
│ │ │ │ -00049f40: 496e 7665 7273 654d 6170 2c20 5072 6576  InverseMap, Prev
│ │ │ │ -00049f50: 3a20 666c 6174 7465 6e5f 6c70 5261 7469  : flatten_lpRati
│ │ │ │ -00049f60: 6f6e 616c 4d61 705f 7270 2c20 5570 3a20  onalMap_rp, Up: 
│ │ │ │ -00049f70: 546f 700a 0a66 6f72 6365 496d 6167 6520  Top..forceImage 
│ │ │ │ -00049f80: 2d2d 2064 6563 6c61 7265 2077 6869 6368  -- declare which
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│ │ │ │ +00049e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +00049e80: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ +00049e90: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ +00049ea0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
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│ │ │ │ +00049f50: 7465 6e5f 6c70 5261 7469 6f6e 616c 4d61  ten_lpRationalMa
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│ │ │ │ +00049f70: 6f72 6365 496d 6167 6520 2d2d 2064 6563  orceImage -- dec
│ │ │ │ +00049f80: 6c61 7265 2077 6869 6368 2069 7320 7468  lare which is th
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│ │ │ │ +00049fa0: 696f 6e61 6c20 6d61 700a 2a2a 2a2a 2a2a  ional map.******
│ │ │ │  00049fb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00049fc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00049fd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00049fe0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -00049ff0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -0004a000: 666f 7263 6549 6d61 6765 2850 6869 2c49  forceImage(Phi,I
│ │ │ │ -0004a010: 290a 2020 2a20 496e 7075 7473 3a0a 2020  ).  * Inputs:.  
│ │ │ │ -0004a020: 2020 2020 2a20 5068 692c 2061 202a 6e6f      * Phi, a *no
│ │ │ │ -0004a030: 7465 2072 6174 696f 6e61 6c20 6d61 703a  te rational map:
│ │ │ │ -0004a040: 2052 6174 696f 6e61 6c4d 6170 2c0a 2020   RationalMap,.  
│ │ │ │ -0004a050: 2020 2020 2a20 492c 2061 6e20 2a6e 6f74      * I, an *not
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│ │ │ │ -0004a080: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -0004a090: 202a 202a 6e6f 7465 206e 756c 6c3a 2028   * *note null: (
│ │ │ │ -0004a0a0: 4d61 6361 756c 6179 3244 6f63 296e 756c  Macaulay2Doc)nul
│ │ │ │ -0004a0b0: 6c2c 0a0a 4465 7363 7269 7074 696f 6e0a  l,..Description.
│ │ │ │ -0004a0c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
│ │ │ │ -0004a0d0: 7320 6d65 7468 6f64 2061 6c6c 6f77 7320  s method allows 
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│ │ │ │ -0004a140: 2049 6e20 7061 7274 6963 756c 6172 2c20   In particular, 
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│ │ │ │ +00049fe0: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ +00049ff0: 200a 2020 2020 2020 2020 666f 7263 6549   .        forceI
│ │ │ │ +0004a000: 6d61 6765 2850 6869 2c49 290a 2020 2a20  mage(Phi,I).  * 
│ │ │ │ +0004a010: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +0004a020: 5068 692c 2061 202a 6e6f 7465 2072 6174  Phi, a *note rat
│ │ │ │ +0004a030: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
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│ │ │ │ +0004a060: 6c3a 2028 4d61 6361 756c 6179 3244 6f63  l: (Macaulay2Doc
│ │ │ │ +0004a070: 2949 6465 616c 2c0a 2020 2a20 4f75 7470  )Ideal,.  * Outp
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│ │ │ │ +0004a090: 7465 206e 756c 6c3a 2028 4d61 6361 756c  te null: (Macaul
│ │ │ │ +0004a0a0: 6179 3244 6f63 296e 756c 6c2c 0a0a 4465  ay2Doc)null,..De
│ │ │ │ +0004a0b0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +0004a0c0: 3d3d 3d3d 3d0a 0a54 6869 7320 6d65 7468  =====..This meth
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│ │ │ │ +0004a100: 6620 6120 6769 7665 6e20 7261 7469 6f6e  f a given ration
│ │ │ │ +0004a110: 616c 206d 6170 0a77 6974 686f 7574 2070  al map.without p
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│ │ │ │  0004a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0004a1f0: 2058 203d 2020 2020 2020 2020 2020 2020   X =            
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│ │ │ │  0004a210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a220: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0004a220: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
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│ │ │ │  0004a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a270: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
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│ │ │ │  0004a2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004a380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004a3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0004a3c0: 3320 3a20 5068 6920 3d20 7261 7469 6f6e  3 : Phi = ration
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│ │ │ │  0004a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a720: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0004a730: 3520 3a20 5261 7469 6f6e 616c 4d61 7020  5 : RationalMap 
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│ │ │ │  0004a780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  0004a830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0004a9c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │  0004ac10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0004ac20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0004ac30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0004ac40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0004ac50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0004ac60: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20  *..  * Usage: . 
│ │ │ │ -0004ac70: 2020 2020 2020 2066 6f72 6365 496e 7665         forceInve
│ │ │ │ -0004ac80: 7273 654d 6170 2850 6869 2c50 7369 290a  rseMap(Phi,Psi).
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│ │ │ │ -0004aca0: 2020 2a20 5068 692c 2061 202a 6e6f 7465    * Phi, a *note
│ │ │ │ -0004acb0: 2072 6174 696f 6e61 6c20 6d61 703a 2052   rational map: R
│ │ │ │ -0004acc0: 6174 696f 6e61 6c4d 6170 2c0a 2020 2020  ationalMap,.    
│ │ │ │ -0004acd0: 2020 2a20 5073 692c 2061 202a 6e6f 7465    * Psi, a *note
│ │ │ │ -0004ace0: 2072 6174 696f 6e61 6c20 6d61 703a 2052   rational map: R
│ │ │ │ -0004acf0: 6174 696f 6e61 6c4d 6170 2c0a 2020 2a20  ationalMap,.  * 
│ │ │ │ -0004ad00: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
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│ │ │ │ -0004aec0: 5261 7469 6f6e 616c 4d61 705f 636d 5261  RationalMap_cmRa
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│ │ │ │ -0004b1a0: 2066 6f72 6365 496e 7665 7273 654d 6170   forceInverseMap
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│ │ │ │ +0004b1a0: 496e 7665 7273 654d 6170 2c20 5570 3a20  InverseMap, Up: 
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│ │ │ │  0004b1f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -0004b2b0: 6974 6820 6f70 7469 6f6e 616c 2069 6e70  ith optional inp
│ │ │ │ -0004b2c0: 7574 732c 3a0a 2020 2020 2020 2a20 2a6e  uts,:.      * *n
│ │ │ │ -0004b2d0: 6f74 6520 426c 6f77 5570 5374 7261 7465  ote BlowUpStrate
│ │ │ │ -0004b2e0: 6779 3a20 426c 6f77 5570 5374 7261 7465  gy: BlowUpStrate
│ │ │ │ -0004b2f0: 6779 2c20 3d3e 202e 2e2e 2c20 6465 6661  gy, => ..., defa
│ │ │ │ -0004b300: 756c 7420 7661 6c75 650a 2020 2020 2020  ult value.      
│ │ │ │ -0004b310: 2020 2245 6c69 6d69 6e61 7465 222c 0a20    "Eliminate",. 
│ │ │ │ -0004b320: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -0004b330: 2020 2a20 6120 2a6e 6f74 6520 7261 7469    * a *note rati
│ │ │ │ -0004b340: 6f6e 616c 206d 6170 3a20 5261 7469 6f6e  onal map: Ration
│ │ │ │ -0004b350: 616c 4d61 702c 2c20 7468 6520 6669 7273  alMap,, the firs
│ │ │ │ -0004b360: 7420 7072 6f6a 6563 7469 6f6e 0a20 2020  t projection.   
│ │ │ │ -0004b370: 2020 202a 2061 202a 6e6f 7465 2072 6174     * a *note rat
│ │ │ │ -0004b380: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ -0004b390: 6e61 6c4d 6170 2c2c 2074 6865 2073 6563  nalMap,, the sec
│ │ │ │ -0004b3a0: 6f6e 6420 7072 6f6a 6563 7469 6f6e 0a0a  ond projection..
│ │ │ │ -0004b3b0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -0004b3c0: 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d  =======..+------
│ │ │ │ +0004b210: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ +0004b220: 200a 2020 2020 2020 2020 6772 6170 6820   .        graph 
│ │ │ │ +0004b230: 7068 690a 2020 2a20 496e 7075 7473 3a0a  phi.  * Inputs:.
│ │ │ │ +0004b240: 2020 2020 2020 2a20 7068 692c 2061 202a        * phi, a *
│ │ │ │ +0004b250: 6e6f 7465 2072 6174 696f 6e61 6c20 6d61  note rational ma
│ │ │ │ +0004b260: 703a 2052 6174 696f 6e61 6c4d 6170 2c0a  p: RationalMap,.
│ │ │ │ +0004b270: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e    * *note Option
│ │ │ │ +0004b280: 616c 2069 6e70 7574 733a 2028 4d61 6361  al inputs: (Maca
│ │ │ │ +0004b290: 756c 6179 3244 6f63 2975 7369 6e67 2066  ulay2Doc)using f
│ │ │ │ +0004b2a0: 756e 6374 696f 6e73 2077 6974 6820 6f70  unctions with op
│ │ │ │ +0004b2b0: 7469 6f6e 616c 2069 6e70 7574 732c 3a0a  tional inputs,:.
│ │ │ │ +0004b2c0: 2020 2020 2020 2a20 2a6e 6f74 6520 426c        * *note Bl
│ │ │ │ +0004b2d0: 6f77 5570 5374 7261 7465 6779 3a20 426c  owUpStrategy: Bl
│ │ │ │ +0004b2e0: 6f77 5570 5374 7261 7465 6779 2c20 3d3e  owUpStrategy, =>
│ │ │ │ +0004b2f0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ +0004b300: 6c75 650a 2020 2020 2020 2020 2245 6c69  lue.        "Eli
│ │ │ │ +0004b310: 6d69 6e61 7465 222c 0a20 202a 204f 7574  minate",.  * Out
│ │ │ │ +0004b320: 7075 7473 3a0a 2020 2020 2020 2a20 6120  puts:.      * a 
│ │ │ │ +0004b330: 2a6e 6f74 6520 7261 7469 6f6e 616c 206d  *note rational m
│ │ │ │ +0004b340: 6170 3a20 5261 7469 6f6e 616c 4d61 702c  ap: RationalMap,
│ │ │ │ +0004b350: 2c20 7468 6520 6669 7273 7420 7072 6f6a  , the first proj
│ │ │ │ +0004b360: 6563 7469 6f6e 0a20 2020 2020 202a 2061  ection.      * a
│ │ │ │ +0004b370: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ +0004b380: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ +0004b390: 2c2c 2074 6865 2073 6563 6f6e 6420 7072  ,, the second pr
│ │ │ │ +0004b3a0: 6f6a 6563 7469 6f6e 0a0a 4465 7363 7269  ojection..Descri
│ │ │ │ +0004b3b0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +0004b3c0: 3d0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  =..+------------
│ │ │ │  0004b3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004b3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004b400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004b410: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2028  -------+.|i1 : (
│ │ │ │ -0004b420: 5a5a 2f31 3930 3138 3129 5b78 5f30 2e2e  ZZ/190181)[x_0..
│ │ │ │ -0004b430: 785f 345d 3b20 7068 6920 3d20 7261 7469  x_4]; phi = rati
│ │ │ │ -0004b440: 6f6e 616c 4d61 7028 6d69 6e6f 7273 2832  onalMap(minors(2
│ │ │ │ -0004b450: 2c6d 6174 7269 787b 7b78 5f30 2e2e 785f  ,matrix{{x_0..x_
│ │ │ │ -0004b460: 337d 2c7b 785f 207c 0a7c 2020 2020 2020  3},{x_ |.|      
│ │ │ │ +0004b410: 2d2b 0a7c 6931 203a 2028 5a5a 2f31 3930  -+.|i1 : (ZZ/190
│ │ │ │ +0004b420: 3138 3129 5b78 5f30 2e2e 785f 345d 3b20  181)[x_0..x_4]; 
│ │ │ │ +0004b430: 7068 6920 3d20 7261 7469 6f6e 616c 4d61  phi = rationalMa
│ │ │ │ +0004b440: 7028 6d69 6e6f 7273 2832 2c6d 6174 7269  p(minors(2,matri
│ │ │ │ +0004b450: 787b 7b78 5f30 2e2e 785f 337d 2c7b 785f  x{{x_0..x_3},{x_
│ │ │ │ +0004b460: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b4b0: 2020 2020 2020 207c 0a7c 6f32 203d 202d         |.|o2 = -
│ │ │ │ -0004b4c0: 2d20 7261 7469 6f6e 616c 206d 6170 202d  - rational map -
│ │ │ │ -0004b4d0: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │ +0004b4b0: 207c 0a7c 6f32 203d 202d 2d20 7261 7469   |.|o2 = -- rati
│ │ │ │ +0004b4c0: 6f6e 616c 206d 6170 202d 2d20 2020 2020  onal map --     
│ │ │ │ +0004b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b500: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004b510: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │ +0004b500: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004b510: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │  0004b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b550: 2020 2020 2020 207c 0a7c 2020 2020 2073         |.|     s
│ │ │ │ -0004b560: 6f75 7263 653a 2050 726f 6a28 2d2d 2d2d  ource: Proj(----
│ │ │ │ -0004b570: 2d2d 5b78 202c 2078 202c 2078 202c 2078  --[x , x , x , x
│ │ │ │ -0004b580: 202c 2078 205d 2920 2020 2020 2020 2020   , x ])         
│ │ │ │ +0004b550: 207c 0a7c 2020 2020 2073 6f75 7263 653a   |.|     source:
│ │ │ │ +0004b560: 2050 726f 6a28 2d2d 2d2d 2d2d 5b78 202c   Proj(------[x ,
│ │ │ │ +0004b570: 2078 202c 2078 202c 2078 202c 2078 205d   x , x , x , x ]
│ │ │ │ +0004b580: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  0004b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b5a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004b5b0: 2020 2020 2020 2020 2020 2020 3139 3031              1901
│ │ │ │ -0004b5c0: 3831 2020 3020 2020 3120 2020 3220 2020  81  0   1   2   
│ │ │ │ -0004b5d0: 3320 2020 3420 2020 2020 2020 2020 2020  3   4           
│ │ │ │ +0004b5a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004b5b0: 2020 2020 2020 3139 3031 3831 2020 3020        190181  0 
│ │ │ │ +0004b5c0: 2020 3120 2020 3220 2020 3320 2020 3420    1   2   3   4 
│ │ │ │ +0004b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b5f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0004b5f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b610: 2020 2020 2020 2020 2020 2020 5a5a 2020              ZZ  
│ │ │ │ +0004b610: 2020 2020 2020 5a5a 2020 2020 2020 2020        ZZ        
│ │ │ │  0004b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b640: 2020 2020 2020 207c 0a7c 2020 2020 2074         |.|     t
│ │ │ │ -0004b650: 6172 6765 743a 2073 7562 7661 7269 6574  arget: subvariet
│ │ │ │ -0004b660: 7920 6f66 2050 726f 6a28 2d2d 2d2d 2d2d  y of Proj(------
│ │ │ │ -0004b670: 5b79 202c 2079 202c 2079 202c 2079 202c  [y , y , y , y ,
│ │ │ │ -0004b680: 2079 202c 2079 205d 2920 6465 6669 6e65   y , y ]) define
│ │ │ │ -0004b690: 6420 6279 2020 207c 0a7c 2020 2020 2020  d by   |.|      
│ │ │ │ +0004b640: 207c 0a7c 2020 2020 2074 6172 6765 743a   |.|     target:
│ │ │ │ +0004b650: 2073 7562 7661 7269 6574 7920 6f66 2050   subvariety of P
│ │ │ │ +0004b660: 726f 6a28 2d2d 2d2d 2d2d 5b79 202c 2079  roj(------[y , y
│ │ │ │ +0004b670: 202c 2079 202c 2079 202c 2079 202c 2079   , y , y , y , y
│ │ │ │ +0004b680: 205d 2920 6465 6669 6e65 6420 6279 2020   ]) defined by  
│ │ │ │ +0004b690: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004b6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b6b0: 2020 2020 2020 2020 2020 3139 3031 3831            190181
│ │ │ │ -0004b6c0: 2020 3020 2020 3120 2020 3220 2020 3320    0   1   2   3 
│ │ │ │ -0004b6d0: 2020 3420 2020 3520 2020 2020 2020 2020    4   5         
│ │ │ │ -0004b6e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004b6f0: 2020 2020 2020 207b 2020 2020 2020 2020         {        
│ │ │ │ +0004b6b0: 2020 2020 3139 3031 3831 2020 3020 2020      190181  0   
│ │ │ │ +0004b6c0: 3120 2020 3220 2020 3320 2020 3420 2020  1   2   3   4   
│ │ │ │ +0004b6d0: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │ +0004b6e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004b6f0: 207b 2020 2020 2020 2020 2020 2020 2020   {              
│ │ │ │  0004b700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b730: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004b740: 2020 2020 2020 2020 7920 7920 202d 2079          y y  - y
│ │ │ │ -0004b750: 2079 2020 2b20 7920 7920 2020 2020 2020   y  + y y       
│ │ │ │ +0004b730: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004b740: 2020 7920 7920 202d 2079 2079 2020 2b20    y y  - y y  + 
│ │ │ │ +0004b750: 7920 7920 2020 2020 2020 2020 2020 2020  y y             
│ │ │ │  0004b760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b780: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004b790: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
│ │ │ │ -0004b7a0: 3120 3420 2020 2030 2035 2020 2020 2020  1 4    0 5      
│ │ │ │ +0004b780: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004b790: 2020 2032 2033 2020 2020 3120 3420 2020     2 3    1 4   
│ │ │ │ +0004b7a0: 2030 2035 2020 2020 2020 2020 2020 2020   0 5            
│ │ │ │  0004b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b7d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004b7e0: 2020 2020 2020 207d 2020 2020 2020 2020         }        
│ │ │ │ +0004b7d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004b7e0: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │  0004b7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b820: 2020 2020 2020 207c 0a7c 2020 2020 2064         |.|     d
│ │ │ │ -0004b830: 6566 696e 696e 6720 666f 726d 733a 207b  efining forms: {
│ │ │ │ +0004b820: 207c 0a7c 2020 2020 2064 6566 696e 696e   |.|     definin
│ │ │ │ +0004b830: 6720 666f 726d 733a 207b 2020 2020 2020  g forms: {      
│ │ │ │  0004b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b870: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004b880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b890: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0004b870: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004b880: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +0004b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b8c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b8e0: 2d20 7820 202b 2078 2078 202c 2020 2020  - x  + x x ,    
│ │ │ │ +0004b8c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004b8d0: 2020 2020 2020 2020 2020 2d20 7820 202b            - x  +
│ │ │ │ +0004b8e0: 2078 2078 202c 2020 2020 2020 2020 2020   x x ,          
│ │ │ │  0004b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b910: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b930: 2020 2031 2020 2020 3020 3220 2020 2020     1    0 2     
│ │ │ │ +0004b910: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004b920: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +0004b930: 2020 3020 3220 2020 2020 2020 2020 2020    0 2           
│ │ │ │  0004b940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b960: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0004b960: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004b970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b9b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b9d0: 2d20 7820 7820 202b 2078 2078 202c 2020  - x x  + x x ,  
│ │ │ │ +0004b9b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004b9c0: 2020 2020 2020 2020 2020 2d20 7820 7820            - x x 
│ │ │ │ +0004b9d0: 202b 2078 2078 202c 2020 2020 2020 2020   + x x ,        
│ │ │ │  0004b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ba00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ba20: 2020 2031 2032 2020 2020 3020 3320 2020     1 2    0 3   
│ │ │ │ +0004ba00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004ba10: 2020 2020 2020 2020 2020 2020 2031 2032               1 2
│ │ │ │ +0004ba20: 2020 2020 3020 3320 2020 2020 2020 2020      0 3         
│ │ │ │  0004ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ba50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0004ba50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004baa0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004bab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bac0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0004baa0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004bab0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +0004bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004baf0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004bb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bb10: 2d20 7820 202b 2078 2078 202c 2020 2020  - x  + x x ,    
│ │ │ │ +0004baf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004bb00: 2020 2020 2020 2020 2020 2d20 7820 202b            - x  +
│ │ │ │ +0004bb10: 2078 2078 202c 2020 2020 2020 2020 2020   x x ,          
│ │ │ │  0004bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bb40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bb60: 2020 2032 2020 2020 3120 3320 2020 2020     2    1 3     
│ │ │ │ +0004bb40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004bb50: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +0004bb60: 2020 3120 3320 2020 2020 2020 2020 2020    1 3           
│ │ │ │  0004bb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bb90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0004bb90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bbe0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bc00: 2d20 7820 7820 202b 2078 2078 202c 2020  - x x  + x x ,  
│ │ │ │ +0004bbe0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004bbf0: 2020 2020 2020 2020 2020 2d20 7820 7820            - x x 
│ │ │ │ +0004bc00: 202b 2078 2078 202c 2020 2020 2020 2020   + x x ,        
│ │ │ │  0004bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bc30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004bc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bc50: 2020 2031 2033 2020 2020 3020 3420 2020     1 3    0 4   
│ │ │ │ +0004bc30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004bc40: 2020 2020 2020 2020 2020 2020 2031 2033               1 3
│ │ │ │ +0004bc50: 2020 2020 3020 3420 2020 2020 2020 2020      0 4         
│ │ │ │  0004bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bc80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0004bc80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004bc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bcd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004bce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bcf0: 2d20 7820 7820 202b 2078 2078 202c 2020  - x x  + x x ,  
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│ │ │ │ -0004cc70: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0004cc20: 207c 0a7c 202d 2d20 7573 6564 2030 2e30   |.| -- used 0.0
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│ │ │ │ +0004cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004cc70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
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│ │ │ │  0004cca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004ccb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004ccc0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2070  -------+.|i4 : p
│ │ │ │ -0004ccd0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +0004ccc0: 2d2b 0a7c 6934 203a 2070 3120 2020 2020  -+.|i4 : p1     
│ │ │ │ +0004ccd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cd10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
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│ │ │ │  0004cd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cd60: 2020 2020 2020 207c 0a7c 6f34 203d 202d         |.|o4 = -
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│ │ │ │ -0004ce70: 2020 2020 2020 2020 2020 3139 3031 3831            190181
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│ │ │ │  0004cee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004cf00: 2020 7920 7920 202d 2079 2079 2020 2b20    y y  - y y  + 
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│ │ │ │  0004cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cf40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
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│ │ │ │ -0004cf60: 3120 3420 2020 2030 2035 2020 2020 2020  1 4    0 5      
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│ │ │ │ +0004cf50: 2020 2032 2033 2020 2020 3120 3420 2020     2 3    1 4   
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│ │ │ │  0004cf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cf90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0004cf90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004cfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cfe0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004cff0: 2020 2020 2020 2020 7820 7920 202d 2078          x y  - x
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│ │ │ │ +0004cff0: 2020 7820 7920 202d 2078 2079 2020 2b20    x y  - x y  + 
│ │ │ │ +0004d000: 7820 7920 2c20 2020 2020 2020 2020 2020  x y ,           
│ │ │ │  0004d010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d030: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004d040: 2020 2020 2020 2020 2034 2032 2020 2020           4 2    
│ │ │ │ -0004d050: 3320 3420 2020 2032 2035 2020 2020 2020  3 4    2 5      
│ │ │ │ +0004d030: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │  0004d060: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004d090: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004d0d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004d0e0: 2020 2020 2020 2020 7820 7920 202d 2078          x y  - x
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│ │ │ │  0004d100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d120: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004d130: 2020 2020 2020 2020 2033 2032 2020 2020           3 2    
│ │ │ │ -0004d140: 3220 3420 2020 2031 2035 2020 2020 2020  2 4    1 5      
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│ │ │ │  0004d150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d170: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
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│ │ │ │  0004d180: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004d1c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004d1d0: 2020 2020 2020 2020 7820 7920 202d 2078          x y  - x
│ │ │ │ -0004d1e0: 2079 2020 2b20 7820 7920 2c20 2020 2020   y  + x y ,     
│ │ │ │ +0004d1c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004d1d0: 2020 7820 7920 202d 2078 2079 2020 2b20    x y  - x y  + 
│ │ │ │ +0004d1e0: 7820 7920 2c20 2020 2020 2020 2020 2020  x y ,           
│ │ │ │  0004d1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d210: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004d220: 2020 2020 2020 2020 2034 2031 2020 2020           4 1    
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│ │ │ │ +0004d210: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004d220: 2020 2034 2031 2020 2020 3320 3320 2020     4 1    3 3   
│ │ │ │ +0004d230: 2031 2035 2020 2020 2020 2020 2020 2020   1 5            
│ │ │ │  0004d240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d260: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
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│ │ │ │ -0004d2b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004d2c0: 2020 2020 2020 2020 7820 7920 202d 2078          x y  - x
│ │ │ │ -0004d2d0: 2079 2020 2b20 7820 7920 2c20 2020 2020   y  + x y ,     
│ │ │ │ +0004d2b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004d2c0: 2020 7820 7920 202d 2078 2079 2020 2b20    x y  - x y  + 
│ │ │ │ +0004d2d0: 7820 7920 2c20 2020 2020 2020 2020 2020  x y ,           
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│ │ │ │  0004d2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004d3b0: 2020 2020 2020 2020 7820 7920 202d 2078          x y  - x
│ │ │ │ -0004d3c0: 2079 2020 2d20 7820 7920 202b 2078 2079   y  - x y  + x y
│ │ │ │ -0004d3d0: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
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│ │ │ │ -0004d400: 2020 2020 2020 2020 2032 2031 2020 2020           2 1    
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│ │ │ │  0004d480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d490: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004d4a0: 2020 2020 2020 2020 7820 7920 202d 2078          x y  - x
│ │ │ │ -0004d4b0: 2079 2020 2b20 7820 7920 2c20 2020 2020   y  + x y ,     
│ │ │ │ +0004d490: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004d4a0: 2020 7820 7920 202d 2078 2079 2020 2b20    x y  - x y  + 
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│ │ │ │  0004d4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d4e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004d4f0: 2020 2020 2020 2020 2034 2030 2020 2020           4 0    
│ │ │ │ -0004d500: 3220 3320 2020 2031 2034 2020 2020 2020  2 3    1 4      
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│ │ │ │ +0004d4f0: 2020 2034 2030 2020 2020 3220 3320 2020     4 0    2 3   
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│ │ │ │ -0004d580: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004d590: 2020 2020 2020 2020 7820 7920 202d 2078          x y  - x
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│ │ │ │ +0004d580: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004d590: 2020 7820 7920 202d 2078 2079 2020 2b20    x y  - x y  + 
│ │ │ │ +0004d5a0: 7820 7920 2c20 2020 2020 2020 2020 2020  x y ,           
│ │ │ │  0004d5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d5d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004d5e0: 2020 2020 2020 2020 2033 2030 2020 2020           3 0    
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│ │ │ │ +0004d5e0: 2020 2033 2030 2020 2020 3120 3320 2020     3 0    1 3   
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│ │ │ │  0004d600: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004d620: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
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│ │ │ │  0004d640: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004d660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d670: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004d680: 2020 2020 2020 2020 7820 7920 202d 2078          x y  - x
│ │ │ │ -0004d690: 2079 2020 2b20 7820 7920 2020 2020 2020   y  + x y       
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│ │ │ │ +0004d680: 2020 7820 7920 202d 2078 2079 2020 2b20    x y  - x y  + 
│ │ │ │ +0004d690: 7820 7920 2020 2020 2020 2020 2020 2020  x y             
│ │ │ │  0004d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d6c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004d6d0: 2020 2020 2020 2020 2032 2030 2020 2020           2 0    
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│ │ │ │ +0004d6d0: 2020 2032 2030 2020 2020 3120 3120 2020     2 0    1 1   
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│ │ │ │  0004d6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004d730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d760: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004d770: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
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│ │ │ │  0004d790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d7b0: 2020 2020 2020 207c 0a7c 2020 2020 2074         |.|     t
│ │ │ │ -0004d7c0: 6172 6765 743a 2050 726f 6a28 2d2d 2d2d  arget: Proj(----
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│ │ │ │ +0004d7b0: 207c 0a7c 2020 2020 2074 6172 6765 743a   |.|     target:
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│ │ │ │ -0004d810: 2020 2020 2020 2020 2020 2020 3139 3031              1901
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│ │ │ │ -0004de50: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │ +0004de40: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
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│ │ │ │  0004de60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004de90: 2020 2020 2020 207c 0a7c 205d 2920 7820         |.| ]) x 
│ │ │ │ -0004dea0: 5072 6f6a 282d 2d2d 2d2d 2d5b 7920 2c20  Proj(------[y , 
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│ │ │ │ +0004de90: 207c 0a7c 205d 2920 7820 5072 6f6a 282d   |.| ]) x Proj(-
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│ │ │ │  0004ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004e250: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004e2a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0004e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e2f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0004e2f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ -0004e340: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0004e340: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │  0004e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e390: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0004e390: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004e3e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │  0004e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e430: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0004e430: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004e480: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0004e480: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004e4d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ -0004e520: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0004e520: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │  0004e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004ee50: 206f 6620 5050 5e34 2078 2050 505e 3520   of PP^4 x PP^5 
│ │ │ │ -0004ee60: 746f 2050 505e 3429 2020 2020 2020 2020  to PP^4)        
│ │ │ │ +0004ee30: 207c 0a7c 6469 6d65 6e73 696f 6e61 6c20   |.|dimensional 
│ │ │ │ +0004ee40: 7375 6276 6172 6965 7479 206f 6620 5050  subvariety of PP
│ │ │ │ +0004ee50: 5e34 2078 2050 505e 3520 746f 2050 505e  ^4 x PP^5 to PP^
│ │ │ │ +0004ee60: 3429 2020 2020 2020 2020 2020 2020 2020  4)              
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│ │ │ │  0004eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0004eee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0004ef20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004ef30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ef40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ef70: 2020 2020 2020 207c 0a7c 6f35 203d 202d         |.|o5 = -
│ │ │ │ -0004ef80: 2d20 7261 7469 6f6e 616c 206d 6170 202d  - rational map -
│ │ │ │ -0004ef90: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │ +0004ef70: 207c 0a7c 6f35 203d 202d 2d20 7261 7469   |.|o5 = -- rati
│ │ │ │ +0004ef80: 6f6e 616c 206d 6170 202d 2d20 2020 2020  onal map --     
│ │ │ │ +0004ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004efc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0004efc0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004efe0: 2020 2020 2020 2020 2020 2020 5a5a 2020              ZZ  
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│ │ │ │  0004eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f010: 2020 2020 2020 207c 0a7c 2020 2020 2073         |.|     s
│ │ │ │ -0004f020: 6f75 7263 653a 2073 7562 7661 7269 6574  ource: subvariet
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│ │ │ │ +0004f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004f080: 2020 2020 2020 2020 2020 3139 3031 3831            190181
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│ │ │ │  0004f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f0b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004f0c0: 2020 2020 2020 207b 2020 2020 2020 2020         {        
│ │ │ │ +0004f0b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004f0c0: 207b 2020 2020 2020 2020 2020 2020 2020   {              
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│ │ │ │  0004f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f100: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004f110: 2020 2020 2020 2020 7920 7920 202d 2079          y y  - y
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│ │ │ │ +0004f120: 7920 7920 2c20 2020 2020 2020 2020 2020  y y ,           
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│ │ │ │  0004f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f150: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004f160: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
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│ │ │ │ +0004f150: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004f160: 2020 2032 2033 2020 2020 3120 3420 2020     2 3    1 4   
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│ │ │ │  0004f190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f1a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0004f1a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004f1f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004f200: 2020 2020 2020 2020 7820 7920 202d 2078          x y  - x
│ │ │ │ -0004f210: 2079 2020 2b20 7820 7920 2c20 2020 2020   y  + x y ,     
│ │ │ │ +0004f1f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004f200: 2020 7820 7920 202d 2078 2079 2020 2b20    x y  - x y  + 
│ │ │ │ +0004f210: 7820 7920 2c20 2020 2020 2020 2020 2020  x y ,           
│ │ │ │  0004f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f240: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004f250: 2020 2020 2020 2020 2034 2032 2020 2020           4 2    
│ │ │ │ -0004f260: 3320 3420 2020 2032 2035 2020 2020 2020  3 4    2 5      
│ │ │ │ +0004f240: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004f250: 2020 2034 2032 2020 2020 3320 3420 2020     4 2    3 4   
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│ │ │ │ -0004f2e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004f2f0: 2020 2020 2020 2020 7820 7920 202d 2078          x y  - x
│ │ │ │ -0004f300: 2079 2020 2b20 7820 7920 2c20 2020 2020   y  + x y ,     
│ │ │ │ +0004f2e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004f2f0: 2020 7820 7920 202d 2078 2079 2020 2b20    x y  - x y  + 
│ │ │ │ +0004f300: 7820 7920 2c20 2020 2020 2020 2020 2020  x y ,           
│ │ │ │  0004f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f330: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004f340: 2020 2020 2020 2020 2033 2032 2020 2020           3 2    
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│ │ │ │ +0004f330: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004f340: 2020 2033 2032 2020 2020 3220 3420 2020     3 2    2 4   
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│ │ │ │  0004f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f380: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0004f380: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ -0004f3d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004f3e0: 2020 2020 2020 2020 7820 7920 202d 2078          x y  - x
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│ │ │ │ +0004f3d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004f3e0: 2020 7820 7920 202d 2078 2079 2020 2b20    x y  - x y  + 
│ │ │ │ +0004f3f0: 7820 7920 2c20 2020 2020 2020 2020 2020  x y ,           
│ │ │ │  0004f400: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004f420: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004f430: 2020 2020 2020 2020 2034 2031 2020 2020           4 1    
│ │ │ │ -0004f440: 3320 3320 2020 2031 2035 2020 2020 2020  3 3    1 5      
│ │ │ │ +0004f420: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0004f430: 2020 2034 2031 2020 2020 3320 3320 2020     4 1    3 3   
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│ │ │ │  0004f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f470: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0004f470: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0004f4c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0004f4d0: 2020 2020 2020 2020 7820 7920 202d 2078          x y  - x
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│ │ │ │ +0004f4c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │  00051290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000512a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000512b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000512c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000512c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000512d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000512e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000512f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051310: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00051310: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00051320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051360: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00051360: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00051370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000513a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000513b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000513b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000513c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000513d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000513e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000513f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051400: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00051400: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00051410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051450: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00051450: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00051460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000514a0: 2020 2020 2020 207c 0a7c 7261 7469 6f6e         |.|ration
│ │ │ │ -000514b0: 616c 206d 6170 2066 726f 6d20 342d 6469  al map from 4-di
│ │ │ │ -000514c0: 6d65 6e73 696f 6e61 6c20 7375 6276 6172  mensional subvar
│ │ │ │ -000514d0: 6965 7479 206f 6620 5050 5e34 2078 2050  iety of PP^4 x P
│ │ │ │ -000514e0: 505e 3520 746f 2068 7970 6572 7375 7266  P^5 to hypersurf
│ │ │ │ -000514f0: 6163 6520 696e 207c 0a7c 2d2d 2d2d 2d2d  ace in |.|------
│ │ │ │ +000514a0: 207c 0a7c 7261 7469 6f6e 616c 206d 6170   |.|rational map
│ │ │ │ +000514b0: 2066 726f 6d20 342d 6469 6d65 6e73 696f   from 4-dimensio
│ │ │ │ +000514c0: 6e61 6c20 7375 6276 6172 6965 7479 206f  nal subvariety o
│ │ │ │ +000514d0: 6620 5050 5e34 2078 2050 505e 3520 746f  f PP^4 x PP^5 to
│ │ │ │ +000514e0: 2068 7970 6572 7375 7266 6163 6520 696e   hypersurface in
│ │ │ │ +000514f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00051500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051540: 2d2d 2d2d 2d2d 2d7c 0a7c 5050 5e35 2920  -------|.|PP^5) 
│ │ │ │ +00051540: 2d7c 0a7c 5050 5e35 2920 2020 2020 2020  -|.|PP^5)       
│ │ │ │  00051550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051590: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00051590: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  000515a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000515b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000515c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000515d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000515e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2061  -------+.|i6 : a
│ │ │ │ -000515f0: 7373 6572 7428 7031 202a 2070 6869 203d  ssert(p1 * phi =
│ │ │ │ -00051600: 3d20 7032 2061 6e64 2070 3220 2a20 7068  = p2 and p2 * ph
│ │ │ │ -00051610: 695e 2d31 203d 3d20 7031 2920 2020 2020  i^-1 == p1)     
│ │ │ │ +000515e0: 2d2b 0a7c 6936 203a 2061 7373 6572 7428  -+.|i6 : assert(
│ │ │ │ +000515f0: 7031 202a 2070 6869 203d 3d20 7032 2061  p1 * phi == p2 a
│ │ │ │ +00051600: 6e64 2070 3220 2a20 7068 695e 2d31 203d  nd p2 * phi^-1 =
│ │ │ │ +00051610: 3d20 7031 2920 2020 2020 2020 2020 2020  = p1)           
│ │ │ │  00051620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051630: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00051630: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00051640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051680: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2064  -------+.|i7 : d
│ │ │ │ -00051690: 6573 6372 6962 6520 7032 2020 2020 2020  escribe p2      
│ │ │ │ +00051680: 2d2b 0a7c 6937 203a 2064 6573 6372 6962  -+.|i7 : describ
│ │ │ │ +00051690: 6520 7032 2020 2020 2020 2020 2020 2020  e p2            
│ │ │ │  000516a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000516b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000516c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000516d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000516d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000516e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000516f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051720: 2020 2020 2020 207c 0a7c 6f37 203d 2072         |.|o7 = r
│ │ │ │ -00051730: 6174 696f 6e61 6c20 6d61 7020 6465 6669  ational map defi
│ │ │ │ -00051740: 6e65 6420 6279 206d 756c 7469 666f 726d  ned by multiform
│ │ │ │ -00051750: 7320 6f66 2064 6567 7265 6520 7b30 2c20  s of degree {0, 
│ │ │ │ -00051760: 317d 2020 2020 2020 2020 2020 2020 2020  1}              
│ │ │ │ -00051770: 2020 2020 2020 207c 0a7c 2020 2020 2073         |.|     s
│ │ │ │ -00051780: 6f75 7263 6520 7661 7269 6574 793a 2034  ource variety: 4
│ │ │ │ -00051790: 2d64 696d 656e 7369 6f6e 616c 2073 7562  -dimensional sub
│ │ │ │ -000517a0: 7661 7269 6574 7920 6f66 2050 505e 3420  variety of PP^4 
│ │ │ │ -000517b0: 7820 5050 5e35 2063 7574 206f 7574 2062  x PP^5 cut out b
│ │ │ │ -000517c0: 7920 3920 2020 207c 0a7c 2020 2020 2074  y 9    |.|     t
│ │ │ │ -000517d0: 6172 6765 7420 7661 7269 6574 793a 2073  arget variety: s
│ │ │ │ -000517e0: 6d6f 6f74 6820 7175 6164 7269 6320 6879  mooth quadric hy
│ │ │ │ -000517f0: 7065 7273 7572 6661 6365 2069 6e20 5050  persurface in PP
│ │ │ │ -00051800: 5e35 2020 2020 2020 2020 2020 2020 2020  ^5              
│ │ │ │ -00051810: 2020 2020 2020 207c 0a7c 2020 2020 2064         |.|     d
│ │ │ │ -00051820: 6f6d 696e 616e 6365 3a20 7472 7565 2020  ominance: true  
│ │ │ │ +00051720: 207c 0a7c 6f37 203d 2072 6174 696f 6e61   |.|o7 = rationa
│ │ │ │ +00051730: 6c20 6d61 7020 6465 6669 6e65 6420 6279  l map defined by
│ │ │ │ +00051740: 206d 756c 7469 666f 726d 7320 6f66 2064   multiforms of d
│ │ │ │ +00051750: 6567 7265 6520 7b30 2c20 317d 2020 2020  egree {0, 1}    
│ │ │ │ +00051760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051770: 207c 0a7c 2020 2020 2073 6f75 7263 6520   |.|     source 
│ │ │ │ +00051780: 7661 7269 6574 793a 2034 2d64 696d 656e  variety: 4-dimen
│ │ │ │ +00051790: 7369 6f6e 616c 2073 7562 7661 7269 6574  sional subvariet
│ │ │ │ +000517a0: 7920 6f66 2050 505e 3420 7820 5050 5e35  y of PP^4 x PP^5
│ │ │ │ +000517b0: 2063 7574 206f 7574 2062 7920 3920 2020   cut out by 9   
│ │ │ │ +000517c0: 207c 0a7c 2020 2020 2074 6172 6765 7420   |.|     target 
│ │ │ │ +000517d0: 7661 7269 6574 793a 2073 6d6f 6f74 6820  variety: smooth 
│ │ │ │ +000517e0: 7175 6164 7269 6320 6879 7065 7273 7572  quadric hypersur
│ │ │ │ +000517f0: 6661 6365 2069 6e20 5050 5e35 2020 2020  face in PP^5    
│ │ │ │ +00051800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051810: 207c 0a7c 2020 2020 2064 6f6d 696e 616e   |.|     dominan
│ │ │ │ +00051820: 6365 3a20 7472 7565 2020 2020 2020 2020  ce: true        
│ │ │ │  00051830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051860: 2020 2020 2020 207c 0a7c 2020 2020 2063         |.|     c
│ │ │ │ -00051870: 6f65 6666 6963 6965 6e74 2072 696e 673a  oefficient ring:
│ │ │ │ -00051880: 205a 5a2f 3139 3031 3831 2020 2020 2020   ZZ/190181      
│ │ │ │ +00051860: 207c 0a7c 2020 2020 2063 6f65 6666 6963   |.|     coeffic
│ │ │ │ +00051870: 6965 6e74 2072 696e 673a 205a 5a2f 3139  ient ring: ZZ/19
│ │ │ │ +00051880: 3031 3831 2020 2020 2020 2020 2020 2020  0181            
│ │ │ │  00051890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000518a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000518b0: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +000518b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000518c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000518d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000518e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000518f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051900: 2d2d 2d2d 2d2d 2d7c 0a7c 6879 7065 7273  -------|.|hypers
│ │ │ │ -00051910: 7572 6661 6365 7320 6f66 2064 6567 7265  urfaces of degre
│ │ │ │ -00051920: 6573 2028 7b30 2c20 327d 2c7b 312c 2031  es ({0, 2},{1, 1
│ │ │ │ -00051930: 7d2c 7b31 2c20 317d 2c7b 312c 2031 7d2c  },{1, 1},{1, 1},
│ │ │ │ -00051940: 7b31 2c20 317d 2c7b 312c 2031 7d2c 7b31  {1, 1},{1, 1},{1
│ │ │ │ -00051950: 2c20 2020 2020 207c 0a7c 2d2d 2d2d 2d2d  ,      |.|------
│ │ │ │ +00051900: 2d7c 0a7c 6879 7065 7273 7572 6661 6365  -|.|hypersurface
│ │ │ │ +00051910: 7320 6f66 2064 6567 7265 6573 2028 7b30  s of degrees ({0
│ │ │ │ +00051920: 2c20 327d 2c7b 312c 2031 7d2c 7b31 2c20  , 2},{1, 1},{1, 
│ │ │ │ +00051930: 317d 2c7b 312c 2031 7d2c 7b31 2c20 317d  1},{1, 1},{1, 1}
│ │ │ │ +00051940: 2c7b 312c 2031 7d2c 7b31 2c20 2020 2020  ,{1, 1},{1,     
│ │ │ │ +00051950: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00051960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000519a0: 2d2d 2d2d 2d2d 2d7c 0a7c 317d 2c7b 312c  -------|.|1},{1,
│ │ │ │ -000519b0: 2031 7d2c 7b31 2c20 317d 2920 2020 2020   1},{1, 1})     
│ │ │ │ +000519a0: 2d7c 0a7c 317d 2c7b 312c 2031 7d2c 7b31  -|.|1},{1, 1},{1
│ │ │ │ +000519b0: 2c20 317d 2920 2020 2020 2020 2020 2020  , 1})           
│ │ │ │  000519c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000519d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000519e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000519f0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000519f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00051a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051a40: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2070  -------+.|i8 : p
│ │ │ │ -00051a50: 726f 6a65 6374 6976 6544 6567 7265 6573  rojectiveDegrees
│ │ │ │ -00051a60: 2070 3220 2020 2020 2020 2020 2020 2020   p2             
│ │ │ │ +00051a40: 2d2b 0a7c 6938 203a 2070 726f 6a65 6374  -+.|i8 : project
│ │ │ │ +00051a50: 6976 6544 6567 7265 6573 2070 3220 2020  iveDegrees p2   
│ │ │ │ +00051a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051a90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00051a90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00051aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051ae0: 2020 2020 2020 207c 0a7c 6f38 203d 207b         |.|o8 = {
│ │ │ │ -00051af0: 3531 2c20 3238 2c20 3134 2c20 362c 2032  51, 28, 14, 6, 2
│ │ │ │ -00051b00: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00051ae0: 207c 0a7c 6f38 203d 207b 3531 2c20 3238   |.|o8 = {51, 28
│ │ │ │ +00051af0: 2c20 3134 2c20 362c 2032 7d20 2020 2020  , 14, 6, 2}     
│ │ │ │ +00051b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051b30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00051b30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00051b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051b80: 2020 2020 2020 207c 0a7c 6f38 203a 204c         |.|o8 : L
│ │ │ │ -00051b90: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +00051b80: 207c 0a7c 6f38 203a 204c 6973 7420 2020   |.|o8 : List   
│ │ │ │ +00051b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051bd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00051bd0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00051be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051c20: 2d2d 2d2d 2d2d 2d2b 0a0a 5768 656e 2074  -------+..When t
│ │ │ │ -00051c30: 6865 2073 6f75 7263 6520 6f66 2074 6865  he source of the
│ │ │ │ -00051c40: 2072 6174 696f 6e61 6c20 6d61 7020 6973   rational map is
│ │ │ │ -00051c50: 2061 206d 756c 7469 2d70 726f 6a65 6374   a multi-project
│ │ │ │ -00051c60: 6976 6520 7661 7269 6574 792c 2074 6865  ive variety, the
│ │ │ │ -00051c70: 206d 6574 686f 640a 7265 7475 726e 7320   method.returns 
│ │ │ │ -00051c80: 616c 6c20 7468 6520 7072 6f6a 6563 7469  all the projecti
│ │ │ │ -00051c90: 6f6e 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ons...+---------
│ │ │ │ +00051c20: 2d2b 0a0a 5768 656e 2074 6865 2073 6f75  -+..When the sou
│ │ │ │ +00051c30: 7263 6520 6f66 2074 6865 2072 6174 696f  rce of the ratio
│ │ │ │ +00051c40: 6e61 6c20 6d61 7020 6973 2061 206d 756c  nal map is a mul
│ │ │ │ +00051c50: 7469 2d70 726f 6a65 6374 6976 6520 7661  ti-projective va
│ │ │ │ +00051c60: 7269 6574 792c 2074 6865 206d 6574 686f  riety, the metho
│ │ │ │ +00051c70: 640a 7265 7475 726e 7320 616c 6c20 7468  d.returns all th
│ │ │ │ +00051c80: 6520 7072 6f6a 6563 7469 6f6e 732e 0a0a  e projections...
│ │ │ │ +00051c90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00051ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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;  
│ │ │ │ +00051cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00051ce0: 7c69 3920 3a20 7469 6d65 2067 203d 2067  |i9 : time g = g
│ │ │ │ +00051cf0: 7261 7068 2070 323b 2020 2020 2020 2020  raph 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 3033 3137 3136 3573 2028 6370 7529  0.0317165s (cpu)
│ │ │ │ -00051d50: 3b20 302e 3033 3132 3632 3473 2028 7468  ; 0.0312624s (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      |.+---------
│ │ │ │ +00051d20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00051d30: 7c20 2d2d 2075 7365 6420 302e 3039 3737  | -- used 0.0977
│ │ │ │ +00051d40: 3432 3373 2028 6370 7529 3b20 302e 3034  423s (cpu); 0.04
│ │ │ │ +00051d50: 3437 3830 3473 2028 7468 7265 6164 293b  47804s (thread);
│ │ │ │ +00051d60: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +00051d70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00051d80: 2b2d 2d2d 2d2d 2d2d 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  ----------------
│ │ │ │ -00051dd0: 2d2d 2d2d 2b0a 7c69 3130 203a 2067 5f30  ----+.|i10 : g_0
│ │ │ │ -00051de0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00051dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00051dd0: 7c69 3130 203a 2067 5f30 3b20 2020 2020  |i10 : g_0;     
│ │ │ │ +00051de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051e20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00051e10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00051e20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00051e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051e70: 2020 2020 7c0a 7c6f 3130 203a 204d 756c      |.|o10 : Mul
│ │ │ │ -00051e80: 7469 686f 6d6f 6765 6e65 6f75 7352 6174  tihomogeneousRat
│ │ │ │ -00051e90: 696f 6e61 6c4d 6170 2028 7261 7469 6f6e  ionalMap (ration
│ │ │ │ -00051ea0: 616c 206d 6170 2066 726f 6d20 342d 6469  al map from 4-di
│ │ │ │ -00051eb0: 6d65 6e73 696f 6e61 6c20 7375 6276 6172  mensional subvar
│ │ │ │ -00051ec0: 6965 7479 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  iety|.|---------
│ │ │ │ +00051e60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00051e70: 7c6f 3130 203a 204d 756c 7469 686f 6d6f  |o10 : Multihomo
│ │ │ │ +00051e80: 6765 6e65 6f75 7352 6174 696f 6e61 6c4d  geneousRationalM
│ │ │ │ +00051e90: 6170 2028 7261 7469 6f6e 616c 206d 6170  ap (rational map
│ │ │ │ +00051ea0: 2066 726f 6d20 342d 6469 6d65 6e73 696f   from 4-dimensio
│ │ │ │ +00051eb0: 6e61 6c20 7375 6276 6172 6965 7479 7c0a  nal subvariety|.
│ │ │ │ +00051ec0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │  00051ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051f10: 2d2d 2d2d 7c0a 7c6f 6620 5050 5e34 2078  ----|.|of PP^4 x
│ │ │ │ -00051f20: 2050 505e 3520 7820 5050 5e35 2074 6f20   PP^5 x PP^5 to 
│ │ │ │ -00051f30: 5050 5e34 2920 2020 2020 2020 2020 2020  PP^4)           
│ │ │ │ +00051f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00051f10: 7c6f 6620 5050 5e34 2078 2050 505e 3520  |of PP^4 x PP^5 
│ │ │ │ +00051f20: 7820 5050 5e35 2074 6f20 5050 5e34 2920  x PP^5 to PP^4) 
│ │ │ │ +00051f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051f60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00051f50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00051f60: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00051f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051fb0: 2d2d 2d2d 2b0a 7c69 3131 203a 2067 5f31  ----+.|i11 : g_1
│ │ │ │ -00051fc0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00051fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00051fb0: 7c69 3131 203a 2067 5f31 3b20 2020 2020  |i11 : g_1;     
│ │ │ │ +00051fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052000: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00051ff0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00052000: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00052010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052050: 2020 2020 7c0a 7c6f 3131 203a 204d 756c      |.|o11 : Mul
│ │ │ │ -00052060: 7469 686f 6d6f 6765 6e65 6f75 7352 6174  tihomogeneousRat
│ │ │ │ -00052070: 696f 6e61 6c4d 6170 2028 7261 7469 6f6e  ionalMap (ration
│ │ │ │ -00052080: 616c 206d 6170 2066 726f 6d20 342d 6469  al map from 4-di
│ │ │ │ -00052090: 6d65 6e73 696f 6e61 6c20 7375 6276 6172  mensional subvar
│ │ │ │ -000520a0: 6965 7479 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  iety|.|---------
│ │ │ │ +00052040: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00052050: 7c6f 3131 203a 204d 756c 7469 686f 6d6f  |o11 : Multihomo
│ │ │ │ +00052060: 6765 6e65 6f75 7352 6174 696f 6e61 6c4d  geneousRationalM
│ │ │ │ +00052070: 6170 2028 7261 7469 6f6e 616c 206d 6170  ap (rational map
│ │ │ │ +00052080: 2066 726f 6d20 342d 6469 6d65 6e73 696f   from 4-dimensio
│ │ │ │ +00052090: 6e61 6c20 7375 6276 6172 6965 7479 7c0a  nal subvariety|.
│ │ │ │ +000520a0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │  000520b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000520c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000520d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000520e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000520f0: 2d2d 2d2d 7c0a 7c6f 6620 5050 5e34 2078  ----|.|of PP^4 x
│ │ │ │ -00052100: 2050 505e 3520 7820 5050 5e35 2074 6f20   PP^5 x PP^5 to 
│ │ │ │ -00052110: 5050 5e35 2920 2020 2020 2020 2020 2020  PP^5)           
│ │ │ │ +000520e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +000520f0: 7c6f 6620 5050 5e34 2078 2050 505e 3520  |of PP^4 x PP^5 
│ │ │ │ +00052100: 7820 5050 5e35 2074 6f20 5050 5e35 2920  x PP^5 to PP^5) 
│ │ │ │ +00052110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052140: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00052130: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00052140: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00052150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052190: 2d2d 2d2d 2b0a 7c69 3132 203a 2067 5f32  ----+.|i12 : g_2
│ │ │ │ -000521a0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00052180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00052190: 7c69 3132 203a 2067 5f32 3b20 2020 2020  |i12 : g_2;     
│ │ │ │ +000521a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000521b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000521c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000521d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000521e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000521d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000521e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000521f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052230: 2020 2020 7c0a 7c6f 3132 203a 204d 756c      |.|o12 : Mul
│ │ │ │ -00052240: 7469 686f 6d6f 6765 6e65 6f75 7352 6174  tihomogeneousRat
│ │ │ │ -00052250: 696f 6e61 6c4d 6170 2028 646f 6d69 6e61  ionalMap (domina
│ │ │ │ -00052260: 6e74 2072 6174 696f 6e61 6c20 6d61 7020  nt rational map 
│ │ │ │ -00052270: 6672 6f6d 2034 2d64 696d 656e 7369 6f6e  from 4-dimension
│ │ │ │ -00052280: 616c 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  al  |.|---------
│ │ │ │ +00052220: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00052230: 7c6f 3132 203a 204d 756c 7469 686f 6d6f  |o12 : Multihomo
│ │ │ │ +00052240: 6765 6e65 6f75 7352 6174 696f 6e61 6c4d  geneousRationalM
│ │ │ │ +00052250: 6170 2028 646f 6d69 6e61 6e74 2072 6174  ap (dominant rat
│ │ │ │ +00052260: 696f 6e61 6c20 6d61 7020 6672 6f6d 2034  ional map from 4
│ │ │ │ +00052270: 2d64 696d 656e 7369 6f6e 616c 2020 7c0a  -dimensional  |.
│ │ │ │ +00052280: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │  00052290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000522a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000522b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000522c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000522d0: 2d2d 2d2d 7c0a 7c73 7562 7661 7269 6574  ----|.|subvariet
│ │ │ │ -000522e0: 7920 6f66 2050 505e 3420 7820 5050 5e35  y of PP^4 x PP^5
│ │ │ │ -000522f0: 2078 2050 505e 3520 746f 2068 7970 6572   x PP^5 to hyper
│ │ │ │ -00052300: 7375 7266 6163 6520 696e 2050 505e 3529  surface in PP^5)
│ │ │ │ -00052310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052320: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000522c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +000522d0: 7c73 7562 7661 7269 6574 7920 6f66 2050  |subvariety of P
│ │ │ │ +000522e0: 505e 3420 7820 5050 5e35 2078 2050 505e  P^4 x PP^5 x PP^
│ │ │ │ +000522f0: 3520 746f 2068 7970 6572 7375 7266 6163  5 to hypersurfac
│ │ │ │ +00052300: 6520 696e 2050 505e 3529 2020 2020 2020  e in PP^5)      
│ │ │ │ +00052310: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00052320: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00052330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052370: 2d2d 2d2d 2b0a 7c69 3133 203a 2064 6573  ----+.|i13 : des
│ │ │ │ -00052380: 6372 6962 6520 675f 3020 2020 2020 2020  cribe g_0       
│ │ │ │ +00052360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00052370: 7c69 3133 203a 2064 6573 6372 6962 6520  |i13 : describe 
│ │ │ │ +00052380: 675f 3020 2020 2020 2020 2020 2020 2020  g_0             
│ │ │ │  00052390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000523a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000523b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000523c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000523b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000523c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000523d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000523e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000523f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052410: 2020 2020 7c0a 7c6f 3133 203d 2072 6174      |.|o13 = rat
│ │ │ │ -00052420: 696f 6e61 6c20 6d61 7020 6465 6669 6e65  ional map define
│ │ │ │ -00052430: 6420 6279 206d 756c 7469 666f 726d 7320  d by multiforms 
│ │ │ │ -00052440: 6f66 2064 6567 7265 6520 7b31 2c20 302c  of degree {1, 0,
│ │ │ │ -00052450: 2030 7d20 2020 2020 2020 2020 2020 2020   0}             
│ │ │ │ -00052460: 2020 2020 7c0a 7c20 2020 2020 2073 6f75      |.|      sou
│ │ │ │ -00052470: 7263 6520 7661 7269 6574 793a 2034 2d64  rce variety: 4-d
│ │ │ │ -00052480: 696d 656e 7369 6f6e 616c 2073 7562 7661  imensional subva
│ │ │ │ -00052490: 7269 6574 7920 6f66 2050 505e 3420 7820  riety of PP^4 x 
│ │ │ │ -000524a0: 5050 5e35 2078 2050 505e 3520 6375 7420  PP^5 x PP^5 cut 
│ │ │ │ -000524b0: 6f75 7420 7c0a 7c20 2020 2020 2074 6172  out |.|      tar
│ │ │ │ -000524c0: 6765 7420 7661 7269 6574 793a 2050 505e  get variety: PP^
│ │ │ │ -000524d0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +00052400: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00052410: 7c6f 3133 203d 2072 6174 696f 6e61 6c20  |o13 = rational 
│ │ │ │ +00052420: 6d61 7020 6465 6669 6e65 6420 6279 206d  map defined by m
│ │ │ │ +00052430: 756c 7469 666f 726d 7320 6f66 2064 6567  ultiforms of deg
│ │ │ │ +00052440: 7265 6520 7b31 2c20 302c 2030 7d20 2020  ree {1, 0, 0}   
│ │ │ │ +00052450: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00052460: 7c20 2020 2020 2073 6f75 7263 6520 7661  |      source va
│ │ │ │ +00052470: 7269 6574 793a 2034 2d64 696d 656e 7369  riety: 4-dimensi
│ │ │ │ +00052480: 6f6e 616c 2073 7562 7661 7269 6574 7920  onal subvariety 
│ │ │ │ +00052490: 6f66 2050 505e 3420 7820 5050 5e35 2078  of PP^4 x PP^5 x
│ │ │ │ +000524a0: 2050 505e 3520 6375 7420 6f75 7420 7c0a   PP^5 cut out |.
│ │ │ │ +000524b0: 7c20 2020 2020 2074 6172 6765 7420 7661  |      target va
│ │ │ │ +000524c0: 7269 6574 793a 2050 505e 3420 2020 2020  riety: PP^4     
│ │ │ │ +000524d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000524e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000524f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052500: 2020 2020 7c0a 7c20 2020 2020 2063 6f65      |.|      coe
│ │ │ │ -00052510: 6666 6963 6965 6e74 2072 696e 673a 205a  fficient ring: Z
│ │ │ │ -00052520: 5a2f 3139 3031 3831 2020 2020 2020 2020  Z/190181        
│ │ │ │ +000524f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00052500: 7c20 2020 2020 2063 6f65 6666 6963 6965  |      coefficie
│ │ │ │ +00052510: 6e74 2072 696e 673a 205a 5a2f 3139 3031  nt ring: ZZ/1901
│ │ │ │ +00052520: 3831 2020 2020 2020 2020 2020 2020 2020  81              
│ │ │ │  00052530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052550: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +00052540: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00052550: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │  00052560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000525a0: 2d2d 2d2d 7c0a 7c62 7920 3334 2068 7970  ----|.|by 34 hyp
│ │ │ │ -000525b0: 6572 7375 7266 6163 6573 206f 6620 6465  ersurfaces of de
│ │ │ │ -000525c0: 6772 6565 7320 287b 302c 2031 2c20 317d  grees ({0, 1, 1}
│ │ │ │ -000525d0: 2c7b 302c 2030 2c20 327d 2c7b 302c 2031  ,{0, 0, 2},{0, 1
│ │ │ │ -000525e0: 2c20 317d 2c7b 302c 2031 2c20 317d 2c7b  , 1},{0, 1, 1},{
│ │ │ │ -000525f0: 302c 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  0,  |.|---------
│ │ │ │ +00052590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +000525a0: 7c62 7920 3334 2068 7970 6572 7375 7266  |by 34 hypersurf
│ │ │ │ +000525b0: 6163 6573 206f 6620 6465 6772 6565 7320  aces of degrees 
│ │ │ │ +000525c0: 287b 302c 2031 2c20 317d 2c7b 302c 2030  ({0, 1, 1},{0, 0
│ │ │ │ +000525d0: 2c20 327d 2c7b 302c 2031 2c20 317d 2c7b  , 2},{0, 1, 1},{
│ │ │ │ +000525e0: 302c 2031 2c20 317d 2c7b 302c 2020 7c0a  0, 1, 1},{0,  |.
│ │ │ │ +000525f0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │  00052600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052640: 2d2d 2d2d 7c0a 7c31 2c20 317d 2c7b 302c  ----|.|1, 1},{0,
│ │ │ │ -00052650: 2031 2c20 317d 2c7b 302c 2031 2c20 317d   1, 1},{0, 1, 1}
│ │ │ │ -00052660: 2c7b 302c 2031 2c20 317d 2c7b 312c 2030  ,{0, 1, 1},{1, 0
│ │ │ │ -00052670: 2c20 317d 2c7b 312c 2030 2c20 317d 2c7b  , 1},{1, 0, 1},{
│ │ │ │ -00052680: 302c 2031 2c20 317d 2c7b 302c 2031 2c20  0, 1, 1},{0, 1, 
│ │ │ │ -00052690: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +00052630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00052640: 7c31 2c20 317d 2c7b 302c 2031 2c20 317d  |1, 1},{0, 1, 1}
│ │ │ │ +00052650: 2c7b 302c 2031 2c20 317d 2c7b 302c 2031  ,{0, 1, 1},{0, 1
│ │ │ │ +00052660: 2c20 317d 2c7b 312c 2030 2c20 317d 2c7b  , 1},{1, 0, 1},{
│ │ │ │ +00052670: 312c 2030 2c20 317d 2c7b 302c 2031 2c20  1, 0, 1},{0, 1, 
│ │ │ │ +00052680: 317d 2c7b 302c 2031 2c20 2020 2020 7c0a  1},{0, 1,     |.
│ │ │ │ +00052690: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │  000526a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000526b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000526c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000526d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000526e0: 2d2d 2d2d 7c0a 7c31 7d2c 7b30 2c20 312c  ----|.|1},{0, 1,
│ │ │ │ -000526f0: 2031 7d2c 7b30 2c20 312c 2031 7d2c 7b31   1},{0, 1, 1},{1
│ │ │ │ -00052700: 2c20 302c 2031 7d2c 7b31 2c20 302c 2031  , 0, 1},{1, 0, 1
│ │ │ │ -00052710: 7d2c 7b31 2c20 302c 2031 7d2c 7b30 2c20  },{1, 0, 1},{0, 
│ │ │ │ -00052720: 312c 2031 7d2c 7b30 2c20 312c 2031 7d2c  1, 1},{0, 1, 1},
│ │ │ │ -00052730: 7b30 2c20 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  {0, |.|---------
│ │ │ │ +000526d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +000526e0: 7c31 7d2c 7b30 2c20 312c 2031 7d2c 7b30  |1},{0, 1, 1},{0
│ │ │ │ +000526f0: 2c20 312c 2031 7d2c 7b31 2c20 302c 2031  , 1, 1},{1, 0, 1
│ │ │ │ +00052700: 7d2c 7b31 2c20 302c 2031 7d2c 7b31 2c20  },{1, 0, 1},{1, 
│ │ │ │ +00052710: 302c 2031 7d2c 7b30 2c20 312c 2031 7d2c  0, 1},{0, 1, 1},
│ │ │ │ +00052720: 7b30 2c20 312c 2031 7d2c 7b30 2c20 7c0a  {0, 1, 1},{0, |.
│ │ │ │ +00052730: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │  00052740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052780: 2d2d 2d2d 7c0a 7c31 2c20 317d 2c7b 302c  ----|.|1, 1},{0,
│ │ │ │ -00052790: 2031 2c20 317d 2c7b 302c 2031 2c20 317d   1, 1},{0, 1, 1}
│ │ │ │ -000527a0: 2c7b 312c 2030 2c20 317d 2c7b 312c 2030  ,{1, 0, 1},{1, 0
│ │ │ │ -000527b0: 2c20 317d 2c7b 312c 2030 2c20 317d 2c7b  , 1},{1, 0, 1},{
│ │ │ │ -000527c0: 302c 2032 2c20 307d 2c7b 312c 2031 2c20  0, 2, 0},{1, 1, 
│ │ │ │ -000527d0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +00052770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00052780: 7c31 2c20 317d 2c7b 302c 2031 2c20 317d  |1, 1},{0, 1, 1}
│ │ │ │ +00052790: 2c7b 302c 2031 2c20 317d 2c7b 312c 2030  ,{0, 1, 1},{1, 0
│ │ │ │ +000527a0: 2c20 317d 2c7b 312c 2030 2c20 317d 2c7b  , 1},{1, 0, 1},{
│ │ │ │ +000527b0: 312c 2030 2c20 317d 2c7b 302c 2032 2c20  1, 0, 1},{0, 2, 
│ │ │ │ +000527c0: 307d 2c7b 312c 2031 2c20 2020 2020 7c0a  0},{1, 1,     |.
│ │ │ │ +000527d0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │  000527e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000527f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052820: 2d2d 2d2d 7c0a 7c30 7d2c 7b31 2c20 312c  ----|.|0},{1, 1,
│ │ │ │ -00052830: 2030 7d2c 7b31 2c20 312c 2030 7d2c 7b31   0},{1, 1, 0},{1
│ │ │ │ -00052840: 2c20 312c 2030 7d2c 7b31 2c20 312c 2030  , 1, 0},{1, 1, 0
│ │ │ │ -00052850: 7d2c 7b31 2c20 312c 2030 7d2c 7b31 2c20  },{1, 1, 0},{1, 
│ │ │ │ -00052860: 312c 2030 7d2c 7b31 2c20 312c 2030 7d29  1, 0},{1, 1, 0})
│ │ │ │ -00052870: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00052810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00052820: 7c30 7d2c 7b31 2c20 312c 2030 7d2c 7b31  |0},{1, 1, 0},{1
│ │ │ │ +00052830: 2c20 312c 2030 7d2c 7b31 2c20 312c 2030  , 1, 0},{1, 1, 0
│ │ │ │ +00052840: 7d2c 7b31 2c20 312c 2030 7d2c 7b31 2c20  },{1, 1, 0},{1, 
│ │ │ │ +00052850: 312c 2030 7d2c 7b31 2c20 312c 2030 7d2c  1, 0},{1, 1, 0},
│ │ │ │ +00052860: 7b31 2c20 312c 2030 7d29 2020 2020 7c0a  {1, 1, 0})    |.
│ │ │ │ +00052870: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00052880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000528a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000528b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000528c0: 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 6f0a  ----+..See also.
│ │ │ │ -000528d0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ -000528e0: 6f74 6520 6772 6170 6828 5269 6e67 4d61  ote graph(RingMa
│ │ │ │ -000528f0: 7029 3a20 6772 6170 685f 6c70 5269 6e67  p): graph_lpRing
│ │ │ │ -00052900: 4d61 705f 7270 2c20 2d2d 2063 6c6f 7375  Map_rp, -- closu
│ │ │ │ -00052910: 7265 206f 6620 7468 6520 6772 6170 6820  re of the graph 
│ │ │ │ -00052920: 6f66 2061 0a20 2020 2072 6174 696f 6e61  of a.    rationa
│ │ │ │ -00052930: 6c20 6d61 700a 2020 2a20 2a6e 6f74 6520  l map.  * *note 
│ │ │ │ -00052940: 6772 6170 6849 6465 616c 3a20 284d 6163  graphIdeal: (Mac
│ │ │ │ -00052950: 6175 6c61 7932 446f 6329 6772 6170 6849  aulay2Doc)graphI
│ │ │ │ -00052960: 6465 616c 5f6c 7052 696e 674d 6170 5f72  deal_lpRingMap_r
│ │ │ │ -00052970: 702c 202d 2d20 7468 6520 6964 6561 6c20  p, -- the ideal 
│ │ │ │ -00052980: 6f66 0a20 2020 2074 6865 2067 7261 7068  of.    the graph
│ │ │ │ -00052990: 206f 6620 7468 6520 7265 6775 6c61 7220   of the regular 
│ │ │ │ -000529a0: 6d61 7020 636f 7272 6573 706f 6e64 696e  map correspondin
│ │ │ │ -000529b0: 6720 746f 2061 2072 696e 6720 6d61 700a  g to a ring map.
│ │ │ │ -000529c0: 0a57 6179 7320 746f 2075 7365 2067 7261  .Ways to use gra
│ │ │ │ -000529d0: 7068 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ph:.============
│ │ │ │ -000529e0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2267 7261  ======..  * "gra
│ │ │ │ -000529f0: 7068 2852 6174 696f 6e61 6c4d 6170 2922  ph(RationalMap)"
│ │ │ │ -00052a00: 0a20 202a 202a 6e6f 7465 2067 7261 7068  .  * *note graph
│ │ │ │ -00052a10: 2852 696e 674d 6170 293a 2067 7261 7068  (RingMap): graph
│ │ │ │ -00052a20: 5f6c 7052 696e 674d 6170 5f72 702c 202d  _lpRingMap_rp, -
│ │ │ │ -00052a30: 2d20 636c 6f73 7572 6520 6f66 2074 6865  - closure of the
│ │ │ │ -00052a40: 2067 7261 7068 206f 6620 610a 2020 2020   graph of a.    
│ │ │ │ -00052a50: 7261 7469 6f6e 616c 206d 6170 0a0a 466f  rational map..Fo
│ │ │ │ -00052a60: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -00052a70: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00052a80: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -00052a90: 2a6e 6f74 6520 6772 6170 683a 2067 7261  *note graph: gra
│ │ │ │ -00052aa0: 7068 2c20 6973 2061 202a 6e6f 7465 206d  ph, is a *note m
│ │ │ │ -00052ab0: 6574 686f 6420 6675 6e63 7469 6f6e 2077  ethod function w
│ │ │ │ -00052ac0: 6974 6820 6f70 7469 6f6e 733a 0a28 4d61  ith options:.(Ma
│ │ │ │ -00052ad0: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ -00052ae0: 6446 756e 6374 696f 6e57 6974 684f 7074  dFunctionWithOpt
│ │ │ │ -00052af0: 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d  ions,...--------
│ │ │ │ +000528b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000528c0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +000528d0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6772  ==..  * *note gr
│ │ │ │ +000528e0: 6170 6828 5269 6e67 4d61 7029 3a20 6772  aph(RingMap): gr
│ │ │ │ +000528f0: 6170 685f 6c70 5269 6e67 4d61 705f 7270  aph_lpRingMap_rp
│ │ │ │ +00052900: 2c20 2d2d 2063 6c6f 7375 7265 206f 6620  , -- closure of 
│ │ │ │ +00052910: 7468 6520 6772 6170 6820 6f66 2061 0a20  the graph of a. 
│ │ │ │ +00052920: 2020 2072 6174 696f 6e61 6c20 6d61 700a     rational map.
│ │ │ │ +00052930: 2020 2a20 2a6e 6f74 6520 6772 6170 6849    * *note graphI
│ │ │ │ +00052940: 6465 616c 3a20 284d 6163 6175 6c61 7932  deal: (Macaulay2
│ │ │ │ +00052950: 446f 6329 6772 6170 6849 6465 616c 5f6c  Doc)graphIdeal_l
│ │ │ │ +00052960: 7052 696e 674d 6170 5f72 702c 202d 2d20  pRingMap_rp, -- 
│ │ │ │ +00052970: 7468 6520 6964 6561 6c20 6f66 0a20 2020  the ideal of.   
│ │ │ │ +00052980: 2074 6865 2067 7261 7068 206f 6620 7468   the graph of th
│ │ │ │ +00052990: 6520 7265 6775 6c61 7220 6d61 7020 636f  e regular map co
│ │ │ │ +000529a0: 7272 6573 706f 6e64 696e 6720 746f 2061  rresponding to a
│ │ │ │ +000529b0: 2072 696e 6720 6d61 700a 0a57 6179 7320   ring map..Ways 
│ │ │ │ +000529c0: 746f 2075 7365 2067 7261 7068 3a0a 3d3d  to use graph:.==
│ │ │ │ +000529d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000529e0: 0a0a 2020 2a20 2267 7261 7068 2852 6174  ..  * "graph(Rat
│ │ │ │ +000529f0: 696f 6e61 6c4d 6170 2922 0a20 202a 202a  ionalMap)".  * *
│ │ │ │ +00052a00: 6e6f 7465 2067 7261 7068 2852 696e 674d  note graph(RingM
│ │ │ │ +00052a10: 6170 293a 2067 7261 7068 5f6c 7052 696e  ap): graph_lpRin
│ │ │ │ +00052a20: 674d 6170 5f72 702c 202d 2d20 636c 6f73  gMap_rp, -- clos
│ │ │ │ +00052a30: 7572 6520 6f66 2074 6865 2067 7261 7068  ure of the graph
│ │ │ │ +00052a40: 206f 6620 610a 2020 2020 7261 7469 6f6e   of a.    ration
│ │ │ │ +00052a50: 616c 206d 6170 0a0a 466f 7220 7468 6520  al map..For the 
│ │ │ │ +00052a60: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ +00052a70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ +00052a80: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ +00052a90: 6772 6170 683a 2067 7261 7068 2c20 6973  graph: graph, is
│ │ │ │ +00052aa0: 2061 202a 6e6f 7465 206d 6574 686f 6420   a *note method 
│ │ │ │ +00052ab0: 6675 6e63 7469 6f6e 2077 6974 6820 6f70  function with op
│ │ │ │ +00052ac0: 7469 6f6e 733a 0a28 4d61 6361 756c 6179  tions:.(Macaulay
│ │ │ │ +00052ad0: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ +00052ae0: 696f 6e57 6974 684f 7074 696f 6e73 2c2e  ionWithOptions,.
│ │ │ │ +00052af0: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..--------------
│ │ │ │  00052b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00052b40: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ -00052b50: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
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│ │ │ │ -00052b70: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ -00052b80: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ -00052b90: 3235 2e31 312b 6473 2f4d 322f 4d61 6361  25.11+ds/M2/Maca
│ │ │ │ -00052ba0: 756c 6179 322f 7061 636b 6167 6573 2f43  ulay2/packages/C
│ │ │ │ -00052bb0: 7265 6d6f 6e61 2f0a 646f 6375 6d65 6e74  remona/.document
│ │ │ │ -00052bc0: 6174 696f 6e2e 6d32 3a32 3533 3a30 2e0a  ation.m2:253:0..
│ │ │ │ -00052bd0: 1f0a 4669 6c65 3a20 4372 656d 6f6e 612e  ..File: Cremona.
│ │ │ │ -00052be0: 696e 666f 2c20 4e6f 6465 3a20 6772 6170  info, Node: grap
│ │ │ │ -00052bf0: 685f 6c70 5269 6e67 4d61 705f 7270 2c20  h_lpRingMap_rp, 
│ │ │ │ -00052c00: 4e65 7874 3a20 6964 6561 6c5f 6c70 5261  Next: ideal_lpRa
│ │ │ │ -00052c10: 7469 6f6e 616c 4d61 705f 7270 2c20 5072  tionalMap_rp, Pr
│ │ │ │ -00052c20: 6576 3a20 6772 6170 682c 2055 703a 2054  ev: graph, Up: T
│ │ │ │ -00052c30: 6f70 0a0a 6772 6170 6828 5269 6e67 4d61  op..graph(RingMa
│ │ │ │ -00052c40: 7029 202d 2d20 636c 6f73 7572 6520 6f66  p) -- closure of
│ │ │ │ -00052c50: 2074 6865 2067 7261 7068 206f 6620 6120   the graph of a 
│ │ │ │ -00052c60: 7261 7469 6f6e 616c 206d 6170 0a2a 2a2a  rational map.***
│ │ │ │ +00052b40: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66  -..The source of
│ │ │ │ +00052b50: 2074 6869 7320 646f 6375 6d65 6e74 2069   this document i
│ │ │ │ +00052b60: 7320 696e 0a2f 6275 696c 642f 7265 7072  s in./build/repr
│ │ │ │ +00052b70: 6f64 7563 6962 6c65 2d70 6174 682f 6d61  oducible-path/ma
│ │ │ │ +00052b80: 6361 756c 6179 322d 312e 3235 2e31 312b  caulay2-1.25.11+
│ │ │ │ +00052b90: 6473 2f4d 322f 4d61 6361 756c 6179 322f  ds/M2/Macaulay2/
│ │ │ │ +00052ba0: 7061 636b 6167 6573 2f43 7265 6d6f 6e61  packages/Cremona
│ │ │ │ +00052bb0: 2f0a 646f 6375 6d65 6e74 6174 696f 6e2e  /.documentation.
│ │ │ │ +00052bc0: 6d32 3a32 3533 3a30 2e0a 1f0a 4669 6c65  m2:253:0....File
│ │ │ │ +00052bd0: 3a20 4372 656d 6f6e 612e 696e 666f 2c20  : Cremona.info, 
│ │ │ │ +00052be0: 4e6f 6465 3a20 6772 6170 685f 6c70 5269  Node: graph_lpRi
│ │ │ │ +00052bf0: 6e67 4d61 705f 7270 2c20 4e65 7874 3a20  ngMap_rp, Next: 
│ │ │ │ +00052c00: 6964 6561 6c5f 6c70 5261 7469 6f6e 616c  ideal_lpRational
│ │ │ │ +00052c10: 4d61 705f 7270 2c20 5072 6576 3a20 6772  Map_rp, Prev: gr
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│ │ │ │  00052c70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00052c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00052c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00052ca0: 2a2a 2a2a 2a0a 0a20 202a 2046 756e 6374  *****..  * Funct
│ │ │ │ -00052cb0: 696f 6e3a 202a 6e6f 7465 2067 7261 7068  ion: *note graph
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│ │ │ │ -00052d00: 6120 2a6e 6f74 6520 7269 6e67 206d 6170  a *note ring map
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│ │ │ │ -00052d70: 7465 204f 7074 696f 6e61 6c20 696e 7075  te Optional inpu
│ │ │ │ -00052d80: 7473 3a20 284d 6163 6175 6c61 7932 446f  ts: (Macaulay2Do
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│ │ │ │ -00052dc0: 202a 6e6f 7465 2042 6c6f 7755 7053 7472   *note BlowUpStr
│ │ │ │ -00052dd0: 6174 6567 793a 2042 6c6f 7755 7053 7472  ategy: BlowUpStr
│ │ │ │ -00052de0: 6174 6567 792c 203d 3e20 2e2e 2e2c 2064  ategy, => ..., d
│ │ │ │ -00052df0: 6566 6175 6c74 2076 616c 7565 0a20 2020  efault value.   
│ │ │ │ -00052e00: 2020 2020 2022 456c 696d 696e 6174 6522       "Eliminate"
│ │ │ │ -00052e10: 2c0a 2020 2a20 4f75 7470 7574 733a 0a20  ,.  * Outputs:. 
│ │ │ │ -00052e20: 2020 2020 202a 2061 202a 6e6f 7465 2072       * a *note r
│ │ │ │ -00052e30: 696e 6720 6d61 703a 2028 4d61 6361 756c  ing map: (Macaul
│ │ │ │ -00052e40: 6179 3244 6f63 2952 696e 674d 6170 2c2c  ay2Doc)RingMap,,
│ │ │ │ -00052e50: 2072 6570 7265 7365 6e74 696e 6720 7468   representing th
│ │ │ │ -00052e60: 6520 7072 6f6a 6563 7469 6f6e 0a20 2020  e projection.   
│ │ │ │ -00052e70: 2020 2020 206f 6e20 7468 6520 6669 7273       on the firs
│ │ │ │ -00052e80: 7420 6661 6374 6f72 2024 5c6d 6174 6862  t factor $\mathb
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│ │ │ │ -00052eb0: 2a6e 6f74 6520 7269 6e67 206d 6170 3a20  *note ring map: 
│ │ │ │ -00052ec0: 284d 6163 6175 6c61 7932 446f 6329 5269  (Macaulay2Doc)Ri
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│ │ │ │ -00052f20: 285c 5068 6929 205c 746f 2059 240a 0a44  (\Phi) \to Y$..D
│ │ │ │ -00052f30: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
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│ │ │ │ +00052c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +00052ca0: 0a20 202a 2046 756e 6374 696f 6e3a 202a  .  * Function: *
│ │ │ │ +00052cb0: 6e6f 7465 2067 7261 7068 3a20 6772 6170  note graph: grap
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│ │ │ │ +00052cd0: 2020 2020 2020 2067 7261 7068 2070 6869         graph phi
│ │ │ │ +00052ce0: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
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│ │ │ │ +00052d50: 5c64 6173 6872 6967 6874 6172 726f 7720  \dashrightarrow 
│ │ │ │ +00052d60: 5924 0a20 202a 202a 6e6f 7465 204f 7074  Y$.  * *note Opt
│ │ │ │ +00052d70: 696f 6e61 6c20 696e 7075 7473 3a20 284d  ional inputs: (M
│ │ │ │ +00052d80: 6163 6175 6c61 7932 446f 6329 7573 696e  acaulay2Doc)usin
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│ │ │ │ +00052db0: 2c3a 0a20 2020 2020 202a 202a 6e6f 7465  ,:.      * *note
│ │ │ │ +00052dc0: 2042 6c6f 7755 7053 7472 6174 6567 793a   BlowUpStrategy:
│ │ │ │ +00052dd0: 2042 6c6f 7755 7053 7472 6174 6567 792c   BlowUpStrategy,
│ │ │ │ +00052de0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ +00052df0: 2076 616c 7565 0a20 2020 2020 2020 2022   value.        "
│ │ │ │ +00052e00: 456c 696d 696e 6174 6522 2c0a 2020 2a20  Eliminate",.  * 
│ │ │ │ +00052e10: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +00052e20: 2061 202a 6e6f 7465 2072 696e 6720 6d61   a *note ring ma
│ │ │ │ +00052e30: 703a 2028 4d61 6361 756c 6179 3244 6f63  p: (Macaulay2Doc
│ │ │ │ +00052e40: 2952 696e 674d 6170 2c2c 2072 6570 7265  )RingMap,, repre
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│ │ │ │ +00052e80: 6f72 2024 5c6d 6174 6862 667b 4772 6170  or $\mathbf{Grap
│ │ │ │ +00052e90: 687d 285c 5068 6929 205c 746f 2058 240a  h}(\Phi) \to X$.
│ │ │ │ +00052ea0: 2020 2020 2020 2a20 6120 2a6e 6f74 6520        * a *note 
│ │ │ │ +00052eb0: 7269 6e67 206d 6170 3a20 284d 6163 6175  ring map: (Macau
│ │ │ │ +00052ec0: 6c61 7932 446f 6329 5269 6e67 4d61 702c  lay2Doc)RingMap,
│ │ │ │ +00052ed0: 2c20 7265 7072 6573 656e 7469 6e67 2074  , representing t
│ │ │ │ +00052ee0: 6865 2070 726f 6a65 6374 696f 6e0a 2020  he projection.  
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│ │ │ │ +00052f00: 6f6e 6420 6661 6374 6f72 2024 5c6d 6174  ond factor $\mat
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│ │ │ │ +00052f20: 205c 746f 2059 240a 0a44 6573 6372 6970   \to Y$..Descrip
│ │ │ │ +00052f30: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ +00052f40: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
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│ │ │ │  00052f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00052f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00052fa0: 6920 3d20 6d61 7028 5151 5b78 5f30 2e2e  i = map(QQ[x_0..
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│ │ │ │ +00052fb0: 515b 795f 302e 2e79 5f32 5d2c 7b2d 785f  Q[y_0..y_2],{-x_
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│ │ │ │ -00053100: 2020 2020 3020 3220 2020 2020 3120 3220      0 2     1 2 
│ │ │ │ -00053110: 2020 2030 2033 2020 2020 2032 2020 2020     0 3     2    
│ │ │ │ -00053120: 3120 3320 2020 7c0a 7c20 2020 2020 2020  1 3   |.|       
│ │ │ │ +00053070: 2020 2020 2032 2020 2020 2020 2020 2020       2          
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│ │ │ │ +000530c0: 2c20 2d20 7820 202b 2078 2078 207d 2920  , - x  + x x }) 
│ │ │ │ +000530d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000530e0: 2030 2020 2033 2020 2020 2020 2030 2020   0   3       0  
│ │ │ │ +000530f0: 2032 2020 2020 2020 2031 2020 2020 3020   2       1    0 
│ │ │ │ +00053100: 3220 2020 2020 3120 3220 2020 2030 2033  2     1 2    0 3
│ │ │ │ +00053110: 2020 2020 2032 2020 2020 3120 3320 2020       2    1 3   
│ │ │ │ +00053120: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00053130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053170: 2020 2020 2020 7c0a 7c6f 3120 3a20 5269        |.|o1 : Ri
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│ │ │ │ +00053180: 5151 5b78 202e 2e78 205d 203c 2d2d 2051  QQ[x ..x ] <-- Q
│ │ │ │ +00053190: 515b 7920 2e2e 7920 5d20 2020 2020 2020  Q[y ..y ]       
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│ │ │ │ -000531c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000531d0: 2020 2020 2020 2020 2020 3020 2020 3320            0   3 
│ │ │ │ -000531e0: 2020 2020 2020 2020 2030 2020 2032 2020           0   2  
│ │ │ │ +000531c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000531d0: 2020 2020 3020 2020 3320 2020 2020 2020      0   3       
│ │ │ │ +000531e0: 2020 2030 2020 2032 2020 2020 2020 2020     0   2        
│ │ │ │  000531f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00053210: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
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│ │ │ │  00053230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00053260: 7c0a 7c32 5e32 2b78 5f31 2a78 5f33 7d29  |.|2^2+x_1*x_3})
│ │ │ │ +00053270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000532a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000532b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000532b0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  000532c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000532d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000532e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00053310: 6920 2020 2020 2020 2020 2020 2020 2020  i               
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│ │ │ │  00053330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053340: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000533a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000533b0: 2020 2020 2020 2020 2020 2020 2020 2051                 Q
│ │ │ │ -000533c0: 515b 7820 2e2e 7820 2c20 7920 2e2e 7920  Q[x ..x , y ..y 
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│ │ │ │ +000533b0: 2020 2020 2020 2020 2051 515b 7820 2e2e           QQ[x ..
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│ │ │ │ -00053410: 2020 2030 2020 2033 2020 2030 2020 2032     0   3   0   2
│ │ │ │ +000533f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00053400: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +00053410: 2033 2020 2030 2020 2032 2020 2020 2020   3   0   2      
│ │ │ │  00053420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053440: 2020 2020 2020 7c0a 7c6f 3220 3d20 286d        |.|o2 = (m
│ │ │ │ -00053450: 6170 2028 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ap (------------
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│ │ │ │  00053460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2c20 5151  ------------, QQ
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│ │ │ │ -00053490: 202c 2078 202c 7c0a 7c20 2020 2020 2020   , x ,|.|       
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│ │ │ │  000538c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000538d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00053a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00053c00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00053c10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00053c20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00053c30: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20  ..  * Function: 
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│ │ │ │ -00053c60: 2c0a 2020 2a20 5573 6167 653a 200a 2020  ,.  * Usage: .  
│ │ │ │ -00053c70: 2020 2020 2020 6964 6561 6c20 7068 690a        ideal phi.
│ │ │ │ -00053c80: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -00053c90: 2020 2a20 7068 692c 2061 202a 6e6f 7465    * phi, a *note
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│ │ │ │ -00053d00: 206f 6620 7468 6520 6261 7365 206c 6f63   of the base loc
│ │ │ │ -00053d10: 7573 206f 660a 2020 2020 2020 2020 7068  us of.        ph
│ │ │ │ -00053d20: 690a 0a44 6573 6372 6970 7469 6f6e 0a3d  i..Description.=
│ │ │ │ -00053d30: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973  ==========..This
│ │ │ │ -00053d40: 2069 7320 6765 6e65 7261 6c6c 7920 6469   is generally di
│ │ │ │ -00053d50: 6666 6963 756c 742c 2062 7574 2069 6e20  fficult, but in 
│ │ │ │ -00053d60: 736f 6d65 2063 6173 6573 2069 7420 6973  some cases it is
│ │ │ │ -00053d70: 2065 7175 6976 616c 656e 7420 746f 2069   equivalent to i
│ │ │ │ -00053d80: 6465 616c 206d 6174 7269 780a 7068 692c  deal matrix.phi,
│ │ │ │ -00053d90: 2077 6869 6368 2064 6f65 7320 6e6f 7420   which does not 
│ │ │ │ -00053da0: 7065 7266 6f72 6d20 616e 7920 636f 6d70  perform any comp
│ │ │ │ -00053db0: 7574 6174 696f 6e2e 0a0a 2b2d 2d2d 2d2d  utation...+-----
│ │ │ │ +00053c20: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +00053c30: 4675 6e63 7469 6f6e 3a20 2a6e 6f74 6520  Function: *note 
│ │ │ │ +00053c40: 6964 6561 6c3a 2028 4d61 6361 756c 6179  ideal: (Macaulay
│ │ │ │ +00053c50: 3244 6f63 2969 6465 616c 2c0a 2020 2a20  2Doc)ideal,.  * 
│ │ │ │ +00053c60: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +00053c70: 6964 6561 6c20 7068 690a 2020 2a20 496e  ideal phi.  * In
│ │ │ │ +00053c80: 7075 7473 3a0a 2020 2020 2020 2a20 7068  puts:.      * ph
│ │ │ │ +00053c90: 692c 2061 202a 6e6f 7465 2072 6174 696f  i, a *note ratio
│ │ │ │ +00053ca0: 6e61 6c20 6d61 703a 2052 6174 696f 6e61  nal map: Rationa
│ │ │ │ +00053cb0: 6c4d 6170 2c0a 2020 2a20 4f75 7470 7574  lMap,.  * Output
│ │ │ │ +00053cc0: 733a 0a20 2020 2020 202a 2061 6e20 2a6e  s:.      * an *n
│ │ │ │ +00053cd0: 6f74 6520 6964 6561 6c3a 2028 4d61 6361  ote ideal: (Maca
│ │ │ │ +00053ce0: 756c 6179 3244 6f63 2949 6465 616c 2c2c  ulay2Doc)Ideal,,
│ │ │ │ +00053cf0: 2074 6865 2069 6465 616c 206f 6620 7468   the ideal of th
│ │ │ │ +00053d00: 6520 6261 7365 206c 6f63 7573 206f 660a  e base locus of.
│ │ │ │ +00053d10: 2020 2020 2020 2020 7068 690a 0a44 6573          phi..Des
│ │ │ │ +00053d20: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +00053d30: 3d3d 3d3d 0a0a 5468 6973 2069 7320 6765  ====..This is ge
│ │ │ │ +00053d40: 6e65 7261 6c6c 7920 6469 6666 6963 756c  nerally difficul
│ │ │ │ +00053d50: 742c 2062 7574 2069 6e20 736f 6d65 2063  t, but in some c
│ │ │ │ +00053d60: 6173 6573 2069 7420 6973 2065 7175 6976  ases it is equiv
│ │ │ │ +00053d70: 616c 656e 7420 746f 2069 6465 616c 206d  alent to ideal m
│ │ │ │ +00053d80: 6174 7269 780a 7068 692c 2077 6869 6368  atrix.phi, which
│ │ │ │ +00053d90: 2064 6f65 7320 6e6f 7420 7065 7266 6f72   does not perfor
│ │ │ │ +00053da0: 6d20 616e 7920 636f 6d70 7574 6174 696f  m any computatio
│ │ │ │ +00053db0: 6e2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  n...+-----------
│ │ │ │  00053dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053e00: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -00053e10: 7920 3d20 6765 6e73 2851 515b 785f 302e  y = gens(QQ[x_0.
│ │ │ │ -00053e20: 2e78 5f35 5d2f 2878 5f32 5e32 2d78 5f32  .x_5]/(x_2^2-x_2
│ │ │ │ -00053e30: 2a78 5f33 2b78 5f31 2a78 5f34 2d78 5f30  *x_3+x_1*x_4-x_0
│ │ │ │ -00053e40: 2a78 5f35 2929 3b20 2020 2020 2020 2020  *x_5));         
│ │ │ │ -00053e50: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00053e00: 2d2d 2b0a 7c69 3120 3a20 7920 3d20 6765  --+.|i1 : y = ge
│ │ │ │ +00053e10: 6e73 2851 515b 785f 302e 2e78 5f35 5d2f  ns(QQ[x_0..x_5]/
│ │ │ │ +00053e20: 2878 5f32 5e32 2d78 5f32 2a78 5f33 2b78  (x_2^2-x_2*x_3+x
│ │ │ │ +00053e30: 5f31 2a78 5f34 2d78 5f30 2a78 5f35 2929  _1*x_4-x_0*x_5))
│ │ │ │ +00053e40: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00053e50: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00053e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053ea0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ -00053eb0: 7068 6920 3d20 7261 7469 6f6e 616c 4d61  phi = rationalMa
│ │ │ │ -00053ec0: 7020 7b79 5f34 5e32 2d79 5f33 2a79 5f35  p {y_4^2-y_3*y_5
│ │ │ │ -00053ed0: 2c2d 795f 322a 795f 342b 795f 332a 795f  ,-y_2*y_4+y_3*y_
│ │ │ │ -00053ee0: 342d 795f 312a 795f 352c 202d 795f 322a  4-y_1*y_5, -y_2*
│ │ │ │ -00053ef0: 795f 332b 795f 335e 7c0a 7c20 2020 2020  y_3+y_3^|.|     
│ │ │ │ +00053ea0: 2d2d 2b0a 7c69 3220 3a20 7068 6920 3d20  --+.|i2 : phi = 
│ │ │ │ +00053eb0: 7261 7469 6f6e 616c 4d61 7020 7b79 5f34  rationalMap {y_4
│ │ │ │ +00053ec0: 5e32 2d79 5f33 2a79 5f35 2c2d 795f 322a  ^2-y_3*y_5,-y_2*
│ │ │ │ +00053ed0: 795f 342b 795f 332a 795f 342d 795f 312a  y_4+y_3*y_4-y_1*
│ │ │ │ +00053ee0: 795f 352c 202d 795f 322a 795f 332b 795f  y_5, -y_2*y_3+y_
│ │ │ │ +00053ef0: 335e 7c0a 7c20 2020 2020 2020 2020 2020  3^|.|           
│ │ │ │  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: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ -00053f50: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ -00053f60: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +00053f40: 2020 7c0a 7c6f 3220 3d20 2d2d 2072 6174    |.|o2 = -- rat
│ │ │ │ +00053f50: 696f 6e61 6c20 6d61 7020 2d2d 2020 2020  ional map --    
│ │ │ │ +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: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00053fa0: 736f 7572 6365 3a20 7375 6276 6172 6965  source: subvarie
│ │ │ │ -00053fb0: 7479 206f 6620 5072 6f6a 2851 515b 7820  ty of Proj(QQ[x 
│ │ │ │ -00053fc0: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ -00053fd0: 2c20 7820 5d29 2064 6566 696e 6564 2062  , x ]) defined b
│ │ │ │ -00053fe0: 7920 2020 2020 2020 7c0a 7c20 2020 2020  y       |.|     
│ │ │ │ +00053f90: 2020 7c0a 7c20 2020 2020 736f 7572 6365    |.|     source
│ │ │ │ +00053fa0: 3a20 7375 6276 6172 6965 7479 206f 6620  : subvariety of 
│ │ │ │ +00053fb0: 5072 6f6a 2851 515b 7820 2c20 7820 2c20  Proj(QQ[x , x , 
│ │ │ │ +00053fc0: 7820 2c20 7820 2c20 7820 2c20 7820 5d29  x , x , x , x ])
│ │ │ │ +00053fd0: 2064 6566 696e 6564 2062 7920 2020 2020   defined by     
│ │ │ │ +00053fe0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00053ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054000: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00054010: 2020 2031 2020 2032 2020 2033 2020 2034     1   2   3   4
│ │ │ │ -00054020: 2020 2035 2020 2020 2020 2020 2020 2020     5            
│ │ │ │ -00054030: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054040: 2020 2020 2020 2020 7b20 2020 2020 2020          {       
│ │ │ │ +00054000: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ +00054010: 2032 2020 2033 2020 2034 2020 2035 2020   2   3   4   5  
│ │ │ │ +00054020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054030: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054040: 2020 7b20 2020 2020 2020 2020 2020 2020    {             
│ │ │ │  00054050: 2020 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 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054090: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00054080: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054090: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  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 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000540e0: 2020 2020 2020 2020 2078 2020 2d20 7820           x  - x 
│ │ │ │ -000540f0: 7820 202b 2078 2078 2020 2d20 7820 7820  x  + x x  - x x 
│ │ │ │ +000540d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000540e0: 2020 2078 2020 2d20 7820 7820 202b 2078     x  - x x  + x
│ │ │ │ +000540f0: 2078 2020 2d20 7820 7820 2020 2020 2020   x  - x x       
│ │ │ │  00054100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054120: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054130: 2020 2020 2020 2020 2020 3220 2020 2032            2    2
│ │ │ │ -00054140: 2033 2020 2020 3120 3420 2020 2030 2035   3    1 4    0 5
│ │ │ │ +00054120: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054130: 2020 2020 3220 2020 2032 2033 2020 2020      2    2 3    
│ │ │ │ +00054140: 3120 3420 2020 2030 2035 2020 2020 2020  1 4    0 5      
│ │ │ │  00054150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054170: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054180: 2020 2020 2020 2020 7d20 2020 2020 2020          }       
│ │ │ │ +00054170: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054180: 2020 7d20 2020 2020 2020 2020 2020 2020    }             
│ │ │ │  00054190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000541a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000541b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000541c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000541d0: 7461 7267 6574 3a20 5072 6f6a 2851 515b  target: Proj(QQ[
│ │ │ │ -000541e0: 7920 2c20 7920 2c20 7920 2c20 7920 2c20  y , y , y , y , 
│ │ │ │ -000541f0: 7920 5d29 2020 2020 2020 2020 2020 2020  y ])            
│ │ │ │ +000541c0: 2020 7c0a 7c20 2020 2020 7461 7267 6574    |.|     target
│ │ │ │ +000541d0: 3a20 5072 6f6a 2851 515b 7920 2c20 7920  : Proj(QQ[y , y 
│ │ │ │ +000541e0: 2c20 7920 2c20 7920 2c20 7920 5d29 2020  , y , y , y ])  
│ │ │ │ +000541f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054210: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054230: 2030 2020 2031 2020 2032 2020 2033 2020   0   1   2   3  
│ │ │ │ -00054240: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +00054210: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054220: 2020 2020 2020 2020 2020 2030 2020 2031             0   1
│ │ │ │ +00054230: 2020 2032 2020 2033 2020 2034 2020 2020     2   3   4    
│ │ │ │ +00054240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054260: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054270: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20  defining forms: 
│ │ │ │ -00054280: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │ +00054260: 2020 7c0a 7c20 2020 2020 6465 6669 6e69    |.|     defini
│ │ │ │ +00054270: 6e67 2066 6f72 6d73 3a20 7b20 2020 2020  ng forms: {     
│ │ │ │ +00054280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000542a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000542b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000542c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000542d0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +000542b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000542c0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +000542d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000542e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000542f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054300: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054320: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
│ │ │ │ +00054300: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054310: 2020 2020 2020 2020 2020 2078 2020 2d20             x  - 
│ │ │ │ +00054320: 7820 7820 2c20 2020 2020 2020 2020 2020  x x ,           
│ │ │ │  00054330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054350: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054370: 2020 3420 2020 2033 2035 2020 2020 2020    4    3 5      
│ │ │ │ +00054350: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054360: 2020 2020 2020 2020 2020 2020 3420 2020              4   
│ │ │ │ +00054370: 2033 2035 2020 2020 2020 2020 2020 2020   3 5            
│ │ │ │  00054380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000543a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000543a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000543b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000543c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000543d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000543e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000543f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054410: 202d 2078 2078 2020 2b20 7820 7820 202d   - x x  + x x  -
│ │ │ │ -00054420: 2078 2078 202c 2020 2020 2020 2020 2020   x x ,          
│ │ │ │ +000543f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054400: 2020 2020 2020 2020 2020 202d 2078 2078             - x x
│ │ │ │ +00054410: 2020 2b20 7820 7820 202d 2078 2078 202c    + x x  - x x ,
│ │ │ │ +00054420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054440: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054460: 2020 2020 3220 3420 2020 2033 2034 2020      2 4    3 4  
│ │ │ │ -00054470: 2020 3120 3520 2020 2020 2020 2020 2020    1 5           
│ │ │ │ +00054440: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054450: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00054460: 3420 2020 2033 2034 2020 2020 3120 3520  4    3 4    1 5 
│ │ │ │ +00054470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054490: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00054490: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000544a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000544e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000544f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054500: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +00054500: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00054510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054530: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054550: 202d 2078 2078 2020 2b20 7820 202d 2078   - x x  + x  - x
│ │ │ │ -00054560: 2078 202c 2020 2020 2020 2020 2020 2020   x ,            
│ │ │ │ +00054530: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054540: 2020 2020 2020 2020 2020 202d 2078 2078             - x x
│ │ │ │ +00054550: 2020 2b20 7820 202d 2078 2078 202c 2020    + x  - x x ,  
│ │ │ │ +00054560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054580: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000545a0: 2020 2020 3220 3320 2020 2033 2020 2020      2 3    3    
│ │ │ │ -000545b0: 3120 3420 2020 2020 2020 2020 2020 2020  1 4             
│ │ │ │ +00054580: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054590: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +000545a0: 3320 2020 2033 2020 2020 3120 3420 2020  3    3    1 4   
│ │ │ │ +000545b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000545c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000545d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000545d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000545e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000545f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054620: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054640: 202d 2078 2078 2020 2b20 7820 7820 202d   - x x  + x x  -
│ │ │ │ -00054650: 2078 2078 202c 2020 2020 2020 2020 2020   x x ,          
│ │ │ │ +00054620: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054630: 2020 2020 2020 2020 2020 202d 2078 2078             - x x
│ │ │ │ +00054640: 2020 2b20 7820 7820 202d 2078 2078 202c    + x x  - x x ,
│ │ │ │ +00054650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054670: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054690: 2020 2020 3120 3220 2020 2031 2033 2020      1 2    1 3  
│ │ │ │ -000546a0: 2020 3020 3420 2020 2020 2020 2020 2020    0 4           
│ │ │ │ +00054670: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054680: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +00054690: 3220 2020 2031 2033 2020 2020 3020 3420  2    1 3    0 4 
│ │ │ │ +000546a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000546b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000546c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000546c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054710: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054730: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00054710: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054720: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00054730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054760: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054780: 2078 2020 2d20 7820 7820 2020 2020 2020   x  - x x       
│ │ │ │ +00054760: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054770: 2020 2020 2020 2020 2020 2078 2020 2d20             x  - 
│ │ │ │ +00054780: 7820 7820 2020 2020 2020 2020 2020 2020  x x             
│ │ │ │  00054790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000547a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000547b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000547c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000547d0: 2020 3120 2020 2030 2033 2020 2020 2020    1    0 3      
│ │ │ │ +000547b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000547c0: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +000547d0: 2030 2033 2020 2020 2020 2020 2020 2020   0 3            
│ │ │ │  000547e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000547f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054800: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054820: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00054800: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054810: 2020 2020 2020 2020 2020 7d20 2020 2020            }     
│ │ │ │ +00054820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054850: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00054850: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00054860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000548a0: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ -000548b0: 5261 7469 6f6e 616c 4d61 7020 2871 7561  RationalMap (qua
│ │ │ │ -000548c0: 6472 6174 6963 2072 6174 696f 6e61 6c20  dratic rational 
│ │ │ │ -000548d0: 6d61 7020 6672 6f6d 2068 7970 6572 7375  map from hypersu
│ │ │ │ -000548e0: 7266 6163 6520 696e 2050 505e 3520 746f  rface in PP^5 to
│ │ │ │ -000548f0: 2050 505e 3429 2020 7c0a 7c2d 2d2d 2d2d   PP^4)  |.|-----
│ │ │ │ +000548a0: 2020 7c0a 7c6f 3220 3a20 5261 7469 6f6e    |.|o2 : Ration
│ │ │ │ +000548b0: 616c 4d61 7020 2871 7561 6472 6174 6963  alMap (quadratic
│ │ │ │ +000548c0: 2072 6174 696f 6e61 6c20 6d61 7020 6672   rational map fr
│ │ │ │ +000548d0: 6f6d 2068 7970 6572 7375 7266 6163 6520  om hypersurface 
│ │ │ │ +000548e0: 696e 2050 505e 3520 746f 2050 505e 3429  in PP^5 to PP^4)
│ │ │ │ +000548f0: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │  00054900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054940: 2d2d 2d2d 2d2d 2d2d 7c0a 7c32 2d79 5f31  --------|.|2-y_1
│ │ │ │ -00054950: 2a79 5f34 2c20 2d79 5f31 2a79 5f32 2b79  *y_4, -y_1*y_2+y
│ │ │ │ -00054960: 5f31 2a79 5f33 2d79 5f30 2a79 5f34 2c20  _1*y_3-y_0*y_4, 
│ │ │ │ -00054970: 795f 315e 322d 795f 302a 795f 337d 2020  y_1^2-y_0*y_3}  
│ │ │ │ +00054940: 2d2d 7c0a 7c32 2d79 5f31 2a79 5f34 2c20  --|.|2-y_1*y_4, 
│ │ │ │ +00054950: 2d79 5f31 2a79 5f32 2b79 5f31 2a79 5f33  -y_1*y_2+y_1*y_3
│ │ │ │ +00054960: 2d79 5f30 2a79 5f34 2c20 795f 315e 322d  -y_0*y_4, y_1^2-
│ │ │ │ +00054970: 795f 302a 795f 337d 2020 2020 2020 2020  y_0*y_3}        
│ │ │ │  00054980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054990: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00054990: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  000549a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000549b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000549c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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  
│ │ │ │ +000549e0: 2d2d 2b0a 7c69 3320 3a20 7469 6d65 2069  --+.|i3 : time i
│ │ │ │ +000549f0: 6465 616c 2070 6869 2020 2020 2020 2020  deal 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 3335 3539 3973 2028  sed 0.0035599s (
│ │ │ │ -00054a50: 6370 7529 3b20 302e 3030 3335 3535 3232  cpu); 0.00355522
│ │ │ │ -00054a60: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -00054a70: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ -00054a80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00054a30: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ +00054a40: 3030 3530 3233 3232 7320 2863 7075 293b  00502322s (cpu);
│ │ │ │ +00054a50: 2030 2e30 3035 3032 3132 3373 2028 7468   0.00502123s (th
│ │ │ │ +00054a60: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +00054a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054a80: 2020 7c0a 7c20 2020 2020 2020 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       
│ │ │ │ +00054ad0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054ae0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  00054af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054b00: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00054b00: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  00054b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054b20: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ -00054b30: 6964 6561 6c20 2878 2020 2d20 7820 7820  ideal (x  - x x 
│ │ │ │ -00054b40: 2c20 7820 7820 202d 2078 2078 2020 2b20  , x x  - x x  + 
│ │ │ │ -00054b50: 7820 7820 2c20 7820 7820 202d 2078 2020  x x , x x  - x  
│ │ │ │ -00054b60: 2b20 7820 7820 2c20 7820 7820 202d 2078  + x x , x x  - x
│ │ │ │ -00054b70: 2078 2020 2b20 2020 7c0a 7c20 2020 2020   x  +   |.|     
│ │ │ │ -00054b80: 2020 2020 2020 2020 3420 2020 2033 2035          4    3 5
│ │ │ │ -00054b90: 2020 2032 2034 2020 2020 3320 3420 2020     2 4    3 4   
│ │ │ │ -00054ba0: 2031 2035 2020 2032 2033 2020 2020 3320   1 5   2 3    3 
│ │ │ │ -00054bb0: 2020 2031 2034 2020 2031 2032 2020 2020     1 4   1 2    
│ │ │ │ -00054bc0: 3120 3320 2020 2020 7c0a 7c20 2020 2020  1 3     |.|     
│ │ │ │ +00054b20: 2020 7c0a 7c6f 3320 3d20 6964 6561 6c20    |.|o3 = ideal 
│ │ │ │ +00054b30: 2878 2020 2d20 7820 7820 2c20 7820 7820  (x  - x x , x x 
│ │ │ │ +00054b40: 202d 2078 2078 2020 2b20 7820 7820 2c20   - x x  + x x , 
│ │ │ │ +00054b50: 7820 7820 202d 2078 2020 2b20 7820 7820  x x  - x  + x x 
│ │ │ │ +00054b60: 2c20 7820 7820 202d 2078 2078 2020 2b20  , x x  - x x  + 
│ │ │ │ +00054b70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054b80: 2020 3420 2020 2033 2035 2020 2032 2034    4    3 5   2 4
│ │ │ │ +00054b90: 2020 2020 3320 3420 2020 2031 2035 2020      3 4    1 5  
│ │ │ │ +00054ba0: 2032 2033 2020 2020 3320 2020 2031 2034   2 3    3    1 4
│ │ │ │ +00054bb0: 2020 2031 2032 2020 2020 3120 3320 2020     1 2    1 3   
│ │ │ │ +00054bc0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d    |.|     ------
│ │ │ │  00054bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054c10: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ -00054c20: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00054c10: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │ +00054c20: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  00054c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054c60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054c70: 7820 7820 2c20 7820 202d 2078 2078 2029  x x , x  - x x )
│ │ │ │ +00054c60: 2020 7c0a 7c20 2020 2020 7820 7820 2c20    |.|     x x , 
│ │ │ │ +00054c70: 7820 202d 2078 2078 2029 2020 2020 2020  x  - x x )      
│ │ │ │  00054c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054cb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054cc0: 2030 2034 2020 2031 2020 2020 3020 3320   0 4   1    0 3 
│ │ │ │ +00054cb0: 2020 7c0a 7c20 2020 2020 2030 2034 2020    |.|      0 4  
│ │ │ │ +00054cc0: 2031 2020 2020 3020 3320 2020 2020 2020   1    0 3       
│ │ │ │  00054cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054d00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00054d00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00054d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054d50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054d70: 5151 5b78 202e 2e78 205d 2020 2020 2020  QQ[x ..x ]      
│ │ │ │ +00054d50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054d60: 2020 2020 2020 2020 2020 5151 5b78 202e            QQ[x .
│ │ │ │ +00054d70: 2e78 205d 2020 2020 2020 2020 2020 2020  .x ]            
│ │ │ │  00054d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054da0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054dc0: 2020 2020 3020 2020 3520 2020 2020 2020      0   5       
│ │ │ │ +00054da0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054db0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00054dc0: 2020 3520 2020 2020 2020 2020 2020 2020    5             
│ │ │ │  00054dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054df0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ -00054e00: 4964 6561 6c20 6f66 202d 2d2d 2d2d 2d2d  Ideal of -------
│ │ │ │ -00054e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054df0: 2020 7c0a 7c6f 3320 3a20 4964 6561 6c20    |.|o3 : Ideal 
│ │ │ │ +00054e00: 6f66 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  of -------------
│ │ │ │ +00054e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020 2020  ----------      
│ │ │ │  00054e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054e40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054e50: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00054e40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054e50: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  00054e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054e90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054ea0: 2020 2020 2020 2020 2078 2020 2d20 7820           x  - x 
│ │ │ │ -00054eb0: 7820 202b 2078 2078 2020 2d20 7820 7820  x  + x x  - x x 
│ │ │ │ +00054e90: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054ea0: 2020 2078 2020 2d20 7820 7820 202b 2078     x  - x x  + x
│ │ │ │ +00054eb0: 2078 2020 2d20 7820 7820 2020 2020 2020   x  - x x       
│ │ │ │  00054ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054ee0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054ef0: 2020 2020 2020 2020 2020 3220 2020 2032            2    2
│ │ │ │ -00054f00: 2033 2020 2020 3120 3420 2020 2030 2035   3    1 4    0 5
│ │ │ │ +00054ee0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00054ef0: 2020 2020 3220 2020 2032 2033 2020 2020      2    2 3    
│ │ │ │ +00054f00: 3120 3420 2020 2030 2035 2020 2020 2020  1 4    0 5      
│ │ │ │  00054f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054f30: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00054f30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00054f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054f80: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ -00054f90: 6173 7365 7274 2869 6465 616c 2070 6869  assert(ideal phi
│ │ │ │ -00054fa0: 203d 3d20 6964 6561 6c20 6d61 7472 6978   == ideal matrix
│ │ │ │ -00054fb0: 2070 6869 2920 2020 2020 2020 2020 2020   phi)           
│ │ │ │ +00054f80: 2d2d 2b0a 7c69 3420 3a20 6173 7365 7274  --+.|i4 : assert
│ │ │ │ +00054f90: 2869 6465 616c 2070 6869 203d 3d20 6964  (ideal phi == id
│ │ │ │ +00054fa0: 6561 6c20 6d61 7472 6978 2070 6869 2920  eal matrix phi) 
│ │ │ │ +00054fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054fd0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00054fd0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00054fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055020: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ -00055030: 7068 6927 203d 206c 6173 7420 6772 6170  phi' = last grap
│ │ │ │ -00055040: 6820 7068 6920 2020 2020 2020 2020 2020  h phi           
│ │ │ │ +00055020: 2d2d 2b0a 7c69 3520 3a20 7068 6927 203d  --+.|i5 : phi' =
│ │ │ │ +00055030: 206c 6173 7420 6772 6170 6820 7068 6920   last graph phi 
│ │ │ │ +00055040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055070: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00055070: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00055080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000550a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000550b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000550c0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ -000550d0: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ -000550e0: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +000550c0: 2020 7c0a 7c6f 3520 3d20 2d2d 2072 6174    |.|o5 = -- rat
│ │ │ │ +000550d0: 696f 6e61 6c20 6d61 7020 2d2d 2020 2020  ional map --    
│ │ │ │ +000550e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000550f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055110: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055120: 736f 7572 6365 3a20 7375 6276 6172 6965  source: subvarie
│ │ │ │ -00055130: 7479 206f 6620 5072 6f6a 2851 515b 7820  ty of Proj(QQ[x 
│ │ │ │ -00055140: 2c20 7820 2c20 7820 2c20 2020 2020 2020  , x , x ,       
│ │ │ │ +00055110: 2020 7c0a 7c20 2020 2020 736f 7572 6365    |.|     source
│ │ │ │ +00055120: 3a20 7375 6276 6172 6965 7479 206f 6620  : subvariety of 
│ │ │ │ +00055130: 5072 6f6a 2851 515b 7820 2c20 7820 2c20  Proj(QQ[x , x , 
│ │ │ │ +00055140: 7820 2c20 2020 2020 2020 2020 2020 2020  x ,             
│ │ │ │  00055150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055160: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00055160: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00055170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055180: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00055190: 2020 2031 2020 2032 2020 2020 2020 2020     1   2        
│ │ │ │ +00055180: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ +00055190: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  000551a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000551b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000551c0: 2020 2020 2020 2020 7b20 2020 2020 2020          {       
│ │ │ │ +000551b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000551c0: 2020 7b20 2020 2020 2020 2020 2020 2020    {             
│ │ │ │  000551d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000551e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000551f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055200: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055210: 2020 2020 2020 2020 2078 2079 2020 2d20           x y  - 
│ │ │ │ -00055220: 7820 7920 202b 2078 2079 202c 2020 2020  x y  + x y ,    
│ │ │ │ +00055200: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00055210: 2020 2078 2079 2020 2d20 7820 7920 202b     x y  - x y  +
│ │ │ │ +00055220: 2078 2079 202c 2020 2020 2020 2020 2020   x y ,          
│ │ │ │  00055230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055250: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055260: 2020 2020 2020 2020 2020 3120 3220 2020            1 2   
│ │ │ │ -00055270: 2033 2033 2020 2020 3420 3420 2020 2020   3 3    4 4     
│ │ │ │ +00055250: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00055260: 2020 2020 3120 3220 2020 2033 2033 2020      1 2    3 3  
│ │ │ │ +00055270: 2020 3420 3420 2020 2020 2020 2020 2020    4 4           
│ │ │ │  00055280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000552a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000552a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000552b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000552c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000552d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000552e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000552f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055300: 2020 2020 2020 2020 2078 2079 2020 2d20           x y  - 
│ │ │ │ -00055310: 7820 7920 202d 2078 2079 2020 2b20 7820  x y  - x y  + x 
│ │ │ │ -00055320: 7920 2c20 2020 2020 2020 2020 2020 2020  y ,             
│ │ │ │ +000552f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00055300: 2020 2078 2079 2020 2d20 7820 7920 202d     x y  - x y  -
│ │ │ │ +00055310: 2078 2079 2020 2b20 7820 7920 2c20 2020   x y  + x y ,   
│ │ │ │ +00055320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055340: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055350: 2020 2020 2020 2020 2020 3020 3220 2020            0 2   
│ │ │ │ -00055360: 2031 2033 2020 2020 3220 3420 2020 2033   1 3    2 4    3
│ │ │ │ -00055370: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +00055340: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00055350: 2020 2020 3020 3220 2020 2031 2033 2020      0 2    1 3  
│ │ │ │ +00055360: 2020 3220 3420 2020 2033 2034 2020 2020    2 4    3 4    
│ │ │ │ +00055370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055390: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00055390: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000553a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000553b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000553c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000553d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000553e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000553f0: 2020 2020 2020 2020 2078 2079 2020 2d20           x y  - 
│ │ │ │ -00055400: 7820 7920 202b 2078 2079 2020 2d20 7820  x y  + x y  - x 
│ │ │ │ -00055410: 7920 2c20 2020 2020 2020 2020 2020 2020  y ,             
│ │ │ │ +000553e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000553f0: 2020 2078 2079 2020 2d20 7820 7920 202b     x y  - x y  +
│ │ │ │ +00055400: 2078 2079 2020 2d20 7820 7920 2c20 2020   x y  - x y ,   
│ │ │ │ +00055410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055430: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055440: 2020 2020 2020 2020 2020 3220 3120 2020            2 1   
│ │ │ │ -00055450: 2033 2031 2020 2020 3420 3220 2020 2035   3 1    4 2    5
│ │ │ │ -00055460: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +00055430: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00055440: 2020 2020 3220 3120 2020 2033 2031 2020      2 1    3 1  
│ │ │ │ +00055450: 2020 3420 3220 2020 2035 2033 2020 2020    4 2    5 3    
│ │ │ │ +00055460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055480: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00055480: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00055490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000554a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000554b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000554c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000554d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000554e0: 2020 2020 2020 2020 2078 2079 2020 2d20           x y  - 
│ │ │ │ -000554f0: 7820 7920 202b 2078 2079 202c 2020 2020  x y  + x y ,    
│ │ │ │ +000554d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000554e0: 2020 2078 2079 2020 2d20 7820 7920 202b     x y  - x y  +
│ │ │ │ +000554f0: 2078 2079 202c 2020 2020 2020 2020 2020   x y ,          
│ │ │ │  00055500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055520: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055530: 2020 2020 2020 2020 2020 3120 3120 2020            1 1   
│ │ │ │ -00055540: 2032 2032 2020 2020 3520 3420 2020 2020   2 2    5 4     
│ │ │ │ +00055520: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00055530: 2020 2020 3120 3120 2020 2032 2032 2020      1 1    2 2  
│ │ │ │ +00055540: 2020 3520 3420 2020 2020 2020 2020 2020    5 4           
│ │ │ │  00055550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055570: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00055570: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00055580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000555a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000555b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000555c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000555d0: 2020 2020 2020 2020 2078 2079 2020 2d20           x y  - 
│ │ │ │ -000555e0: 7820 7920 202b 2078 2079 202c 2020 2020  x y  + x y ,    
│ │ │ │ +000555c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000555d0: 2020 2078 2079 2020 2d20 7820 7920 202b     x y  - x y  +
│ │ │ │ +000555e0: 2078 2079 202c 2020 2020 2020 2020 2020   x y ,          
│ │ │ │  000555f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055610: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055620: 2020 2020 2020 2020 2020 3020 3120 2020            0 1   
│ │ │ │ -00055630: 2032 2033 2020 2020 3420 3420 2020 2020   2 3    4 4     
│ │ │ │ +00055610: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00055620: 2020 2020 3020 3120 2020 2032 2033 2020      0 1    2 3  
│ │ │ │ +00055630: 2020 3420 3420 2020 2020 2020 2020 2020    4 4           
│ │ │ │  00055640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055660: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00055660: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
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│ │ │ │ -000556b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000556c0: 2020 2020 2020 2020 2078 2079 2020 2d20           x y  - 
│ │ │ │ -000556d0: 7820 7920 202b 2078 2079 2020 2d20 7820  x y  + x y  - x 
│ │ │ │ -000556e0: 7920 2c20 2020 2020 2020 2020 2020 2020  y ,             
│ │ │ │ +000556b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000556c0: 2020 2078 2079 2020 2d20 7820 7920 202b     x y  - x y  +
│ │ │ │ +000556d0: 2078 2079 2020 2d20 7820 7920 2c20 2020   x y  - x y ,   
│ │ │ │ +000556e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000556f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055700: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055710: 2020 2020 2020 2020 2020 3220 3020 2020            2 0   
│ │ │ │ -00055720: 2033 2030 2020 2020 3420 3120 2020 2035   3 0    4 1    5
│ │ │ │ -00055730: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00055700: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00055710: 2020 2020 3220 3020 2020 2033 2030 2020      2 0    3 0  
│ │ │ │ +00055720: 2020 3420 3120 2020 2035 2032 2020 2020    4 1    5 2    
│ │ │ │ +00055730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055750: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00055750: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
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│ │ │ │ -000557a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000557b0: 2020 2020 2020 2020 2078 2079 2020 2d20           x y  - 
│ │ │ │ -000557c0: 7820 7920 202b 2078 2079 202c 2020 2020  x y  + x y ,    
│ │ │ │ +000557a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000557b0: 2020 2078 2079 2020 2d20 7820 7920 202b     x y  - x y  +
│ │ │ │ +000557c0: 2078 2079 202c 2020 2020 2020 2020 2020   x y ,          
│ │ │ │  000557d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000557e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000557f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055800: 2020 2020 2020 2020 2020 3120 3020 2020            1 0   
│ │ │ │ -00055810: 2033 2031 2020 2020 3420 3220 2020 2020   3 1    4 2     
│ │ │ │ +000557f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00055800: 2020 2020 3120 3020 2020 2033 2031 2020      1 0    3 1  
│ │ │ │ +00055810: 2020 3420 3220 2020 2020 2020 2020 2020    4 2           
│ │ │ │  00055820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055840: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00055840: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00055850: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00055890: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000558a0: 2020 2020 2020 2020 2078 2079 2020 2d20           x y  - 
│ │ │ │ -000558b0: 7820 7920 202b 2078 2079 202c 2020 2020  x y  + x y ,    
│ │ │ │ +00055890: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000558a0: 2020 2078 2079 2020 2d20 7820 7920 202b     x y  - x y  +
│ │ │ │ +000558b0: 2078 2079 202c 2020 2020 2020 2020 2020   x y ,          
│ │ │ │  000558c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000558d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000558e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000558f0: 2020 2020 2020 2020 2020 3020 3020 2020            0 0   
│ │ │ │ -00055900: 2032 2032 2020 2020 3420 3320 2020 2020   2 2    4 3     
│ │ │ │ +000558e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000558f0: 2020 2020 3020 3020 2020 2032 2032 2020      0 0    2 2  
│ │ │ │ +00055900: 2020 3420 3320 2020 2020 2020 2020 2020    4 3           
│ │ │ │  00055910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055930: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │  000559a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000559d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000559e0: 2020 2020 2020 2020 2078 2020 2d20 7820           x  - x 
│ │ │ │ -000559f0: 7820 202b 2078 2078 2020 2d20 7820 7820  x  + x x  - x x 
│ │ │ │ +000559d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000559e0: 2020 2078 2020 2d20 7820 7820 202b 2078     x  - x x  + x
│ │ │ │ +000559f0: 2078 2020 2d20 7820 7820 2020 2020 2020   x  - x x       
│ │ │ │  00055a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00055a20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055a30: 2020 2020 2020 2020 2020 3220 2020 2032            2    2
│ │ │ │ -00055a40: 2033 2020 2020 3120 3420 2020 2030 2035   3    1 4    0 5
│ │ │ │ +00055a20: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00055a30: 2020 2020 3220 2020 2032 2033 2020 2020      2    2 3    
│ │ │ │ +00055a40: 3120 3420 2020 2030 2035 2020 2020 2020  1 4    0 5      
│ │ │ │  00055a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055a70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055a80: 2020 2020 2020 2020 7d20 2020 2020 2020          }       
│ │ │ │ +00055a70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00055a80: 2020 7d20 2020 2020 2020 2020 2020 2020    }             
│ │ │ │  00055a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00055ac0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055ad0: 7461 7267 6574 3a20 5072 6f6a 2851 515b  target: Proj(QQ[
│ │ │ │ -00055ae0: 7920 2c20 7920 2c20 7920 2c20 7920 2c20  y , y , y , y , 
│ │ │ │ -00055af0: 7920 5d29 2020 2020 2020 2020 2020 2020  y ])            
│ │ │ │ +00055ac0: 2020 7c0a 7c20 2020 2020 7461 7267 6574    |.|     target
│ │ │ │ +00055ad0: 3a20 5072 6f6a 2851 515b 7920 2c20 7920  : Proj(QQ[y , y 
│ │ │ │ +00055ae0: 2c20 7920 2c20 7920 2c20 7920 5d29 2020  , y , y , y ])  
│ │ │ │ +00055af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055b10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055b30: 2030 2020 2031 2020 2032 2020 2033 2020   0   1   2   3  
│ │ │ │ -00055b40: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +00055b10: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00055b20: 2020 2020 2020 2020 2020 2030 2020 2031             0   1
│ │ │ │ +00055b30: 2020 2032 2020 2033 2020 2034 2020 2020     2   3   4    
│ │ │ │ +00055b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055b60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055b70: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20  defining forms: 
│ │ │ │ -00055b80: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │ +00055b60: 2020 7c0a 7c20 2020 2020 6465 6669 6e69    |.|     defini
│ │ │ │ +00055b70: 6e67 2066 6f72 6d73 3a20 7b20 2020 2020  ng forms: {     
│ │ │ │ +00055b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055bb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055bd0: 2079 202c 2020 2020 2020 2020 2020 2020   y ,            
│ │ │ │ +00055bb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00055bc0: 2020 2020 2020 2020 2020 2079 202c 2020             y ,  
│ │ │ │ +00055bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055c00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055c20: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00055c00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00055c10: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +00055c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055c50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00055c50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00055c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055ca0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055cc0: 2079 202c 2020 2020 2020 2020 2020 2020   y ,            
│ │ │ │ +00055ca0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00055cb0: 2020 2020 2020 2020 2020 2079 202c 2020             y ,  
│ │ │ │ +00055cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00055ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055cf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00055d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00055d10: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +00055cf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00055d00: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +00055d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00056c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056c90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00056c90: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00056ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ce0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00056ce0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00056cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056d30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00056d30: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00056d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056d80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00056d80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00056d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056dd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00056dd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00056de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056e20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00056e20: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00056e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056e70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00056e70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00056e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ec0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00056ec0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00056ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056f10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00056f10: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00056f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056f60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00056f60: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00056f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056fb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00056fb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00056fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057000: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00057000: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00057010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057050: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00057050: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00057060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000570a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000570a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000570b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000570c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000570d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000570e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000570f0: 2020 2020 2020 2020 7c0a 7c66 726f 6d20          |.|from 
│ │ │ │ -00057100: 342d 6469 6d65 6e73 696f 6e61 6c20 7375  4-dimensional su
│ │ │ │ -00057110: 6276 6172 6965 7479 206f 6620 5050 5e35  bvariety of PP^5
│ │ │ │ -00057120: 2078 2050 505e 3420 746f 2050 505e 3429   x PP^4 to PP^4)
│ │ │ │ +000570f0: 2020 7c0a 7c66 726f 6d20 342d 6469 6d65    |.|from 4-dime
│ │ │ │ +00057100: 6e73 696f 6e61 6c20 7375 6276 6172 6965  nsional subvarie
│ │ │ │ +00057110: 7479 206f 6620 5050 5e35 2078 2050 505e  ty of PP^5 x PP^
│ │ │ │ +00057120: 3420 746f 2050 505e 3429 2020 2020 2020  4 to PP^4)      
│ │ │ │  00057130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057140: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00057140: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00057150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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' 
│ │ │ │ +00057190: 2d2d 2b0a 7c69 3620 3a20 7469 6d65 2069  --+.|i6 : time i
│ │ │ │ +000571a0: 6465 616c 2070 6869 2720 2020 2020 2020  deal 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 3330 3639 3173 2028  sed 0.0930691s (
│ │ │ │ -00057200: 6370 7529 3b20 302e 3039 3330 3438 3873  cpu); 0.0930488s
│ │ │ │ -00057210: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -00057220: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │ -00057230: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000571e0: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ +000571f0: 3131 3432 3639 7320 2863 7075 293b 2030  114269s (cpu); 0
│ │ │ │ +00057200: 2e31 3134 3236 3873 2028 7468 7265 6164  .114268s (thread
│ │ │ │ +00057210: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +00057220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057230: 2020 7c0a 7c20 2020 2020 2020 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         
│ │ │ │ +00057280: 2020 7c0a 7c6f 3620 3d20 6964 6561 6c20    |.|o6 = ideal 
│ │ │ │ +00057290: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  000572a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000572b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000572c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000572d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000572d0: 2020 7c0a 7c20 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 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00057320: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00057330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057370: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00057370: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00057380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000573a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000573b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000573c0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ -000573d0: 4964 6561 6c20 6f66 202d 2d2d 2d2d 2d2d  Ideal of -------
│ │ │ │ +000573c0: 2020 7c0a 7c6f 3620 3a20 4964 6561 6c20    |.|o6 : Ideal 
│ │ │ │ +000573d0: 6f66 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  of -------------
│ │ │ │  000573e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000573f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057410: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +00057410: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │  00057420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057460: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00057470: 2020 2020 2020 2020 2028 7820 7920 202d           (x y  -
│ │ │ │ -00057480: 2078 2079 2020 2b20 7820 7920 2c20 7820   x y  + x y , x 
│ │ │ │ -00057490: 7920 202d 2078 2079 2020 2d20 7820 7920  y  - x y  - x y 
│ │ │ │ -000574a0: 202b 2078 2079 202c 2078 2079 2020 2d20   + x y , x y  - 
│ │ │ │ -000574b0: 7820 7920 202b 2078 7c0a 7c20 2020 2020  x y  + x|.|     
│ │ │ │ -000574c0: 2020 2020 2020 2020 2020 2031 2032 2020             1 2  
│ │ │ │ -000574d0: 2020 3320 3320 2020 2034 2034 2020 2030    3 3    4 4   0
│ │ │ │ -000574e0: 2032 2020 2020 3120 3320 2020 2032 2034   2    1 3    2 4
│ │ │ │ -000574f0: 2020 2020 3320 3420 2020 3220 3120 2020      3 4   2 1   
│ │ │ │ -00057500: 2033 2031 2020 2020 7c0a 7c2d 2d2d 2d2d   3 1    |.|-----
│ │ │ │ +00057460: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00057470: 2020 2028 7820 7920 202d 2078 2079 2020     (x y  - x y  
│ │ │ │ +00057480: 2b20 7820 7920 2c20 7820 7920 202d 2078  + x y , x y  - x
│ │ │ │ +00057490: 2079 2020 2d20 7820 7920 202b 2078 2079   y  - x y  + x y
│ │ │ │ +000574a0: 202c 2078 2079 2020 2d20 7820 7920 202b   , x y  - x y  +
│ │ │ │ +000574b0: 2078 7c0a 7c20 2020 2020 2020 2020 2020   x|.|           
│ │ │ │ +000574c0: 2020 2020 2031 2032 2020 2020 3320 3320       1 2    3 3 
│ │ │ │ +000574d0: 2020 2034 2034 2020 2030 2032 2020 2020     4 4   0 2    
│ │ │ │ +000574e0: 3120 3320 2020 2032 2034 2020 2020 3320  1 3    2 4    3 
│ │ │ │ +000574f0: 3420 2020 3220 3120 2020 2033 2031 2020  4   2 1    3 1  
│ │ │ │ +00057500: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │  00057510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057550: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +00057550: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │  00057560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057570: 2020 2020 2020 2020 2020 5151 5b78 202e            QQ[x .
│ │ │ │ -00057580: 2e78 202c 2079 202e 2e79 205d 2020 2020  .x , y ..y ]    
│ │ │ │ +00057570: 2020 2020 5151 5b78 202e 2e78 202c 2079      QQ[x ..x , y
│ │ │ │ +00057580: 202e 2e79 205d 2020 2020 2020 2020 2020   ..y ]          
│ │ │ │  00057590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000575a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000575a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000575b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000575c0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -000575d0: 2020 3520 2020 3020 2020 3420 2020 2020    5   0   4     
│ │ │ │ +000575c0: 2020 2020 2020 2020 3020 2020 3520 2020          0   5   
│ │ │ │ +000575d0: 3020 2020 3420 2020 2020 2020 2020 2020  0   4           
│ │ │ │  000575e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000575f0: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d          |.|-----
│ │ │ │ +000575f0: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │  00057600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057640: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +00057640: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │  00057650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057690: 2020 2020 2020 2020 7c0a 7c20 7920 202d          |.| y  -
│ │ │ │ -000576a0: 2078 2079 202c 2078 2079 2020 2d20 7820   x y , x y  - x 
│ │ │ │ -000576b0: 7920 202b 2078 2079 202c 2078 2079 2020  y  + x y , x y  
│ │ │ │ -000576c0: 2d20 7820 7920 202b 2078 2079 202c 2078  - x y  + x y , x
│ │ │ │ -000576d0: 2079 2020 2d20 7820 7920 202b 2078 2079   y  - x y  + x y
│ │ │ │ -000576e0: 2020 2d20 7820 7920 7c0a 7c34 2032 2020    - x y |.|4 2  
│ │ │ │ -000576f0: 2020 3520 3320 2020 3120 3120 2020 2032    5 3   1 1    2
│ │ │ │ -00057700: 2032 2020 2020 3520 3420 2020 3020 3120   2    5 4   0 1 
│ │ │ │ -00057710: 2020 2032 2033 2020 2020 3420 3420 2020     2 3    4 4   
│ │ │ │ -00057720: 3220 3020 2020 2033 2030 2020 2020 3420  2 0    3 0    4 
│ │ │ │ -00057730: 3120 2020 2035 2032 7c0a 7c2d 2d2d 2d2d  1    5 2|.|-----
│ │ │ │ +00057690: 2020 7c0a 7c20 7920 202d 2078 2079 202c    |.| y  - x y ,
│ │ │ │ +000576a0: 2078 2079 2020 2d20 7820 7920 202b 2078   x y  - x y  + x
│ │ │ │ +000576b0: 2079 202c 2078 2079 2020 2d20 7820 7920   y , x y  - x y 
│ │ │ │ +000576c0: 202b 2078 2079 202c 2078 2079 2020 2d20   + x y , x y  - 
│ │ │ │ +000576d0: 7820 7920 202b 2078 2079 2020 2d20 7820  x y  + x y  - x 
│ │ │ │ +000576e0: 7920 7c0a 7c34 2032 2020 2020 3520 3320  y |.|4 2    5 3 
│ │ │ │ +000576f0: 2020 3120 3120 2020 2032 2032 2020 2020    1 1    2 2    
│ │ │ │ +00057700: 3520 3420 2020 3020 3120 2020 2032 2033  5 4   0 1    2 3
│ │ │ │ +00057710: 2020 2020 3420 3420 2020 3220 3020 2020      4 4   2 0   
│ │ │ │ +00057720: 2033 2030 2020 2020 3420 3120 2020 2035   3 0    4 1    5
│ │ │ │ +00057730: 2032 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d   2|.|-----------
│ │ │ │  00057740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057780: 2d2d 2d2d 2d2d 2d2d 7c0a 7c2d 2d2d 2d2d  --------|.|-----
│ │ │ │ +00057780: 2d2d 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d  --|.|-----------
│ │ │ │  00057790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000577a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000577b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000577c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d20 2020  -------------   
│ │ │ │ -000577d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000577c0: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020  -------         
│ │ │ │ +000577d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000577e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000577f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057800: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00057800: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  00057810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057820: 2020 2020 2020 2020 7c0a 7c2c 2078 2079          |.|, x y
│ │ │ │ -00057830: 2020 2d20 7820 7920 202b 2078 2079 202c    - x y  + x y ,
│ │ │ │ -00057840: 2078 2079 2020 2d20 7820 7920 202b 2078   x y  - x y  + x
│ │ │ │ -00057850: 2079 202c 2078 2020 2d20 7820 7820 202b   y , x  - x x  +
│ │ │ │ -00057860: 2078 2078 2020 2d20 7820 7820 2920 2020   x x  - x x )   
│ │ │ │ -00057870: 2020 2020 2020 2020 7c0a 7c20 2020 3120          |.|   1 
│ │ │ │ -00057880: 3020 2020 2033 2031 2020 2020 3420 3220  0    3 1    4 2 
│ │ │ │ -00057890: 2020 3020 3020 2020 2032 2032 2020 2020    0 0    2 2    
│ │ │ │ -000578a0: 3420 3320 2020 3220 2020 2032 2033 2020  4 3   2    2 3  
│ │ │ │ -000578b0: 2020 3120 3420 2020 2030 2035 2020 2020    1 4    0 5    
│ │ │ │ -000578c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
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│ │ │ │ +00057840: 2d20 7820 7920 202b 2078 2079 202c 2078  - x y  + x y , x
│ │ │ │ +00057850: 2020 2d20 7820 7820 202b 2078 2078 2020    - x x  + x x  
│ │ │ │ +00057860: 2d20 7820 7820 2920 2020 2020 2020 2020  - x x )         
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│ │ │ │ +00057890: 2020 2032 2032 2020 2020 3420 3320 2020     2 2    4 3   
│ │ │ │ +000578a0: 3220 2020 2032 2033 2020 2020 3120 3420  2    2 3    1 4 
│ │ │ │ +000578b0: 2020 2030 2035 2020 2020 2020 2020 2020     0 5          
│ │ │ │ +000578c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  000578d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000578e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000578f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057910: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20  --------+.|i7 : 
│ │ │ │ -00057920: 6173 7365 7274 2869 6465 616c 2070 6869  assert(ideal phi
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│ │ │ │ +00057910: 2d2d 2b0a 7c69 3720 3a20 6173 7365 7274  --+.|i7 : assert
│ │ │ │ +00057920: 2869 6465 616c 2070 6869 2720 213d 2069  (ideal phi' != i
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│ │ │ │ +00057940: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00057950: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00057970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000579a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000579b0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ -000579c0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -000579d0: 2a20 2a6e 6f74 6520 6973 4d6f 7270 6869  * *note isMorphi
│ │ │ │ -000579e0: 736d 3a20 6973 4d6f 7270 6869 736d 2c20  sm: isMorphism, 
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│ │ │ │ -00057a00: 696f 6e61 6c20 6d61 7020 6973 2061 206d  ional map is a m
│ │ │ │ -00057a10: 6f72 7068 6973 6d0a 2020 2a20 2a6e 6f74  orphism.  * *not
│ │ │ │ -00057a20: 6520 6d61 7472 6978 2852 6174 696f 6e61  e matrix(Rationa
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│ │ │ │ -00057a40: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ -00057a50: 2d2d 2074 6865 206d 6174 7269 780a 2020  -- the matrix.  
│ │ │ │ -00057a60: 2020 6173 736f 6369 6174 6564 2074 6f20    associated to 
│ │ │ │ -00057a70: 6120 7261 7469 6f6e 616c 206d 6170 0a0a  a rational map..
│ │ │ │ -00057a80: 5761 7973 2074 6f20 7573 6520 7468 6973  Ways to use this
│ │ │ │ -00057a90: 206d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d   method:.=======
│ │ │ │ -00057aa0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00057ab0: 3d0a 0a20 202a 202a 6e6f 7465 2069 6465  =..  * *note ide
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│ │ │ │ -00057ad0: 2069 6465 616c 5f6c 7052 6174 696f 6e61   ideal_lpRationa
│ │ │ │ -00057ae0: 6c4d 6170 5f72 702c 202d 2d20 6261 7365  lMap_rp, -- base
│ │ │ │ -00057af0: 206c 6f63 7573 206f 6620 610a 2020 2020   locus of a.    
│ │ │ │ -00057b00: 7261 7469 6f6e 616c 206d 6170 0a2d 2d2d  rational map.---
│ │ │ │ +000579b0: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d  --+..See also.==
│ │ │ │ +000579c0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ +000579d0: 6520 6973 4d6f 7270 6869 736d 3a20 6973  e isMorphism: is
│ │ │ │ +000579e0: 4d6f 7270 6869 736d 2c20 2d2d 2077 6865  Morphism, -- whe
│ │ │ │ +000579f0: 7468 6572 2061 2072 6174 696f 6e61 6c20  ther a rational 
│ │ │ │ +00057a00: 6d61 7020 6973 2061 206d 6f72 7068 6973  map is a morphis
│ │ │ │ +00057a10: 6d0a 2020 2a20 2a6e 6f74 6520 6d61 7472  m.  * *note matr
│ │ │ │ +00057a20: 6978 2852 6174 696f 6e61 6c4d 6170 293a  ix(RationalMap):
│ │ │ │ +00057a30: 206d 6174 7269 785f 6c70 5261 7469 6f6e   matrix_lpRation
│ │ │ │ +00057a40: 616c 4d61 705f 7270 2c20 2d2d 2074 6865  alMap_rp, -- the
│ │ │ │ +00057a50: 206d 6174 7269 780a 2020 2020 6173 736f   matrix.    asso
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│ │ │ │ +00057a90: 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  d:.=============
│ │ │ │ +00057aa0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +00057ab0: 202a 6e6f 7465 2069 6465 616c 2852 6174   *note ideal(Rat
│ │ │ │ +00057ac0: 696f 6e61 6c4d 6170 293a 2069 6465 616c  ionalMap): ideal
│ │ │ │ +00057ad0: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ +00057ae0: 702c 202d 2d20 6261 7365 206c 6f63 7573  p, -- base locus
│ │ │ │ +00057af0: 206f 6620 610a 2020 2020 7261 7469 6f6e   of a.    ration
│ │ │ │ +00057b00: 616c 206d 6170 0a2d 2d2d 2d2d 2d2d 2d2d  al map.---------
│ │ │ │  00057b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
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│ │ │ │ -00057bf0: 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 653a  mona.info, Node:
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│ │ │ │ -00057c10: 6c4d 6170 5f63 6d53 7472 696e 675f 7270  lMap_cmString_rp
│ │ │ │ -00057c20: 2c20 4e65 7874 3a20 696d 6167 655f 6c70  , Next: image_lp
│ │ │ │ -00057c30: 5261 7469 6f6e 616c 4d61 705f 636d 5a5a  RationalMap_cmZZ
│ │ │ │ -00057c40: 5f72 702c 2050 7265 763a 2069 6465 616c  _rp, Prev: ideal
│ │ │ │ -00057c50: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ -00057c60: 702c 2055 703a 2054 6f70 0a0a 696d 6167  p, Up: Top..imag
│ │ │ │ -00057c70: 6528 5261 7469 6f6e 616c 4d61 702c 5374  e(RationalMap,St
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│ │ │ │ -00057c90: 206f 6620 7468 6520 696d 6167 6520 6f66   of the image of
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│ │ │ │ -00057cb0: 7573 696e 6720 7468 6520 4634 2061 6c67  using the F4 alg
│ │ │ │ -00057cc0: 6f72 6974 686d 2028 6578 7065 7269 6d65  orithm (experime
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│ │ │ │ +00057c00: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f63  _lpRationalMap_c
│ │ │ │ +00057c10: 6d53 7472 696e 675f 7270 2c20 4e65 7874  mString_rp, Next
│ │ │ │ +00057c20: 3a20 696d 6167 655f 6c70 5261 7469 6f6e  : image_lpRation
│ │ │ │ +00057c30: 616c 4d61 705f 636d 5a5a 5f72 702c 2050  alMap_cmZZ_rp, P
│ │ │ │ +00057c40: 7265 763a 2069 6465 616c 5f6c 7052 6174  rev: ideal_lpRat
│ │ │ │ +00057c50: 696f 6e61 6c4d 6170 5f72 702c 2055 703a  ionalMap_rp, Up:
│ │ │ │ +00057c60: 2054 6f70 0a0a 696d 6167 6528 5261 7469   Top..image(Rati
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│ │ │ │ +00057ca0: 696f 6e61 6c20 6d61 7020 7573 696e 6720  ional map using 
│ │ │ │ +00057cb0: 7468 6520 4634 2061 6c67 6f72 6974 686d  the F4 algorithm
│ │ │ │ +00057cc0: 2028 6578 7065 7269 6d65 6e74 616c 290a   (experimental).
│ │ │ │ +00057cd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00057ce0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00057cf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00057d00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00057d10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00057d20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00057d30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ -00057d40: 0a20 202a 2046 756e 6374 696f 6e3a 202a  .  * Function: *
│ │ │ │ -00057d50: 6e6f 7465 2069 6d61 6765 3a20 284d 6163  note image: (Mac
│ │ │ │ -00057d60: 6175 6c61 7932 446f 6329 696d 6167 652c  aulay2Doc)image,
│ │ │ │ -00057d70: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ -00057d80: 2020 2020 2069 6d61 6765 2850 6869 2c22       image(Phi,"
│ │ │ │ -00057d90: 4634 2229 0a20 2020 2020 2020 2069 6d61  F4").        ima
│ │ │ │ -00057da0: 6765 2850 6869 2c22 4d47 4222 290a 2020  ge(Phi,"MGB").  
│ │ │ │ -00057db0: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -00057dc0: 2a20 5068 692c 2061 202a 6e6f 7465 2072  * Phi, a *note r
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│ │ │ │ -00057de0: 696f 6e61 6c4d 6170 2c0a 2020 2020 2020  ionalMap,.      
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│ │ │ │ -00057e50: 6361 756c 6179 3244 6f63 2949 6465 616c  caulay2Doc)Ideal
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│ │ │ │ -00057ed0: 284d 6163 6175 6c61 7932 446f 6329 6772  (Macaulay2Doc)gr
│ │ │ │ -00057ee0: 6f65 626e 6572 4261 7369 732c 282e 2e2e  oebnerBasis,(...
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│ │ │ │ -00057f00: 206f 7220 2a6e 6f74 650a 2020 2020 2020   or *note.      
│ │ │ │ -00057f10: 2020 6772 6f65 626e 6572 4261 7369 733a    groebnerBasis:
│ │ │ │ -00057f20: 2028 4d61 6361 756c 6179 3244 6f63 2967   (Macaulay2Doc)g
│ │ │ │ -00057f30: 726f 6562 6e65 7242 6173 6973 2c28 2e2e  roebnerBasis,(..
│ │ │ │ -00057f40: 2e2c 5374 7261 7465 6779 3d3e 224d 4742  .,Strategy=>"MGB
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│ │ │ │ -00057f60: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ -00057f70: 6520 696d 6167 6528 5261 7469 6f6e 616c  e image(Rational
│ │ │ │ -00057f80: 4d61 7029 3a20 696d 6167 655f 6c70 5261  Map): image_lpRa
│ │ │ │ -00057f90: 7469 6f6e 616c 4d61 705f 636d 5a5a 5f72  tionalMap_cmZZ_r
│ │ │ │ -00057fa0: 702c 202d 2d20 636c 6f73 7572 6520 6f66  p, -- closure of
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│ │ │ │ -00057fe0: 2852 6174 696f 6e61 6c4d 6170 2c5a 5a29  (RationalMap,ZZ)
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│ │ │ │ -00058000: 616c 4d61 705f 636d 5a5a 5f72 702c 202d  alMap_cmZZ_rp, -
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│ │ │ │ -00058040: 202a 6e6f 7465 2067 726f 6562 6e65 7242   *note groebnerB
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│ │ │ │ -000580a0: 7365 2074 6869 7320 6d65 7468 6f64 3a0a  se this method:.
│ │ │ │ +00057d30: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046  *********..  * F
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│ │ │ │ +00057d50: 6d61 6765 3a20 284d 6163 6175 6c61 7932  mage: (Macaulay2
│ │ │ │ +00057d60: 446f 6329 696d 6167 652c 0a20 202a 2055  Doc)image,.  * U
│ │ │ │ +00057d70: 7361 6765 3a20 0a20 2020 2020 2020 2069  sage: .        i
│ │ │ │ +00057d80: 6d61 6765 2850 6869 2c22 4634 2229 0a20  mage(Phi,"F4"). 
│ │ │ │ +00057d90: 2020 2020 2020 2069 6d61 6765 2850 6869         image(Phi
│ │ │ │ +00057da0: 2c22 4d47 4222 290a 2020 2a20 496e 7075  ,"MGB").  * Inpu
│ │ │ │ +00057db0: 7473 3a0a 2020 2020 2020 2a20 5068 692c  ts:.      * Phi,
│ │ │ │ +00057dc0: 2061 202a 6e6f 7465 2072 6174 696f 6e61   a *note rationa
│ │ │ │ +00057dd0: 6c20 6d61 703a 2052 6174 696f 6e61 6c4d  l map: RationalM
│ │ │ │ +00057de0: 6170 2c0a 2020 2020 2020 2a20 6120 2a6e  ap,.      * a *n
│ │ │ │ +00057df0: 6f74 6520 7374 7269 6e67 3a20 284d 6163  ote string: (Mac
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│ │ │ │ +00057e10: 2c2c 2022 4634 2220 6f72 2022 4d47 4222  ,, "F4" or "MGB"
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│ │ │ │ +00057e30: 2020 2020 2a20 7468 6520 2a6e 6f74 6520      * the *note 
│ │ │ │ +00057e40: 6964 6561 6c3a 2028 4d61 6361 756c 6179  ideal: (Macaulay
│ │ │ │ +00057e50: 3244 6f63 2949 6465 616c 2c20 6465 6669  2Doc)Ideal, defi
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│ │ │ │ +00057e80: 2020 2020 2020 6f66 2050 6869 3b20 7468        of Phi; th
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│ │ │ │ +00057eb0: 7465 2067 726f 6562 6e65 7242 6173 6973  te groebnerBasis
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│ │ │ │ +00057ee0: 4261 7369 732c 282e 2e2e 2c53 7472 6174  Basis,(...,Strat
│ │ │ │ +00057ef0: 6567 793d 3e22 4634 2229 206f 7220 2a6e  egy=>"F4") or *n
│ │ │ │ +00057f00: 6f74 650a 2020 2020 2020 2020 6772 6f65  ote.        groe
│ │ │ │ +00057f10: 626e 6572 4261 7369 733a 2028 4d61 6361  bnerBasis: (Maca
│ │ │ │ +00057f20: 756c 6179 3244 6f63 2967 726f 6562 6e65  ulay2Doc)groebne
│ │ │ │ +00057f30: 7242 6173 6973 2c28 2e2e 2e2c 5374 7261  rBasis,(...,Stra
│ │ │ │ +00057f40: 7465 6779 3d3e 224d 4742 2229 2e0a 0a53  tegy=>"MGB")...S
│ │ │ │ +00057f50: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ +00057f60: 0a0a 2020 2a20 2a6e 6f74 6520 696d 6167  ..  * *note imag
│ │ │ │ +00057f70: 6528 5261 7469 6f6e 616c 4d61 7029 3a20  e(RationalMap): 
│ │ │ │ +00057f80: 696d 6167 655f 6c70 5261 7469 6f6e 616c  image_lpRational
│ │ │ │ +00057f90: 4d61 705f 636d 5a5a 5f72 702c 202d 2d20  Map_cmZZ_rp, -- 
│ │ │ │ +00057fa0: 636c 6f73 7572 6520 6f66 2074 6865 0a20  closure of the. 
│ │ │ │ +00057fb0: 2020 2069 6d61 6765 206f 6620 6120 7261     image of a ra
│ │ │ │ +00057fc0: 7469 6f6e 616c 206d 6170 0a20 202a 202a  tional map.  * *
│ │ │ │ +00057fd0: 6e6f 7465 2069 6d61 6765 2852 6174 696f  note image(Ratio
│ │ │ │ +00057fe0: 6e61 6c4d 6170 2c5a 5a29 3a20 696d 6167  nalMap,ZZ): imag
│ │ │ │ +00057ff0: 655f 6c70 5261 7469 6f6e 616c 4d61 705f  e_lpRationalMap_
│ │ │ │ +00058000: 636d 5a5a 5f72 702c 202d 2d20 636c 6f73  cmZZ_rp, -- clos
│ │ │ │ +00058010: 7572 6520 6f66 2074 6865 0a20 2020 2069  ure of the.    i
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│ │ │ │ +00058030: 616c 206d 6170 0a20 202a 202a 6e6f 7465  al map.  * *note
│ │ │ │ +00058040: 2067 726f 6562 6e65 7242 6173 6973 3a20   groebnerBasis: 
│ │ │ │ +00058050: 284d 6163 6175 6c61 7932 446f 6329 6772  (Macaulay2Doc)gr
│ │ │ │ +00058060: 6f65 626e 6572 4261 7369 732c 202d 2d20  oebnerBasis, -- 
│ │ │ │ +00058070: 4772 c3b6 626e 6572 2062 6173 6973 2c20  Gr..bner basis, 
│ │ │ │ +00058080: 6173 2061 0a20 2020 206d 6174 7269 780a  as a.    matrix.
│ │ │ │ +00058090: 0a57 6179 7320 746f 2075 7365 2074 6869  .Ways to use thi
│ │ │ │ +000580a0: 7320 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d  s method:.======
│ │ │ │  000580b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000580c0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ -000580d0: 6f74 6520 696d 6167 6528 5261 7469 6f6e  ote image(Ration
│ │ │ │ -000580e0: 616c 4d61 702c 5374 7269 6e67 293a 2069  alMap,String): i
│ │ │ │ -000580f0: 6d61 6765 5f6c 7052 6174 696f 6e61 6c4d  mage_lpRationalM
│ │ │ │ -00058100: 6170 5f63 6d53 7472 696e 675f 7270 2c20  ap_cmString_rp, 
│ │ │ │ -00058110: 2d2d 0a20 2020 2063 6c6f 7375 7265 206f  --.    closure o
│ │ │ │ -00058120: 6620 7468 6520 696d 6167 6520 6f66 2061  f the image of a
│ │ │ │ -00058130: 2072 6174 696f 6e61 6c20 6d61 7020 7573   rational map us
│ │ │ │ -00058140: 696e 6720 7468 6520 4634 2061 6c67 6f72  ing the F4 algor
│ │ │ │ -00058150: 6974 686d 0a20 2020 2028 6578 7065 7269  ithm.    (experi
│ │ │ │ -00058160: 6d65 6e74 616c 290a 2d2d 2d2d 2d2d 2d2d  mental).--------
│ │ │ │ +000580c0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 696d  ==..  * *note im
│ │ │ │ +000580d0: 6167 6528 5261 7469 6f6e 616c 4d61 702c  age(RationalMap,
│ │ │ │ +000580e0: 5374 7269 6e67 293a 2069 6d61 6765 5f6c  String): image_l
│ │ │ │ +000580f0: 7052 6174 696f 6e61 6c4d 6170 5f63 6d53  pRationalMap_cmS
│ │ │ │ +00058100: 7472 696e 675f 7270 2c20 2d2d 0a20 2020  tring_rp, --.   
│ │ │ │ +00058110: 2063 6c6f 7375 7265 206f 6620 7468 6520   closure of the 
│ │ │ │ +00058120: 696d 6167 6520 6f66 2061 2072 6174 696f  image of a ratio
│ │ │ │ +00058130: 6e61 6c20 6d61 7020 7573 696e 6720 7468  nal map using th
│ │ │ │ +00058140: 6520 4634 2061 6c67 6f72 6974 686d 0a20  e F4 algorithm. 
│ │ │ │ +00058150: 2020 2028 6578 7065 7269 6d65 6e74 616c     (experimental
│ │ │ │ +00058160: 290a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ).--------------
│ │ │ │  00058170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000581a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000581b0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
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│ │ │ │ -000581e0: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ -000581f0: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ -00058200: 3235 2e31 312b 6473 2f4d 322f 4d61 6361  25.11+ds/M2/Maca
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│ │ │ │ -00058220: 7265 6d6f 6e61 2f0a 646f 6375 6d65 6e74  remona/.document
│ │ │ │ -00058230: 6174 696f 6e2e 6d32 3a36 3831 3a30 2e0a  ation.m2:681:0..
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│ │ │ │ -00058250: 696e 666f 2c20 4e6f 6465 3a20 696d 6167  info, Node: imag
│ │ │ │ -00058260: 655f 6c70 5261 7469 6f6e 616c 4d61 705f  e_lpRationalMap_
│ │ │ │ -00058270: 636d 5a5a 5f72 702c 204e 6578 743a 2069  cmZZ_rp, Next: i
│ │ │ │ -00058280: 6e76 6572 7365 5f6c 7052 6174 696f 6e61  nverse_lpRationa
│ │ │ │ -00058290: 6c4d 6170 5f72 702c 2050 7265 763a 2069  lMap_rp, Prev: i
│ │ │ │ -000582a0: 6d61 6765 5f6c 7052 6174 696f 6e61 6c4d  mage_lpRationalM
│ │ │ │ -000582b0: 6170 5f63 6d53 7472 696e 675f 7270 2c20  ap_cmString_rp, 
│ │ │ │ -000582c0: 5570 3a20 546f 700a 0a69 6d61 6765 2852  Up: Top..image(R
│ │ │ │ -000582d0: 6174 696f 6e61 6c4d 6170 2c5a 5a29 202d  ationalMap,ZZ) -
│ │ │ │ -000582e0: 2d20 636c 6f73 7572 6520 6f66 2074 6865  - closure of the
│ │ │ │ -000582f0: 2069 6d61 6765 206f 6620 6120 7261 7469   image of a rati
│ │ │ │ -00058300: 6f6e 616c 206d 6170 0a2a 2a2a 2a2a 2a2a  onal map.*******
│ │ │ │ +000581b0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66  -..The source of
│ │ │ │ +000581c0: 2074 6869 7320 646f 6375 6d65 6e74 2069   this document i
│ │ │ │ +000581d0: 7320 696e 0a2f 6275 696c 642f 7265 7072  s in./build/repr
│ │ │ │ +000581e0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61  oducible-path/ma
│ │ │ │ +000581f0: 6361 756c 6179 322d 312e 3235 2e31 312b  caulay2-1.25.11+
│ │ │ │ +00058200: 6473 2f4d 322f 4d61 6361 756c 6179 322f  ds/M2/Macaulay2/
│ │ │ │ +00058210: 7061 636b 6167 6573 2f43 7265 6d6f 6e61  packages/Cremona
│ │ │ │ +00058220: 2f0a 646f 6375 6d65 6e74 6174 696f 6e2e  /.documentation.
│ │ │ │ +00058230: 6d32 3a36 3831 3a30 2e0a 1f0a 4669 6c65  m2:681:0....File
│ │ │ │ +00058240: 3a20 4372 656d 6f6e 612e 696e 666f 2c20  : Cremona.info, 
│ │ │ │ +00058250: 4e6f 6465 3a20 696d 6167 655f 6c70 5261  Node: image_lpRa
│ │ │ │ +00058260: 7469 6f6e 616c 4d61 705f 636d 5a5a 5f72  tionalMap_cmZZ_r
│ │ │ │ +00058270: 702c 204e 6578 743a 2069 6e76 6572 7365  p, Next: inverse
│ │ │ │ +00058280: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ +00058290: 702c 2050 7265 763a 2069 6d61 6765 5f6c  p, Prev: image_l
│ │ │ │ +000582a0: 7052 6174 696f 6e61 6c4d 6170 5f63 6d53  pRationalMap_cmS
│ │ │ │ +000582b0: 7472 696e 675f 7270 2c20 5570 3a20 546f  tring_rp, Up: To
│ │ │ │ +000582c0: 700a 0a69 6d61 6765 2852 6174 696f 6e61  p..image(Rationa
│ │ │ │ +000582d0: 6c4d 6170 2c5a 5a29 202d 2d20 636c 6f73  lMap,ZZ) -- clos
│ │ │ │ +000582e0: 7572 6520 6f66 2074 6865 2069 6d61 6765  ure of the image
│ │ │ │ +000582f0: 206f 6620 6120 7261 7469 6f6e 616c 206d   of a rational m
│ │ │ │ +00058300: 6170 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ap.*************
│ │ │ │  00058310: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00058320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00058330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00058340: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675  ********..  * Fu
│ │ │ │ -00058350: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 696d  nction: *note im
│ │ │ │ -00058360: 6167 653a 2028 4d61 6361 756c 6179 3244  age: (Macaulay2D
│ │ │ │ -00058370: 6f63 2969 6d61 6765 2c0a 2020 2a20 5573  oc)image,.  * Us
│ │ │ │ -00058380: 6167 653a 200a 2020 2020 2020 2020 696d  age: .        im
│ │ │ │ -00058390: 6167 6520 5068 6920 0a20 2020 2020 2020  age Phi .       
│ │ │ │ -000583a0: 2069 6d61 6765 2850 6869 2c64 290a 2020   image(Phi,d).  
│ │ │ │ -000583b0: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -000583c0: 2a20 5068 692c 2061 202a 6e6f 7465 2072  * Phi, a *note r
│ │ │ │ -000583d0: 6174 696f 6e61 6c20 6d61 703a 2052 6174  ational map: Rat
│ │ │ │ -000583e0: 696f 6e61 6c4d 6170 2c0a 2020 2020 2020  ionalMap,.      
│ │ │ │ -000583f0: 2a20 642c 2061 6e20 2a6e 6f74 6520 696e  * d, an *note in
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│ │ │ │ -00058430: 202a 6e6f 7465 2069 6465 616c 3a20 284d   *note ideal: (M
│ │ │ │ -00058440: 6163 6175 6c61 7932 446f 6329 4964 6561  acaulay2Doc)Idea
│ │ │ │ -00058450: 6c2c 2c20 7468 6520 6964 6561 6c20 6465  l,, the ideal de
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│ │ │ │ -00058480: 6520 696d 6167 6520 6f66 2050 6869 2c20  e image of Phi, 
│ │ │ │ -00058490: 6f72 2069 7473 2064 6567 7265 6520 6420  or its degree d 
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│ │ │ │ -000584d0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  n.===========..T
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│ │ │ │ -00058530: 616c 206d 6170 2e20 5365 6520 2a6e 6f74  al map. See *not
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│ │ │ │ -00058560: 296b 6572 6e65 6c5f 6c70 5269 6e67 4d61  )kernel_lpRingMa
│ │ │ │ -00058570: 705f 7270 2c20 616e 640a 2a6e 6f74 6520  p_rp, and.*note 
│ │ │ │ -00058580: 6b65 726e 656c 2852 696e 674d 6170 2c5a  kernel(RingMap,Z
│ │ │ │ -00058590: 5a29 3a20 6b65 726e 656c 5f6c 7052 696e  Z): kernel_lpRin
│ │ │ │ -000585a0: 674d 6170 5f63 6d5a 5a5f 7270 2c20 666f  gMap_cmZZ_rp, fo
│ │ │ │ -000585b0: 7220 6d6f 7265 2064 6574 6169 6c73 2e0a  r more details..
│ │ │ │ -000585c0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -000585d0: 3d3d 0a0a 2020 2a20 6b65 726e 656c 2852  ==..  * kernel(R
│ │ │ │ -000585e0: 696e 674d 6170 2c5a 5a29 2028 6d69 7373  ingMap,ZZ) (miss
│ │ │ │ -000585f0: 696e 6720 646f 6375 6d65 6e74 6174 696f  ing documentatio
│ │ │ │ -00058600: 6e29 0a20 202a 202a 6e6f 7465 206b 6572  n).  * *note ker
│ │ │ │ -00058610: 6e65 6c28 5269 6e67 4d61 7029 3a20 284d  nel(RingMap): (M
│ │ │ │ -00058620: 6163 6175 6c61 7932 446f 6329 6b65 726e  acaulay2Doc)kern
│ │ │ │ -00058630: 656c 5f6c 7052 696e 674d 6170 5f72 702c  el_lpRingMap_rp,
│ │ │ │ -00058640: 202d 2d20 6b65 726e 656c 206f 6620 610a   -- kernel of a.
│ │ │ │ -00058650: 2020 2020 7269 6e67 6d61 700a 2020 2a20      ringmap.  * 
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│ │ │ │ -00058670: 6f6e 616c 4d61 702c 5374 7269 6e67 293a  onalMap,String):
│ │ │ │ -00058680: 2069 6d61 6765 5f6c 7052 6174 696f 6e61   image_lpRationa
│ │ │ │ -00058690: 6c4d 6170 5f63 6d53 7472 696e 675f 7270  lMap_cmString_rp
│ │ │ │ -000586a0: 2c20 2d2d 0a20 2020 2063 6c6f 7375 7265  , --.    closure
│ │ │ │ -000586b0: 206f 6620 7468 6520 696d 6167 6520 6f66   of the image of
│ │ │ │ -000586c0: 2061 2072 6174 696f 6e61 6c20 6d61 7020   a rational map 
│ │ │ │ -000586d0: 7573 696e 6720 7468 6520 4634 2061 6c67  using the F4 alg
│ │ │ │ -000586e0: 6f72 6974 686d 0a20 2020 2028 6578 7065  orithm.    (expe
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│ │ │ │ -00058700: 746f 2075 7365 2074 6869 7320 6d65 7468  to use this meth
│ │ │ │ -00058710: 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  od:.============
│ │ │ │ -00058720: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ -00058730: 2a20 2269 6d61 6765 2852 6174 696f 6e61  * "image(Rationa
│ │ │ │ -00058740: 6c4d 6170 2922 0a20 202a 202a 6e6f 7465  lMap)".  * *note
│ │ │ │ -00058750: 2069 6d61 6765 2852 6174 696f 6e61 6c4d   image(RationalM
│ │ │ │ -00058760: 6170 2c5a 5a29 3a20 696d 6167 655f 6c70  ap,ZZ): image_lp
│ │ │ │ -00058770: 5261 7469 6f6e 616c 4d61 705f 636d 5a5a  RationalMap_cmZZ
│ │ │ │ -00058780: 5f72 702c 202d 2d20 636c 6f73 7572 6520  _rp, -- closure 
│ │ │ │ -00058790: 6f66 2074 6865 0a20 2020 2069 6d61 6765  of the.    image
│ │ │ │ -000587a0: 206f 6620 6120 7261 7469 6f6e 616c 206d   of a rational m
│ │ │ │ -000587b0: 6170 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ap.-------------
│ │ │ │ +00058340: 2a2a 0a0a 2020 2a20 4675 6e63 7469 6f6e  **..  * Function
│ │ │ │ +00058350: 3a20 2a6e 6f74 6520 696d 6167 653a 2028  : *note image: (
│ │ │ │ +00058360: 4d61 6361 756c 6179 3244 6f63 2969 6d61  Macaulay2Doc)ima
│ │ │ │ +00058370: 6765 2c0a 2020 2a20 5573 6167 653a 200a  ge,.  * Usage: .
│ │ │ │ +00058380: 2020 2020 2020 2020 696d 6167 6520 5068          image Ph
│ │ │ │ +00058390: 6920 0a20 2020 2020 2020 2069 6d61 6765  i .        image
│ │ │ │ +000583a0: 2850 6869 2c64 290a 2020 2a20 496e 7075  (Phi,d).  * Inpu
│ │ │ │ +000583b0: 7473 3a0a 2020 2020 2020 2a20 5068 692c  ts:.      * Phi,
│ │ │ │ +000583c0: 2061 202a 6e6f 7465 2072 6174 696f 6e61   a *note rationa
│ │ │ │ +000583d0: 6c20 6d61 703a 2052 6174 696f 6e61 6c4d  l map: RationalM
│ │ │ │ +000583e0: 6170 2c0a 2020 2020 2020 2a20 642c 2061  ap,.      * d, a
│ │ │ │ +000583f0: 6e20 2a6e 6f74 6520 696e 7465 6765 723a  n *note integer:
│ │ │ │ +00058400: 2028 4d61 6361 756c 6179 3244 6f63 295a   (Macaulay2Doc)Z
│ │ │ │ +00058410: 5a2c 0a20 202a 204f 7574 7075 7473 3a0a  Z,.  * Outputs:.
│ │ │ │ +00058420: 2020 2020 2020 2a20 616e 202a 6e6f 7465        * an *note
│ │ │ │ +00058430: 2069 6465 616c 3a20 284d 6163 6175 6c61   ideal: (Macaula
│ │ │ │ +00058440: 7932 446f 6329 4964 6561 6c2c 2c20 7468  y2Doc)Ideal,, th
│ │ │ │ +00058450: 6520 6964 6561 6c20 6465 6669 6e69 6e67  e ideal defining
│ │ │ │ +00058460: 2074 6865 2063 6c6f 7375 7265 206f 660a   the closure of.
│ │ │ │ +00058470: 2020 2020 2020 2020 7468 6520 696d 6167          the imag
│ │ │ │ +00058480: 6520 6f66 2050 6869 2c20 6f72 2069 7473  e of Phi, or its
│ │ │ │ +00058490: 2064 6567 7265 6520 6420 686f 6d6f 6765   degree d homoge
│ │ │ │ +000584a0: 6e65 6f75 7320 636f 6d70 6f6e 656e 7420  neous component 
│ │ │ │ +000584b0: 6966 2064 2069 7320 7061 7373 6564 0a0a  if d is passed..
│ │ │ │ +000584c0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +000584d0: 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 636f  =======..This co
│ │ │ │ +000584e0: 6d70 7574 6174 696f 6e20 6973 2064 6f6e  mputation is don
│ │ │ │ +000584f0: 6520 7468 726f 7567 6820 7468 6520 6b65  e through the ke
│ │ │ │ +00058500: 726e 656c 206f 6620 6120 7269 6e67 206d  rnel of a ring m
│ │ │ │ +00058510: 6170 2072 6570 7265 7365 6e74 696e 6720  ap representing 
│ │ │ │ +00058520: 7468 650a 7261 7469 6f6e 616c 206d 6170  the.rational map
│ │ │ │ +00058530: 2e20 5365 6520 2a6e 6f74 6520 6b65 726e  . See *note kern
│ │ │ │ +00058540: 656c 2852 696e 674d 6170 293a 2028 4d61  el(RingMap): (Ma
│ │ │ │ +00058550: 6361 756c 6179 3244 6f63 296b 6572 6e65  caulay2Doc)kerne
│ │ │ │ +00058560: 6c5f 6c70 5269 6e67 4d61 705f 7270 2c20  l_lpRingMap_rp, 
│ │ │ │ +00058570: 616e 640a 2a6e 6f74 6520 6b65 726e 656c  and.*note kernel
│ │ │ │ +00058580: 2852 696e 674d 6170 2c5a 5a29 3a20 6b65  (RingMap,ZZ): ke
│ │ │ │ +00058590: 726e 656c 5f6c 7052 696e 674d 6170 5f63  rnel_lpRingMap_c
│ │ │ │ +000585a0: 6d5a 5a5f 7270 2c20 666f 7220 6d6f 7265  mZZ_rp, for more
│ │ │ │ +000585b0: 2064 6574 6169 6c73 2e0a 0a53 6565 2061   details...See a
│ │ │ │ +000585c0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ +000585d0: 2a20 6b65 726e 656c 2852 696e 674d 6170  * kernel(RingMap
│ │ │ │ +000585e0: 2c5a 5a29 2028 6d69 7373 696e 6720 646f  ,ZZ) (missing do
│ │ │ │ +000585f0: 6375 6d65 6e74 6174 696f 6e29 0a20 202a  cumentation).  *
│ │ │ │ +00058600: 202a 6e6f 7465 206b 6572 6e65 6c28 5269   *note kernel(Ri
│ │ │ │ +00058610: 6e67 4d61 7029 3a20 284d 6163 6175 6c61  ngMap): (Macaula
│ │ │ │ +00058620: 7932 446f 6329 6b65 726e 656c 5f6c 7052  y2Doc)kernel_lpR
│ │ │ │ +00058630: 696e 674d 6170 5f72 702c 202d 2d20 6b65  ingMap_rp, -- ke
│ │ │ │ +00058640: 726e 656c 206f 6620 610a 2020 2020 7269  rnel of a.    ri
│ │ │ │ +00058650: 6e67 6d61 700a 2020 2a20 2a6e 6f74 6520  ngmap.  * *note 
│ │ │ │ +00058660: 696d 6167 6528 5261 7469 6f6e 616c 4d61  image(RationalMa
│ │ │ │ +00058670: 702c 5374 7269 6e67 293a 2069 6d61 6765  p,String): image
│ │ │ │ +00058680: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f63  _lpRationalMap_c
│ │ │ │ +00058690: 6d53 7472 696e 675f 7270 2c20 2d2d 0a20  mString_rp, --. 
│ │ │ │ +000586a0: 2020 2063 6c6f 7375 7265 206f 6620 7468     closure of th
│ │ │ │ +000586b0: 6520 696d 6167 6520 6f66 2061 2072 6174  e image of a rat
│ │ │ │ +000586c0: 696f 6e61 6c20 6d61 7020 7573 696e 6720  ional map using 
│ │ │ │ +000586d0: 7468 6520 4634 2061 6c67 6f72 6974 686d  the F4 algorithm
│ │ │ │ +000586e0: 0a20 2020 2028 6578 7065 7269 6d65 6e74  .    (experiment
│ │ │ │ +000586f0: 616c 290a 0a57 6179 7320 746f 2075 7365  al)..Ways to use
│ │ │ │ +00058700: 2074 6869 7320 6d65 7468 6f64 3a0a 3d3d   this method:.==
│ │ │ │ +00058710: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00058720: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2269 6d61  ======..  * "ima
│ │ │ │ +00058730: 6765 2852 6174 696f 6e61 6c4d 6170 2922  ge(RationalMap)"
│ │ │ │ +00058740: 0a20 202a 202a 6e6f 7465 2069 6d61 6765  .  * *note image
│ │ │ │ +00058750: 2852 6174 696f 6e61 6c4d 6170 2c5a 5a29  (RationalMap,ZZ)
│ │ │ │ +00058760: 3a20 696d 6167 655f 6c70 5261 7469 6f6e  : image_lpRation
│ │ │ │ +00058770: 616c 4d61 705f 636d 5a5a 5f72 702c 202d  alMap_cmZZ_rp, -
│ │ │ │ +00058780: 2d20 636c 6f73 7572 6520 6f66 2074 6865  - closure of the
│ │ │ │ +00058790: 0a20 2020 2069 6d61 6765 206f 6620 6120  .    image of a 
│ │ │ │ +000587a0: 7261 7469 6f6e 616c 206d 6170 0a2d 2d2d  rational map.---
│ │ │ │ +000587b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058800: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ -00058810: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ -00058820: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ -00058830: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ -00058840: 6163 6175 6c61 7932 2d31 2e32 352e 3131  acaulay2-1.25.11
│ │ │ │ -00058850: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ -00058860: 2f70 6163 6b61 6765 732f 4372 656d 6f6e  /packages/Cremon
│ │ │ │ -00058870: 612f 0a64 6f63 756d 656e 7461 7469 6f6e  a/.documentation
│ │ │ │ -00058880: 2e6d 323a 3637 323a 302e 0a1f 0a46 696c  .m2:672:0....Fil
│ │ │ │ -00058890: 653a 2043 7265 6d6f 6e61 2e69 6e66 6f2c  e: Cremona.info,
│ │ │ │ -000588a0: 204e 6f64 653a 2069 6e76 6572 7365 5f6c   Node: inverse_l
│ │ │ │ -000588b0: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ -000588c0: 204e 6578 743a 2069 6e76 6572 7365 4d61   Next: inverseMa
│ │ │ │ -000588d0: 702c 2050 7265 763a 2069 6d61 6765 5f6c  p, Prev: image_l
│ │ │ │ -000588e0: 7052 6174 696f 6e61 6c4d 6170 5f63 6d5a  pRationalMap_cmZ
│ │ │ │ -000588f0: 5a5f 7270 2c20 5570 3a20 546f 700a 0a69  Z_rp, Up: Top..i
│ │ │ │ -00058900: 6e76 6572 7365 2852 6174 696f 6e61 6c4d  nverse(RationalM
│ │ │ │ -00058910: 6170 2920 2d2d 2069 6e76 6572 7365 206f  ap) -- inverse o
│ │ │ │ -00058920: 6620 6120 6269 7261 7469 6f6e 616c 206d  f a birational m
│ │ │ │ -00058930: 6170 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ap.*************
│ │ │ │ +000587f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
│ │ │ │ +00058800: 6520 736f 7572 6365 206f 6620 7468 6973  e source of this
│ │ │ │ +00058810: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
│ │ │ │ +00058820: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ +00058830: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ +00058840: 7932 2d31 2e32 352e 3131 2b64 732f 4d32  y2-1.25.11+ds/M2
│ │ │ │ +00058850: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ +00058860: 6765 732f 4372 656d 6f6e 612f 0a64 6f63  ges/Cremona/.doc
│ │ │ │ +00058870: 756d 656e 7461 7469 6f6e 2e6d 323a 3637  umentation.m2:67
│ │ │ │ +00058880: 323a 302e 0a1f 0a46 696c 653a 2043 7265  2:0....File: Cre
│ │ │ │ +00058890: 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 653a  mona.info, Node:
│ │ │ │ +000588a0: 2069 6e76 6572 7365 5f6c 7052 6174 696f   inverse_lpRatio
│ │ │ │ +000588b0: 6e61 6c4d 6170 5f72 702c 204e 6578 743a  nalMap_rp, Next:
│ │ │ │ +000588c0: 2069 6e76 6572 7365 4d61 702c 2050 7265   inverseMap, Pre
│ │ │ │ +000588d0: 763a 2069 6d61 6765 5f6c 7052 6174 696f  v: image_lpRatio
│ │ │ │ +000588e0: 6e61 6c4d 6170 5f63 6d5a 5a5f 7270 2c20  nalMap_cmZZ_rp, 
│ │ │ │ +000588f0: 5570 3a20 546f 700a 0a69 6e76 6572 7365  Up: Top..inverse
│ │ │ │ +00058900: 2852 6174 696f 6e61 6c4d 6170 2920 2d2d  (RationalMap) --
│ │ │ │ +00058910: 2069 6e76 6572 7365 206f 6620 6120 6269   inverse of a bi
│ │ │ │ +00058920: 7261 7469 6f6e 616c 206d 6170 0a2a 2a2a  rational map.***
│ │ │ │ +00058930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00058940: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00058950: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00058960: 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675 6e63  ******..  * Func
│ │ │ │ -00058970: 7469 6f6e 3a20 2a6e 6f74 6520 696e 7665  tion: *note inve
│ │ │ │ -00058980: 7273 653a 2028 4d61 6361 756c 6179 3244  rse: (Macaulay2D
│ │ │ │ -00058990: 6f63 2969 6e76 6572 7365 2c0a 2020 2a20  oc)inverse,.  * 
│ │ │ │ -000589a0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -000589b0: 696e 7665 7273 6520 7068 690a 2020 2020  inverse phi.    
│ │ │ │ -000589c0: 2020 2020 696e 7665 7273 6528 7068 692c      inverse(phi,
│ │ │ │ -000589d0: 4365 7274 6966 793d 3e62 290a 2020 2a20  Certify=>b).  * 
│ │ │ │ -000589e0: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -000589f0: 7068 692c 2061 202a 6e6f 7465 2072 6174  phi, a *note rat
│ │ │ │ -00058a00: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ -00058a10: 6e61 6c4d 6170 2c2c 2077 6869 6368 2068  nalMap,, which h
│ │ │ │ -00058a20: 6173 2074 6f20 6265 2062 6972 6174 696f  as to be biratio
│ │ │ │ -00058a30: 6e61 6c2c 0a20 2020 2020 2020 2061 6e64  nal,.        and
│ │ │ │ -00058a40: 2062 2069 7320 6120 2a6e 6f74 6520 626f   b is a *note bo
│ │ │ │ -00058a50: 6f6c 6561 6e20 7661 6c75 653a 2028 4d61  olean value: (Ma
│ │ │ │ -00058a60: 6361 756c 6179 3244 6f63 2942 6f6f 6c65  caulay2Doc)Boole
│ │ │ │ -00058a70: 616e 2c2c 2074 6861 7420 6973 2c20 7472  an,, that is, tr
│ │ │ │ -00058a80: 7565 0a20 2020 2020 2020 206f 7220 6661  ue.        or fa
│ │ │ │ -00058a90: 6c73 6520 2874 6865 2064 6566 6175 6c74  lse (the default
│ │ │ │ -00058aa0: 2076 616c 7565 2069 7320 6661 6c73 6529   value is false)
│ │ │ │ -00058ab0: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -00058ac0: 2020 2020 2a20 6120 2a6e 6f74 6520 7261      * a *note ra
│ │ │ │ -00058ad0: 7469 6f6e 616c 206d 6170 3a20 5261 7469  tional map: Rati
│ │ │ │ -00058ae0: 6f6e 616c 4d61 702c 2c20 7468 6520 696e  onalMap,, the in
│ │ │ │ -00058af0: 7665 7273 6520 6d61 7020 6f66 2070 6869  verse map of phi
│ │ │ │ -00058b00: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ -00058b10: 3d3d 3d3d 3d3d 3d3d 3d0a 0a69 6e76 6572  =========..inver
│ │ │ │ -00058b20: 7365 2870 6869 2c43 6572 7469 6679 3d3e  se(phi,Certify=>
│ │ │ │ -00058b30: 7472 7565 2920 6973 2074 6865 2073 616d  true) is the sam
│ │ │ │ -00058b40: 6520 6173 202a 6e6f 7465 2069 6e76 6572  e as *note inver
│ │ │ │ -00058b50: 7365 4d61 703a 0a69 6e76 6572 7365 4d61  seMap:.inverseMa
│ │ │ │ -00058b60: 702c 2870 6869 2c43 6572 7469 6679 3d3e  p,(phi,Certify=>
│ │ │ │ -00058b70: 7472 7565 2c56 6572 626f 7365 3d3e 6661  true,Verbose=>fa
│ │ │ │ -00058b80: 6c73 6529 2c20 7768 696c 650a 696e 7665  lse), while.inve
│ │ │ │ -00058b90: 7273 6528 7068 692c 4365 7274 6966 793d  rse(phi,Certify=
│ │ │ │ -00058ba0: 3e66 616c 7365 2920 6170 706c 6965 7320  >false) applies 
│ │ │ │ -00058bb0: 6120 6d69 6464 6c65 2067 726f 756e 6420  a middle ground 
│ │ │ │ -00058bc0: 6170 7072 6f61 6368 2062 6574 7765 656e  approach between
│ │ │ │ -00058bd0: 202a 6e6f 7465 0a69 6e76 6572 7365 4d61   *note.inverseMa
│ │ │ │ -00058be0: 703a 2069 6e76 6572 7365 4d61 702c 2870  p: inverseMap,(p
│ │ │ │ -00058bf0: 6869 2c43 6572 7469 6679 3d3e 7472 7565  hi,Certify=>true
│ │ │ │ -00058c00: 2920 616e 6420 2a6e 6f74 6520 696e 7665  ) and *note inve
│ │ │ │ -00058c10: 7273 654d 6170 3a0a 696e 7665 7273 654d  rseMap:.inverseM
│ │ │ │ -00058c20: 6170 2c28 7068 692c 4365 7274 6966 793d  ap,(phi,Certify=
│ │ │ │ -00058c30: 3e66 616c 7365 292e 2054 6865 2070 726f  >false). The pro
│ │ │ │ -00058c40: 6365 6475 7265 2066 6f72 2074 6865 206c  cedure for the l
│ │ │ │ -00058c50: 6174 7465 7220 6973 2061 7320 666f 6c6c  atter is as foll
│ │ │ │ -00058c60: 6f77 733a 2049 740a 6669 7273 7420 636f  ows: It.first co
│ │ │ │ -00058c70: 6d70 7574 6573 2074 6865 2069 6e76 6572  mputes the inver
│ │ │ │ -00058c80: 7365 206d 6170 206f 6620 7068 6920 7573  se map of phi us
│ │ │ │ -00058c90: 696e 6720 7073 6920 3d20 2a6e 6f74 6520  ing psi = *note 
│ │ │ │ -00058ca0: 696e 7665 7273 654d 6170 3a20 696e 7665  inverseMap: inve
│ │ │ │ -00058cb0: 7273 654d 6170 2c0a 7068 692e 2054 6865  rseMap,.phi. The
│ │ │ │ -00058cc0: 6e20 6974 2069 7320 6368 6563 6b65 6420  n it is checked 
│ │ │ │ -00058cd0: 7468 6174 202a 6e6f 7465 2070 7369 2070  that *note psi p
│ │ │ │ -00058ce0: 6869 2070 3a20 5261 7469 6f6e 616c 4d61  hi p: RationalMa
│ │ │ │ -00058cf0: 7020 5f75 735f 7374 2c20 3d3d 2070 2061  p _us_st, == p a
│ │ │ │ -00058d00: 6e64 0a2a 6e6f 7465 2070 6869 2070 7369  nd.*note phi psi
│ │ │ │ -00058d10: 2071 3a20 5261 7469 6f6e 616c 4d61 7020   q: RationalMap 
│ │ │ │ -00058d20: 5f75 735f 7374 2c20 3d3d 2071 2c20 7768  _us_st, == q, wh
│ │ │ │ -00058d30: 6572 6520 702c 7120 6172 652c 2072 6573  ere p,q are, res
│ │ │ │ -00058d40: 7065 6374 6976 656c 792c 2061 202a 6e6f  pectively, a *no
│ │ │ │ -00058d50: 7465 0a72 616e 646f 6d20 706f 696e 743a  te.random point:
│ │ │ │ -00058d60: 2070 6f69 6e74 2c20 6f6e 2074 6865 202a   point, on the *
│ │ │ │ -00058d70: 6e6f 7465 2073 6f75 7263 653a 2073 6f75  note source: sou
│ │ │ │ -00058d80: 7263 655f 6c70 5261 7469 6f6e 616c 4d61  rce_lpRationalMa
│ │ │ │ -00058d90: 705f 7270 2c20 616e 6420 7468 650a 2a6e  p_rp, and the.*n
│ │ │ │ -00058da0: 6f74 6520 7461 7267 6574 3a20 7461 7267  ote target: targ
│ │ │ │ -00058db0: 6574 5f6c 7052 6174 696f 6e61 6c4d 6170  et_lpRationalMap
│ │ │ │ -00058dc0: 5f72 702c 206f 6620 7068 692e 2046 696e  _rp, of phi. Fin
│ │ │ │ -00058dd0: 616c 6c79 2c20 6966 2074 6865 2074 6573  ally, if the tes
│ │ │ │ -00058de0: 7473 2070 6173 732c 2074 6865 0a63 6f6d  ts pass, the.com
│ │ │ │ -00058df0: 6d61 6e64 202a 6e6f 7465 2066 6f72 6365  mand *note force
│ │ │ │ -00058e00: 496e 7665 7273 654d 6170 3a20 666f 7263  InverseMap: forc
│ │ │ │ -00058e10: 6549 6e76 6572 7365 4d61 702c 2870 6869  eInverseMap,(phi
│ │ │ │ -00058e20: 2c70 7369 2920 6973 2069 6e76 6f6b 6564  ,psi) is invoked
│ │ │ │ -00058e30: 2061 6e64 2070 7369 2069 730a 7265 7475   and psi is.retu
│ │ │ │ -00058e40: 726e 6564 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  rned...+--------
│ │ │ │ +00058960: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20  ..  * Function: 
│ │ │ │ +00058970: 2a6e 6f74 6520 696e 7665 7273 653a 2028  *note inverse: (
│ │ │ │ +00058980: 4d61 6361 756c 6179 3244 6f63 2969 6e76  Macaulay2Doc)inv
│ │ │ │ +00058990: 6572 7365 2c0a 2020 2a20 5573 6167 653a  erse,.  * Usage:
│ │ │ │ +000589a0: 200a 2020 2020 2020 2020 696e 7665 7273   .        invers
│ │ │ │ +000589b0: 6520 7068 690a 2020 2020 2020 2020 696e  e phi.        in
│ │ │ │ +000589c0: 7665 7273 6528 7068 692c 4365 7274 6966  verse(phi,Certif
│ │ │ │ +000589d0: 793d 3e62 290a 2020 2a20 496e 7075 7473  y=>b).  * Inputs
│ │ │ │ +000589e0: 3a0a 2020 2020 2020 2a20 7068 692c 2061  :.      * phi, a
│ │ │ │ +000589f0: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ +00058a00: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ +00058a10: 2c2c 2077 6869 6368 2068 6173 2074 6f20  ,, which has to 
│ │ │ │ +00058a20: 6265 2062 6972 6174 696f 6e61 6c2c 0a20  be birational,. 
│ │ │ │ +00058a30: 2020 2020 2020 2061 6e64 2062 2069 7320         and b is 
│ │ │ │ +00058a40: 6120 2a6e 6f74 6520 626f 6f6c 6561 6e20  a *note boolean 
│ │ │ │ +00058a50: 7661 6c75 653a 2028 4d61 6361 756c 6179  value: (Macaulay
│ │ │ │ +00058a60: 3244 6f63 2942 6f6f 6c65 616e 2c2c 2074  2Doc)Boolean,, t
│ │ │ │ +00058a70: 6861 7420 6973 2c20 7472 7565 0a20 2020  hat is, true.   
│ │ │ │ +00058a80: 2020 2020 206f 7220 6661 6c73 6520 2874       or false (t
│ │ │ │ +00058a90: 6865 2064 6566 6175 6c74 2076 616c 7565  he default value
│ │ │ │ +00058aa0: 2069 7320 6661 6c73 6529 0a20 202a 204f   is false).  * O
│ │ │ │ +00058ab0: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ +00058ac0: 6120 2a6e 6f74 6520 7261 7469 6f6e 616c  a *note rational
│ │ │ │ +00058ad0: 206d 6170 3a20 5261 7469 6f6e 616c 4d61   map: RationalMa
│ │ │ │ +00058ae0: 702c 2c20 7468 6520 696e 7665 7273 6520  p,, the inverse 
│ │ │ │ +00058af0: 6d61 7020 6f66 2070 6869 0a0a 4465 7363  map of phi..Desc
│ │ │ │ +00058b00: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +00058b10: 3d3d 3d0a 0a69 6e76 6572 7365 2870 6869  ===..inverse(phi
│ │ │ │ +00058b20: 2c43 6572 7469 6679 3d3e 7472 7565 2920  ,Certify=>true) 
│ │ │ │ +00058b30: 6973 2074 6865 2073 616d 6520 6173 202a  is the same as *
│ │ │ │ +00058b40: 6e6f 7465 2069 6e76 6572 7365 4d61 703a  note inverseMap:
│ │ │ │ +00058b50: 0a69 6e76 6572 7365 4d61 702c 2870 6869  .inverseMap,(phi
│ │ │ │ +00058b60: 2c43 6572 7469 6679 3d3e 7472 7565 2c56  ,Certify=>true,V
│ │ │ │ +00058b70: 6572 626f 7365 3d3e 6661 6c73 6529 2c20  erbose=>false), 
│ │ │ │ +00058b80: 7768 696c 650a 696e 7665 7273 6528 7068  while.inverse(ph
│ │ │ │ +00058b90: 692c 4365 7274 6966 793d 3e66 616c 7365  i,Certify=>false
│ │ │ │ +00058ba0: 2920 6170 706c 6965 7320 6120 6d69 6464  ) applies a midd
│ │ │ │ +00058bb0: 6c65 2067 726f 756e 6420 6170 7072 6f61  le ground approa
│ │ │ │ +00058bc0: 6368 2062 6574 7765 656e 202a 6e6f 7465  ch between *note
│ │ │ │ +00058bd0: 0a69 6e76 6572 7365 4d61 703a 2069 6e76  .inverseMap: inv
│ │ │ │ +00058be0: 6572 7365 4d61 702c 2870 6869 2c43 6572  erseMap,(phi,Cer
│ │ │ │ +00058bf0: 7469 6679 3d3e 7472 7565 2920 616e 6420  tify=>true) and 
│ │ │ │ +00058c00: 2a6e 6f74 6520 696e 7665 7273 654d 6170  *note inverseMap
│ │ │ │ +00058c10: 3a0a 696e 7665 7273 654d 6170 2c28 7068  :.inverseMap,(ph
│ │ │ │ +00058c20: 692c 4365 7274 6966 793d 3e66 616c 7365  i,Certify=>false
│ │ │ │ +00058c30: 292e 2054 6865 2070 726f 6365 6475 7265  ). The procedure
│ │ │ │ +00058c40: 2066 6f72 2074 6865 206c 6174 7465 7220   for the latter 
│ │ │ │ +00058c50: 6973 2061 7320 666f 6c6c 6f77 733a 2049  is as follows: I
│ │ │ │ +00058c60: 740a 6669 7273 7420 636f 6d70 7574 6573  t.first computes
│ │ │ │ +00058c70: 2074 6865 2069 6e76 6572 7365 206d 6170   the inverse map
│ │ │ │ +00058c80: 206f 6620 7068 6920 7573 696e 6720 7073   of phi using ps
│ │ │ │ +00058c90: 6920 3d20 2a6e 6f74 6520 696e 7665 7273  i = *note invers
│ │ │ │ +00058ca0: 654d 6170 3a20 696e 7665 7273 654d 6170  eMap: inverseMap
│ │ │ │ +00058cb0: 2c0a 7068 692e 2054 6865 6e20 6974 2069  ,.phi. Then it i
│ │ │ │ +00058cc0: 7320 6368 6563 6b65 6420 7468 6174 202a  s checked that *
│ │ │ │ +00058cd0: 6e6f 7465 2070 7369 2070 6869 2070 3a20  note psi phi p: 
│ │ │ │ +00058ce0: 5261 7469 6f6e 616c 4d61 7020 5f75 735f  RationalMap _us_
│ │ │ │ +00058cf0: 7374 2c20 3d3d 2070 2061 6e64 0a2a 6e6f  st, == p and.*no
│ │ │ │ +00058d00: 7465 2070 6869 2070 7369 2071 3a20 5261  te phi psi q: Ra
│ │ │ │ +00058d10: 7469 6f6e 616c 4d61 7020 5f75 735f 7374  tionalMap _us_st
│ │ │ │ +00058d20: 2c20 3d3d 2071 2c20 7768 6572 6520 702c  , == q, where p,
│ │ │ │ +00058d30: 7120 6172 652c 2072 6573 7065 6374 6976  q are, respectiv
│ │ │ │ +00058d40: 656c 792c 2061 202a 6e6f 7465 0a72 616e  ely, a *note.ran
│ │ │ │ +00058d50: 646f 6d20 706f 696e 743a 2070 6f69 6e74  dom point: point
│ │ │ │ +00058d60: 2c20 6f6e 2074 6865 202a 6e6f 7465 2073  , on the *note s
│ │ │ │ +00058d70: 6f75 7263 653a 2073 6f75 7263 655f 6c70  ource: source_lp
│ │ │ │ +00058d80: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ +00058d90: 616e 6420 7468 650a 2a6e 6f74 6520 7461  and the.*note ta
│ │ │ │ +00058da0: 7267 6574 3a20 7461 7267 6574 5f6c 7052  rget: target_lpR
│ │ │ │ +00058db0: 6174 696f 6e61 6c4d 6170 5f72 702c 206f  ationalMap_rp, o
│ │ │ │ +00058dc0: 6620 7068 692e 2046 696e 616c 6c79 2c20  f phi. Finally, 
│ │ │ │ +00058dd0: 6966 2074 6865 2074 6573 7473 2070 6173  if the tests pas
│ │ │ │ +00058de0: 732c 2074 6865 0a63 6f6d 6d61 6e64 202a  s, the.command *
│ │ │ │ +00058df0: 6e6f 7465 2066 6f72 6365 496e 7665 7273  note forceInvers
│ │ │ │ +00058e00: 654d 6170 3a20 666f 7263 6549 6e76 6572  eMap: forceInver
│ │ │ │ +00058e10: 7365 4d61 702c 2870 6869 2c70 7369 2920  seMap,(phi,psi) 
│ │ │ │ +00058e20: 6973 2069 6e76 6f6b 6564 2061 6e64 2070  is invoked and p
│ │ │ │ +00058e30: 7369 2069 730a 7265 7475 726e 6564 2e0a  si is.returned..
│ │ │ │ +00058e40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00058e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058e90: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d  -----+.|i1 : R =
│ │ │ │ -00058ea0: 2051 515b 785f 302e 2e78 5f34 5d3b 2070   QQ[x_0..x_4]; p
│ │ │ │ -00058eb0: 6869 203d 2072 6174 696f 6e61 6c4d 6170  hi = rationalMap
│ │ │ │ -00058ec0: 206d 696e 6f72 7328 342c 7261 6e64 6f6d   minors(4,random
│ │ │ │ -00058ed0: 2852 5e7b 343a 317d 2c52 5e20 2020 2020  (R^{4:1},R^     
│ │ │ │ -00058ee0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00058e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00058e90: 0a7c 6931 203a 2052 203d 2051 515b 785f  .|i1 : R = QQ[x_
│ │ │ │ +00058ea0: 302e 2e78 5f34 5d3b 2070 6869 203d 2072  0..x_4]; phi = r
│ │ │ │ +00058eb0: 6174 696f 6e61 6c4d 6170 206d 696e 6f72  ationalMap minor
│ │ │ │ +00058ec0: 7328 342c 7261 6e64 6f6d 2852 5e7b 343a  s(4,random(R^{4:
│ │ │ │ +00058ed0: 317d 2c52 5e20 2020 2020 2020 2020 207c  1},R^          |
│ │ │ │ +00058ee0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00058ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058f30: 2020 2020 207c 0a7c 6f32 203d 202d 2d20       |.|o2 = -- 
│ │ │ │ -00058f40: 7261 7469 6f6e 616c 206d 6170 202d 2d20  rational map -- 
│ │ │ │ +00058f20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00058f30: 0a7c 6f32 203d 202d 2d20 7261 7469 6f6e  .|o2 = -- ration
│ │ │ │ +00058f40: 616c 206d 6170 202d 2d20 2020 2020 2020  al map --       
│ │ │ │  00058f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058f80: 2020 2020 207c 0a7c 2020 2020 2073 6f75       |.|     sou
│ │ │ │ -00058f90: 7263 653a 2050 726f 6a28 5151 5b78 202c  rce: Proj(QQ[x ,
│ │ │ │ -00058fa0: 2078 202c 2078 202c 2078 202c 2078 205d   x , x , x , x ]
│ │ │ │ -00058fb0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00058fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058fd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00058fe0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00058ff0: 2020 3120 2020 3220 2020 3320 2020 3420    1   2   3   4 
│ │ │ │ +00058f70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00058f80: 0a7c 2020 2020 2073 6f75 7263 653a 2050  .|     source: P
│ │ │ │ +00058f90: 726f 6a28 5151 5b78 202c 2078 202c 2078  roj(QQ[x , x , x
│ │ │ │ +00058fa0: 202c 2078 202c 2078 205d 2920 2020 2020   , x , x ])     
│ │ │ │ +00058fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058fc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00058fd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00058fe0: 2020 2020 2020 2020 3020 2020 3120 2020          0   1   
│ │ │ │ +00058ff0: 3220 2020 3320 2020 3420 2020 2020 2020  2   3   4       
│ │ │ │  00059000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059020: 2020 2020 207c 0a7c 2020 2020 2074 6172       |.|     tar
│ │ │ │ -00059030: 6765 743a 2050 726f 6a28 5151 5b78 202c  get: Proj(QQ[x ,
│ │ │ │ -00059040: 2078 202c 2078 202c 2078 202c 2078 205d   x , x , x , x ]
│ │ │ │ -00059050: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00059060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059070: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00059080: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00059090: 2020 3120 2020 3220 2020 3320 2020 3420    1   2   3   4 
│ │ │ │ +00059010: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059020: 0a7c 2020 2020 2074 6172 6765 743a 2050  .|     target: P
│ │ │ │ +00059030: 726f 6a28 5151 5b78 202c 2078 202c 2078  roj(QQ[x , x , x
│ │ │ │ +00059040: 202c 2078 202c 2078 205d 2920 2020 2020   , x , x ])     
│ │ │ │ +00059050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059060: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059070: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00059080: 2020 2020 2020 2020 3020 2020 3120 2020          0   1   
│ │ │ │ +00059090: 3220 2020 3320 2020 3420 2020 2020 2020  2   3   4       
│ │ │ │  000590a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000590b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000590c0: 2020 2020 207c 0a7c 2020 2020 2064 6566       |.|     def
│ │ │ │ -000590d0: 696e 696e 6720 666f 726d 733a 207b 2020  ining forms: {  
│ │ │ │ +000590b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000590c0: 0a7c 2020 2020 2064 6566 696e 696e 6720  .|     defining 
│ │ │ │ +000590d0: 666f 726d 733a 207b 2020 2020 2020 2020  forms: {        
│ │ │ │  000590e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000590f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059110: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00059120: 2020 2020 2020 2020 2020 2020 2020 3134                14
│ │ │ │ -00059130: 3938 3639 2034 2020 2037 3832 3439 3036  9869 4   7824906
│ │ │ │ -00059140: 3133 2033 2020 2020 2031 3834 3832 3337  13 3     1848237
│ │ │ │ -00059150: 3136 3031 2032 2032 2020 2020 2020 2020  1601 2 2        
│ │ │ │ -00059160: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00059170: 2020 2020 2020 2020 2020 2020 2020 2d2d                --
│ │ │ │ -00059180: 2d2d 2d2d 7820 202b 202d 2d2d 2d2d 2d2d  ----x  + -------
│ │ │ │ -00059190: 2d2d 7820 7820 202d 202d 2d2d 2d2d 2d2d  --x x  - -------
│ │ │ │ -000591a0: 2d2d 2d2d 7820 7820 202d 2020 2020 2020  ----x x  -      
│ │ │ │ -000591b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000591c0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -000591d0: 3031 3630 2030 2020 2020 3136 3933 3434  0160 0    169344
│ │ │ │ -000591e0: 3030 2030 2031 2020 2020 3131 3433 3037  00 0 1    114307
│ │ │ │ -000591f0: 3230 3020 2030 2031 2020 2020 2020 2020  200  0 1        
│ │ │ │ -00059200: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00059100: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059110: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00059120: 2020 2020 2020 2020 3134 3938 3639 2034          149869 4
│ │ │ │ +00059130: 2020 2037 3832 3439 3036 3133 2033 2020     782490613 3  
│ │ │ │ +00059140: 2020 2031 3834 3832 3337 3136 3031 2032     18482371601 2
│ │ │ │ +00059150: 2032 2020 2020 2020 2020 2020 2020 207c   2             |
│ │ │ │ +00059160: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00059170: 2020 2020 2020 2020 2d2d 2d2d 2d2d 7820          ------x 
│ │ │ │ +00059180: 202b 202d 2d2d 2d2d 2d2d 2d2d 7820 7820   + ---------x x 
│ │ │ │ +00059190: 202d 202d 2d2d 2d2d 2d2d 2d2d 2d2d 7820   - -----------x 
│ │ │ │ +000591a0: 7820 202d 2020 2020 2020 2020 2020 207c  x  -           |
│ │ │ │ +000591b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000591c0: 2020 2020 2020 2020 2032 3031 3630 2030           20160 0
│ │ │ │ +000591d0: 2020 2020 3136 3933 3434 3030 2030 2031      16934400 0 1
│ │ │ │ +000591e0: 2020 2020 3131 3433 3037 3230 3020 2030      114307200  0
│ │ │ │ +000591f0: 2031 2020 2020 2020 2020 2020 2020 207c   1             |
│ │ │ │ +00059200: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00059210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059250: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00059260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059270: 3432 3036 3431 2034 2020 2032 3738 3638  420641 4   27868
│ │ │ │ -00059280: 3630 3733 3320 3320 2020 2020 3131 3935  60733 3     1195
│ │ │ │ -00059290: 3037 3834 3537 2032 2032 2020 2020 2020  078457 2 2      
│ │ │ │ -000592a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000592b0: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -000592c0: 2d2d 2d2d 2d2d 7820 202b 202d 2d2d 2d2d  ------x  + -----
│ │ │ │ -000592d0: 2d2d 2d2d 2d78 2078 2020 2d20 2d2d 2d2d  -----x x  - ----
│ │ │ │ -000592e0: 2d2d 2d2d 2d2d 7820 7820 2020 2020 2020  ------x x       
│ │ │ │ -000592f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00059300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059310: 2032 3832 3234 2030 2020 2020 3633 3530   28224 0    6350
│ │ │ │ -00059320: 3430 3030 2020 3020 3120 2020 2033 3137  4000  0 1    317
│ │ │ │ -00059330: 3532 3030 3020 2030 2031 2020 2020 2020  52000  0 1      
│ │ │ │ -00059340: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00059240: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059250: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00059260: 2020 2020 2020 2020 2020 3432 3036 3431            420641
│ │ │ │ +00059270: 2034 2020 2032 3738 3638 3630 3733 3320   4   2786860733 
│ │ │ │ +00059280: 3320 2020 2020 3131 3935 3037 3834 3537  3     1195078457
│ │ │ │ +00059290: 2032 2032 2020 2020 2020 2020 2020 207c   2 2           |
│ │ │ │ +000592a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000592b0: 2020 2020 2020 2020 2d20 2d2d 2d2d 2d2d          - ------
│ │ │ │ +000592c0: 7820 202b 202d 2d2d 2d2d 2d2d 2d2d 2d78  x  + ----------x
│ │ │ │ +000592d0: 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d   x  - ----------
│ │ │ │ +000592e0: 7820 7820 2020 2020 2020 2020 2020 207c  x x            |
│ │ │ │ +000592f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00059300: 2020 2020 2020 2020 2020 2032 3832 3234             28224
│ │ │ │ +00059310: 2030 2020 2020 3633 3530 3430 3030 2020   0    63504000  
│ │ │ │ +00059320: 3020 3120 2020 2033 3137 3532 3030 3020  0 1    31752000 
│ │ │ │ +00059330: 2030 2031 2020 2020 2020 2020 2020 207c   0 1           |
│ │ │ │ +00059340: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00059350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059390: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000593a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000593b0: 3536 3130 3739 2034 2020 2035 3030 3333  561079 4   50033
│ │ │ │ -000593c0: 3233 3330 3720 3320 2020 2020 3937 3432  23307 3     9742
│ │ │ │ -000593d0: 3733 3435 3332 3720 3220 3220 2020 2020  7345327 2 2     
│ │ │ │ -000593e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000593f0: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -00059400: 2d2d 2d2d 2d2d 7820 202b 202d 2d2d 2d2d  ------x  + -----
│ │ │ │ -00059410: 2d2d 2d2d 2d78 2078 2020 2d20 2d2d 2d2d  -----x x  - ----
│ │ │ │ -00059420: 2d2d 2d2d 2d2d 2d78 2078 2020 2020 2020  -------x x      
│ │ │ │ -00059430: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00059440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059450: 2035 3634 3438 2030 2020 2020 3139 3035   56448 0    1905
│ │ │ │ -00059460: 3132 3030 2020 3020 3120 2020 2031 3930  1200  0 1    190
│ │ │ │ -00059470: 3531 3230 3030 2020 3020 3120 2020 2020  512000  0 1     
│ │ │ │ -00059480: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00059380: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059390: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000593a0: 2020 2020 2020 2020 2020 3536 3130 3739            561079
│ │ │ │ +000593b0: 2034 2020 2035 3030 3333 3233 3330 3720   4   5003323307 
│ │ │ │ +000593c0: 3320 2020 2020 3937 3432 3733 3435 3332  3     9742734532
│ │ │ │ +000593d0: 3720 3220 3220 2020 2020 2020 2020 207c  7 2 2          |
│ │ │ │ +000593e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000593f0: 2020 2020 2020 2020 2d20 2d2d 2d2d 2d2d          - ------
│ │ │ │ +00059400: 7820 202b 202d 2d2d 2d2d 2d2d 2d2d 2d78  x  + ----------x
│ │ │ │ +00059410: 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d   x  - ----------
│ │ │ │ +00059420: 2d78 2078 2020 2020 2020 2020 2020 207c  -x x           |
│ │ │ │ +00059430: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00059440: 2020 2020 2020 2020 2020 2035 3634 3438             56448
│ │ │ │ +00059450: 2030 2020 2020 3139 3035 3132 3030 2020   0    19051200  
│ │ │ │ +00059460: 3020 3120 2020 2031 3930 3531 3230 3030  0 1    190512000
│ │ │ │ +00059470: 2020 3020 3120 2020 2020 2020 2020 207c    0 1          |
│ │ │ │ +00059480: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00059490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000594a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000594b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000594c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000594d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000594e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000594f0: 3831 3532 3537 2034 2020 2037 3036 3035  815257 4   70605
│ │ │ │ -00059500: 3639 3432 3120 3320 2020 2020 3139 3833  69421 3     1983
│ │ │ │ -00059510: 3130 3033 3039 3120 3220 3220 2020 2020  1003091 2 2     
│ │ │ │ -00059520: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00059530: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -00059540: 2d2d 2d2d 2d2d 7820 202b 202d 2d2d 2d2d  ------x  + -----
│ │ │ │ -00059550: 2d2d 2d2d 2d78 2078 2020 2d20 2d2d 2d2d  -----x x  - ----
│ │ │ │ -00059560: 2d2d 2d2d 2d2d 2d78 2078 2020 2020 2020  -------x x      
│ │ │ │ -00059570: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00059580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059590: 3136 3933 3434 2030 2020 2020 3534 3433  169344 0    5443
│ │ │ │ -000595a0: 3230 3030 2020 3020 3120 2020 2031 3432  2000  0 1    142
│ │ │ │ -000595b0: 3838 3430 3030 2020 3020 3120 2020 2020  884000  0 1     
│ │ │ │ -000595c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000594c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000594d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000594e0: 2020 2020 2020 2020 2020 3831 3532 3537            815257
│ │ │ │ +000594f0: 2034 2020 2037 3036 3035 3639 3432 3120   4   7060569421 
│ │ │ │ +00059500: 3320 2020 2020 3139 3833 3130 3033 3039  3     1983100309
│ │ │ │ +00059510: 3120 3220 3220 2020 2020 2020 2020 207c  1 2 2          |
│ │ │ │ +00059520: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00059530: 2020 2020 2020 2020 2d20 2d2d 2d2d 2d2d          - ------
│ │ │ │ +00059540: 7820 202b 202d 2d2d 2d2d 2d2d 2d2d 2d78  x  + ----------x
│ │ │ │ +00059550: 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d   x  - ----------
│ │ │ │ +00059560: 2d78 2078 2020 2020 2020 2020 2020 207c  -x x           |
│ │ │ │ +00059570: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00059580: 2020 2020 2020 2020 2020 3136 3933 3434            169344
│ │ │ │ +00059590: 2030 2020 2020 3534 3433 3230 3030 2020   0    54432000  
│ │ │ │ +000595a0: 3020 3120 2020 2031 3432 3838 3430 3030  0 1    142884000
│ │ │ │ +000595b0: 2020 3020 3120 2020 2020 2020 2020 207c    0 1          |
│ │ │ │ +000595c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000595d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000595e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000595f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059610: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00059620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059630: 3934 3834 3531 3320 3420 2020 3230 3334  9484513 4   2034
│ │ │ │ -00059640: 3734 3930 3320 3320 2020 2020 3134 3636  74903 3     1466
│ │ │ │ -00059650: 3832 3534 3832 3531 2032 2020 2020 2020  82548251 2      
│ │ │ │ -00059660: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00059670: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -00059680: 2d2d 2d2d 2d2d 2d78 2020 2b20 2d2d 2d2d  -------x  + ----
│ │ │ │ -00059690: 2d2d 2d2d 2d78 2078 2020 2b20 2d2d 2d2d  -----x x  + ----
│ │ │ │ -000596a0: 2d2d 2d2d 2d2d 2d2d 7820 7820 2020 2020  --------x x     
│ │ │ │ -000596b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000596c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000596d0: 2032 3930 3330 3420 3020 2020 2033 3838   290304 0    388
│ │ │ │ -000596e0: 3830 3030 2020 3020 3120 2020 2032 3933  8000  0 1    293
│ │ │ │ -000596f0: 3933 3238 3030 3020 2030 2020 2020 2020  9328000  0      
│ │ │ │ -00059700: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00059710: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ +00059600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059610: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00059620: 2020 2020 2020 2020 2020 3934 3834 3531            948451
│ │ │ │ +00059630: 3320 3420 2020 3230 3334 3734 3930 3320  3 4   203474903 
│ │ │ │ +00059640: 3320 2020 2020 3134 3636 3832 3534 3832  3     1466825482
│ │ │ │ +00059650: 3531 2032 2020 2020 2020 2020 2020 207c  51 2           |
│ │ │ │ +00059660: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00059670: 2020 2020 2020 2020 2d20 2d2d 2d2d 2d2d          - ------
│ │ │ │ +00059680: 2d78 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d78  -x  + ---------x
│ │ │ │ +00059690: 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d2d   x  + ----------
│ │ │ │ +000596a0: 2d2d 7820 7820 2020 2020 2020 2020 207c  --x x          |
│ │ │ │ +000596b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000596c0: 2020 2020 2020 2020 2020 2032 3930 3330             29030
│ │ │ │ +000596d0: 3420 3020 2020 2033 3838 3830 3030 2020  4 0    3888000  
│ │ │ │ +000596e0: 3020 3120 2020 2032 3933 3933 3238 3030  0 1    293932800
│ │ │ │ +000596f0: 3020 2030 2020 2020 2020 2020 2020 207c  0  0           |
│ │ │ │ +00059700: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00059710: 2020 2020 2020 207d 2020 2020 2020 2020         }        
│ │ │ │  00059720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059750: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00059740: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059750: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00059760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000597a0: 2020 2020 207c 0a7c 6f32 203a 2052 6174       |.|o2 : Rat
│ │ │ │ -000597b0: 696f 6e61 6c4d 6170 2028 7261 7469 6f6e  ionalMap (ration
│ │ │ │ -000597c0: 616c 206d 6170 2066 726f 6d20 5050 5e34  al map from PP^4
│ │ │ │ -000597d0: 2074 6f20 5050 5e34 2920 2020 2020 2020   to PP^4)       
│ │ │ │ -000597e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000597f0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00059790: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000597a0: 0a7c 6f32 203a 2052 6174 696f 6e61 6c4d  .|o2 : RationalM
│ │ │ │ +000597b0: 6170 2028 7261 7469 6f6e 616c 206d 6170  ap (rational map
│ │ │ │ +000597c0: 2066 726f 6d20 5050 5e34 2074 6f20 5050   from PP^4 to PP
│ │ │ │ +000597d0: 5e34 2920 2020 2020 2020 2020 2020 2020  ^4)             
│ │ │ │ +000597e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000597f0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00059800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059840: 2d2d 2d2d 2d7c 0a7c 3529 2920 2d2d 2073  -----|.|5)) -- s
│ │ │ │ -00059850: 7065 6369 616c 2043 7265 6d6f 6e61 2074  pecial Cremona t
│ │ │ │ -00059860: 7261 6e73 666f 726d 6174 696f 6e20 6f66  ransformation of
│ │ │ │ -00059870: 2050 5e34 206f 6620 7479 7065 2028 342c   P^4 of type (4,
│ │ │ │ -00059880: 3429 2020 2020 2020 2020 2020 2020 2020  4)              
│ │ │ │ -00059890: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00059830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00059840: 0a7c 3529 2920 2d2d 2073 7065 6369 616c  .|5)) -- special
│ │ │ │ +00059850: 2043 7265 6d6f 6e61 2074 7261 6e73 666f   Cremona transfo
│ │ │ │ +00059860: 726d 6174 696f 6e20 6f66 2050 5e34 206f  rmation of P^4 o
│ │ │ │ +00059870: 6620 7479 7065 2028 342c 3429 2020 2020  f type (4,4)    
│ │ │ │ +00059880: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059890: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000598a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000598b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000598c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000598d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000598e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000598d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000598e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000598f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059930: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00059920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059930: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00059940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059980: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00059970: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059980: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00059990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000599a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000599b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000599c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000599d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000599c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000599d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000599e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000599f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059a20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00059a10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059a20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00059a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059a70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00059a60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059a70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00059a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059ac0: 2020 2020 207c 0a7c 3831 3930 3733 3037       |.|81907307
│ │ │ │ -00059ad0: 3920 2020 3320 2020 3138 3737 3839 2034  9   3   187789 4
│ │ │ │ -00059ae0: 2020 2031 3336 3437 3438 3936 3720 3320     1364748967 3 
│ │ │ │ -00059af0: 2020 2020 3636 3131 3432 3935 3839 3337      661142958937
│ │ │ │ -00059b00: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00059b10: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -00059b20: 2d78 2078 2020 2b20 2d2d 2d2d 2d2d 7820  -x x  + ------x 
│ │ │ │ -00059b30: 202b 202d 2d2d 2d2d 2d2d 2d2d 2d78 2078   + ----------x x
│ │ │ │ -00059b40: 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d    + ------------
│ │ │ │ -00059b50: 7820 7820 7820 202d 2020 2020 2020 2020  x x x  -        
│ │ │ │ -00059b60: 2020 2020 207c 0a7c 2036 3533 3138 3430       |.| 6531840
│ │ │ │ -00059b70: 3020 3020 3120 2020 2032 3136 3030 2031  0 0 1    21600 1
│ │ │ │ -00059b80: 2020 2020 3637 3733 3736 3030 2020 3020      67737600  0 
│ │ │ │ -00059b90: 3220 2020 2031 3031 3630 3634 3030 3020  2    1016064000 
│ │ │ │ -00059ba0: 2030 2031 2032 2020 2020 2020 2020 2020   0 1 2          
│ │ │ │ -00059bb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00059ab0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059ac0: 0a7c 3831 3930 3733 3037 3920 2020 3320  .|819073079   3 
│ │ │ │ +00059ad0: 2020 3138 3737 3839 2034 2020 2031 3336    187789 4   136
│ │ │ │ +00059ae0: 3437 3438 3936 3720 3320 2020 2020 3636  4748967 3     66
│ │ │ │ +00059af0: 3131 3432 3935 3839 3337 2032 2020 2020  1142958937 2    
│ │ │ │ +00059b00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059b10: 0a7c 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020  .|---------x x  
│ │ │ │ +00059b20: 2b20 2d2d 2d2d 2d2d 7820 202b 202d 2d2d  + ------x  + ---
│ │ │ │ +00059b30: 2d2d 2d2d 2d2d 2d78 2078 2020 2b20 2d2d  -------x x  + --
│ │ │ │ +00059b40: 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820 7820  ----------x x x 
│ │ │ │ +00059b50: 202d 2020 2020 2020 2020 2020 2020 207c   -             |
│ │ │ │ +00059b60: 0a7c 2036 3533 3138 3430 3020 3020 3120  .| 65318400 0 1 
│ │ │ │ +00059b70: 2020 2032 3136 3030 2031 2020 2020 3637     21600 1    67
│ │ │ │ +00059b80: 3733 3736 3030 2020 3020 3220 2020 2031  737600  0 2    1
│ │ │ │ +00059b90: 3031 3630 3634 3030 3020 2030 2031 2032  016064000  0 1 2
│ │ │ │ +00059ba0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059bb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00059bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059c00: 2020 2020 207c 0a7c 2020 3737 3130 3930       |.|  771090
│ │ │ │ -00059c10: 3835 3120 2020 3320 2020 3634 3033 3939  851   3   640399
│ │ │ │ -00059c20: 3920 3420 2020 3633 3439 3833 3339 3937  9 4   6349833997
│ │ │ │ -00059c30: 2033 2020 2020 2031 3437 3730 3736 3134   3     147707614
│ │ │ │ -00059c40: 3531 2032 2020 2020 2020 2020 2020 2020  51 2            
│ │ │ │ -00059c50: 2020 2020 207c 0a7c 2b20 2d2d 2d2d 2d2d       |.|+ ------
│ │ │ │ -00059c60: 2d2d 2d78 2078 2020 2b20 2d2d 2d2d 2d2d  ---x x  + ------
│ │ │ │ -00059c70: 2d78 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d  -x  - ----------
│ │ │ │ -00059c80: 7820 7820 202b 202d 2d2d 2d2d 2d2d 2d2d  x x  + ---------
│ │ │ │ -00059c90: 2d2d 7820 7820 7820 2020 2020 2020 2020  --x x x         
│ │ │ │ -00059ca0: 2020 2020 207c 0a7c 2020 2032 3732 3136       |.|   27216
│ │ │ │ -00059cb0: 3030 3020 3020 3120 2020 2033 3234 3030  000 0 1    32400
│ │ │ │ -00059cc0: 3020 3120 2020 2037 3632 3034 3830 3020  0 1    76204800 
│ │ │ │ -00059cd0: 2030 2032 2020 2020 3334 3239 3231 3630   0 2    34292160
│ │ │ │ -00059ce0: 3020 2030 2031 2032 2020 2020 2020 2020  0  0 1 2        
│ │ │ │ -00059cf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00059bf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059c00: 0a7c 2020 3737 3130 3930 3835 3120 2020  .|  771090851   
│ │ │ │ +00059c10: 3320 2020 3634 3033 3939 3920 3420 2020  3   6403999 4   
│ │ │ │ +00059c20: 3633 3439 3833 3339 3937 2033 2020 2020  6349833997 3    
│ │ │ │ +00059c30: 2031 3437 3730 3736 3134 3531 2032 2020   14770761451 2  
│ │ │ │ +00059c40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059c50: 0a7c 2b20 2d2d 2d2d 2d2d 2d2d 2d78 2078  .|+ ---------x x
│ │ │ │ +00059c60: 2020 2b20 2d2d 2d2d 2d2d 2d78 2020 2d20    + -------x  - 
│ │ │ │ +00059c70: 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820 202b  ----------x x  +
│ │ │ │ +00059c80: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820   -----------x x 
│ │ │ │ +00059c90: 7820 2020 2020 2020 2020 2020 2020 207c  x              |
│ │ │ │ +00059ca0: 0a7c 2020 2032 3732 3136 3030 3020 3020  .|   27216000 0 
│ │ │ │ +00059cb0: 3120 2020 2033 3234 3030 3020 3120 2020  1    324000 1   
│ │ │ │ +00059cc0: 2037 3632 3034 3830 3020 2030 2032 2020   76204800  0 2  
│ │ │ │ +00059cd0: 2020 3334 3239 3231 3630 3020 2030 2031    342921600  0 1
│ │ │ │ +00059ce0: 2032 2020 2020 2020 2020 2020 2020 207c   2             |
│ │ │ │ +00059cf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00059d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059d40: 2020 2020 207c 0a7c 2020 2031 3436 3739       |.|   14679
│ │ │ │ -00059d50: 3439 3934 3920 2020 3320 2020 3836 3937  49949   3   8697
│ │ │ │ -00059d60: 3720 3420 2020 3130 3530 3833 3533 2033  7 4   10508353 3
│ │ │ │ -00059d70: 2020 2020 2033 3631 3432 3136 3735 3239       36142167529
│ │ │ │ -00059d80: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00059d90: 2020 2020 207c 0a7c 202b 202d 2d2d 2d2d       |.| + -----
│ │ │ │ -00059da0: 2d2d 2d2d 2d78 2078 2020 2b20 2d2d 2d2d  -----x x  + ----
│ │ │ │ -00059db0: 2d78 2020 2b20 2d2d 2d2d 2d2d 2d2d 7820  -x  + --------x 
│ │ │ │ -00059dc0: 7820 202d 202d 2d2d 2d2d 2d2d 2d2d 2d2d  x  - -----------
│ │ │ │ -00059dd0: 7820 7820 7820 202d 2020 2020 2020 2020  x x x  -        
│ │ │ │ -00059de0: 2020 2020 207c 0a7c 2020 2020 3133 3630       |.|    1360
│ │ │ │ -00059df0: 3830 3030 2020 3020 3120 2020 2034 3035  8000  0 1    405
│ │ │ │ -00059e00: 3020 3120 2020 2037 3134 3432 3020 2030  0 1    714420  0
│ │ │ │ -00059e10: 2032 2020 2020 3638 3538 3433 3230 3020   2    685843200 
│ │ │ │ -00059e20: 2030 2031 2032 2020 2020 2020 2020 2020   0 1 2          
│ │ │ │ -00059e30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00059d30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059d40: 0a7c 2020 2031 3436 3739 3439 3934 3920  .|   1467949949 
│ │ │ │ +00059d50: 2020 3320 2020 3836 3937 3720 3420 2020    3   86977 4   
│ │ │ │ +00059d60: 3130 3530 3833 3533 2033 2020 2020 2033  10508353 3     3
│ │ │ │ +00059d70: 3631 3432 3136 3735 3239 2032 2020 2020  6142167529 2    
│ │ │ │ +00059d80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059d90: 0a7c 202b 202d 2d2d 2d2d 2d2d 2d2d 2d78  .| + ----------x
│ │ │ │ +00059da0: 2078 2020 2b20 2d2d 2d2d 2d78 2020 2b20   x  + -----x  + 
│ │ │ │ +00059db0: 2d2d 2d2d 2d2d 2d2d 7820 7820 202d 202d  --------x x  - -
│ │ │ │ +00059dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820 7820  ----------x x x 
│ │ │ │ +00059dd0: 202d 2020 2020 2020 2020 2020 2020 207c   -             |
│ │ │ │ +00059de0: 0a7c 2020 2020 3133 3630 3830 3030 2020  .|    13608000  
│ │ │ │ +00059df0: 3020 3120 2020 2034 3035 3020 3120 2020  0 1    4050 1   
│ │ │ │ +00059e00: 2037 3134 3432 3020 2030 2032 2020 2020   714420  0 2    
│ │ │ │ +00059e10: 3638 3538 3433 3230 3020 2030 2031 2032  685843200  0 1 2
│ │ │ │ +00059e20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059e30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00059e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059e80: 2020 2020 207c 0a7c 2020 2035 3433 3637       |.|   54367
│ │ │ │ -00059e90: 3835 3938 3120 2020 3320 2020 3737 3631  85981   3   7761
│ │ │ │ -00059ea0: 3232 3120 3420 2020 3136 3930 3939 3837  221 4   16909987
│ │ │ │ -00059eb0: 3433 3320 3320 2020 2020 3234 3034 3832  433 3     240482
│ │ │ │ -00059ec0: 3139 3535 3439 2032 2020 2020 2020 2020  195549 2        
│ │ │ │ -00059ed0: 2020 2020 207c 0a7c 202d 202d 2d2d 2d2d       |.| - -----
│ │ │ │ -00059ee0: 2d2d 2d2d 2d78 2078 2020 2d20 2d2d 2d2d  -----x x  - ----
│ │ │ │ -00059ef0: 2d2d 2d78 2020 2d20 2d2d 2d2d 2d2d 2d2d  ---x  - --------
│ │ │ │ -00059f00: 2d2d 2d78 2078 2020 2b20 2d2d 2d2d 2d2d  ---x x  + ------
│ │ │ │ -00059f10: 2d2d 2d2d 2d2d 7820 7820 2020 2020 2020  ------x x       
│ │ │ │ -00059f20: 2020 2020 207c 0a7c 2020 2020 3138 3134       |.|    1814
│ │ │ │ -00059f30: 3430 3030 2020 3020 3120 2020 2033 3234  4000  0 1    324
│ │ │ │ -00059f40: 3030 3020 3120 2020 2034 3537 3232 3838  000 1    4572288
│ │ │ │ -00059f50: 3030 2020 3020 3220 2020 2020 3431 3939  00  0 2     4199
│ │ │ │ -00059f60: 3034 3030 3020 2030 2020 2020 2020 2020  04000  0        
│ │ │ │ -00059f70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00059e70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059e80: 0a7c 2020 2035 3433 3637 3835 3938 3120  .|   5436785981 
│ │ │ │ +00059e90: 2020 3320 2020 3737 3631 3232 3120 3420    3   7761221 4 
│ │ │ │ +00059ea0: 2020 3136 3930 3939 3837 3433 3320 3320    16909987433 3 
│ │ │ │ +00059eb0: 2020 2020 3234 3034 3832 3139 3535 3439      240482195549
│ │ │ │ +00059ec0: 2032 2020 2020 2020 2020 2020 2020 207c   2             |
│ │ │ │ +00059ed0: 0a7c 202d 202d 2d2d 2d2d 2d2d 2d2d 2d78  .| - ----------x
│ │ │ │ +00059ee0: 2078 2020 2d20 2d2d 2d2d 2d2d 2d78 2020   x  - -------x  
│ │ │ │ +00059ef0: 2d20 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078  - -----------x x
│ │ │ │ +00059f00: 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d    + ------------
│ │ │ │ +00059f10: 7820 7820 2020 2020 2020 2020 2020 207c  x x            |
│ │ │ │ +00059f20: 0a7c 2020 2020 3138 3134 3430 3030 2020  .|    18144000  
│ │ │ │ +00059f30: 3020 3120 2020 2033 3234 3030 3020 3120  0 1    324000 1 
│ │ │ │ +00059f40: 2020 2034 3537 3232 3838 3030 2020 3020     457228800  0 
│ │ │ │ +00059f50: 3220 2020 2020 3431 3939 3034 3030 3020  2     419904000 
│ │ │ │ +00059f60: 2030 2020 2020 2020 2020 2020 2020 207c   0             |
│ │ │ │ +00059f70: 0a7c 2020 2020 2020 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059fc0: 2020 2020 207c 0a7c 3220 2020 3533 3035       |.|2   5305
│ │ │ │ -00059fd0: 3134 3034 3937 2020 2033 2020 2037 3937  140497   3   797
│ │ │ │ -00059fe0: 3237 2034 2020 2034 3639 3939 3637 3230  27 4   469996720
│ │ │ │ -00059ff0: 3733 2033 2020 2020 2039 3835 3435 3535  73 3     9854555
│ │ │ │ -0005a000: 3134 3035 3839 2032 2020 2020 2020 2020  140589 2        
│ │ │ │ -0005a010: 2020 2020 207c 0a7c 2020 2d20 2d2d 2d2d       |.|  - ----
│ │ │ │ -0005a020: 2d2d 2d2d 2d2d 7820 7820 202d 202d 2d2d  ------x x  - ---
│ │ │ │ -0005a030: 2d2d 7820 202d 202d 2d2d 2d2d 2d2d 2d2d  --x  - ---------
│ │ │ │ -0005a040: 2d2d 7820 7820 202b 202d 2d2d 2d2d 2d2d  --x x  + -------
│ │ │ │ -0005a050: 2d2d 2d2d 2d2d 7820 7820 2020 2020 2020  ------x x       
│ │ │ │ -0005a060: 2020 2020 207c 0a7c 3120 2020 2031 3633       |.|1    163
│ │ │ │ -0005a070: 3239 3630 3030 2030 2031 2020 2032 3032  296000 0 1   202
│ │ │ │ -0005a080: 3530 2031 2020 2020 3232 3836 3134 3430  50 1    22861440
│ │ │ │ -0005a090: 3020 2030 2032 2020 2020 3431 3135 3035  0  0 2    411505
│ │ │ │ -0005a0a0: 3932 3030 3020 2030 2020 2020 2020 2020  92000  0        
│ │ │ │ -0005a0b0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00059fb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059fc0: 0a7c 3220 2020 3533 3035 3134 3034 3937  .|2   5305140497
│ │ │ │ +00059fd0: 2020 2033 2020 2037 3937 3237 2034 2020     3   79727 4  
│ │ │ │ +00059fe0: 2034 3639 3939 3637 3230 3733 2033 2020   46999672073 3  
│ │ │ │ +00059ff0: 2020 2039 3835 3435 3535 3134 3035 3839     9854555140589
│ │ │ │ +0005a000: 2032 2020 2020 2020 2020 2020 2020 207c   2             |
│ │ │ │ +0005a010: 0a7c 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d  .|  - ----------
│ │ │ │ +0005a020: 7820 7820 202d 202d 2d2d 2d2d 7820 202d  x x  - -----x  -
│ │ │ │ +0005a030: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820   -----------x x 
│ │ │ │ +0005a040: 202b 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   + -------------
│ │ │ │ +0005a050: 7820 7820 2020 2020 2020 2020 2020 207c  x x            |
│ │ │ │ +0005a060: 0a7c 3120 2020 2031 3633 3239 3630 3030  .|1    163296000
│ │ │ │ +0005a070: 2030 2031 2020 2032 3032 3530 2031 2020   0 1   20250 1  
│ │ │ │ +0005a080: 2020 3232 3836 3134 3430 3020 2030 2032    228614400  0 2
│ │ │ │ +0005a090: 2020 2020 3431 3135 3035 3932 3030 3020      41150592000 
│ │ │ │ +0005a0a0: 2030 2020 2020 2020 2020 2020 2020 207c   0             |
│ │ │ │ +0005a0b0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0005a0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a100: 2d2d 2d2d 2d7c 0a7c 3239 3533 3039 3034  -----|.|29530904
│ │ │ │ -0005a110: 3733 3239 2020 2032 2020 2020 2032 3436  7329   2     246
│ │ │ │ -0005a120: 3333 3831 3120 3320 2020 2020 2020 2020  33811 3         
│ │ │ │ +0005a0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0005a100: 0a7c 3239 3533 3039 3034 3733 3239 2020  .|295309047329  
│ │ │ │ +0005a110: 2032 2020 2020 2032 3436 3333 3831 3120   2     24633811 
│ │ │ │ +0005a120: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │  0005a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a150: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -0005a160: 2d2d 2d2d 7820 7820 7820 202b 202d 2d2d  ----x x x  + ---
│ │ │ │ -0005a170: 2d2d 2d2d 2d78 2078 2020 2b20 2020 2020  -----x x  +     
│ │ │ │ +0005a140: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a150: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7820  .|------------x 
│ │ │ │ +0005a160: 7820 7820 202b 202d 2d2d 2d2d 2d2d 2d78  x x  + --------x
│ │ │ │ +0005a170: 2078 2020 2b20 2020 2020 2020 2020 2020   x  +           
│ │ │ │  0005a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a1a0: 2020 2020 207c 0a7c 2033 3432 3932 3136       |.| 3429216
│ │ │ │ -0005a1b0: 3030 3020 2030 2031 2032 2020 2020 3534  000  0 1 2    54
│ │ │ │ -0005a1c0: 3433 3230 2020 3120 3220 2020 2020 2020  4320  1 2       
│ │ │ │ +0005a190: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a1a0: 0a7c 2033 3432 3932 3136 3030 3020 2030  .| 3429216000  0
│ │ │ │ +0005a1b0: 2031 2032 2020 2020 3534 3433 3230 2020   1 2    544320  
│ │ │ │ +0005a1c0: 3120 3220 2020 2020 2020 2020 2020 2020  1 2             
│ │ │ │  0005a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a1f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005a1e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a1f0: 0a7c 2020 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a240: 2020 2020 207c 0a7c 2020 3830 3939 3539       |.|  809959
│ │ │ │ -0005a250: 3438 3120 2020 3220 2020 2020 3835 3430  481   2     8540
│ │ │ │ -0005a260: 3032 3132 3320 3320 2020 2020 2020 2020  02123 3         
│ │ │ │ +0005a230: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a240: 0a7c 2020 3830 3939 3539 3438 3120 2020  .|  809959481   
│ │ │ │ +0005a250: 3220 2020 2020 3835 3430 3032 3132 3320  2     854002123 
│ │ │ │ +0005a260: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │  0005a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a290: 2020 2020 207c 0a7c 2d20 2d2d 2d2d 2d2d       |.|- ------
│ │ │ │ -0005a2a0: 2d2d 2d78 2078 2078 2020 2b20 2d2d 2d2d  ---x x x  + ----
│ │ │ │ -0005a2b0: 2d2d 2d2d 2d78 2078 2020 2d20 2020 2020  -----x x  -     
│ │ │ │ +0005a280: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a290: 0a7c 2d20 2d2d 2d2d 2d2d 2d2d 2d78 2078  .|- ---------x x
│ │ │ │ +0005a2a0: 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d78   x  + ---------x
│ │ │ │ +0005a2b0: 2078 2020 2d20 2020 2020 2020 2020 2020   x  -           
│ │ │ │  0005a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a2e0: 2020 2020 207c 0a7c 2020 2031 3035 3834       |.|   10584
│ │ │ │ -0005a2f0: 3030 3020 3020 3120 3220 2020 2037 3737  000 0 1 2    777
│ │ │ │ -0005a300: 3630 3030 2020 3120 3220 2020 2020 2020  6000  1 2       
│ │ │ │ +0005a2d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a2e0: 0a7c 2020 2031 3035 3834 3030 3020 3020  .|   10584000 0 
│ │ │ │ +0005a2f0: 3120 3220 2020 2037 3737 3630 3030 2020  1 2    7776000  
│ │ │ │ +0005a300: 3120 3220 2020 2020 2020 2020 2020 2020  1 2             
│ │ │ │  0005a310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a330: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005a320: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a330: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a380: 2020 2020 207c 0a7c 3237 3033 3837 3737       |.|27038777
│ │ │ │ -0005a390: 3734 3331 2020 2032 2020 2020 2036 3536  7431   2     656
│ │ │ │ -0005a3a0: 3232 3430 3537 2033 2020 2020 2020 2020  224057 3        
│ │ │ │ +0005a370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a380: 0a7c 3237 3033 3837 3737 3734 3331 2020  .|270387777431  
│ │ │ │ +0005a390: 2032 2020 2020 2036 3536 3232 3430 3537   2     656224057
│ │ │ │ +0005a3a0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  0005a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a3d0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -0005a3e0: 2d2d 2d2d 7820 7820 7820 202b 202d 2d2d  ----x x x  + ---
│ │ │ │ -0005a3f0: 2d2d 2d2d 2d2d 7820 7820 202b 2020 2020  ------x x  +    
│ │ │ │ +0005a3c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a3d0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7820  .|------------x 
│ │ │ │ +0005a3e0: 7820 7820 202b 202d 2d2d 2d2d 2d2d 2d2d  x x  + ---------
│ │ │ │ +0005a3f0: 7820 7820 202b 2020 2020 2020 2020 2020  x x  +          
│ │ │ │  0005a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a420: 2020 2020 207c 0a7c 2020 3338 3130 3234       |.|  381024
│ │ │ │ -0005a430: 3030 3020 2030 2031 2032 2020 2020 3638  000  0 1 2    68
│ │ │ │ -0005a440: 3034 3030 3020 2031 2032 2020 2020 2020  04000  1 2      
│ │ │ │ +0005a410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a420: 0a7c 2020 3338 3130 3234 3030 3020 2030  .|  381024000  0
│ │ │ │ +0005a430: 2031 2032 2020 2020 3638 3034 3030 3020   1 2    6804000 
│ │ │ │ +0005a440: 2031 2032 2020 2020 2020 2020 2020 2020   1 2            
│ │ │ │  0005a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a470: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005a460: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a470: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a4c0: 2020 2020 207c 0a7c 2020 2020 2020 3335       |.|      35
│ │ │ │ -0005a4d0: 3739 3931 3838 3635 3120 2020 3220 2020  799188651   2   
│ │ │ │ -0005a4e0: 2020 3631 3736 3934 3830 3920 2020 2020    617694809     
│ │ │ │ +0005a4b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a4c0: 0a7c 2020 2020 2020 3335 3739 3931 3838  .|      35799188
│ │ │ │ +0005a4d0: 3635 3120 2020 3220 2020 2020 3631 3736  651   2     6176
│ │ │ │ +0005a4e0: 3934 3830 3920 2020 2020 2020 2020 2020  94809           
│ │ │ │  0005a4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a510: 2020 2020 207c 0a7c 2078 2020 2b20 2d2d       |.| x  + --
│ │ │ │ -0005a520: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078 2020  ---------x x x  
│ │ │ │ -0005a530: 2d20 2d2d 2d2d 2d2d 2d2d 2d78 2020 2020  - ---------x    
│ │ │ │ +0005a500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a510: 0a7c 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d  .| x  + --------
│ │ │ │ +0005a520: 2d2d 2d78 2078 2078 2020 2d20 2d2d 2d2d  ---x x x  - ----
│ │ │ │ +0005a530: 2d2d 2d2d 2d78 2020 2020 2020 2020 2020  -----x          
│ │ │ │  0005a540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a560: 2020 2020 207c 0a7c 3120 3220 2020 2020       |.|1 2     
│ │ │ │ -0005a570: 3437 3632 3830 3030 2020 3020 3120 3220  47628000  0 1 2 
│ │ │ │ -0005a580: 2020 2035 3434 3332 3030 3020 2020 2020     54432000     
│ │ │ │ +0005a550: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a560: 0a7c 3120 3220 2020 2020 3437 3632 3830  .|1 2     476280
│ │ │ │ +0005a570: 3030 2020 3020 3120 3220 2020 2035 3434  00  0 1 2    544
│ │ │ │ +0005a580: 3332 3030 3020 2020 2020 2020 2020 2020  32000           
│ │ │ │  0005a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a5b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005a5a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a5b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a600: 2020 2020 207c 0a7c 2020 2020 2020 3336       |.|      36
│ │ │ │ -0005a610: 3631 3930 3131 3336 3737 3920 2020 3220  61901136779   2 
│ │ │ │ -0005a620: 2020 2020 3639 3239 3232 3433 2020 2020      69292243    
│ │ │ │ +0005a5f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a600: 0a7c 2020 2020 2020 3336 3631 3930 3131  .|      36619011
│ │ │ │ +0005a610: 3336 3737 3920 2020 3220 2020 2020 3639  36779   2     69
│ │ │ │ +0005a620: 3239 3232 3433 2020 2020 2020 2020 2020  292243          
│ │ │ │  0005a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a650: 2020 2020 207c 0a7c 2078 2020 2b20 2d2d       |.| x  + --
│ │ │ │ -0005a660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078  -----------x x x
│ │ │ │ -0005a670: 2020 2d20 2d2d 2d2d 2d2d 2d2d 2020 2020    - --------    
│ │ │ │ +0005a640: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a650: 0a7c 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d  .| x  + --------
│ │ │ │ +0005a660: 2d2d 2d2d 2d78 2078 2078 2020 2d20 2d2d  -----x x x  - --
│ │ │ │ +0005a670: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020  ------          
│ │ │ │  0005a680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a6a0: 2020 2020 207c 0a7c 3120 3220 2020 2034       |.|1 2    4
│ │ │ │ -0005a6b0: 3131 3530 3539 3230 3030 2020 3020 3120  1150592000  0 1 
│ │ │ │ -0005a6c0: 3220 2020 2033 3838 3830 3030 2020 2020  2    3888000    
│ │ │ │ +0005a690: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a6a0: 0a7c 3120 3220 2020 2034 3131 3530 3539  .|1 2    4115059
│ │ │ │ +0005a6b0: 3230 3030 2020 3020 3120 3220 2020 2033  2000  0 1 2    3
│ │ │ │ +0005a6c0: 3838 3830 3030 2020 2020 2020 2020 2020  888000          
│ │ │ │  0005a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a6f0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +0005a6e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a6f0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0005a700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a740: 2d2d 2d2d 2d7c 0a7c 3830 3630 3933 3837  -----|.|80609387
│ │ │ │ -0005a750: 2032 2032 2020 2036 3237 3333 3530 3933   2 2   627335093
│ │ │ │ -0005a760: 3137 3120 2020 2020 3220 2020 3634 3131  171     2   6411
│ │ │ │ -0005a770: 3139 3038 3334 3720 3220 3220 2020 3130  1908347 2 2   10
│ │ │ │ -0005a780: 3734 3234 3937 2020 2033 2020 2031 3033  742497   3   103
│ │ │ │ -0005a790: 3830 3633 377c 0a7c 2d2d 2d2d 2d2d 2d2d  80637|.|--------
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│ │ │ │ -0005a7e0: 2d2d 2d2d 2d7c 0a7c 2035 3038 3033 3230  -----|.| 5080320
│ │ │ │ -0005a7f0: 2030 2032 2020 2020 3137 3134 3630 3830   0 2    17146080
│ │ │ │ -0005a800: 3030 2020 3020 3120 3220 2020 2032 3835  00  0 1 2    285
│ │ │ │ -0005a810: 3736 3830 3030 2020 3120 3220 2020 2031  768000  1 2    1
│ │ │ │ -0005a820: 3538 3736 3030 2030 2032 2020 2020 3131  587600 0 2    11
│ │ │ │ -0005a830: 3433 3037 327c 0a7c 2020 2020 2020 2020  43072|.|        
│ │ │ │ +0005a730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
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│ │ │ │ +0005a750: 2036 3237 3333 3530 3933 3137 3120 2020   627335093171   
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│ │ │ │ +0005a7e0: 0a7c 2035 3038 3033 3230 2030 2032 2020  .| 5080320 0 2  
│ │ │ │ +0005a7f0: 2020 3137 3134 3630 3830 3030 2020 3020    1714608000  0 
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│ │ │ │ -0005a9e0: 3237 3439 2020 2020 2032 2020 2031 3133  2749     2   113
│ │ │ │ -0005a9f0: 3334 3134 3737 2032 2032 2020 2032 3938  341477 2 2   298
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│ │ │ │ -0005aa70: 3630 2020 3020 3220 2020 2034 3238 3635  60  0 2    42865
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│ │ │ │ -0005aa90: 3431 3230 3020 2031 2032 2020 2020 3635  41200  1 2    65
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│ │ │ │ -0005aab0: 3336 3030 207c 0a7c 2020 2020 2020 2020  3600 |.|        
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│ │ │ │ +0005aa00: 2020 3320 2020 3135 3535 3331 3633 207c    3   15553163 |
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│ │ │ │ +0005aa70: 3220 2020 2034 3238 3635 3230 3020 2030  2    42865200  0
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│ │ │ │ +0005aaa0: 3020 3220 2020 2036 3132 3336 3030 207c  0 2    6123600 |
│ │ │ │ +0005aab0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ -0005abb0: 3832 3839 3135 3230 3020 2030 2032 2020  828915200  0 2  
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│ │ │ │ -0005abf0: 3336 3030 207c 0a7c 2020 2020 2020 2020  3600 |.|        
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│ │ │ │ -0005acf0: 2033 3537 3231 3030 2020 3020 3220 2020   3572100  0 2   
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│ │ │ │ -0005ae20: 2078 2078 207c 0a7c 3020 2031 2032 2020   x x |.|0  1 2  
│ │ │ │ -0005ae30: 2020 3539 3533 3520 3220 2020 2031 3438    59535 2    148
│ │ │ │ -0005ae40: 3137 3630 3030 2020 3020 3320 2020 2020  176000  0 3     
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│ │ │ │ -0005ae60: 2020 2020 2037 3632 3034 3830 3030 2020       762048000  
│ │ │ │ -0005ae70: 3020 3120 337c 0a7c 2020 2020 2020 2020  0 1 3|.|        
│ │ │ │ +0005ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0005ad80: 0a7c 3237 2020 2033 2020 2034 3633 3836  .|27   3   46386
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│ │ │ │ +0005adf0: 2d78 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d  -x x  - --------
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│ │ │ │ +0005ae30: 3520 3220 2020 2031 3438 3137 3630 3030  5 2    148176000
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│ │ │ │ -0005af70: 3220 2020 2032 3732 3136 3020 2032 2020  2    272160  2  
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│ │ │ │ -0005b710: 3033 3338 3430 3030 2020 3020 3120 3220  03384000  0 1 2 
│ │ │ │ -0005b720: 3320 2020 2020 3336 3734 3136 3030 3020  3     367416000 
│ │ │ │ -0005b730: 2031 2032 207c 0a7c 2020 2020 2020 2020   1 2 |.|        
│ │ │ │ +0005b630: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005b640: 0a7c 3236 3530 3332 3937 3320 3320 2020  .|265032973 3   
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│ │ │ │ +0005b660: 2020 2020 2033 3636 3336 3537 3737 3734       36636577774
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│ │ │ │ +0005b680: 3736 3830 3432 3732 3837 2032 2020 207c  7680427287 2   |
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│ │ │ │ +0005b7c0: 3837 3532 3320 2020 2020 2020 2020 207c  87523          |
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│ │ │ │ -0005bd30: 3030 3930 3234 3030 2020 3020 3220 3320  00902400  0 2 3 
│ │ │ │ -0005bd40: 2020 2032 3537 3139 3132 3030 2031 2032     257191200 1 2
│ │ │ │ -0005bd50: 2033 2020 2020 3130 3731 3633 2032 2033   3    107163 2 3
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│ │ │ │ -0005bd70: 3320 2020 207c 0a7c 2020 2020 2020 2020  3    |.|        
│ │ │ │ +0005bc70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +0005bcb0: 3639 3335 3031 2033 2020 2020 2039 3934  693501 3     994
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│ │ │ │ +0005bcd0: 0a7c 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d2d  .|  + ----------
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│ │ │ │ +0005bd10: 2d2d 2d2d 2d2d 2d78 2078 2020 2d20 2d7c  -------x x  - -|
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│ │ │ │  0005bda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0005be20: 2d2d 2d2d 2d78 2078 2078 2020 2b20 2d2d  -----x x x  + --
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│ │ │ │ -0005be50: 2078 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d   x x  + --------
│ │ │ │ -0005be60: 7820 7820 207c 0a7c 2020 3235 3731 3931  x x  |.|  257191
│ │ │ │ -0005be70: 3230 3030 2020 3120 3220 3320 2020 2031  2000  1 2 3    1
│ │ │ │ -0005be80: 3036 3638 3637 3230 3020 2030 2032 2033  066867200  0 2 3
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│ │ │ │ -0005beb0: 2032 2033 207c 0a7c 2020 2020 2020 2020   2 3 |.|        
│ │ │ │ +0005bdb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005bdc0: 0a7c 2036 3930 3531 3138 3731 3531 3320  .| 690511871513 
│ │ │ │ +0005bdd0: 3220 2020 2020 2020 3732 3635 3232 3738  2       72652278
│ │ │ │ +0005bde0: 3635 3431 2020 2032 2020 2020 2033 3438  6541   2     348
│ │ │ │ +0005bdf0: 3833 3632 3938 3930 3120 2020 3220 2020  836298901   2   
│ │ │ │ +0005be00: 2020 3234 3238 3438 3433 2033 2020 207c    24284843 3   |
│ │ │ │ +0005be10: 0a7c 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d78  .| ------------x
│ │ │ │ +0005be20: 2078 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d   x x  + --------
│ │ │ │ +0005be30: 2d2d 2d2d 7820 7820 7820 202b 202d 2d2d  ----x x x  + ---
│ │ │ │ +0005be40: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078 2020  ---------x x x  
│ │ │ │ +0005be50: 2b20 2d2d 2d2d 2d2d 2d2d 7820 7820 207c  + --------x x  |
│ │ │ │ +0005be60: 0a7c 2020 3235 3731 3931 3230 3030 2020  .|  2571912000  
│ │ │ │ +0005be70: 3120 3220 3320 2020 2031 3036 3638 3637  1 2 3    1066867
│ │ │ │ +0005be80: 3230 3020 2030 2032 2033 2020 2020 3133  200  0 2 3    13
│ │ │ │ +0005be90: 3731 3638 3634 3030 2020 3120 3220 3320  71686400  1 2 3 
│ │ │ │ +0005bea0: 2020 2032 3338 3134 3020 2032 2033 207c     238140  2 3 |
│ │ │ │ +0005beb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005bee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005bef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005bf00: 2020 2020 207c 0a7c 3932 3432 3320 3220       |.|92423 2 
│ │ │ │ -0005bf10: 2020 2020 2020 3332 3331 3830 3636 3030        3231806600
│ │ │ │ -0005bf20: 3320 2020 3220 2020 2020 3339 3536 3936  3   2     395696
│ │ │ │ -0005bf30: 3436 3133 3031 2020 2032 2020 2020 2036  461301   2     6
│ │ │ │ -0005bf40: 3539 3535 3936 3632 3920 3320 2020 2020  595596629 3     
│ │ │ │ -0005bf50: 3436 3031 357c 0a7c 2d2d 2d2d 2d78 2078  46015|.|-----x x
│ │ │ │ -0005bf60: 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d2d   x  + ----------
│ │ │ │ -0005bf70: 2d78 2078 2078 2020 2d20 2d2d 2d2d 2d2d  -x x x  - ------
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│ │ │ │ -0005bfa0: 2d2d 2d2d 2d7c 0a7c 3830 3030 2020 3120  -----|.|8000  1 
│ │ │ │ -0005bfb0: 3220 3320 2020 2032 3232 3236 3430 3030  2 3    222264000
│ │ │ │ -0005bfc0: 2020 3020 3220 3320 2020 2031 3230 3032    0 2 3    12002
│ │ │ │ -0005bfd0: 3235 3630 3030 2031 2032 2033 2020 2020  256000 1 2 3    
│ │ │ │ -0005bfe0: 3636 3637 3932 3030 2020 3220 3320 2020  66679200  2 3   
│ │ │ │ -0005bff0: 2036 3636 377c 0a7c 2d2d 2d2d 2d2d 2d2d   6667|.|--------
│ │ │ │ +0005bef0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005bf00: 0a7c 3932 3432 3320 3220 2020 2020 2020  .|92423 2       
│ │ │ │ +0005bf10: 3332 3331 3830 3636 3030 3320 2020 3220  32318066003   2 
│ │ │ │ +0005bf20: 2020 2020 3339 3536 3936 3436 3133 3031      395696461301
│ │ │ │ +0005bf30: 2020 2032 2020 2020 2036 3539 3535 3936     2     6595596
│ │ │ │ +0005bf40: 3632 3920 3320 2020 2020 3436 3031 357c  629 3     46015|
│ │ │ │ +0005bf50: 0a7c 2d2d 2d2d 2d78 2078 2078 2020 2b20  .|-----x x x  + 
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│ │ │ │ +0005bf70: 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d    - ------------
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│ │ │ │ +0005bfa0: 0a7c 3830 3030 2020 3120 3220 3320 2020  .|8000  1 2 3   
│ │ │ │ +0005bfb0: 2032 3232 3236 3430 3030 2020 3020 3220   222264000  0 2 
│ │ │ │ +0005bfc0: 3320 2020 2031 3230 3032 3235 3630 3030  3    12002256000
│ │ │ │ +0005bfd0: 2031 2032 2033 2020 2020 3636 3637 3932   1 2 3    666792
│ │ │ │ +0005bfe0: 3030 2020 3220 3320 2020 2036 3636 377c  00  2 3    6667|
│ │ │ │ +0005bff0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0005c000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c040: 2d2d 2d2d 2d7c 0a7c 3220 2020 3135 3132  -----|.|2   1512
│ │ │ │ -0005c050: 3530 3430 3430 3720 2020 2020 3220 2020  5040407     2   
│ │ │ │ -0005c060: 3838 3538 3132 3638 3831 2032 2032 2020  8858126881 2 2  
│ │ │ │ -0005c070: 2031 3631 3038 3137 3339 3839 2020 2020   16108173989    
│ │ │ │ -0005c080: 2032 2020 2036 3431 3932 3436 3536 3720   2   6419246567 
│ │ │ │ -0005c090: 2020 2020 327c 0a7c 2020 2d20 2d2d 2d2d      2|.|  - ----
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│ │ │ │ -0005c0c0: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820   -----------x x 
│ │ │ │ -0005c0d0: 7820 202b 202d 2d2d 2d2d 2d2d 2d2d 2d78  x  + ----------x
│ │ │ │ -0005c0e0: 2078 2078 207c 0a7c 3320 2020 2020 3736   x x |.|3     76
│ │ │ │ -0005c0f0: 3230 3438 3030 2020 3020 3120 3320 2020  204800  0 1 3   
│ │ │ │ -0005c100: 2038 3136 3438 3030 3020 2031 2033 2020   81648000  1 3  
│ │ │ │ -0005c110: 2020 2032 3936 3335 3230 3020 2030 2032     29635200  0 2
│ │ │ │ -0005c120: 2033 2020 2020 3231 3136 3830 3030 2020   3    21168000  
│ │ │ │ -0005c130: 3120 3220 337c 0a7c 2020 2020 2020 2020  1 2 3|.|        
│ │ │ │ +0005c030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0005c040: 0a7c 3220 2020 3135 3132 3530 3430 3430  .|2   1512504040
│ │ │ │ +0005c050: 3720 2020 2020 3220 2020 3838 3538 3132  7     2   885812
│ │ │ │ +0005c060: 3638 3831 2032 2032 2020 2031 3631 3038  6881 2 2   16108
│ │ │ │ +0005c070: 3137 3339 3839 2020 2020 2032 2020 2036  173989     2   6
│ │ │ │ +0005c080: 3431 3932 3436 3536 3720 2020 2020 327c  419246567     2|
│ │ │ │ +0005c090: 0a7c 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d  .|  - ----------
│ │ │ │ +0005c0a0: 2d78 2078 2078 2020 2b20 2d2d 2d2d 2d2d  -x x x  + ------
│ │ │ │ +0005c0b0: 2d2d 2d2d 7820 7820 202b 202d 2d2d 2d2d  ----x x  + -----
│ │ │ │ +0005c0c0: 2d2d 2d2d 2d2d 7820 7820 7820 202b 202d  ------x x x  + -
│ │ │ │ +0005c0d0: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078 207c  ---------x x x |
│ │ │ │ +0005c0e0: 0a7c 3320 2020 2020 3736 3230 3438 3030  .|3     76204800
│ │ │ │ +0005c0f0: 2020 3020 3120 3320 2020 2038 3136 3438    0 1 3    81648
│ │ │ │ +0005c100: 3030 3020 2031 2033 2020 2020 2032 3936  000  1 3     296
│ │ │ │ +0005c110: 3335 3230 3020 2030 2032 2033 2020 2020  35200  0 2 3    
│ │ │ │ +0005c120: 3231 3136 3830 3030 2020 3120 3220 337c  21168000  1 2 3|
│ │ │ │ +0005c130: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c180: 2020 2020 207c 0a7c 2032 2032 2020 2033       |.| 2 2   3
│ │ │ │ -0005c190: 3333 3332 3831 3035 3735 3320 2020 2020  33328105753     
│ │ │ │ -0005c1a0: 3220 2020 3331 3334 3930 3239 3920 3220  2   313490299 2 
│ │ │ │ -0005c1b0: 3220 2020 3436 3137 3435 3534 3536 3320  2   46174554563 
│ │ │ │ -0005c1c0: 2020 2020 3220 2020 3137 3639 3537 3631      2   17695761
│ │ │ │ -0005c1d0: 3534 3539 207c 0a7c 7820 7820 202d 202d  5459 |.|x x  - -
│ │ │ │ -0005c1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078  -----------x x x
│ │ │ │ -0005c1f0: 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d78 2078    + ---------x x
│ │ │ │ -0005c200: 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d 2d78    - -----------x
│ │ │ │ -0005c210: 2078 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d   x x  + --------
│ │ │ │ -0005c220: 2d2d 2d2d 787c 0a7c 2030 2033 2020 2020  ----x|.| 0 3    
│ │ │ │ -0005c230: 3130 3030 3138 3830 3030 2020 3020 3120  1000188000  0 1 
│ │ │ │ -0005c240: 3320 2020 2031 3237 3537 3530 2020 3120  3    1275750  1 
│ │ │ │ -0005c250: 3320 2020 2031 3435 3836 3037 3530 2020  3    145860750  
│ │ │ │ -0005c260: 3020 3220 3320 2020 2020 3232 3232 3634  0 2 3     222264
│ │ │ │ -0005c270: 3030 3020 207c 0a7c 2020 2020 2020 2020  000  |.|        
│ │ │ │ +0005c170: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005c180: 0a7c 2032 2032 2020 2033 3333 3332 3831  .| 2 2   3333281
│ │ │ │ +0005c190: 3035 3735 3320 2020 2020 3220 2020 3331  05753     2   31
│ │ │ │ +0005c1a0: 3334 3930 3239 3920 3220 3220 2020 3436  3490299 2 2   46
│ │ │ │ +0005c1b0: 3137 3435 3534 3536 3320 2020 2020 3220  174554563     2 
│ │ │ │ +0005c1c0: 2020 3137 3639 3537 3631 3534 3539 207c    176957615459 |
│ │ │ │ +0005c1d0: 0a7c 7820 7820 202d 202d 2d2d 2d2d 2d2d  .|x x  - -------
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│ │ │ │ +0005c1f0: 2d2d 2d2d 2d2d 2d78 2078 2020 2d20 2d2d  -------x x  - --
│ │ │ │ +0005c200: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078 2020  ---------x x x  
│ │ │ │ +0005c210: 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 787c  + ------------x|
│ │ │ │ +0005c220: 0a7c 2030 2033 2020 2020 3130 3030 3138  .| 0 3    100018
│ │ │ │ +0005c230: 3830 3030 2020 3020 3120 3320 2020 2031  8000  0 1 3    1
│ │ │ │ +0005c240: 3237 3537 3530 2020 3120 3320 2020 2031  275750  1 3    1
│ │ │ │ +0005c250: 3435 3836 3037 3530 2020 3020 3220 3320  45860750  0 2 3 
│ │ │ │ +0005c260: 2020 2020 3232 3232 3634 3030 3020 207c      222264000  |
│ │ │ │ +0005c270: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c2c0: 2020 2020 207c 0a7c 3631 3432 3832 3730       |.|61428270
│ │ │ │ -0005c2d0: 3337 2020 2020 2032 2020 2034 3436 3534  37     2   44654
│ │ │ │ -0005c2e0: 3730 3538 3120 3220 3220 2020 3431 3636  70581 2 2   4166
│ │ │ │ -0005c2f0: 3331 3139 3735 3920 2020 2020 3220 2020  3119759     2   
│ │ │ │ -0005c300: 3639 3433 3331 3732 3938 3720 2020 2020  69433172987     
│ │ │ │ -0005c310: 3220 2020 327c 0a7c 2d2d 2d2d 2d2d 2d2d  2   2|.|--------
│ │ │ │ -0005c320: 2d2d 7820 7820 7820 202b 202d 2d2d 2d2d  --x x x  + -----
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│ │ │ │ -0005c370: 3020 2030 2031 2033 2020 2020 3335 3732  0  0 1 3    3572
│ │ │ │ -0005c380: 3130 3030 2020 3120 3320 2020 2020 3334  1000  1 3     34
│ │ │ │ -0005c390: 3537 3434 3030 2020 3020 3220 3320 2020  574400  0 2 3   
│ │ │ │ -0005c3a0: 2031 3030 3031 3838 3030 2020 3120 3220   100018800  1 2 
│ │ │ │ -0005c3b0: 3320 2020 207c 0a7c 2020 2020 2020 2020  3    |.|        
│ │ │ │ +0005c2b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005c2c0: 0a7c 3631 3432 3832 3730 3337 2020 2020  .|6142827037    
│ │ │ │ +0005c2d0: 2032 2020 2034 3436 3534 3730 3538 3120   2   4465470581 
│ │ │ │ +0005c2e0: 3220 3220 2020 3431 3636 3331 3139 3735  2 2   4166311975
│ │ │ │ +0005c2f0: 3920 2020 2020 3220 2020 3639 3433 3331  9     2   694331
│ │ │ │ +0005c300: 3732 3938 3720 2020 2020 3220 2020 327c  72987     2   2|
│ │ │ │ +0005c310: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820  .|----------x x 
│ │ │ │ +0005c320: 7820 202b 202d 2d2d 2d2d 2d2d 2d2d 2d78  x  + ----------x
│ │ │ │ +0005c330: 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d   x  - ----------
│ │ │ │ +0005c340: 2d78 2078 2078 2020 2b20 2d2d 2d2d 2d2d  -x x x  + ------
│ │ │ │ +0005c350: 2d2d 2d2d 2d78 2078 2078 2020 2d20 2d7c  -----x x x  - -|
│ │ │ │ +0005c360: 0a7c 2034 3736 3238 3030 3020 2030 2031  .| 47628000  0 1
│ │ │ │ +0005c370: 2033 2020 2020 3335 3732 3130 3030 2020   3    35721000  
│ │ │ │ +0005c380: 3120 3320 2020 2020 3334 3537 3434 3030  1 3     34574400
│ │ │ │ +0005c390: 2020 3020 3220 3320 2020 2031 3030 3031    0 2 3    10001
│ │ │ │ +0005c3a0: 3838 3030 2020 3120 3220 3320 2020 207c  8800  1 2 3    |
│ │ │ │ +0005c3b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005c3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c400: 2020 2020 207c 0a7c 2020 3332 3234 3736       |.|  322476
│ │ │ │ -0005c410: 3632 3238 3120 3220 3220 2020 3537 3838  62281 2 2   5788
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│ │ │ │ -0005c430: 2036 3137 3637 3739 3937 2032 2032 2020   617677997 2 2  
│ │ │ │ -0005c440: 2032 3739 3738 3736 3336 3633 3120 2020   279787636631   
│ │ │ │ -0005c450: 2020 3220 207c 0a7c 2b20 2d2d 2d2d 2d2d    2  |.|+ ------
│ │ │ │ -0005c460: 2d2d 2d2d 2d78 2078 2020 2d20 2d2d 2d2d  -----x x  - ----
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│ │ │ │ -0005c480: 202d 2d2d 2d2d 2d2d 2d2d 7820 7820 202b   ---------x x  +
│ │ │ │ -0005c490: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078   ------------x x
│ │ │ │ -0005c4a0: 2078 2020 2d7c 0a7c 2020 2020 3838 3930   x  -|.|    8890
│ │ │ │ -0005c4b0: 3536 3030 2020 3020 3320 2020 2020 3234  5600  0 3     24
│ │ │ │ -0005c4c0: 3439 3434 3030 3020 2030 2031 2033 2020  4944000  0 1 3  
│ │ │ │ -0005c4d0: 2020 3430 3832 3430 3030 2031 2033 2020    40824000 1 3  
│ │ │ │ -0005c4e0: 2020 2034 3030 3037 3532 3030 2020 3020     400075200  0 
│ │ │ │ -0005c4f0: 3220 3320 207c 0a7c 2020 2020 2020 2020  2 3  |.|        
│ │ │ │ +0005c3f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005c400: 0a7c 2020 3332 3234 3736 3632 3238 3120  .|  32247662281 
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│ │ │ │ +0005c420: 3433 2020 2020 2032 2020 2036 3137 3637  43     2   61767
│ │ │ │ +0005c430: 3739 3937 2032 2032 2020 2032 3739 3738  7997 2 2   27978
│ │ │ │ +0005c440: 3736 3336 3633 3120 2020 2020 3220 207c  7636631     2  |
│ │ │ │ +0005c450: 0a7c 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2d78  .|+ -----------x
│ │ │ │ +0005c460: 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d   x  - ----------
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│ │ │ │ +0005c480: 2d2d 2d2d 7820 7820 202b 202d 2d2d 2d2d  ----x x  + -----
│ │ │ │ +0005c490: 2d2d 2d2d 2d2d 2d78 2078 2078 2020 2d7c  -------x x x  -|
│ │ │ │ +0005c4a0: 0a7c 2020 2020 3838 3930 3536 3030 2020  .|    88905600  
│ │ │ │ +0005c4b0: 3020 3320 2020 2020 3234 3439 3434 3030  0 3     24494400
│ │ │ │ +0005c4c0: 3020 2030 2031 2033 2020 2020 3430 3832  0  0 1 3    4082
│ │ │ │ +0005c4d0: 3430 3030 2031 2033 2020 2020 2034 3030  4000 1 3     400
│ │ │ │ +0005c4e0: 3037 3532 3030 2020 3020 3220 3320 207c  075200  0 2 3  |
│ │ │ │ +0005c4f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005c520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c540: 2020 2020 207c 0a7c 3630 3834 3933 2032       |.|608493 2
│ │ │ │ -0005c550: 2032 2020 2031 3037 3037 3636 3530 3335   2   10707665035
│ │ │ │ -0005c560: 3320 2020 2020 3220 2020 3630 3436 3331  3     2   604631
│ │ │ │ -0005c570: 3439 3720 3220 3220 2020 3233 3930 3734  497 2 2   239074
│ │ │ │ -0005c580: 3638 3331 3337 2020 2020 2032 2020 2032  683137     2   2
│ │ │ │ -0005c590: 3731 3835 387c 0a7c 2d2d 2d2d 2d2d 7820  71858|.|------x 
│ │ │ │ -0005c5a0: 7820 202d 202d 2d2d 2d2d 2d2d 2d2d 2d2d  x  - -----------
│ │ │ │ -0005c5b0: 2d78 2078 2078 2020 2d20 2d2d 2d2d 2d2d  -x x x  - ------
│ │ │ │ -0005c5c0: 2d2d 2d78 2078 2020 2d20 2d2d 2d2d 2d2d  ---x x  - ------
│ │ │ │ -0005c5d0: 2d2d 2d2d 2d2d 7820 7820 7820 202d 202d  ------x x x  - -
│ │ │ │ -0005c5e0: 2d2d 2d2d 2d7c 0a7c 3932 3030 3020 2030  -----|.|92000  0
│ │ │ │ -0005c5f0: 2033 2020 2020 2036 3636 3739 3230 3030   3     666792000
│ │ │ │ -0005c600: 2020 3020 3120 3320 2020 2034 3736 3238    0 1 3    47628
│ │ │ │ -0005c610: 3030 3020 3120 3320 2020 2031 3333 3335  000 1 3    13335
│ │ │ │ -0005c620: 3834 3030 3020 2030 2032 2033 2020 2020  84000  0 2 3    
│ │ │ │ -0005c630: 3434 3435 327c 0a7c 2d2d 2d2d 2d2d 2d2d  44452|.|--------
│ │ │ │ +0005c530: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005c540: 0a7c 3630 3834 3933 2032 2032 2020 2031  .|608493 2 2   1
│ │ │ │ +0005c550: 3037 3037 3636 3530 3335 3320 2020 2020  07076650353     
│ │ │ │ +0005c560: 3220 2020 3630 3436 3331 3439 3720 3220  2   604631497 2 
│ │ │ │ +0005c570: 3220 2020 3233 3930 3734 3638 3331 3337  2   239074683137
│ │ │ │ +0005c580: 2020 2020 2032 2020 2032 3731 3835 387c       2   271858|
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│ │ │ │ +0005c5b0: 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d78 2078    - ---------x x
│ │ │ │ +0005c5c0: 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d    - ------------
│ │ │ │ +0005c5d0: 7820 7820 7820 202d 202d 2d2d 2d2d 2d7c  x x x  - ------|
│ │ │ │ +0005c5e0: 0a7c 3932 3030 3020 2030 2033 2020 2020  .|92000  0 3    
│ │ │ │ +0005c5f0: 2036 3636 3739 3230 3030 2020 3020 3120   666792000  0 1 
│ │ │ │ +0005c600: 3320 2020 2034 3736 3238 3030 3020 3120  3    47628000 1 
│ │ │ │ +0005c610: 3320 2020 2031 3333 3335 3834 3030 3020  3    1333584000 
│ │ │ │ +0005c620: 2030 2032 2033 2020 2020 3434 3435 327c   0 2 3    44452|
│ │ │ │ +0005c630: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0005c640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005c650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0005c670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0005c690: 3036 3931 3920 3220 3220 2020 3237 3232  06919 2 2   2722
│ │ │ │ -0005c6a0: 3735 3639 3320 2020 3320 2020 3531 3230  75693   3   5120
│ │ │ │ -0005c6b0: 3039 3639 2020 2033 2020 2038 3031 3234  0969   3   80124
│ │ │ │ -0005c6c0: 3037 2020 2033 2020 2032 3336 3333 3320  07   3   236333 
│ │ │ │ -0005c6d0: 2020 2020 207c 0a7c 202d 202d 2d2d 2d2d       |.| - -----
│ │ │ │ -0005c6e0: 2d2d 2d2d 2d78 2078 2020 2b20 2d2d 2d2d  -----x x  + ----
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│ │ │ │ -0005c730: 3536 3030 2020 3220 3320 2020 2033 3532  5600  2 3    352
│ │ │ │ -0005c740: 3830 3030 2020 3020 3320 2020 2031 3934  8000  0 3    194
│ │ │ │ -0005c750: 3430 3030 2031 2033 2020 2020 3232 3638  4000 1 3    2268
│ │ │ │ -0005c760: 3030 2032 2033 2020 2020 3934 3530 2020  00 2 3    9450  
│ │ │ │ -0005c770: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005c670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0005c680: 0a7c 2020 2032 3138 3931 3036 3931 3920  .|   2189106919 
│ │ │ │ +0005c690: 3220 3220 2020 3237 3232 3735 3639 3320  2 2   272275693 
│ │ │ │ +0005c6a0: 2020 3320 2020 3531 3230 3039 3639 2020    3   51200969  
│ │ │ │ +0005c6b0: 2033 2020 2038 3031 3234 3037 2020 2033   3   8012407   3
│ │ │ │ +0005c6c0: 2020 2032 3336 3333 3320 2020 2020 207c     236333      |
│ │ │ │ +0005c6d0: 0a7c 202d 202d 2d2d 2d2d 2d2d 2d2d 2d78  .| - ----------x
│ │ │ │ +0005c6e0: 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d78   x  + ---------x
│ │ │ │ +0005c6f0: 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d 7820   x  + --------x 
│ │ │ │ +0005c700: 7820 202d 202d 2d2d 2d2d 2d2d 7820 7820  x  - -------x x 
│ │ │ │ +0005c710: 202d 202d 2d2d 2d2d 2d20 2020 2020 207c   - ------      |
│ │ │ │ +0005c720: 0a7c 2020 2020 2039 3532 3536 3030 2020  .|     9525600  
│ │ │ │ +0005c730: 3220 3320 2020 2033 3532 3830 3030 2020  2 3    3528000  
│ │ │ │ +0005c740: 3020 3320 2020 2031 3934 3430 3030 2031  0 3    1944000 1
│ │ │ │ +0005c750: 2033 2020 2020 3232 3638 3030 2032 2033   3    226800 2 3
│ │ │ │ +0005c760: 2020 2020 3934 3530 2020 2020 2020 207c      9450       |
│ │ │ │ +0005c770: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0005c7c0: 2020 2020 207c 0a7c 2020 2020 3220 2020       |.|    2   
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│ │ │ │ -0005c7f0: 2020 2031 3039 3835 3639 3237 2020 2033     109856927   3
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│ │ │ │ -0005c860: 2020 2020 207c 0a7c 3120 3220 3320 2020       |.|1 2 3   
│ │ │ │ -0005c870: 2020 3236 3637 3136 3830 2020 3220 3320    26671680  2 3 
│ │ │ │ -0005c880: 2020 2031 3139 3037 3030 3020 2030 2033     11907000  0 3
│ │ │ │ -0005c890: 2020 2020 3236 3436 3030 3020 2031 2033      2646000  1 3
│ │ │ │ -0005c8a0: 2020 2020 3333 3333 3936 3030 2020 3220      33339600  2 
│ │ │ │ -0005c8b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005c7b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005c7c0: 0a7c 2020 2020 3220 2020 3130 3930 3434  .|    2   109044
│ │ │ │ +0005c7d0: 3233 3938 3720 3220 3220 2020 3231 3834  23987 2 2   2184
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│ │ │ │ +0005c7f0: 3835 3639 3237 2020 2033 2020 2035 3231  856927   3   521
│ │ │ │ +0005c800: 3337 3930 3331 3120 2020 2020 2020 207c  3790311        |
│ │ │ │ +0005c810: 0a7c 2078 2078 2020 2d20 2d2d 2d2d 2d2d  .| x x  - ------
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│ │ │ │ +0005c860: 0a7c 3120 3220 3320 2020 2020 3236 3637  .|1 2 3     2667
│ │ │ │ +0005c870: 3136 3830 2020 3220 3320 2020 2031 3139  1680  2 3    119
│ │ │ │ +0005c880: 3037 3030 3020 2030 2033 2020 2020 3236  07000  0 3    26
│ │ │ │ +0005c890: 3436 3030 3020 2031 2033 2020 2020 3333  46000  1 3    33
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│ │ │ │ +0005c8b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0005c920: 3134 3720 2020 3320 2020 3138 3830 3935  147   3   188095
│ │ │ │ -0005c930: 3320 2020 3320 2020 3732 3132 3030 3930  3   3   72120090
│ │ │ │ -0005c940: 3137 2020 2033 2020 2033 3931 3639 2020  17   3   39169  
│ │ │ │ -0005c950: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
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│ │ │ │ -0005c9e0: 3020 2032 2033 2020 2020 3337 3830 2020  0  2 3    3780  
│ │ │ │ -0005c9f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ +0005c900: 0a7c 3931 3734 3735 3038 3831 2032 2032  .|9174750881 2 2
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│ │ │ │ +0005c930: 2020 3732 3132 3030 3930 3137 2020 2033    7212009017   3
│ │ │ │ +0005c940: 2020 2033 3931 3639 2020 2020 2020 207c     39169       |
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│ │ │ │ +0005c980: 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820  + ----------x x 
│ │ │ │ +0005c990: 202b 202d 2d2d 2d2d 7820 2020 2020 207c   + -----x      |
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│ │ │ │ +0005c9b0: 2020 2020 3333 3333 3936 3030 2020 3020      33339600  0 
│ │ │ │ +0005c9c0: 3320 2020 2033 3738 3030 2020 3120 3320  3    37800  1 3 
│ │ │ │ +0005c9d0: 2020 2033 3333 3339 3630 3020 2032 2033     33339600  2 3
│ │ │ │ +0005c9e0: 2020 2020 3337 3830 2020 2020 2020 207c      3780       |
│ │ │ │ +0005c9f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ -0005ca40: 2020 2020 207c 0a7c 2033 3831 3032 3733       |.| 3810273
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│ │ │ │ -0005ca60: 3931 3632 3331 3920 3220 3220 2020 3232  9162319 2 2   22
│ │ │ │ -0005ca70: 3433 3931 3935 3131 2020 2033 2020 2033  43919511   3   3
│ │ │ │ -0005ca80: 3435 3433 3438 3139 2020 2033 2020 2020  45434819   3    
│ │ │ │ -0005ca90: 2020 2020 207c 0a7c 202d 2d2d 2d2d 2d2d       |.| -------
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│ │ │ │ -0005cad0: 2d2d 2d2d 2d2d 2d2d 7820 7820 202d 2020  --------x x  -  
│ │ │ │ -0005cae0: 2020 2020 207c 0a7c 2020 3137 3134 3630       |.|  171460
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│ │ │ │ -0005cb00: 3135 3336 3030 2020 3220 3320 2020 2020  153600  2 3     
│ │ │ │ -0005cb10: 3331 3735 3230 3020 2030 2033 2020 2020  3175200  0 3    
│ │ │ │ -0005cb20: 3133 3630 3830 3020 2031 2033 2020 2020  1360800  1 3    
│ │ │ │ -0005cb30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005ca30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005ca40: 0a7c 2033 3831 3032 3733 3736 3231 2020  .| 38102737621  
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│ │ │ │ +0005ca60: 3920 3220 3220 2020 3232 3433 3931 3935  9 2 2   22439195
│ │ │ │ +0005ca70: 3131 2020 2033 2020 2033 3435 3433 3438  11   3   3454348
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│ │ │ │ +0005ca90: 0a7c 202d 2d2d 2d2d 2d2d 2d2d 2d2d 7820  .| -----------x 
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│ │ │ │ +0005cb20: 3020 2031 2033 2020 2020 2020 2020 207c  0  1 3         |
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│ │ │ │ -0005cb90: 2020 3220 2020 3134 3232 3133 3031 3833    2   1422130183
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│ │ │ │ -0005cbb0: 3239 2020 2033 2020 2031 3636 3839 3439  29   3   1668949
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│ │ │ │ -0005cbd0: 2020 2020 207c 0a7c 2d2d 2d2d 2d78 2078       |.|-----x x
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│ │ │ │ -0005cc20: 2020 2020 207c 0a7c 3830 3030 2020 3120       |.|8000  1 
│ │ │ │ -0005cc30: 3220 3320 2020 2037 3737 3932 3430 3030  2 3    777924000
│ │ │ │ -0005cc40: 2020 3220 3320 2020 2035 3535 3636 3030    2 3    5556600
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│ │ │ │ -0005cc60: 2031 2033 2020 2020 3334 3732 3837 3520   1 3    3472875 
│ │ │ │ -0005cc70: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
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│ │ │ │ +0005cb80: 0a7c 3837 3432 3920 2020 2020 3220 2020  .|87429     2   
│ │ │ │ +0005cb90: 3134 3232 3133 3031 3833 3320 3220 3220  14221301833 2 2 
│ │ │ │ +0005cba0: 2020 3538 3234 3931 3438 3239 2020 2033    5824914829   3
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│ │ │ │ +0005cc00: 202b 202d 2d2d 2d2d 2d2d 7820 7820 202b   + -------x x  +
│ │ │ │ +0005cc10: 202d 2d2d 2d2d 2d2d 2d20 2020 2020 207c   --------      |
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│ │ │ │ +0005cc30: 2037 3737 3932 3430 3030 2020 3220 3320   777924000  2 3 
│ │ │ │ +0005cc40: 2020 2035 3535 3636 3030 3020 2030 2033     55566000  0 3
│ │ │ │ +0005cc50: 2020 2020 3130 3538 3430 2031 2033 2020      105840 1 3  
│ │ │ │ +0005cc60: 2020 3334 3732 3837 3520 2020 2020 207c    3472875      |
│ │ │ │ +0005cc70: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
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│ │ │ │ -0005ccd0: 3837 3731 3733 2033 2020 2020 2033 3230  877173 3     320
│ │ │ │ -0005cce0: 3533 3139 3134 3320 3220 2020 2020 2020  5319143 2       
│ │ │ │ -0005ccf0: 3237 3131 3535 3634 3531 3920 2020 3220  27115564519   2 
│ │ │ │ -0005cd00: 2020 2020 3431 3132 3739 3135 3120 3320      411279151 3 
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│ │ │ │ -0005cd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078  -----------x x x
│ │ │ │ -0005cd50: 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d78 2078    - ---------x x
│ │ │ │ -0005cd60: 2020 2d20 2d7c 0a7c 2033 2020 2031 3639    - -|.| 3   169
│ │ │ │ -0005cd70: 3334 3430 3030 2030 2034 2020 2020 3537  344000 0 4    57
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│ │ │ │ -0005cd90: 2039 3134 3435 3736 3030 2020 3020 3120   914457600  0 1 
│ │ │ │ -0005cda0: 3420 2020 2031 3336 3038 3030 3020 3120  4    13608000 1 
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│ │ │ │ +0005ccc0: 0a7c 2034 2020 2039 3133 3837 3731 3733  .| 4   913877173
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│ │ │ │ +0005cd20: 7820 7820 202d 202d 2d2d 2d2d 2d2d 2d2d  x x  - ---------
│ │ │ │ +0005cd30: 2d78 2078 2078 2020 2d20 2d2d 2d2d 2d2d  -x x x  - ------
│ │ │ │ +0005cd40: 2d2d 2d2d 2d78 2078 2078 2020 2d20 2d2d  -----x x x  - --
│ │ │ │ +0005cd50: 2d2d 2d2d 2d2d 2d78 2078 2020 2d20 2d7c  -------x x  - -|
│ │ │ │ +0005cd60: 0a7c 2033 2020 2031 3639 3334 3430 3030  .| 3   169344000
│ │ │ │ +0005cd70: 2030 2034 2020 2020 3537 3135 3336 3030   0 4    57153600
│ │ │ │ +0005cd80: 2020 3020 3120 3420 2020 2039 3134 3435    0 1 4    91445
│ │ │ │ +0005cd90: 3736 3030 2020 3020 3120 3420 2020 2031  7600  0 1 4    1
│ │ │ │ +0005cda0: 3336 3038 3030 3020 3120 3420 2020 207c  3608000 1 4    |
│ │ │ │ +0005cdb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005cdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ce00: 2020 2020 207c 0a7c 2033 2020 2033 3230       |.| 3   320
│ │ │ │ -0005ce10: 3338 3339 2034 2020 2033 3439 3238 3631  3839 4   3492861
│ │ │ │ -0005ce20: 3037 2033 2020 2020 2035 3831 3935 3732  07 3     5819572
│ │ │ │ -0005ce30: 3337 2032 2020 2020 2020 2031 3133 3933  37 2       11393
│ │ │ │ -0005ce40: 3930 3931 3334 3720 2020 3220 2020 2020  9091347   2     
│ │ │ │ -0005ce50: 3138 3630 377c 0a7c 7820 202d 202d 2d2d  18607|.|x  - ---
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│ │ │ │ -0005ce80: 2d2d 7820 7820 7820 202d 202d 2d2d 2d2d  --x x x  - -----
│ │ │ │ -0005ce90: 2d2d 2d2d 2d2d 2d78 2078 2078 2020 2d20  -------x x x  - 
│ │ │ │ -0005cea0: 2d2d 2d2d 2d7c 0a7c 2033 2020 2020 3133  -----|.| 3    13
│ │ │ │ -0005ceb0: 3233 3030 2033 2020 2020 3139 3035 3132  2300 3    190512
│ │ │ │ -0005cec0: 3030 2030 2034 2020 2020 3930 3732 3030  00 0 4    907200
│ │ │ │ -0005ced0: 3020 2030 2031 2034 2020 2020 3238 3537  0  0 1 4    2857
│ │ │ │ -0005cee0: 3638 3030 3030 2020 3020 3120 3420 2020  680000  0 1 4   
│ │ │ │ -0005cef0: 2033 3234 307c 0a7c 2020 2020 2020 2020   3240|.|        
│ │ │ │ +0005cdf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005ce00: 0a7c 2033 2020 2033 3230 3338 3339 2034  .| 3   3203839 4
│ │ │ │ +0005ce10: 2020 2033 3439 3238 3631 3037 2033 2020     349286107 3  
│ │ │ │ +0005ce20: 2020 2035 3831 3935 3732 3337 2032 2020     581957237 2  
│ │ │ │ +0005ce30: 2020 2020 2031 3133 3933 3930 3931 3334       11393909134
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│ │ │ │ +0005ce80: 7820 202d 202d 2d2d 2d2d 2d2d 2d2d 2d2d  x  - -----------
│ │ │ │ +0005ce90: 2d78 2078 2078 2020 2d20 2d2d 2d2d 2d7c  -x x x  - -----|
│ │ │ │ +0005cea0: 0a7c 2033 2020 2020 3133 3233 3030 2033  .| 3    132300 3
│ │ │ │ +0005ceb0: 2020 2020 3139 3035 3132 3030 2030 2034      19051200 0 4
│ │ │ │ +0005cec0: 2020 2020 3930 3732 3030 3020 2030 2031      9072000  0 1
│ │ │ │ +0005ced0: 2034 2020 2020 3238 3537 3638 3030 3030   4    2857680000
│ │ │ │ +0005cee0: 2020 3020 3120 3420 2020 2033 3234 307c    0 1 4    3240|
│ │ │ │ +0005cef0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005cf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005cf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cf40: 2020 2020 207c 0a7c 3420 2020 3136 3931       |.|4   1691
│ │ │ │ -0005cf50: 3239 3138 3320 3320 2020 2020 3934 3236  29183 3     9426
│ │ │ │ -0005cf60: 3538 3237 3539 3037 2032 2020 2020 2020  58275907 2      
│ │ │ │ -0005cf70: 2031 3732 3132 3639 3133 3839 2020 2032   17212691389   2
│ │ │ │ -0005cf80: 2020 2020 2033 3435 3232 3235 3920 3320       34522259 3 
│ │ │ │ -0005cf90: 2020 2020 317c 0a7c 2020 2b20 2d2d 2d2d      1|.|  + ----
│ │ │ │ -0005cfa0: 2d2d 2d2d 2d78 2078 2020 2b20 2d2d 2d2d  -----x x  + ----
│ │ │ │ -0005cfb0: 2d2d 2d2d 2d2d 2d2d 7820 7820 7820 202b  --------x x x  +
│ │ │ │ -0005cfc0: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820   -----------x x 
│ │ │ │ -0005cfd0: 7820 202b 202d 2d2d 2d2d 2d2d 2d78 2078  x  + --------x x
│ │ │ │ -0005cfe0: 2020 2d20 2d7c 0a7c 3320 2020 2037 3134    - -|.|3    714
│ │ │ │ -0005cff0: 3432 3030 2020 3020 3420 2020 2035 3134  4200  0 4    514
│ │ │ │ -0005d000: 3338 3234 3030 3020 2030 2031 2034 2020  3824000  0 1 4  
│ │ │ │ -0005d010: 2020 2037 3632 3034 3830 2020 2030 2031     7620480   0 1
│ │ │ │ -0005d020: 2034 2020 2020 3136 3230 3030 2020 3120   4    162000  1 
│ │ │ │ -0005d030: 3420 2020 207c 0a7c 2020 2020 2020 2020  4    |.|        
│ │ │ │ +0005cf30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005cf40: 0a7c 3420 2020 3136 3931 3239 3138 3320  .|4   169129183 
│ │ │ │ +0005cf50: 3320 2020 2020 3934 3236 3538 3237 3539  3     9426582759
│ │ │ │ +0005cf60: 3037 2032 2020 2020 2020 2031 3732 3132  07 2       17212
│ │ │ │ +0005cf70: 3639 3133 3839 2020 2032 2020 2020 2033  691389   2     3
│ │ │ │ +0005cf80: 3435 3232 3235 3920 3320 2020 2020 317c  4522259 3     1|
│ │ │ │ +0005cf90: 0a7c 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d78  .|  + ---------x
│ │ │ │ +0005cfa0: 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d2d   x  + ----------
│ │ │ │ +0005cfb0: 2d2d 7820 7820 7820 202b 202d 2d2d 2d2d  --x x x  + -----
│ │ │ │ +0005cfc0: 2d2d 2d2d 2d2d 7820 7820 7820 202b 202d  ------x x x  + -
│ │ │ │ +0005cfd0: 2d2d 2d2d 2d2d 2d78 2078 2020 2d20 2d7c  -------x x  - -|
│ │ │ │ +0005cfe0: 0a7c 3320 2020 2037 3134 3432 3030 2020  .|3    7144200  
│ │ │ │ +0005cff0: 3020 3420 2020 2035 3134 3338 3234 3030  0 4    514382400
│ │ │ │ +0005d000: 3020 2030 2031 2034 2020 2020 2037 3632  0  0 1 4     762
│ │ │ │ +0005d010: 3034 3830 2020 2030 2031 2034 2020 2020  0480   0 1 4    
│ │ │ │ +0005d020: 3136 3230 3030 2020 3120 3420 2020 207c  162000  1 4    |
│ │ │ │ +0005d030: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005d040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d080: 2020 2020 207c 0a7c 3136 3930 3534 3539       |.|16905459
│ │ │ │ -0005d090: 3720 2020 3320 2020 3338 3033 3531 2034  7   3   380351 4
│ │ │ │ -0005d0a0: 2020 2035 3538 3439 3531 3133 2033 2020     558495113 3  
│ │ │ │ -0005d0b0: 2020 2032 3230 3334 3338 3131 3839 2032     22034381189 2
│ │ │ │ -0005d0c0: 2020 2020 2020 2036 3133 3538 3336 3336         613583636
│ │ │ │ -0005d0d0: 3837 3920 207c 0a7c 2d2d 2d2d 2d2d 2d2d  879  |.|--------
│ │ │ │ -0005d0e0: 2d78 2078 2020 2b20 2d2d 2d2d 2d2d 7820  -x x  + ------x 
│ │ │ │ -0005d0f0: 202b 202d 2d2d 2d2d 2d2d 2d2d 7820 7820   + ---------x x 
│ │ │ │ -0005d100: 202b 202d 2d2d 2d2d 2d2d 2d2d 2d2d 7820   + -----------x 
│ │ │ │ -0005d110: 7820 7820 202b 202d 2d2d 2d2d 2d2d 2d2d  x x  + ---------
│ │ │ │ -0005d120: 2d2d 2d78 207c 0a7c 2031 3538 3736 3030  ---x |.| 1587600
│ │ │ │ -0005d130: 2020 3220 3320 2020 2033 3738 3020 2033    2 3    3780  3
│ │ │ │ -0005d140: 2020 2020 3435 3732 3238 3830 2030 2034      45722880 0 4
│ │ │ │ -0005d150: 2020 2020 3131 3433 3037 3230 3030 2030      1143072000 0
│ │ │ │ -0005d160: 2031 2034 2020 2020 3134 3238 3834 3030   1 4    14288400
│ │ │ │ -0005d170: 3030 2020 307c 0a7c 2020 2020 2020 2020  00  0|.|        
│ │ │ │ +0005d070: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d080: 0a7c 3136 3930 3534 3539 3720 2020 3320  .|169054597   3 
│ │ │ │ +0005d090: 2020 3338 3033 3531 2034 2020 2035 3538    380351 4   558
│ │ │ │ +0005d0a0: 3439 3531 3133 2033 2020 2020 2032 3230  495113 3     220
│ │ │ │ +0005d0b0: 3334 3338 3131 3839 2032 2020 2020 2020  34381189 2      
│ │ │ │ +0005d0c0: 2036 3133 3538 3336 3336 3837 3920 207c   613583636879  |
│ │ │ │ +0005d0d0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020  .|---------x x  
│ │ │ │ +0005d0e0: 2b20 2d2d 2d2d 2d2d 7820 202b 202d 2d2d  + ------x  + ---
│ │ │ │ +0005d0f0: 2d2d 2d2d 2d2d 7820 7820 202b 202d 2d2d  ------x x  + ---
│ │ │ │ +0005d100: 2d2d 2d2d 2d2d 2d2d 7820 7820 7820 202b  --------x x x  +
│ │ │ │ +0005d110: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 207c   ------------x |
│ │ │ │ +0005d120: 0a7c 2031 3538 3736 3030 2020 3220 3320  .| 1587600  2 3 
│ │ │ │ +0005d130: 2020 2033 3738 3020 2033 2020 2020 3435     3780  3    45
│ │ │ │ +0005d140: 3732 3238 3830 2030 2034 2020 2020 3131  722880 0 4    11
│ │ │ │ +0005d150: 3433 3037 3230 3030 2030 2031 2034 2020  43072000 0 1 4  
│ │ │ │ +0005d160: 2020 3134 3238 3834 3030 3030 2020 307c    1428840000  0|
│ │ │ │ +0005d170: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005d180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d1c0: 2020 2020 207c 0a7c 2020 2033 2020 2031       |.|   3   1
│ │ │ │ -0005d1d0: 3732 3332 3420 3420 2020 3134 3936 3639  72324 4   149669
│ │ │ │ -0005d1e0: 3438 3320 3320 2020 2020 3439 3735 3737  483 3     497577
│ │ │ │ -0005d1f0: 3235 3130 3633 2032 2020 2020 2020 2032  251063 2       2
│ │ │ │ -0005d200: 3535 3232 3431 3433 3334 3831 2020 2032  552241433481   2
│ │ │ │ -0005d210: 2020 2020 207c 0a7c 7820 7820 202b 202d       |.|x x  + -
│ │ │ │ -0005d220: 2d2d 2d2d 2d78 2020 2b20 2d2d 2d2d 2d2d  -----x  + ------
│ │ │ │ -0005d230: 2d2d 2d78 2078 2020 2d20 2d2d 2d2d 2d2d  ---x x  - ------
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│ │ │ │ -0005d250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820  ------------x x 
│ │ │ │ -0005d260: 7820 202d 207c 0a7c 2032 2033 2020 2020  x  - |.| 2 3    
│ │ │ │ -0005d270: 3131 3032 3520 3320 2020 2034 3636 3536  11025 3    46656
│ │ │ │ -0005d280: 3030 2020 3020 3420 2020 2032 3933 3933  00  0 4    29393
│ │ │ │ -0005d290: 3238 3030 3020 2030 2031 2034 2020 2020  28000  0 1 4    
│ │ │ │ -0005d2a0: 2034 3839 3838 3830 3030 3020 2030 2031   4898880000  0 1
│ │ │ │ -0005d2b0: 2034 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d   4   |.|--------
│ │ │ │ +0005d1b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d1c0: 0a7c 2020 2033 2020 2031 3732 3332 3420  .|   3   172324 
│ │ │ │ +0005d1d0: 3420 2020 3134 3936 3639 3438 3320 3320  4   149669483 3 
│ │ │ │ +0005d1e0: 2020 2020 3439 3735 3737 3235 3130 3633      497577251063
│ │ │ │ +0005d1f0: 2032 2020 2020 2020 2032 3535 3232 3431   2       2552241
│ │ │ │ +0005d200: 3433 3334 3831 2020 2032 2020 2020 207c  433481   2     |
│ │ │ │ +0005d210: 0a7c 7820 7820 202b 202d 2d2d 2d2d 2d78  .|x x  + ------x
│ │ │ │ +0005d220: 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d78 2078    + ---------x x
│ │ │ │ +0005d230: 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d    - ------------
│ │ │ │ +0005d240: 7820 7820 7820 202d 202d 2d2d 2d2d 2d2d  x x x  - -------
│ │ │ │ +0005d250: 2d2d 2d2d 2d2d 7820 7820 7820 202d 207c  ------x x x  - |
│ │ │ │ +0005d260: 0a7c 2032 2033 2020 2020 3131 3032 3520  .| 2 3    11025 
│ │ │ │ +0005d270: 3320 2020 2034 3636 3536 3030 2020 3020  3    4665600  0 
│ │ │ │ +0005d280: 3420 2020 2032 3933 3933 3238 3030 3020  4    2939328000 
│ │ │ │ +0005d290: 2030 2031 2034 2020 2020 2034 3839 3838   0 1 4     48988
│ │ │ │ +0005d2a0: 3830 3030 3020 2030 2031 2034 2020 207c  80000  0 1 4   |
│ │ │ │ +0005d2b0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0005d2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d300: 2d2d 2d2d 2d7c 0a7c 3332 3331 3831 3436  -----|.|32318146
│ │ │ │ -0005d310: 3931 2032 2020 2020 2020 2037 3339 3633  91 2       73963
│ │ │ │ -0005d320: 3630 3634 3638 3430 3320 2020 2020 2020  606468403       
│ │ │ │ -0005d330: 2020 2020 3234 3635 3036 3036 3920 3220      246506069 2 
│ │ │ │ -0005d340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d350: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -0005d360: 2d2d 7820 7820 7820 202d 202d 2d2d 2d2d  --x x x  - -----
│ │ │ │ -0005d370: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078 2078  ---------x x x x
│ │ │ │ -0005d380: 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d78 2078    - ---------x x
│ │ │ │ -0005d390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d3a0: 2020 2020 207c 0a7c 3830 3031 3530 3430       |.|80015040
│ │ │ │ -0005d3b0: 3030 2030 2032 2034 2020 2020 3134 3430  00 0 2 4    1440
│ │ │ │ -0005d3c0: 3237 3037 3230 3030 2020 3020 3120 3220  27072000  0 1 2 
│ │ │ │ -0005d3d0: 3420 2020 2033 3430 3230 3030 2020 3120  4    3402000  1 
│ │ │ │ -0005d3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d3f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005d2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0005d300: 0a7c 3332 3331 3831 3436 3931 2032 2020  .|3231814691 2  
│ │ │ │ +0005d310: 2020 2020 2037 3339 3633 3630 3634 3638       73963606468
│ │ │ │ +0005d320: 3430 3320 2020 2020 2020 2020 2020 3234  403           24
│ │ │ │ +0005d330: 3635 3036 3036 3920 3220 2020 2020 2020  6506069 2       
│ │ │ │ +0005d340: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d350: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820  .|----------x x 
│ │ │ │ +0005d360: 7820 202d 202d 2d2d 2d2d 2d2d 2d2d 2d2d  x  - -----------
│ │ │ │ +0005d370: 2d2d 2d78 2078 2078 2078 2020 2d20 2d2d  ---x x x x  - --
│ │ │ │ +0005d380: 2d2d 2d2d 2d2d 2d78 2078 2020 2020 2020  -------x x      
│ │ │ │ +0005d390: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d3a0: 0a7c 3830 3031 3530 3430 3030 2030 2032  .|8001504000 0 2
│ │ │ │ +0005d3b0: 2034 2020 2020 3134 3430 3237 3037 3230   4    1440270720
│ │ │ │ +0005d3c0: 3030 2020 3020 3120 3220 3420 2020 2033  00  0 1 2 4    3
│ │ │ │ +0005d3d0: 3430 3230 3030 2020 3120 2020 2020 2020  402000  1       
│ │ │ │ +0005d3e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d3f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005d400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d440: 2020 2020 207c 0a7c 3135 3931 2033 2020       |.|1591 3  
│ │ │ │ -0005d450: 2020 2031 3839 3638 3937 3838 3238 3237     1896897882827
│ │ │ │ -0005d460: 2032 2020 2020 2020 2038 3134 3933 3432   2       8149342
│ │ │ │ -0005d470: 3831 3630 3720 2020 2020 2020 2020 2020  81607           
│ │ │ │ -0005d480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d490: 2020 2020 207c 0a7c 2d2d 2d2d 7820 7820       |.|----x x 
│ │ │ │ -0005d4a0: 202b 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   + -------------
│ │ │ │ -0005d4b0: 7820 7820 7820 202b 202d 2d2d 2d2d 2d2d  x x x  + -------
│ │ │ │ -0005d4c0: 2d2d 2d2d 2d78 2078 2078 2078 2020 2b20  -----x x x x  + 
│ │ │ │ -0005d4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d4e0: 2020 2020 207c 0a7c 3030 3020 2031 2034       |.|000  1 4
│ │ │ │ -0005d4f0: 2020 2020 3132 3030 3232 3536 3030 3020      12002256000 
│ │ │ │ -0005d500: 2030 2032 2034 2020 2020 3236 3637 3136   0 2 4    266716
│ │ │ │ -0005d510: 3830 3030 2020 3020 3120 3220 3420 2020  8000  0 1 2 4   
│ │ │ │ -0005d520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d530: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005d430: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d440: 0a7c 3135 3931 2033 2020 2020 2031 3839  .|1591 3     189
│ │ │ │ +0005d450: 3638 3937 3838 3238 3237 2032 2020 2020  6897882827 2    
│ │ │ │ +0005d460: 2020 2038 3134 3933 3432 3831 3630 3720     814934281607 
│ │ │ │ +0005d470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005d480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d490: 0a7c 2d2d 2d2d 7820 7820 202b 202d 2d2d  .|----x x  + ---
│ │ │ │ +0005d4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820 7820  ----------x x x 
│ │ │ │ +0005d4b0: 202b 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d78   + ------------x
│ │ │ │ +0005d4c0: 2078 2078 2078 2020 2b20 2020 2020 2020   x x x  +       
│ │ │ │ +0005d4d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d4e0: 0a7c 3030 3020 2031 2034 2020 2020 3132  .|000  1 4    12
│ │ │ │ +0005d4f0: 3030 3232 3536 3030 3020 2030 2032 2034  002256000  0 2 4
│ │ │ │ +0005d500: 2020 2020 3236 3637 3136 3830 3030 2020      2667168000  
│ │ │ │ +0005d510: 3020 3120 3220 3420 2020 2020 2020 2020  0 1 2 4         
│ │ │ │ +0005d520: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d530: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d580: 2020 2020 207c 0a7c 3833 3930 3534 3239       |.|83905429
│ │ │ │ -0005d590: 3331 3120 3220 2020 2020 2020 3137 3837  311 2       1787
│ │ │ │ -0005d5a0: 3137 3735 3935 3437 2020 2020 2020 2020  17759547        
│ │ │ │ -0005d5b0: 2020 2036 3235 3536 3532 3533 3437 2032     62556525347 2
│ │ │ │ -0005d5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d5d0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -0005d5e0: 2d2d 2d78 2078 2078 2020 2b20 2d2d 2d2d  ---x x x  + ----
│ │ │ │ -0005d5f0: 2d2d 2d2d 2d2d 2d2d 7820 7820 7820 7820  --------x x x x 
│ │ │ │ -0005d600: 202b 202d 2d2d 2d2d 2d2d 2d2d 2d2d 7820   + -----------x 
│ │ │ │ -0005d610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d620: 2020 2020 207c 0a7c 3130 3238 3736 3438       |.|10287648
│ │ │ │ -0005d630: 3030 2020 3020 3220 3420 2020 2031 3436  00  0 2 4    146
│ │ │ │ -0005d640: 3936 3634 3030 3020 2030 2031 2032 2034  9664000  0 1 2 4
│ │ │ │ -0005d650: 2020 2020 2038 3136 3438 3030 3020 2031       81648000  1
│ │ │ │ -0005d660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d670: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005d570: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d580: 0a7c 3833 3930 3534 3239 3331 3120 3220  .|83905429311 2 
│ │ │ │ +0005d590: 2020 2020 2020 3137 3837 3137 3735 3935        1787177595
│ │ │ │ +0005d5a0: 3437 2020 2020 2020 2020 2020 2036 3235  47           625
│ │ │ │ +0005d5b0: 3536 3532 3533 3437 2032 2020 2020 2020  56525347 2      
│ │ │ │ +0005d5c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d5d0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078  .|-----------x x
│ │ │ │ +0005d5e0: 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d2d   x  + ----------
│ │ │ │ +0005d5f0: 2d2d 7820 7820 7820 7820 202b 202d 2d2d  --x x x x  + ---
│ │ │ │ +0005d600: 2d2d 2d2d 2d2d 2d2d 7820 2020 2020 2020  --------x       
│ │ │ │ +0005d610: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d620: 0a7c 3130 3238 3736 3438 3030 2020 3020  .|1028764800  0 
│ │ │ │ +0005d630: 3220 3420 2020 2031 3436 3936 3634 3030  2 4    146966400
│ │ │ │ +0005d640: 3020 2030 2031 2032 2034 2020 2020 2038  0  0 1 2 4     8
│ │ │ │ +0005d650: 3136 3438 3030 3020 2031 2020 2020 2020  1648000  1      
│ │ │ │ +0005d660: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d670: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005d680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d6c0: 2020 2020 207c 0a7c 2032 2020 2020 2033       |.| 2     3
│ │ │ │ -0005d6d0: 3233 3130 3634 3320 3320 2020 2020 3933  2310643 3     93
│ │ │ │ -0005d6e0: 3833 3238 3430 3132 3438 3920 3220 2020  83284012489 2   
│ │ │ │ -0005d6f0: 2020 2020 3730 3934 3832 3238 3931 3833      709482289183
│ │ │ │ -0005d700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d710: 2020 2020 207c 0a7c 7820 7820 202b 202d       |.|x x  + -
│ │ │ │ -0005d720: 2d2d 2d2d 2d2d 2d78 2078 2020 2b20 2d2d  -------x x  + --
│ │ │ │ -0005d730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078  -----------x x x
│ │ │ │ -0005d740: 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d    + ------------
│ │ │ │ -0005d750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d760: 2020 2020 207c 0a7c 2031 2034 2020 2020       |.| 1 4    
│ │ │ │ -0005d770: 3634 3830 3030 3020 3120 3420 2020 2032  6480000 1 4    2
│ │ │ │ -0005d780: 3838 3035 3431 3434 3030 3020 3020 3220  88054144000 0 2 
│ │ │ │ -0005d790: 3420 2020 2035 3333 3433 3336 3030 3020  4    5334336000 
│ │ │ │ -0005d7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d7b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005d6b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d6c0: 0a7c 2032 2020 2020 2033 3233 3130 3634  .| 2     3231064
│ │ │ │ +0005d6d0: 3320 3320 2020 2020 3933 3833 3238 3430  3 3     93832840
│ │ │ │ +0005d6e0: 3132 3438 3920 3220 2020 2020 2020 3730  12489 2       70
│ │ │ │ +0005d6f0: 3934 3832 3238 3931 3833 2020 2020 2020  9482289183      
│ │ │ │ +0005d700: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d710: 0a7c 7820 7820 202b 202d 2d2d 2d2d 2d2d  .|x x  + -------
│ │ │ │ +0005d720: 2d78 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d  -x x  + --------
│ │ │ │ +0005d730: 2d2d 2d2d 2d78 2078 2078 2020 2b20 2d2d  -----x x x  + --
│ │ │ │ +0005d740: 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020 2020  ----------      
│ │ │ │ +0005d750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d760: 0a7c 2031 2034 2020 2020 3634 3830 3030  .| 1 4    648000
│ │ │ │ +0005d770: 3020 3120 3420 2020 2032 3838 3035 3431  0 1 4    2880541
│ │ │ │ +0005d780: 3434 3030 3020 3020 3220 3420 2020 2035  44000 0 2 4    5
│ │ │ │ +0005d790: 3333 3433 3336 3030 3020 2020 2020 2020  334336000       
│ │ │ │ +0005d7a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d7b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005d7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005d7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d800: 2020 2020 207c 0a7c 3134 3534 3037 3231       |.|14540721
│ │ │ │ -0005d810: 3133 2033 2020 2020 2031 3633 3438 3431  13 3     1634841
│ │ │ │ -0005d820: 3837 3031 3930 3720 3220 2020 2020 2020  8701907 2       
│ │ │ │ -0005d830: 3837 3238 3131 3636 3139 3534 3320 2020  8728116619543   
│ │ │ │ -0005d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d850: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -0005d860: 2d2d 7820 7820 202b 202d 2d2d 2d2d 2d2d  --x x  + -------
│ │ │ │ -0005d870: 2d2d 2d2d 2d2d 2d78 2078 2078 2020 2b20  -------x x x  + 
│ │ │ │ -0005d880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078  -------------x x
│ │ │ │ -0005d890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d8a0: 2020 2020 207c 0a7c 2032 3931 3630 3030       |.| 2916000
│ │ │ │ -0005d8b0: 3020 2031 2034 2020 2020 2034 3131 3530  0  1 4     41150
│ │ │ │ -0005d8c0: 3539 3230 3030 2020 3020 3220 3420 2020  592000  0 2 4   
│ │ │ │ -0005d8d0: 2032 3035 3735 3239 3630 3030 2020 3020   20575296000  0 
│ │ │ │ -0005d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d8f0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +0005d7f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d800: 0a7c 3134 3534 3037 3231 3133 2033 2020  .|1454072113 3  
│ │ │ │ +0005d810: 2020 2031 3633 3438 3431 3837 3031 3930     1634841870190
│ │ │ │ +0005d820: 3720 3220 2020 2020 2020 3837 3238 3131  7 2       872811
│ │ │ │ +0005d830: 3636 3139 3534 3320 2020 2020 2020 2020  6619543         
│ │ │ │ +0005d840: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d850: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820  .|----------x x 
│ │ │ │ +0005d860: 202b 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   + -------------
│ │ │ │ +0005d870: 2d78 2078 2078 2020 2b20 2d2d 2d2d 2d2d  -x x x  + ------
│ │ │ │ +0005d880: 2d2d 2d2d 2d2d 2d78 2078 2020 2020 2020  -------x x      
│ │ │ │ +0005d890: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d8a0: 0a7c 2032 3931 3630 3030 3020 2031 2034  .| 29160000  1 4
│ │ │ │ +0005d8b0: 2020 2020 2034 3131 3530 3539 3230 3030       41150592000
│ │ │ │ +0005d8c0: 2020 3020 3220 3420 2020 2032 3035 3735    0 2 4    20575
│ │ │ │ +0005d8d0: 3239 3630 3030 2020 3020 2020 2020 2020  296000  0       
│ │ │ │ +0005d8e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d8f0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0005d900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d940: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 3438  -----|.|      48
│ │ │ │ -0005d950: 3037 3233 3939 3539 3033 2020 2032 2020  0723995903   2  
│ │ │ │ -0005d960: 2020 2031 3431 3630 3734 3637 3230 3920     141607467209 
│ │ │ │ -0005d970: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0005d980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d990: 2020 2020 207c 0a7c 2078 2020 2d20 2d2d       |.| x  - --
│ │ │ │ -0005d9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820 7820  ----------x x x 
│ │ │ │ -0005d9b0: 202d 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d78   - ------------x
│ │ │ │ -0005d9c0: 2078 2078 2020 2d20 2020 2020 2020 2020   x x  -         
│ │ │ │ -0005d9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d9e0: 2020 2020 207c 0a7c 3220 3420 2020 2034       |.|2 4    4
│ │ │ │ -0005d9f0: 3830 3039 3032 3430 3020 2030 2032 2034  800902400  0 2 4
│ │ │ │ -0005da00: 2020 2020 2038 3839 3035 3630 3030 2020       889056000  
│ │ │ │ -0005da10: 3120 3220 3420 2020 2020 2020 2020 2020  1 2 4           
│ │ │ │ -0005da20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005da30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0005d940: 0a7c 2020 2020 2020 3438 3037 3233 3939  .|      48072399
│ │ │ │ +0005d950: 3539 3033 2020 2032 2020 2020 2031 3431  5903   2     141
│ │ │ │ +0005d960: 3630 3734 3637 3230 3920 2020 3220 2020  607467209   2   
│ │ │ │ +0005d970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005d980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d990: 0a7c 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d  .| x  - --------
│ │ │ │ +0005d9a0: 2d2d 2d2d 7820 7820 7820 202d 202d 2d2d  ----x x x  - ---
│ │ │ │ +0005d9b0: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078 2020  ---------x x x  
│ │ │ │ +0005d9c0: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │ +0005d9d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005d9e0: 0a7c 3220 3420 2020 2034 3830 3039 3032  .|2 4    4800902
│ │ │ │ +0005d9f0: 3430 3020 2030 2032 2034 2020 2020 2038  400  0 2 4     8
│ │ │ │ +0005da00: 3839 3035 3630 3030 2020 3120 3220 3420  89056000  1 2 4 
│ │ │ │ +0005da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005da20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005da30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005da40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005da50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005da60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005da80: 2020 2020 207c 0a7c 3237 3739 3035 3833       |.|27790583
│ │ │ │ -0005da90: 2032 2020 2020 2020 2033 3037 3936 3431   2       3079641
│ │ │ │ -0005daa0: 3839 2020 2032 2020 2020 2033 3930 3230  89   2     39020
│ │ │ │ -0005dab0: 3333 3039 3435 3920 2020 2020 2020 2020  3309459         
│ │ │ │ -0005dac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dad0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -0005dae0: 7820 7820 7820 202b 202d 2d2d 2d2d 2d2d  x x x  + -------
│ │ │ │ -0005daf0: 2d2d 7820 7820 7820 202b 202d 2d2d 2d2d  --x x x  + -----
│ │ │ │ -0005db00: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020  -------         
│ │ │ │ -0005db10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005db20: 2020 2020 207c 0a7c 2035 3833 3230 3030       |.| 5832000
│ │ │ │ -0005db30: 2031 2032 2034 2020 2020 3334 3032 3030   1 2 4    340200
│ │ │ │ -0005db40: 3020 2030 2032 2034 2020 2020 3334 3239  0  0 2 4    3429
│ │ │ │ -0005db50: 3231 3630 3030 2020 2020 2020 2020 2020  216000          
│ │ │ │ -0005db60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005db70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005da70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005da80: 0a7c 3237 3739 3035 3833 2032 2020 2020  .|27790583 2    
│ │ │ │ +0005da90: 2020 2033 3037 3936 3431 3839 2020 2032     307964189   2
│ │ │ │ +0005daa0: 2020 2020 2033 3930 3230 3333 3039 3435       39020330945
│ │ │ │ +0005dab0: 3920 2020 2020 2020 2020 2020 2020 2020  9               
│ │ │ │ +0005dac0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005dad0: 0a7c 2d2d 2d2d 2d2d 2d2d 7820 7820 7820  .|--------x x x 
│ │ │ │ +0005dae0: 202b 202d 2d2d 2d2d 2d2d 2d2d 7820 7820   + ---------x x 
│ │ │ │ +0005daf0: 7820 202b 202d 2d2d 2d2d 2d2d 2d2d 2d2d  x  + -----------
│ │ │ │ +0005db00: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │ +0005db10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005db20: 0a7c 2035 3833 3230 3030 2031 2032 2034  .| 5832000 1 2 4
│ │ │ │ +0005db30: 2020 2020 3334 3032 3030 3020 2030 2032      3402000  0 2
│ │ │ │ +0005db40: 2034 2020 2020 3334 3239 3231 3630 3030   4    3429216000
│ │ │ │ +0005db50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005db60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005db70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005db90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dbc0: 2020 2020 207c 0a7c 2020 2020 2020 2031       |.|       1
│ │ │ │ -0005dbd0: 3534 3935 3932 3032 3931 2020 2032 2020  5495920291   2  
│ │ │ │ -0005dbe0: 2020 2033 3538 3737 3639 3538 3337 2020     35877695837  
│ │ │ │ -0005dbf0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0005dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dc10: 2020 2020 207c 0a7c 7820 7820 202d 202d       |.|x x  - -
│ │ │ │ -0005dc20: 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820 7820  ----------x x x 
│ │ │ │ -0005dc30: 202b 202d 2d2d 2d2d 2d2d 2d2d 2d2d 7820   + -----------x 
│ │ │ │ -0005dc40: 7820 7820 202d 2020 2020 2020 2020 2020  x x  -          
│ │ │ │ -0005dc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dc60: 2020 2020 207c 0a7c 2032 2034 2020 2020       |.| 2 4    
│ │ │ │ -0005dc70: 2038 3233 3031 3138 3420 2030 2032 2034   82301184  0 2 4
│ │ │ │ -0005dc80: 2020 2020 3139 3035 3132 3030 3020 2031      190512000  1
│ │ │ │ -0005dc90: 2032 2034 2020 2020 2020 2020 2020 2020   2 4            
│ │ │ │ -0005dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dcb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005dbb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005dbc0: 0a7c 2020 2020 2020 2031 3534 3935 3932  .|       1549592
│ │ │ │ +0005dbd0: 3032 3931 2020 2032 2020 2020 2033 3538  0291   2     358
│ │ │ │ +0005dbe0: 3737 3639 3538 3337 2020 2032 2020 2020  77695837   2    
│ │ │ │ +0005dbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005dc00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005dc10: 0a7c 7820 7820 202d 202d 2d2d 2d2d 2d2d  .|x x  - -------
│ │ │ │ +0005dc20: 2d2d 2d2d 7820 7820 7820 202b 202d 2d2d  ----x x x  + ---
│ │ │ │ +0005dc30: 2d2d 2d2d 2d2d 2d2d 7820 7820 7820 202d  --------x x x  -
│ │ │ │ +0005dc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005dc50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005dc60: 0a7c 2032 2034 2020 2020 2038 3233 3031  .| 2 4     82301
│ │ │ │ +0005dc70: 3138 3420 2030 2032 2034 2020 2020 3139  184  0 2 4    19
│ │ │ │ +0005dc80: 3035 3132 3030 3020 2031 2032 2034 2020  0512000  1 2 4  
│ │ │ │ +0005dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005dca0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005dcb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dd00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005dd10: 2020 2031 3330 3033 3432 3831 3131 3720     130034281117 
│ │ │ │ -0005dd20: 3220 2020 2020 2020 3933 3938 3339 3739  2       93983979
│ │ │ │ -0005dd30: 3331 3033 2020 2020 2020 2020 2020 2020  3103            
│ │ │ │ -0005dd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dd50: 2020 2020 207c 0a7c 7820 7820 7820 7820       |.|x x x x 
│ │ │ │ -0005dd60: 202b 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d78   + ------------x
│ │ │ │ -0005dd70: 2078 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d   x x  - --------
│ │ │ │ -0005dd80: 2d2d 2d2d 7820 7820 2020 2020 2020 2020  ----x x         
│ │ │ │ -0005dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dda0: 2020 2020 207c 0a7c 2030 2031 2032 2034       |.| 0 1 2 4
│ │ │ │ -0005ddb0: 2020 2020 2036 3533 3138 3430 3030 2020       653184000  
│ │ │ │ -0005ddc0: 3120 3220 3420 2020 3134 3430 3237 3037  1 2 4   14402707
│ │ │ │ -0005ddd0: 3230 3030 2030 2020 2020 2020 2020 2020  2000 0          
│ │ │ │ -0005dde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ddf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005dcf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005dd00: 0a7c 2020 2020 2020 2020 2020 2031 3330  .|           130
│ │ │ │ +0005dd10: 3033 3432 3831 3131 3720 3220 2020 2020  034281117 2     
│ │ │ │ +0005dd20: 2020 3933 3938 3339 3739 3331 3033 2020    939839793103  
│ │ │ │ +0005dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005dd40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005dd50: 0a7c 7820 7820 7820 7820 202b 202d 2d2d  .|x x x x  + ---
│ │ │ │ +0005dd60: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078 2020  ---------x x x  
│ │ │ │ +0005dd70: 2d20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7820  - ------------x 
│ │ │ │ +0005dd80: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ +0005dd90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005dda0: 0a7c 2030 2031 2032 2034 2020 2020 2036  .| 0 1 2 4     6
│ │ │ │ +0005ddb0: 3533 3138 3430 3030 2020 3120 3220 3420  53184000  1 2 4 
│ │ │ │ +0005ddc0: 2020 3134 3430 3237 3037 3230 3030 2030    144027072000 0
│ │ │ │ +0005ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005dde0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005ddf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005de00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005de10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005de40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005de50: 3637 3635 3730 3133 3532 3331 2032 2020  676570135231 2  
│ │ │ │ -0005de60: 2020 2020 2037 3539 3638 3039 3131 3737       75968091177
│ │ │ │ -0005de70: 3031 2020 2032 2020 2020 2020 2020 2020  01   2          
│ │ │ │ -0005de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005de90: 2020 2020 207c 0a7c 2078 2078 2020 2d20       |.| x x  - 
│ │ │ │ -0005dea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820  ------------x x 
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│ │ │ │ -0005dec0: 2d2d 7820 7820 7820 2020 2020 2020 2020  --x x x         
│ │ │ │ -0005ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dee0: 2020 2020 207c 0a7c 3120 3220 3420 2020       |.|1 2 4   
│ │ │ │ -0005def0: 2034 3839 3838 3830 3030 3020 2031 2032   4898880000  1 2
│ │ │ │ -0005df00: 2034 2020 2020 3230 3537 3532 3936 3030   4    2057529600
│ │ │ │ -0005df10: 3020 2030 2032 2020 2020 2020 2020 2020  0  0 2          
│ │ │ │ -0005df20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005df30: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +0005de30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005de40: 0a7c 2020 2020 2020 2020 3637 3635 3730  .|        676570
│ │ │ │ +0005de50: 3133 3532 3331 2032 2020 2020 2020 2037  135231 2       7
│ │ │ │ +0005de60: 3539 3638 3039 3131 3737 3031 2020 2032  596809117701   2
│ │ │ │ +0005de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005de80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005de90: 0a7c 2078 2078 2020 2d20 2d2d 2d2d 2d2d  .| x x  - ------
│ │ │ │ +0005dea0: 2d2d 2d2d 2d2d 7820 7820 7820 202b 202d  ------x x x  + -
│ │ │ │ +0005deb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820  ------------x x 
│ │ │ │ +0005dec0: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ +0005ded0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005dee0: 0a7c 3120 3220 3420 2020 2034 3839 3838  .|1 2 4    48988
│ │ │ │ +0005def0: 3830 3030 3020 2031 2032 2034 2020 2020  80000  1 2 4    
│ │ │ │ +0005df00: 3230 3537 3532 3936 3030 3020 2030 2032  20575296000  0 2
│ │ │ │ +0005df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005df20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005df30: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0005df40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005df50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005df60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005df70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005df80: 2d2d 2d2d 2d7c 0a7c 2037 3133 3935 3835  -----|.| 7139585
│ │ │ │ -0005df90: 3839 2033 2020 2020 2036 3138 3638 3233  89 3     6186823
│ │ │ │ -0005dfa0: 3231 3031 2032 2020 2020 2020 2037 3131  2101 2       711
│ │ │ │ -0005dfb0: 3731 3331 3337 3735 3438 3720 2020 2020  71313775487     
│ │ │ │ -0005dfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dfd0: 2020 2020 207c 0a7c 202d 2d2d 2d2d 2d2d       |.| -------
│ │ │ │ -0005dfe0: 2d2d 7820 7820 202b 202d 2d2d 2d2d 2d2d  --x x  + -------
│ │ │ │ -0005dff0: 2d2d 2d2d 7820 7820 7820 202d 202d 2d2d  ----x x x  - ---
│ │ │ │ -0005e000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078  -----------x x x
│ │ │ │ -0005e010: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │ -0005e020: 2020 2020 207c 0a7c 2020 3132 3334 3830       |.|  123480
│ │ │ │ -0005e030: 3030 2032 2034 2020 2020 3533 3334 3333  00 2 4    533433
│ │ │ │ -0005e040: 3630 3030 2030 2033 2034 2020 2020 2038  6000 0 3 4     8
│ │ │ │ -0005e050: 3030 3135 3034 3030 3030 2020 3020 3120  0015040000  0 1 
│ │ │ │ -0005e060: 3320 3420 2020 2020 2020 2020 2020 2020  3 4             
│ │ │ │ -0005e070: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005df70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0005df80: 0a7c 2037 3133 3935 3835 3839 2033 2020  .| 713958589 3  
│ │ │ │ +0005df90: 2020 2036 3138 3638 3233 3231 3031 2032     61868232101 2
│ │ │ │ +0005dfa0: 2020 2020 2020 2037 3131 3731 3331 3337         711713137
│ │ │ │ +0005dfb0: 3735 3438 3720 2020 2020 2020 2020 2020  75487           
│ │ │ │ +0005dfc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005dfd0: 0a7c 202d 2d2d 2d2d 2d2d 2d2d 7820 7820  .| ---------x x 
│ │ │ │ +0005dfe0: 202b 202d 2d2d 2d2d 2d2d 2d2d 2d2d 7820   + -----------x 
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│ │ │ │ +0005e000: 2d2d 2d2d 2d78 2078 2078 2078 2020 2020  -----x x x x    
│ │ │ │ +0005e010: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e020: 0a7c 2020 3132 3334 3830 3030 2032 2034  .|  12348000 2 4
│ │ │ │ +0005e030: 2020 2020 3533 3334 3333 3630 3030 2030      5334336000 0
│ │ │ │ +0005e040: 2033 2034 2020 2020 2038 3030 3135 3034   3 4     8001504
│ │ │ │ +0005e050: 3030 3030 2020 3020 3120 3320 3420 2020  0000  0 1 3 4   
│ │ │ │ +0005e060: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e070: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005e080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e0c0: 2020 2020 207c 0a7c 2020 2032 2020 2020       |.|   2    
│ │ │ │ -0005e0d0: 2031 3238 3133 3835 3531 3637 2033 2020   12813855167 3  
│ │ │ │ -0005e0e0: 2020 2031 3730 3034 3532 3330 3120 3220     1700452301 2 
│ │ │ │ -0005e0f0: 2020 2020 2020 3731 3832 3039 3337 3239        7182093729
│ │ │ │ -0005e100: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │ -0005e110: 2020 2020 207c 0a7c 7820 7820 7820 202b       |.|x x x  +
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│ │ │ │ -0005e140: 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d   x  - ----------
│ │ │ │ -0005e150: 2d78 2078 2020 2020 2020 2020 2020 2020  -x x            
│ │ │ │ -0005e160: 2020 2020 207c 0a7c 2031 2032 2034 2020       |.| 1 2 4  
│ │ │ │ -0005e170: 2020 3835 3733 3034 3030 3020 2032 2034    857304000  2 4
│ │ │ │ -0005e180: 2020 2020 3632 3531 3137 3530 2020 3020      62511750  0 
│ │ │ │ -0005e190: 3320 3420 2020 2032 3632 3434 3030 3030  3 4    262440000
│ │ │ │ -0005e1a0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ -0005e1b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005e0b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e0c0: 0a7c 2020 2032 2020 2020 2031 3238 3133  .|   2     12813
│ │ │ │ +0005e0d0: 3835 3531 3637 2033 2020 2020 2031 3730  855167 3     170
│ │ │ │ +0005e0e0: 3034 3532 3330 3120 3220 2020 2020 2020  0452301 2       
│ │ │ │ +0005e0f0: 3731 3832 3039 3337 3239 3720 2020 2020  71820937297     
│ │ │ │ +0005e100: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e110: 0a7c 7820 7820 7820 202b 202d 2d2d 2d2d  .|x x x  + -----
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│ │ │ │ +0005e140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020  -----------x x  
│ │ │ │ +0005e150: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e160: 0a7c 2031 2032 2034 2020 2020 3835 3733  .| 1 2 4    8573
│ │ │ │ +0005e170: 3034 3030 3020 2032 2034 2020 2020 3632  04000  2 4    62
│ │ │ │ +0005e180: 3531 3137 3530 2020 3020 3320 3420 2020  511750  0 3 4   
│ │ │ │ +0005e190: 2032 3632 3434 3030 3030 2020 3020 2020   262440000  0   
│ │ │ │ +0005e1a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e1b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e200: 2020 2020 207c 0a7c 3732 3834 3330 3034       |.|72843004
│ │ │ │ -0005e210: 3120 3320 2020 2020 3532 3537 3037 3835  1 3     52570785
│ │ │ │ -0005e220: 3836 3039 2032 2020 2020 2020 2032 3330  8609 2       230
│ │ │ │ -0005e230: 3831 3737 3437 3735 3120 2020 2020 2020  817747751       
│ │ │ │ -0005e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e250: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -0005e260: 2d78 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d  -x x  - --------
│ │ │ │ -0005e270: 2d2d 2d2d 7820 7820 7820 202b 202d 2d2d  ----x x x  + ---
│ │ │ │ -0005e280: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078 2078  ---------x x x x
│ │ │ │ -0005e290: 2020 2b20 2020 2020 2020 2020 2020 2020    +             
│ │ │ │ -0005e2a0: 2020 2020 207c 0a7c 2031 3432 3838 3430       |.| 1428840
│ │ │ │ -0005e2b0: 3020 3220 3420 2020 2031 3030 3031 3838  0 2 4    1000188
│ │ │ │ -0005e2c0: 3030 3020 2030 2033 2034 2020 2020 2033  000  0 3 4     3
│ │ │ │ -0005e2d0: 3030 3035 3634 3030 2020 3020 3120 3320  00056400  0 1 3 
│ │ │ │ -0005e2e0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -0005e2f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005e1f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e200: 0a7c 3732 3834 3330 3034 3120 3320 2020  .|728430041 3   
│ │ │ │ +0005e210: 2020 3532 3537 3037 3835 3836 3039 2032    525707858609 2
│ │ │ │ +0005e220: 2020 2020 2020 2032 3330 3831 3737 3437         230817747
│ │ │ │ +0005e230: 3735 3120 2020 2020 2020 2020 2020 2020  751             
│ │ │ │ +0005e240: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e250: 0a7c 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020  .|---------x x  
│ │ │ │ +0005e260: 2d20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7820  - ------------x 
│ │ │ │ +0005e270: 7820 7820 202b 202d 2d2d 2d2d 2d2d 2d2d  x x  + ---------
│ │ │ │ +0005e280: 2d2d 2d78 2078 2078 2078 2020 2b20 2020  ---x x x x  +   
│ │ │ │ +0005e290: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e2a0: 0a7c 2031 3432 3838 3430 3020 3220 3420  .| 14288400 2 4 
│ │ │ │ +0005e2b0: 2020 2031 3030 3031 3838 3030 3020 2030     1000188000  0
│ │ │ │ +0005e2c0: 2033 2034 2020 2020 2033 3030 3035 3634   3 4     3000564
│ │ │ │ +0005e2d0: 3030 2020 3020 3120 3320 3420 2020 2020  00  0 1 3 4     
│ │ │ │ +0005e2e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e2f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e340: 2020 2020 207c 0a7c 3220 2020 2020 3833       |.|2     83
│ │ │ │ -0005e350: 3138 3030 3633 3436 3720 2020 3220 2020  180063467   2   
│ │ │ │ -0005e360: 2020 3838 3331 3637 3338 3137 2033 2020    8831673817 3  
│ │ │ │ -0005e370: 2020 2033 3430 3231 3035 3638 3533 3120     340210568531 
│ │ │ │ -0005e380: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0005e390: 2020 2020 207c 0a7c 2078 2020 2b20 2d2d       |.| x  + --
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│ │ │ │ -0005e3c0: 202d 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d78   - ------------x
│ │ │ │ -0005e3d0: 2078 2078 2020 2020 2020 2020 2020 2020   x x            
│ │ │ │ -0005e3e0: 2020 2020 207c 0a7c 3220 3420 2020 2032       |.|2 4    2
│ │ │ │ -0005e3f0: 3238 3631 3434 3030 3020 3120 3220 3420  286144000 1 2 4 
│ │ │ │ -0005e400: 2020 2036 3636 3739 3230 3030 2032 2034     666792000 2 4
│ │ │ │ -0005e410: 2020 2020 3330 3030 3536 3430 3030 2020      3000564000  
│ │ │ │ -0005e420: 3020 3320 2020 2020 2020 2020 2020 2020  0 3             
│ │ │ │ -0005e430: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005e330: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e340: 0a7c 3220 2020 2020 3833 3138 3030 3633  .|2     83180063
│ │ │ │ +0005e350: 3436 3720 2020 3220 2020 2020 3838 3331  467   2     8831
│ │ │ │ +0005e360: 3637 3338 3137 2033 2020 2020 2033 3430  673817 3     340
│ │ │ │ +0005e370: 3231 3035 3638 3533 3120 3220 2020 2020  210568531 2     
│ │ │ │ +0005e380: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e390: 0a7c 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d  .| x  + --------
│ │ │ │ +0005e3a0: 2d2d 2d78 2078 2078 2020 2d20 2d2d 2d2d  ---x x x  - ----
│ │ │ │ +0005e3b0: 2d2d 2d2d 2d2d 7820 7820 202d 202d 2d2d  ------x x  - ---
│ │ │ │ +0005e3c0: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078 2020  ---------x x x  
│ │ │ │ +0005e3d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e3e0: 0a7c 3220 3420 2020 2032 3238 3631 3434  .|2 4    2286144
│ │ │ │ +0005e3f0: 3030 3020 3120 3220 3420 2020 2036 3636  000 1 2 4    666
│ │ │ │ +0005e400: 3739 3230 3030 2032 2034 2020 2020 3330  792000 2 4    30
│ │ │ │ +0005e410: 3030 3536 3430 3030 2020 3020 3320 2020  00564000  0 3   
│ │ │ │ +0005e420: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e430: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e450: 2020 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 207c 0a7c 2020 2020 3138 3637       |.|    1867
│ │ │ │ -0005e490: 3536 3837 3539 3531 2020 2032 2020 2020  56875951   2    
│ │ │ │ -0005e4a0: 2039 3636 3636 3039 3439 3920 3320 2020   9666609499 3   
│ │ │ │ -0005e4b0: 2020 3131 3930 3737 3030 3136 3920 3220    11907700169 2 
│ │ │ │ -0005e4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e4d0: 2020 2020 207c 0a7c 2020 2b20 2d2d 2d2d       |.|  + ----
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│ │ │ │ -0005e4f0: 202d 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020   ----------x x  
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│ │ │ │ -0005e510: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │ -0005e520: 2020 2020 207c 0a7c 3420 2020 2031 3337       |.|4    137
│ │ │ │ -0005e530: 3136 3836 3430 3020 2031 2032 2034 2020  1686400  1 2 4  
│ │ │ │ -0005e540: 2020 3935 3235 3630 3030 2020 3220 3420    95256000  2 4 
│ │ │ │ -0005e550: 2020 2020 3430 3832 3430 3030 2020 3020      40824000  0 
│ │ │ │ -0005e560: 3320 3420 2020 2020 2020 2020 2020 2020  3 4             
│ │ │ │ -0005e570: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +0005e470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e480: 0a7c 2020 2020 3138 3637 3536 3837 3539  .|    1867568759
│ │ │ │ +0005e490: 3531 2020 2032 2020 2020 2039 3636 3636  51   2     96666
│ │ │ │ +0005e4a0: 3039 3439 3920 3320 2020 2020 3131 3930  09499 3     1190
│ │ │ │ +0005e4b0: 3737 3030 3136 3920 3220 2020 2020 2020  7700169 2       
│ │ │ │ +0005e4c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e4d0: 0a7c 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d2d  .|  + ----------
│ │ │ │ +0005e4e0: 2d2d 7820 7820 7820 202b 202d 2d2d 2d2d  --x x x  + -----
│ │ │ │ +0005e4f0: 2d2d 2d2d 2d78 2078 2020 2b20 2d2d 2d2d  -----x x  + ----
│ │ │ │ +0005e500: 2d2d 2d2d 2d2d 2d78 2078 2078 2020 2020  -------x x x    
│ │ │ │ +0005e510: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e520: 0a7c 3420 2020 2031 3337 3136 3836 3430  .|4    137168640
│ │ │ │ +0005e530: 3020 2031 2032 2034 2020 2020 3935 3235  0  1 2 4    9525
│ │ │ │ +0005e540: 3630 3030 2020 3220 3420 2020 2020 3430  6000  2 4     40
│ │ │ │ +0005e550: 3832 3430 3030 2020 3020 3320 3420 2020  824000  0 3 4   
│ │ │ │ +0005e560: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e570: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
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│ │ │ │ -0005e5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e5c0: 2d2d 2d2d 2d7c 0a7c 2020 3633 3831 3836  -----|.|  638186
│ │ │ │ -0005e5d0: 3036 3330 3537 2032 2020 2020 2020 2032  063057 2       2
│ │ │ │ -0005e5e0: 3137 3633 3335 3537 3133 3231 2020 2020  176335571321    
│ │ │ │ +0005e5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0005e5c0: 0a7c 2020 3633 3831 3836 3036 3330 3537  .|  638186063057
│ │ │ │ +0005e5d0: 2032 2020 2020 2020 2032 3137 3633 3335   2       2176335
│ │ │ │ +0005e5e0: 3537 3133 3231 2020 2020 2020 2020 2020  571321          
│ │ │ │  0005e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e610: 2020 2020 207c 0a7c 2b20 2d2d 2d2d 2d2d       |.|+ ------
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│ │ │ │ -0005e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820  ------------x x 
│ │ │ │ -0005e640: 7820 7820 2020 2020 2020 2020 2020 2020  x x             
│ │ │ │ -0005e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e660: 2020 2020 207c 0a7c 2020 2033 3637 3431       |.|   36741
│ │ │ │ -0005e670: 3630 3030 3020 2031 2033 2034 2020 2020  60000  1 3 4    
│ │ │ │ -0005e680: 2038 3030 3135 3034 3030 3020 2030 2032   8001504000  0 2
│ │ │ │ -0005e690: 2033 2034 2020 2020 2020 2020 2020 2020   3 4            
│ │ │ │ -0005e6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e6b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005e600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e610: 0a7c 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|+ ------------
│ │ │ │ +0005e620: 7820 7820 7820 202b 202d 2d2d 2d2d 2d2d  x x x  + -------
│ │ │ │ +0005e630: 2d2d 2d2d 2d2d 7820 7820 7820 7820 2020  ------x x x x   
│ │ │ │ +0005e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e650: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e660: 0a7c 2020 2033 3637 3431 3630 3030 3020  .|   3674160000 
│ │ │ │ +0005e670: 2031 2033 2034 2020 2020 2038 3030 3135   1 3 4     80015
│ │ │ │ +0005e680: 3034 3030 3020 2030 2032 2033 2034 2020  04000  0 2 3 4  
│ │ │ │ +0005e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e6a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e6b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e700: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005e710: 3232 3332 3938 3533 3235 3536 3920 3220  2232985325569 2 
│ │ │ │ -0005e720: 2020 2020 2020 3437 3631 3836 3637 3031        4761866701
│ │ │ │ -0005e730: 3938 3720 2020 2020 2020 2020 2020 2020  987             
│ │ │ │ -0005e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e750: 2020 2020 207c 0a7c 2078 2078 2020 2b20       |.| x x  + 
│ │ │ │ -0005e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078  -------------x x
│ │ │ │ -0005e770: 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d2d   x  + ----------
│ │ │ │ -0005e780: 2d2d 2d78 2020 2020 2020 2020 2020 2020  ---x            
│ │ │ │ -0005e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e7a0: 2020 2020 207c 0a7c 3120 3320 3420 2020       |.|1 3 4   
│ │ │ │ -0005e7b0: 2020 3432 3836 3532 3030 3030 2020 3120    4286520000  1 
│ │ │ │ -0005e7c0: 3320 3420 2020 2035 3034 3039 3437 3532  3 4    504094752
│ │ │ │ -0005e7d0: 3030 2020 2020 2020 2020 2020 2020 2020  00              
│ │ │ │ -0005e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e7f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005e6f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e700: 0a7c 2020 2020 2020 2020 3232 3332 3938  .|        223298
│ │ │ │ +0005e710: 3533 3235 3536 3920 3220 2020 2020 2020  5325569 2       
│ │ │ │ +0005e720: 3437 3631 3836 3637 3031 3938 3720 2020  4761866701987   
│ │ │ │ +0005e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e740: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e750: 0a7c 2078 2078 2020 2b20 2d2d 2d2d 2d2d  .| x x  + ------
│ │ │ │ +0005e760: 2d2d 2d2d 2d2d 2d78 2078 2078 2020 2b20  -------x x x  + 
│ │ │ │ +0005e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2020  -------------x  
│ │ │ │ +0005e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e790: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e7a0: 0a7c 3120 3320 3420 2020 2020 3432 3836  .|1 3 4     4286
│ │ │ │ +0005e7b0: 3532 3030 3030 2020 3120 3320 3420 2020  520000  1 3 4   
│ │ │ │ +0005e7c0: 2035 3034 3039 3437 3532 3030 2020 2020   50409475200    
│ │ │ │ +0005e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e7e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e7f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e840: 2020 2020 207c 0a7c 3932 3932 3639 3632       |.|92926962
│ │ │ │ -0005e850: 3037 3720 3220 2020 2020 2020 3338 3831  077 2       3881
│ │ │ │ -0005e860: 3433 3536 3738 3233 3320 2020 2020 2020  435678233       
│ │ │ │ +0005e830: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e840: 0a7c 3932 3932 3639 3632 3037 3720 3220  .|92926962077 2 
│ │ │ │ +0005e850: 2020 2020 2020 3338 3831 3433 3536 3738        3881435678
│ │ │ │ +0005e860: 3233 3320 2020 2020 2020 2020 2020 2020  233             
│ │ │ │  0005e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e890: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -0005e8a0: 2d2d 2d78 2078 2078 2020 2d20 2d2d 2d2d  ---x x x  - ----
│ │ │ │ -0005e8b0: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078 2078  ---------x x x x
│ │ │ │ -0005e8c0: 2020 2b20 2020 2020 2020 2020 2020 2020    +             
│ │ │ │ -0005e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e8e0: 2020 2020 207c 0a7c 2034 3238 3635 3230       |.| 4286520
│ │ │ │ -0005e8f0: 3030 2020 3120 3320 3420 2020 2020 3630  00  1 3 4     60
│ │ │ │ -0005e900: 3031 3132 3830 3030 2020 3020 3220 3320  01128000  0 2 3 
│ │ │ │ -0005e910: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -0005e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e930: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005e880: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e890: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078  .|-----------x x
│ │ │ │ +0005e8a0: 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d   x  - ----------
│ │ │ │ +0005e8b0: 2d2d 2d78 2078 2078 2078 2020 2b20 2020  ---x x x x  +   
│ │ │ │ +0005e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e8d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e8e0: 0a7c 2034 3238 3635 3230 3030 2020 3120  .| 428652000  1 
│ │ │ │ +0005e8f0: 3320 3420 2020 2020 3630 3031 3132 3830  3 4     60011280
│ │ │ │ +0005e900: 3030 2020 3020 3220 3320 3420 2020 2020  00  0 2 3 4     
│ │ │ │ +0005e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e930: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e980: 2020 2020 207c 0a7c 2020 2020 3631 3531       |.|    6151
│ │ │ │ -0005e990: 3837 3036 3637 3539 3720 2020 2020 2020  870667597       
│ │ │ │ -0005e9a0: 2020 2020 3234 3635 3832 3538 3920 3220      246582589 2 
│ │ │ │ +0005e970: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e980: 0a7c 2020 2020 3631 3531 3837 3036 3637  .|    6151870667
│ │ │ │ +0005e990: 3539 3720 2020 2020 2020 2020 2020 3234  597           24
│ │ │ │ +0005e9a0: 3635 3832 3538 3920 3220 2020 2020 2020  6582589 2       
│ │ │ │  0005e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e9d0: 2020 2020 207c 0a7c 2020 2d20 2d2d 2d2d       |.|  - ----
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│ │ │ │ -0005e9f0: 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d78 2078    - ---------x x
│ │ │ │ -0005ea00: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │ -0005ea10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ea20: 2020 2020 207c 0a7c 3420 2020 2031 3032       |.|4    102
│ │ │ │ -0005ea30: 3837 3634 3830 3030 2020 3020 3120 3320  87648000  0 1 3 
│ │ │ │ -0005ea40: 3420 2020 2031 3639 3334 3430 2020 3120  4    1693440  1 
│ │ │ │ -0005ea50: 3320 3420 2020 2020 2020 2020 2020 2020  3 4             
│ │ │ │ -0005ea60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ea70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005e9c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005e9d0: 0a7c 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d  .|  - ----------
│ │ │ │ +0005e9e0: 2d2d 2d78 2078 2078 2078 2020 2d20 2d2d  ---x x x x  - --
│ │ │ │ +0005e9f0: 2d2d 2d2d 2d2d 2d78 2078 2078 2020 2020  -------x x x    
│ │ │ │ +0005ea00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ea10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005ea20: 0a7c 3420 2020 2031 3032 3837 3634 3830  .|4    102876480
│ │ │ │ +0005ea30: 3030 2020 3020 3120 3320 3420 2020 2031  00  0 1 3 4    1
│ │ │ │ +0005ea40: 3639 3334 3430 2020 3120 3320 3420 2020  693440  1 3 4   
│ │ │ │ +0005ea50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ea60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005ea70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005ea80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ea90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005eaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eac0: 2020 2020 207c 0a7c 2020 3734 3839 3137       |.|  748917
│ │ │ │ -0005ead0: 3634 3238 3037 3120 2020 2020 2020 2020  6428071         
│ │ │ │ -0005eae0: 2020 3130 3530 3234 3434 3837 2032 2020    1050244487 2  
│ │ │ │ +0005eab0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005eac0: 0a7c 2020 3734 3839 3137 3634 3238 3037  .|  748917642807
│ │ │ │ +0005ead0: 3120 2020 2020 2020 2020 2020 3130 3530  1           1050
│ │ │ │ +0005eae0: 3234 3434 3837 2032 2020 2020 2020 2020  244487 2        
│ │ │ │  0005eaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eb10: 2020 2020 207c 0a7c 2d20 2d2d 2d2d 2d2d       |.|- ------
│ │ │ │ -0005eb20: 2d2d 2d2d 2d2d 2d78 2078 2078 2078 2020  -------x x x x  
│ │ │ │ -0005eb30: 2d20 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820  - ----------x x 
│ │ │ │ -0005eb40: 7820 202b 2020 2020 2020 2020 2020 2020  x  +            
│ │ │ │ -0005eb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eb60: 2020 2020 207c 0a7c 2020 2020 3835 3733       |.|    8573
│ │ │ │ -0005eb70: 3034 3030 3030 2020 3020 3120 3320 3420  040000  0 1 3 4 
│ │ │ │ -0005eb80: 2020 2031 3136 3634 3030 3020 2031 2033     11664000  1 3
│ │ │ │ -0005eb90: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -0005eba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ebb0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
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│ │ │ │ +0005eb10: 0a7c 2d20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|- ------------
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│ │ │ │ +0005eb30: 2d2d 2d2d 2d2d 7820 7820 7820 202b 2020  ------x x x  +  
│ │ │ │ +0005eb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005eb50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005eb60: 0a7c 2020 2020 3835 3733 3034 3030 3030  .|    8573040000
│ │ │ │ +0005eb70: 2020 3020 3120 3320 3420 2020 2031 3136    0 1 3 4    116
│ │ │ │ +0005eb80: 3634 3030 3020 2031 2033 2034 2020 2020  64000  1 3 4    
│ │ │ │ +0005eb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005eba0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005ebb0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0005ebc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ebd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ebe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ebf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ec00: 2d2d 2d2d 2d7c 0a7c 2020 2032 3134 3137  -----|.|   21417
│ │ │ │ -0005ec10: 3734 3632 3931 2020 2020 2020 2020 2020  746291          
│ │ │ │ -0005ec20: 2033 3834 3330 3237 3032 3320 3220 2020   3843027023 2   
│ │ │ │ -0005ec30: 2020 2020 3536 3834 3033 3231 3338 3320      56840321383 
│ │ │ │ -0005ec40: 2020 3220 2020 2020 3137 3838 3837 3533    2     17888753
│ │ │ │ -0005ec50: 3831 3920 207c 0a7c 202d 202d 2d2d 2d2d  819  |.| - -----
│ │ │ │ -0005ec60: 2d2d 2d2d 2d2d 7820 7820 7820 7820 202d  ------x x x x  -
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│ │ │ │ -0005ec80: 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2d78    + -----------x
│ │ │ │ -0005ec90: 2078 2078 2020 2b20 2d2d 2d2d 2d2d 2d2d   x x  + --------
│ │ │ │ -0005eca0: 2d2d 2d78 207c 0a7c 2020 2020 3230 3030  ---x |.|    2000
│ │ │ │ -0005ecb0: 3337 3630 3030 2031 2032 2033 2034 2020  376000 1 2 3 4  
│ │ │ │ -0005ecc0: 2020 3734 3038 3830 3030 2020 3220 3320    74088000  2 3 
│ │ │ │ -0005ecd0: 3420 2020 2020 3833 3334 3930 3030 2020  4     83349000  
│ │ │ │ -0005ece0: 3020 3320 3420 2020 2020 3639 3938 3430  0 3 4     699840
│ │ │ │ -0005ecf0: 3030 2020 317c 0a7c 2020 2020 2020 2020  00  1|.|        
│ │ │ │ +0005ebf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0005ec00: 0a7c 2020 2032 3134 3137 3734 3632 3931  .|   21417746291
│ │ │ │ +0005ec10: 2020 2020 2020 2020 2020 2033 3834 3330             38430
│ │ │ │ +0005ec20: 3237 3032 3320 3220 2020 2020 2020 3536  27023 2       56
│ │ │ │ +0005ec30: 3834 3033 3231 3338 3320 2020 3220 2020  840321383   2   
│ │ │ │ +0005ec40: 2020 3137 3838 3837 3533 3831 3920 207c    17888753819  |
│ │ │ │ +0005ec50: 0a7c 202d 202d 2d2d 2d2d 2d2d 2d2d 2d2d  .| - -----------
│ │ │ │ +0005ec60: 7820 7820 7820 7820 202d 202d 2d2d 2d2d  x x x x  - -----
│ │ │ │ +0005ec70: 2d2d 2d2d 2d78 2078 2078 2020 2b20 2d2d  -----x x x  + --
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│ │ │ │ +0005eca0: 0a7c 2020 2020 3230 3030 3337 3630 3030  .|    2000376000
│ │ │ │ +0005ecb0: 2031 2032 2033 2034 2020 2020 3734 3038   1 2 3 4    7408
│ │ │ │ +0005ecc0: 3830 3030 2020 3220 3320 3420 2020 2020  8000  2 3 4     
│ │ │ │ +0005ecd0: 3833 3334 3930 3030 2020 3020 3320 3420  83349000  0 3 4 
│ │ │ │ +0005ece0: 2020 2020 3639 3938 3430 3030 2020 317c      69984000  1|
│ │ │ │ +0005ecf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005ed00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ed10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ed20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ed30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ed40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005ed50: 2020 3136 3033 3032 3032 3037 3239 3831    16030202072981
│ │ │ │ -0005ed60: 2020 2020 2020 2020 2020 2031 3237 3235             12725
│ │ │ │ -0005ed70: 3432 3835 3733 3031 2032 2020 2020 2020  42857301 2      
│ │ │ │ -0005ed80: 2031 3135 3938 3031 3035 3934 2020 2032   11598010594   2
│ │ │ │ -0005ed90: 2020 2020 207c 0a7c 2078 2078 2078 2020       |.| x x x  
│ │ │ │ -0005eda0: 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  + --------------
│ │ │ │ -0005edb0: 7820 7820 7820 7820 202d 202d 2d2d 2d2d  x x x x  - -----
│ │ │ │ -0005edc0: 2d2d 2d2d 2d2d 2d2d 7820 7820 7820 202d  --------x x x  -
│ │ │ │ -0005edd0: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820   -----------x x 
│ │ │ │ -0005ede0: 7820 202b 207c 0a7c 3020 3220 3320 3420  x  + |.|0 2 3 4 
│ │ │ │ -0005edf0: 2020 2020 3336 3030 3637 3638 3030 3020      36006768000 
│ │ │ │ -0005ee00: 2031 2032 2033 2034 2020 2020 3138 3030   1 2 3 4    1800
│ │ │ │ -0005ee10: 3333 3834 3030 3020 2032 2033 2034 2020  3384000  2 3 4  
│ │ │ │ -0005ee20: 2020 2032 3433 3130 3132 3520 2030 2033     24310125  0 3
│ │ │ │ -0005ee30: 2034 2020 207c 0a7c 2020 2020 2020 2020   4   |.|        
│ │ │ │ +0005ed30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005ed40: 0a7c 2020 2020 2020 2020 2020 3136 3033  .|          1603
│ │ │ │ +0005ed50: 3032 3032 3037 3239 3831 2020 2020 2020  0202072981      
│ │ │ │ +0005ed60: 2020 2020 2031 3237 3235 3432 3835 3733       12725428573
│ │ │ │ +0005ed70: 3031 2032 2020 2020 2020 2031 3135 3938  01 2       11598
│ │ │ │ +0005ed80: 3031 3035 3934 2020 2032 2020 2020 207c  010594   2     |
│ │ │ │ +0005ed90: 0a7c 2078 2078 2078 2020 2b20 2d2d 2d2d  .| x x x  + ----
│ │ │ │ +0005eda0: 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820 7820  ----------x x x 
│ │ │ │ +0005edb0: 7820 202d 202d 2d2d 2d2d 2d2d 2d2d 2d2d  x  - -----------
│ │ │ │ +0005edc0: 2d2d 7820 7820 7820 202d 202d 2d2d 2d2d  --x x x  - -----
│ │ │ │ +0005edd0: 2d2d 2d2d 2d2d 7820 7820 7820 202b 207c  ------x x x  + |
│ │ │ │ +0005ede0: 0a7c 3020 3220 3320 3420 2020 2020 3336  .|0 2 3 4     36
│ │ │ │ +0005edf0: 3030 3637 3638 3030 3020 2031 2032 2033  006768000  1 2 3
│ │ │ │ +0005ee00: 2034 2020 2020 3138 3030 3333 3834 3030   4    1800338400
│ │ │ │ +0005ee10: 3020 2032 2033 2034 2020 2020 2032 3433  0  2 3 4     243
│ │ │ │ +0005ee20: 3130 3132 3520 2030 2033 2034 2020 207c  10125  0 3 4   |
│ │ │ │ +0005ee30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ee50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ee60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ee70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ee80: 2020 2020 207c 0a7c 3330 3935 3235 3434       |.|30952544
│ │ │ │ -0005ee90: 3932 3120 2020 2020 2020 2020 2020 3534  921           54
│ │ │ │ -0005eea0: 3436 3132 3537 3032 3937 2032 2020 2020  4612570297 2    
│ │ │ │ -0005eeb0: 2020 2031 3438 3936 3632 3733 3133 2020     14896627313  
│ │ │ │ -0005eec0: 2032 2020 2020 2031 3931 3739 3834 3530   2     191798450
│ │ │ │ -0005eed0: 3131 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d  11   |.|--------
│ │ │ │ -0005eee0: 2d2d 2d78 2078 2078 2078 2020 2d20 2d2d  ---x x x x  - --
│ │ │ │ -0005eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820 7820  ----------x x x 
│ │ │ │ -0005ef00: 202d 202d 2d2d 2d2d 2d2d 2d2d 2d2d 7820   - -----------x 
│ │ │ │ -0005ef10: 7820 7820 202b 202d 2d2d 2d2d 2d2d 2d2d  x x  + ---------
│ │ │ │ -0005ef20: 2d2d 7820 787c 0a7c 2020 3535 3536 3630  --x x|.|  555660
│ │ │ │ -0005ef30: 3030 2020 3120 3220 3320 3420 2020 2036  00  1 2 3 4    6
│ │ │ │ -0005ef40: 3030 3131 3238 3030 3020 2032 2033 2034  001128000  2 3 4
│ │ │ │ -0005ef50: 2020 2020 2031 3636 3639 3830 3020 2030       16669800  0
│ │ │ │ -0005ef60: 2033 2034 2020 2020 3130 3030 3138 3830   3 4    10001880
│ │ │ │ -0005ef70: 3030 2031 207c 0a7c 2020 2020 2020 2020  00 1 |.|        
│ │ │ │ +0005ee70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005ee80: 0a7c 3330 3935 3235 3434 3932 3120 2020  .|30952544921   
│ │ │ │ +0005ee90: 2020 2020 2020 2020 3534 3436 3132 3537          54461257
│ │ │ │ +0005eea0: 3032 3937 2032 2020 2020 2020 2031 3438  0297 2       148
│ │ │ │ +0005eeb0: 3936 3632 3733 3133 2020 2032 2020 2020  96627313   2    
│ │ │ │ +0005eec0: 2031 3931 3739 3834 3530 3131 2020 207c   19179845011   |
│ │ │ │ +0005eed0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078  .|-----------x x
│ │ │ │ +0005eee0: 2078 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d   x x  - --------
│ │ │ │ +0005eef0: 2d2d 2d2d 7820 7820 7820 202d 202d 2d2d  ----x x x  - ---
│ │ │ │ +0005ef00: 2d2d 2d2d 2d2d 2d2d 7820 7820 7820 202b  --------x x x  +
│ │ │ │ +0005ef10: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 7820 787c   -----------x x|
│ │ │ │ +0005ef20: 0a7c 2020 3535 3536 3630 3030 2020 3120  .|  55566000  1 
│ │ │ │ +0005ef30: 3220 3320 3420 2020 2036 3030 3131 3238  2 3 4    6001128
│ │ │ │ +0005ef40: 3030 3020 2032 2033 2034 2020 2020 2031  000  2 3 4     1
│ │ │ │ +0005ef50: 3636 3639 3830 3020 2030 2033 2034 2020  6669800  0 3 4  
│ │ │ │ +0005ef60: 2020 3130 3030 3138 3830 3030 2031 207c    1000188000 1 |
│ │ │ │ +0005ef70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005ef80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005efc0: 2020 2020 207c 0a7c 2020 3130 3036 3437       |.|  100647
│ │ │ │ -0005efd0: 3639 3933 3234 3720 2020 2020 2020 2020  6993247         
│ │ │ │ -0005efe0: 2020 3834 3433 3138 3932 3536 3934 3720    8443189256947 
│ │ │ │ -0005eff0: 2020 2020 2020 2020 2020 3331 3933 3639            319369
│ │ │ │ -0005f000: 3835 3436 3034 3120 3220 2020 2020 2020  8546041 2       
│ │ │ │ -0005f010: 3136 3634 367c 0a7c 2b20 2d2d 2d2d 2d2d  16646|.|+ ------
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│ │ │ │ -0005f060: 2d2d 2d2d 2d7c 0a7c 2020 2020 3830 3031  -----|.|    8001
│ │ │ │ -0005f070: 3530 3430 3030 2020 3020 3220 3320 3420  504000  0 2 3 4 
│ │ │ │ -0005f080: 2020 2031 3434 3032 3730 3732 3030 3020     144027072000 
│ │ │ │ -0005f090: 3120 3220 3320 3420 2020 2038 3430 3135  1 2 3 4    84015
│ │ │ │ -0005f0a0: 3739 3230 3030 2020 3220 3320 3420 2020  792000  2 3 4   
│ │ │ │ -0005f0b0: 2020 3933 377c 0a7c 2020 2020 2020 2020    937|.|        
│ │ │ │ +0005efb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005efc0: 0a7c 2020 3130 3036 3437 3639 3933 3234  .|  100647699324
│ │ │ │ +0005efd0: 3720 2020 2020 2020 2020 2020 3834 3433  7           8443
│ │ │ │ +0005efe0: 3138 3932 3536 3934 3720 2020 2020 2020  189256947       
│ │ │ │ +0005eff0: 2020 2020 3331 3933 3639 3835 3436 3034      319369854604
│ │ │ │ +0005f000: 3120 3220 2020 2020 2020 3136 3634 367c  1 2       16646|
│ │ │ │ +0005f010: 0a7c 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|+ ------------
│ │ │ │ +0005f020: 2d78 2078 2078 2078 2020 2d20 2d2d 2d2d  -x x x x  - ----
│ │ │ │ +0005f030: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078 2078  ---------x x x x
│ │ │ │ +0005f040: 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d    + ------------
│ │ │ │ +0005f050: 2d78 2078 2078 2020 2d20 2d2d 2d2d 2d7c  -x x x  - -----|
│ │ │ │ +0005f060: 0a7c 2020 2020 3830 3031 3530 3430 3030  .|    8001504000
│ │ │ │ +0005f070: 2020 3020 3220 3320 3420 2020 2031 3434    0 2 3 4    144
│ │ │ │ +0005f080: 3032 3730 3732 3030 3020 3120 3220 3320  027072000 1 2 3 
│ │ │ │ +0005f090: 3420 2020 2038 3430 3135 3739 3230 3030  4    84015792000
│ │ │ │ +0005f0a0: 2020 3220 3320 3420 2020 2020 3933 377c    2 3 4     937|
│ │ │ │ +0005f0b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005f0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f100: 2020 2020 207c 0a7c 2031 3232 3635 3938       |.| 1226598
│ │ │ │ -0005f110: 3735 3337 3138 3720 2020 2020 2020 2020  7537187         
│ │ │ │ -0005f120: 2020 3731 3231 3632 3739 3238 3438 3120    7121627928481 
│ │ │ │ -0005f130: 2020 2020 2020 2020 2020 3232 3032 3437            220247
│ │ │ │ -0005f140: 3039 3638 3932 3120 3220 2020 2020 2020  0968921 2       
│ │ │ │ -0005f150: 3234 3631 347c 0a7c 202d 2d2d 2d2d 2d2d  24614|.| -------
│ │ │ │ -0005f160: 2d2d 2d2d 2d2d 2d78 2078 2078 2078 2020  -------x x x x  
│ │ │ │ -0005f170: 2d20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d78  - -------------x
│ │ │ │ -0005f180: 2078 2078 2078 2020 2b20 2d2d 2d2d 2d2d   x x x  + ------
│ │ │ │ -0005f190: 2d2d 2d2d 2d2d 2d78 2078 2078 2020 2d20  -------x x x  - 
│ │ │ │ -0005f1a0: 2d2d 2d2d 2d7c 0a7c 2020 2032 3430 3034  -----|.|   24004
│ │ │ │ -0005f1b0: 3531 3230 3030 2020 3020 3220 3320 3420  512000  0 2 3 4 
│ │ │ │ -0005f1c0: 2020 2033 3432 3932 3136 3030 3030 2020     34292160000  
│ │ │ │ -0005f1d0: 3120 3220 3320 3420 2020 2031 3230 3032  1 2 3 4    12002
│ │ │ │ -0005f1e0: 3235 3630 3030 2020 3220 3320 3420 2020  256000  2 3 4   
│ │ │ │ -0005f1f0: 2020 3333 337c 0a7c 2d2d 2d2d 2d2d 2d2d    333|.|--------
│ │ │ │ +0005f0f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005f100: 0a7c 2031 3232 3635 3938 3735 3337 3138  .| 1226598753718
│ │ │ │ +0005f110: 3720 2020 2020 2020 2020 2020 3731 3231  7           7121
│ │ │ │ +0005f120: 3632 3739 3238 3438 3120 2020 2020 2020  627928481       
│ │ │ │ +0005f130: 2020 2020 3232 3032 3437 3039 3638 3932      220247096892
│ │ │ │ +0005f140: 3120 3220 2020 2020 2020 3234 3631 347c  1 2       24614|
│ │ │ │ +0005f150: 0a7c 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .| -------------
│ │ │ │ +0005f160: 2d78 2078 2078 2078 2020 2d20 2d2d 2d2d  -x x x x  - ----
│ │ │ │ +0005f170: 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078 2078  ---------x x x x
│ │ │ │ +0005f180: 2020 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d    + ------------
│ │ │ │ +0005f190: 2d78 2078 2078 2020 2d20 2d2d 2d2d 2d7c  -x x x  - -----|
│ │ │ │ +0005f1a0: 0a7c 2020 2032 3430 3034 3531 3230 3030  .|   24004512000
│ │ │ │ +0005f1b0: 2020 3020 3220 3320 3420 2020 2033 3432    0 2 3 4    342
│ │ │ │ +0005f1c0: 3932 3136 3030 3030 2020 3120 3220 3320  92160000  1 2 3 
│ │ │ │ +0005f1d0: 3420 2020 2031 3230 3032 3235 3630 3030  4    12002256000
│ │ │ │ +0005f1e0: 2020 3220 3320 3420 2020 2020 3333 337c    2 3 4     333|
│ │ │ │ +0005f1f0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0005f200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f240: 2d2d 2d2d 2d7c 0a7c 2032 2020 2020 2039  -----|.| 2     9
│ │ │ │ -0005f250: 3030 3130 3938 3535 2020 2032 2020 2020  00109855   2    
│ │ │ │ -0005f260: 2034 3038 3232 3338 3120 3320 2020 2020   40822381 3     
│ │ │ │ -0005f270: 3136 3634 3335 3035 3137 3320 3220 2020  16643505173 2   
│ │ │ │ -0005f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f290: 2020 2020 207c 0a7c 7820 7820 202b 202d       |.|x x  + -
│ │ │ │ -0005f2a0: 2d2d 2d2d 2d2d 2d2d 7820 7820 7820 202d  --------x x x  -
│ │ │ │ -0005f2b0: 202d 2d2d 2d2d 2d2d 2d78 2078 2020 2d20   --------x x  - 
│ │ │ │ -0005f2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020  -----------x x  
│ │ │ │ -0005f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f2e0: 2020 2020 207c 0a7c 2033 2034 2020 2020       |.| 3 4    
│ │ │ │ -0005f2f0: 3830 3031 3530 3420 2032 2033 2034 2020  8001504  2 3 4  
│ │ │ │ -0005f300: 2020 3338 3838 3030 2020 3320 3420 2020    388800  3 4   
│ │ │ │ -0005f310: 2034 3537 3232 3838 3030 3020 3020 2020   4572288000 0   
│ │ │ │ -0005f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f330: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0005f240: 0a7c 2032 2020 2020 2039 3030 3130 3938  .| 2     9001098
│ │ │ │ +0005f250: 3535 2020 2032 2020 2020 2034 3038 3232  55   2     40822
│ │ │ │ +0005f260: 3338 3120 3320 2020 2020 3136 3634 3335  381 3     166435
│ │ │ │ +0005f270: 3035 3137 3320 3220 2020 2020 2020 2020  05173 2         
│ │ │ │ +0005f280: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005f290: 0a7c 7820 7820 202b 202d 2d2d 2d2d 2d2d  .|x x  + -------
│ │ │ │ +0005f2a0: 2d2d 7820 7820 7820 202d 202d 2d2d 2d2d  --x x x  - -----
│ │ │ │ +0005f2b0: 2d2d 2d78 2078 2020 2d20 2d2d 2d2d 2d2d  ---x x  - ------
│ │ │ │ +0005f2c0: 2d2d 2d2d 2d78 2078 2020 2020 2020 2020  -----x x        
│ │ │ │ +0005f2d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005f2e0: 0a7c 2033 2034 2020 2020 3830 3031 3530  .| 3 4    800150
│ │ │ │ +0005f2f0: 3420 2032 2033 2034 2020 2020 3338 3838  4  2 3 4    3888
│ │ │ │ +0005f300: 3030 2020 3320 3420 2020 2034 3537 3232  00  3 4    45722
│ │ │ │ +0005f310: 3838 3030 3020 3020 2020 2020 2020 2020  88000 0         
│ │ │ │ +0005f320: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005f330: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005f340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f380: 2020 2020 207c 0a7c 3236 3433 3233 3039       |.|26432309
│ │ │ │ -0005f390: 3935 2020 2032 2020 2020 2033 3433 3530  95   2     34350
│ │ │ │ -0005f3a0: 3637 3336 3333 2020 2032 2020 2020 2031  673633   2     1
│ │ │ │ -0005f3b0: 3534 3930 3432 3933 2033 2020 2020 2020  54904293 3      
│ │ │ │ -0005f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f3d0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
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│ │ │ │ -0005f400: 2d2d 2d2d 2d2d 2d2d 7820 7820 202d 2020  --------x x  -  
│ │ │ │ -0005f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f420: 2020 2020 207c 0a7c 2020 3830 3031 3530       |.|  800150
│ │ │ │ -0005f430: 3420 2031 2033 2034 2020 2020 3730 3031  4  1 3 4    7001
│ │ │ │ -0005f440: 3331 3630 3020 2032 2033 2034 2020 2020  31600  2 3 4    
│ │ │ │ -0005f450: 3136 3636 3938 3030 2033 2034 2020 2020  16669800 3 4    
│ │ │ │ -0005f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f470: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005f380: 0a7c 3236 3433 3233 3039 3935 2020 2032  .|2643230995   2
│ │ │ │ +0005f390: 2020 2020 2033 3433 3530 3637 3336 3333       34350673633
│ │ │ │ +0005f3a0: 2020 2032 2020 2020 2031 3534 3930 3432     2     1549042
│ │ │ │ +0005f3b0: 3933 2033 2020 2020 2020 2020 2020 2020  93 3            
│ │ │ │ +0005f3c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005f3d0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 7820 7820  .|----------x x 
│ │ │ │ +0005f3e0: 7820 202d 202d 2d2d 2d2d 2d2d 2d2d 2d2d  x  - -----------
│ │ │ │ +0005f3f0: 7820 7820 7820 202d 202d 2d2d 2d2d 2d2d  x x x  - -------
│ │ │ │ +0005f400: 2d2d 7820 7820 202d 2020 2020 2020 2020  --x x  -        
│ │ │ │ +0005f410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005f420: 0a7c 2020 3830 3031 3530 3420 2031 2033  .|  8001504  1 3
│ │ │ │ +0005f430: 2034 2020 2020 3730 3031 3331 3630 3020   4    700131600 
│ │ │ │ +0005f440: 2032 2033 2034 2020 2020 3136 3636 3938   2 3 4    166698
│ │ │ │ +0005f450: 3030 2033 2034 2020 2020 2020 2020 2020  00 3 4          
│ │ │ │ +0005f460: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005f470: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f4c0: 2020 2020 207c 0a7c 3220 2020 2020 3434       |.|2     44
│ │ │ │ -0005f4d0: 3839 3838 3834 3939 3239 2020 2032 2020  8988849929   2  
│ │ │ │ -0005f4e0: 2020 2038 3132 3536 3438 3620 3320 2020     81256486 3   
│ │ │ │ -0005f4f0: 2020 3939 3732 3833 3436 3720 3220 2020    997283467 2   
│ │ │ │ -0005f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f510: 2020 2020 207c 0a7c 2078 2020 2d20 2d2d       |.| x  - --
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│ │ │ │ -0005f530: 202b 202d 2d2d 2d2d 2d2d 2d78 2078 2020   + --------x x  
│ │ │ │ -0005f540: 2b20 2d2d 2d2d 2d2d 2d2d 2d78 2078 2020  + ---------x x  
│ │ │ │ -0005f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f560: 2020 2020 207c 0a7c 3320 3420 2020 2033       |.|3 4    3
│ │ │ │ -0005f570: 3530 3036 3538 3030 3020 2032 2033 2034  500658000  2 3 4
│ │ │ │ -0005f580: 2020 2020 3230 3833 3732 3520 3320 3420      2083725 3 4 
│ │ │ │ -0005f590: 2020 2033 3137 3532 3030 3020 3020 2020     31752000 0   
│ │ │ │ -0005f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f5b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f4b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005f4c0: 0a7c 3220 2020 2020 3434 3839 3838 3834  .|2     44898884
│ │ │ │ +0005f4d0: 3939 3239 2020 2032 2020 2020 2038 3132  9929   2     812
│ │ │ │ +0005f4e0: 3536 3438 3620 3320 2020 2020 3939 3732  56486 3     9972
│ │ │ │ +0005f4f0: 3833 3436 3720 3220 2020 2020 2020 2020  83467 2         
│ │ │ │ +0005f500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005f510: 0a7c 2078 2020 2d20 2d2d 2d2d 2d2d 2d2d  .| x  - --------
│ │ │ │ +0005f520: 2d2d 2d2d 7820 7820 7820 202b 202d 2d2d  ----x x x  + ---
│ │ │ │ +0005f530: 2d2d 2d2d 2d78 2078 2020 2b20 2d2d 2d2d  -----x x  + ----
│ │ │ │ +0005f540: 2d2d 2d2d 2d78 2078 2020 2020 2020 2020  -----x x        
│ │ │ │ +0005f550: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005f560: 0a7c 3320 3420 2020 2033 3530 3036 3538  .|3 4    3500658
│ │ │ │ +0005f570: 3030 3020 2032 2033 2034 2020 2020 3230  000  2 3 4    20
│ │ │ │ +0005f580: 3833 3732 3520 3320 3420 2020 2033 3137  83725 3 4    317
│ │ │ │ +0005f590: 3532 3030 3020 3020 2020 2020 2020 2020  52000 0         
│ │ │ │ +0005f5a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005f5b0: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f600: 2020 2020 207c 0a7c 3837 3736 3238 2020       |.|877628  
│ │ │ │ -0005f610: 2032 2020 2020 2039 3235 3432 3830 3031   2     925428001
│ │ │ │ -0005f620: 3320 2020 3220 2020 2020 3133 3932 3633  3   2     139263
│ │ │ │ -0005f630: 3933 3031 3120 2020 3220 2020 2020 2020  93011   2       
│ │ │ │ -0005f640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f650: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 7820       |.|------x 
│ │ │ │ -0005f660: 7820 7820 202d 202d 2d2d 2d2d 2d2d 2d2d  x x  - ---------
│ │ │ │ -0005f670: 2d78 2078 2078 2020 2d20 2d2d 2d2d 2d2d  -x x x  - ------
│ │ │ │ -0005f680: 2d2d 2d2d 2d78 2078 2078 2020 2d20 2020  -----x x x  -   
│ │ │ │ -0005f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f6a0: 2020 2020 207c 0a7c 3637 3632 3520 2030       |.|67625  0
│ │ │ │ -0005f6b0: 2033 2034 2020 2020 3731 3434 3230 3030   3 4    71442000
│ │ │ │ -0005f6c0: 2020 3120 3320 3420 2020 2034 3830 3039    1 3 4    48009
│ │ │ │ -0005f6d0: 3032 3430 2020 3220 3320 3420 2020 2020  0240  2 3 4     
│ │ │ │ -0005f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f6f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005f5f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005f600: 0a7c 3837 3736 3238 2020 2032 2020 2020  .|877628   2    
│ │ │ │ +0005f610: 2039 3235 3432 3830 3031 3320 2020 3220   9254280013   2 
│ │ │ │ +0005f620: 2020 2020 3133 3932 3633 3933 3031 3120      13926393011 
│ │ │ │ +0005f630: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0005f640: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005f650: 0a7c 2d2d 2d2d 2d2d 7820 7820 7820 202d  .|------x x x  -
│ │ │ │ +0005f660: 202d 2d2d 2d2d 2d2d 2d2d 2d78 2078 2078   ----------x x x
│ │ │ │ +0005f670: 2020 2d20 2d2d 2d2d 2d2d 2d2d 2d2d 2d78    - -----------x
│ │ │ │ +0005f680: 2078 2078 2020 2d20 2020 2020 2020 2020   x x  -         
│ │ │ │ +0005f690: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005f6a0: 0a7c 3637 3632 3520 2030 2033 2034 2020  .|67625  0 3 4  
│ │ │ │ +0005f6b0: 2020 3731 3434 3230 3030 2020 3120 3320    71442000  1 3 
│ │ │ │ +0005f6c0: 3420 2020 2034 3830 3039 3032 3430 2020  4    480090240  
│ │ │ │ +0005f6d0: 3220 3320 3420 2020 2020 2020 2020 2020  2 3 4           
│ │ │ │ +0005f6e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005f6f0: 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f740: 2020 2020 207c 0a7c 3232 3839 3839 3920       |.|2289899 
│ │ │ │ -0005f750: 2020 3220 2020 2020 3137 3930 3631 3439    2     17906149
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│ │ │ │ -0005f770: 3431 3633 3433 2020 2032 2020 2020 2020  416343   2      
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│ │ │ │ -00060c00: 3020 3420 2020 2020 2020 2020 2020 2020  0 4             
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│ │ │ │ +00060be0: 0a7c 2020 3432 3836 3532 3030 3030 2033  .|  4286520000 3
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│ │ │ │ +00060c80: 0a7c 2033 3033 3531 3730 3720 2020 3320  .| 30351707   3 
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│ │ │ │  00060cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00060e20: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
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│ │ │ │ -00060f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00060fd0: 3020 3420 2020 2020 2020 2020 2020 2020  0 4             
│ │ │ │ -00060fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00060f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │  00061000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061040: 2020 2020 207c 0a7c 2020 2020 3733 3237       |.|    7327
│ │ │ │ -00061050: 3733 3920 2020 3320 2020 3132 3838 3034  739   3   128804
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│ │ │ │ -00061080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061090: 2020 2020 207c 0a7c 202d 202d 2d2d 2d2d       |.| - -----
│ │ │ │ -000610a0: 2d2d 2d78 2078 2020 2d20 2d2d 2d2d 2d2d  ---x x  - ------
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│ │ │ │ -000610c0: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ -000610d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00061130: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
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│ │ │ │ +00061040: 0a7c 2020 2020 3733 3237 3733 3920 2020  .|    7327739   
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│ │ │ │ +000610c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000610d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +00061110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061120: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061130: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
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│ │ │ │  00061150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00061180: 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 696d  -----+.|i3 : tim
│ │ │ │ -00061190: 6520 696e 7665 7273 6520 7068 6920 2020  e inverse phi   
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│ │ │ │ +00061180: 0a7c 6933 203a 2074 696d 6520 696e 7665  .|i3 : time inve
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│ │ │ │  000611a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000611d0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -000611e0: 2030 2e30 3536 3936 3973 2028 6370 7529   0.056969s (cpu)
│ │ │ │ -000611f0: 3b20 302e 3035 3639 3638 3773 2028 7468  ; 0.0569687s (th
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│ │ │ │  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                  
│ │ │ │ -00061270: 2020 2020 207c 0a7c 6f33 203d 202d 2d20       |.|o3 = -- 
│ │ │ │ -00061280: 7261 7469 6f6e 616c 206d 6170 202d 2d20  rational map -- 
│ │ │ │ +00061260: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ -00061400: 2020 2020 207c 0a7c 2020 2020 2064 6566       |.|     def
│ │ │ │ -00061410: 696e 696e 6720 666f 726d 733a 207b 2020  ining forms: {  
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│ │ │ │ -000614a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ -000614c0: 3330 3931 3438 3338 3435 3237 3336 3030  3091483845273600
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│ │ │ │ +000614b0: 2020 2020 2020 2020 2d20 3330 3931 3438          - 309148
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│ │ │ │ -00061520: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00061530: 2020 2020 2020 3020 3120 2020 2020 2020        0 1       
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│ │ │ │  00061570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061590: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │  00061640: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00061700: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00061730: 2020 2020 2020 2020 2020 2020 2020 3232                22
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│ │ │ │ -00061790: 2020 2020 2020 2020 2020 2020 2020 2030                 0
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│ │ │ │ -000617b0: 2020 2020 3020 3120 2020 2020 2020 2020      0 1         
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│ │ │ │  000617d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000617f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061800: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00061800: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │  00061820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061840: 2034 2020 2020 2020 2020 2020 2020 2020   4              
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│ │ │ │ -00061870: 2020 2020 2020 2020 2020 2020 2020 2d20                - 
│ │ │ │ -00061880: 3337 3537 3931 3035 3032 3532 3830 3030  3757910502528000
│ │ │ │ -00061890: 7820 202b 2031 3439 3237 3436 3730 3233  x  + 14927467023
│ │ │ │ -000618a0: 3436 3336 3830 7820 7820 2020 2020 2020  463680x x       
│ │ │ │ -000618b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00061830: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ +00061840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061850: 2033 2020 2020 2020 2020 2020 2020 207c   3             |
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│ │ │ │ +00061870: 2020 2020 2020 2020 2d20 3337 3537 3931          - 375791
│ │ │ │ +00061880: 3035 3032 3532 3830 3030 7820 202b 2031  0502528000x  + 1
│ │ │ │ +00061890: 3439 3237 3436 3730 3233 3436 3336 3830  4927467023463680
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│ │ │ │  000618c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000618e0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
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│ │ │ │ -00061900: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │  00061910: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00061970: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │ -00061980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061990: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ -000619a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000619b0: 2020 2020 2020 2020 2020 2020 2020 3137                17
│ │ │ │ -000619c0: 3936 3831 3837 3534 3330 3430 3030 7820  96818754304000x 
│ │ │ │ -000619d0: 202d 2031 3339 3138 3331 3332 3835 3435   - 1391831328545
│ │ │ │ -000619e0: 3238 3030 7820 7820 202b 2020 2020 2020  2800x x  +      
│ │ │ │ -000619f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00061970: 2020 2020 2020 2020 2034 2020 2020 2020           4      
│ │ │ │ +00061980: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ +00061990: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000619a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000619b0: 2020 2020 2020 2020 3137 3936 3831 3837          17968187
│ │ │ │ +000619c0: 3534 3330 3430 3030 7820 202d 2031 3339  54304000x  - 139
│ │ │ │ +000619d0: 3138 3331 3332 3835 3435 3238 3030 7820  18313285452800x 
│ │ │ │ +000619e0: 7820 202b 2020 2020 2020 2020 2020 207c  x  +           |
│ │ │ │ +000619f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00061a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061a10: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00061a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061a30: 2020 2020 2030 2031 2020 2020 2020 2020       0 1        
│ │ │ │ -00061a40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00061a50: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ +00061a10: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ +00061a20: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00061a30: 2031 2020 2020 2020 2020 2020 2020 207c   1             |
│ │ │ │ +00061a40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00061a50: 2020 2020 2020 207d 2020 2020 2020 2020         }        
│ │ │ │  00061a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061a90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00061a80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061a90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00061aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00061ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061ae0: 2020 2020 207c 0a7c 6f33 203a 2052 6174       |.|o3 : Rat
│ │ │ │ -00061af0: 696f 6e61 6c4d 6170 2028 4372 656d 6f6e  ionalMap (Cremon
│ │ │ │ -00061b00: 6120 7472 616e 7366 6f72 6d61 7469 6f6e  a transformation
│ │ │ │ -00061b10: 206f 6620 5050 5e34 2920 2020 2020 2020   of PP^4)       
│ │ │ │ -00061b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061b30: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00061ad0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061ae0: 0a7c 6f33 203a 2052 6174 696f 6e61 6c4d  .|o3 : RationalM
│ │ │ │ +00061af0: 6170 2028 4372 656d 6f6e 6120 7472 616e  ap (Cremona tran
│ │ │ │ +00061b00: 7366 6f72 6d61 7469 6f6e 206f 6620 5050  sformation of PP
│ │ │ │ +00061b10: 5e34 2920 2020 2020 2020 2020 2020 2020  ^4)             
│ │ │ │ +00061b20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061b30: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00061b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061b80: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -00061b90: 2020 2020 2020 2020 2020 2032 2032 2020             2 2  
│ │ │ │ -00061ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061bb0: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ -00061bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061bd0: 2020 2020 207c 0a7c 2d20 3233 3232 3830       |.|- 232280
│ │ │ │ -00061be0: 3033 3230 3037 3532 3030 7820 7820 202b  0320075200x x  +
│ │ │ │ -00061bf0: 2036 3032 3037 3732 3037 3437 3535 3230   602077207475520
│ │ │ │ -00061c00: 7820 7820 2020 2020 2020 2020 2020 2020  x x             
│ │ │ │ -00061c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061c20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00061c30: 2020 2020 2020 2020 2020 2030 2031 2020             0 1  
│ │ │ │ -00061c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061c50: 2030 2031 2020 2020 2020 2020 2020 2020   0 1            
│ │ │ │ -00061c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061c70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00061b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00061b80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00061b90: 2020 2020 2032 2032 2020 2020 2020 2020       2 2        
│ │ │ │ +00061ba0: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ +00061bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061bc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061bd0: 0a7c 2d20 3233 3232 3830 3033 3230 3037  .|- 232280032007
│ │ │ │ +00061be0: 3532 3030 7820 7820 202b 2036 3032 3037  5200x x  + 60207
│ │ │ │ +00061bf0: 3732 3037 3437 3535 3230 7820 7820 2020  7207475520x x   
│ │ │ │ +00061c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061c10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061c20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00061c30: 2020 2020 2030 2031 2020 2020 2020 2020       0 1        
│ │ │ │ +00061c40: 2020 2020 2020 2020 2020 2030 2031 2020             0 1  
│ │ │ │ +00061c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061c60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061c70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ -00061cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061cc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00061cd0: 2020 2020 2020 2020 2020 2032 2032 2020             2 2  
│ │ │ │ -00061ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061cf0: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ -00061d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061d10: 2020 2020 207c 0a7c 2031 3734 3334 3731       |.| 1743471
│ │ │ │ -00061d20: 3233 3930 3936 3136 3030 7820 7820 202d  2390961600x x  -
│ │ │ │ -00061d30: 2038 3336 3137 3433 3531 3537 3338 3230   836174351573820
│ │ │ │ -00061d40: 3078 2078 2020 2020 2020 2020 2020 2020  0x x            
│ │ │ │ -00061d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061d60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00061d70: 2020 2020 2020 2020 2020 2030 2031 2020             0 1  
│ │ │ │ -00061d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061d90: 2020 3020 3120 2020 2020 2020 2020 2020    0 1           
│ │ │ │ -00061da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061db0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00061cb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061cc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00061cd0: 2020 2020 2032 2032 2020 2020 2020 2020       2 2        
│ │ │ │ +00061ce0: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ +00061cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061d00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061d10: 0a7c 2031 3734 3334 3731 3233 3930 3936  .| 1743471239096
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│ │ │ │ +00061d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061d50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061d60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00061d70: 2020 2020 2030 2031 2020 2020 2020 2020       0 1        
│ │ │ │ +00061d80: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ +00061d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061db0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ -00061e00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00061e10: 2020 2020 2020 2020 2032 2032 2020 2020           2 2    
│ │ │ │ -00061e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061e30: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -00061e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061e50: 2020 2020 207c 0a7c 3133 3135 3039 3332       |.|13150932
│ │ │ │ -00061e60: 3034 3435 3434 3030 7820 7820 202d 2033  04454400x x  - 3
│ │ │ │ -00061e70: 3332 3635 3631 3237 3130 3238 3830 7820  32656127102880x 
│ │ │ │ -00061e80: 7820 202b 2020 2020 2020 2020 2020 2020  x  +            
│ │ │ │ -00061e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061ea0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00061eb0: 2020 2020 2020 2020 2030 2031 2020 2020           0 1    
│ │ │ │ -00061ec0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00061ed0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ -00061ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061ef0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00061df0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061e00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00061e10: 2020 2032 2032 2020 2020 2020 2020 2020     2 2          
│ │ │ │ +00061e20: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ +00061e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061e40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061e50: 0a7c 3133 3135 3039 3332 3034 3435 3434  .|13150932044544
│ │ │ │ +00061e60: 3030 7820 7820 202d 2033 3332 3635 3631  00x x  - 3326561
│ │ │ │ +00061e70: 3237 3130 3238 3830 7820 7820 202b 2020  27102880x x  +  
│ │ │ │ +00061e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061e90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061ea0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00061eb0: 2020 2030 2031 2020 2020 2020 2020 2020     0 1          
│ │ │ │ +00061ec0: 2020 2020 2020 2020 2030 2031 2020 2020           0 1    
│ │ │ │ +00061ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061ee0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061ef0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00061f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061f40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00061f50: 2020 2020 2020 2020 2020 2020 2032 2032               2 2
│ │ │ │ +00061f30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061f40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00061f50: 2020 2020 2020 2032 2032 2020 2020 2020         2 2      
│ │ │ │  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 207c 0a7c 202d 2031 3732 3235       |.| - 17225
│ │ │ │ -00061fa0: 3030 3834 3938 3031 3432 3430 7820 7820  008498014240x x 
│ │ │ │ -00061fb0: 202b 2037 3138 3132 3238 3335 3633 3432   + 7181228356342
│ │ │ │ -00061fc0: 3332 3078 2020 2020 2020 2020 2020 2020  320x            
│ │ │ │ -00061fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061fe0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00061ff0: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
│ │ │ │ -00062000: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00061f90: 0a7c 202d 2031 3732 3235 3030 3834 3938  .| - 17225008498
│ │ │ │ +00061fa0: 3031 3432 3430 7820 7820 202b 2037 3138  014240x x  + 718
│ │ │ │ +00061fb0: 3132 3238 3335 3633 3432 3332 3078 2020  1228356342320x  
│ │ │ │ +00061fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061fd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061fe0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00061ff0: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ +00062000: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00062010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062020: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062030: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00062040: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00062080: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00062090: 2020 2020 2020 2020 2020 2032 2032 2020             2 2  
│ │ │ │ -000620a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000620c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000620d0: 2020 2020 207c 0a7c 2031 3130 3537 3833       |.| 1105783
│ │ │ │ -000620e0: 3731 3333 3036 3732 3030 7820 7820 202d  7133067200x x  -
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│ │ │ │ -00062100: 3078 2078 2020 2020 2020 2020 2020 2020  0x x            
│ │ │ │ -00062110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062120: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ +000620a0: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ +000620b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000620c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000620d0: 0a7c 2031 3130 3537 3833 3731 3333 3036  .| 1105783713306
│ │ │ │ +000620e0: 3732 3030 7820 7820 202d 2031 3336 3738  7200x x  - 13678
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│ │ │ │ +00062100: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00062120: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062130: 2020 2020 2030 2031 2020 2020 2020 2020       0 1        
│ │ │ │ +00062140: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
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│ │ │ │ +00062160: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062170: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00062180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000621a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000621b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000621c0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -000621d0: 2020 2020 2020 2020 2034 2020 2020 2020           4      
│ │ │ │ -000621e0: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ -000621f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062200: 2020 2020 2032 2020 2020 2020 2020 2020       2          
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│ │ │ │ -00062270: 2020 2020 2020 2020 2031 2020 2020 2020           1      
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│ │ │ │ -000622b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ +000621d0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
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│ │ │ │ +00062210: 0a7c 2d20 3332 3337 3738 3534 3738 3134  .|- 323778547814
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│ │ │ │ +00062270: 2020 2031 2020 2020 2020 2020 2020 2020     1            
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│ │ │ │  000622c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000622d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000622e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000622f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062300: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00062310: 2020 2020 2020 2020 2020 2020 3420 2020              4   
│ │ │ │ -00062320: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00062340: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -00062350: 2020 2020 207c 0a7c 202b 2031 3232 3534       |.| + 12254
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│ │ │ │ +00062320: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ +00062330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062340: 2020 2032 2020 2020 2020 2020 2020 207c     2           |
│ │ │ │ +00062350: 0a7c 202b 2031 3232 3534 3535 3930 3930  .| + 12254559090
│ │ │ │ +00062360: 3733 3230 3078 2020 2b20 3431 3531 3431  73200x  + 415141
│ │ │ │ +00062370: 3533 3432 3831 3630 3030 7820 7820 202d  5342816000x x  -
│ │ │ │ +00062380: 2031 3836 3230 3435 3533 3937 3938 3735   186204553979875
│ │ │ │ +00062390: 3230 7820 7820 7820 2020 2020 2020 207c  20x x x        |
│ │ │ │ +000623a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000623b0: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +000623c0: 2020 2020 2020 2020 2020 2030 2032 2020             0 2  
│ │ │ │ +000623d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000623e0: 2020 2030 2031 2020 2020 2020 2020 207c     0 1         |
│ │ │ │ +000623f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  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 2020 2020 2020 2020       |.|        
│ │ │ │ -00062450: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ -00062460: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ -00062470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062480: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -00062490: 2020 2020 207c 0a7c 3136 3633 3930 3831       |.|16639081
│ │ │ │ -000624a0: 3738 3136 3136 3078 2020 2d20 3636 3532  7816160x  - 6652
│ │ │ │ -000624b0: 3739 3032 3331 3237 3034 3030 7820 7820  790231270400x x 
│ │ │ │ -000624c0: 202b 2031 3331 3536 3837 3733 3831 3331   + 1315687738131
│ │ │ │ -000624d0: 3634 3830 7820 7820 7820 202d 2020 2020  6480x x x  -    
│ │ │ │ -000624e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000624f0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -00062500: 2020 2020 2020 2020 2020 2020 2030 2032               0 2
│ │ │ │ -00062510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062520: 2020 2020 2030 2031 2032 2020 2020 2020       0 1 2      
│ │ │ │ -00062530: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00062430: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062440: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062450: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +00062460: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +00062470: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00062480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062490: 0a7c 3136 3633 3930 3831 3738 3136 3136  .|16639081781616
│ │ │ │ +000624a0: 3078 2020 2d20 3636 3532 3739 3032 3331  0x  - 6652790231
│ │ │ │ +000624b0: 3237 3034 3030 7820 7820 202b 2031 3331  270400x x  + 131
│ │ │ │ +000624c0: 3536 3837 3733 3831 3331 3634 3830 7820  56877381316480x 
│ │ │ │ +000624d0: 7820 7820 202d 2020 2020 2020 2020 207c  x x  -         |
│ │ │ │ +000624e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000624f0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +00062500: 2020 2020 2020 2030 2032 2020 2020 2020         0 2      
│ │ │ │ +00062510: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00062520: 2031 2032 2020 2020 2020 2020 2020 207c   1 2           |
│ │ │ │ +00062530: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00062540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2033 2020 2020 2020       |.| 3      
│ │ │ │ -00062590: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ -000625a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000625b0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -000625c0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -000625d0: 2020 2020 207c 0a7c 7820 202d 2039 3634       |.|x  - 964
│ │ │ │ -000625e0: 3037 3530 3133 3531 3631 3630 7820 202d  075013516160x  -
│ │ │ │ -000625f0: 2033 3136 3732 3134 3531 3435 3331 3230   316721451453120
│ │ │ │ -00062600: 3078 2078 2020 2b20 3836 3936 3437 3334  0x x  + 86964734
│ │ │ │ -00062610: 3733 3039 3436 3830 7820 7820 7820 2020  73094680x x x   
│ │ │ │ -00062620: 2020 2020 207c 0a7c 2031 2020 2020 2020       |.| 1      
│ │ │ │ -00062630: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -00062640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062650: 2020 3020 3220 2020 2020 2020 2020 2020    0 2           
│ │ │ │ -00062660: 2020 2020 2020 2020 2030 2031 2020 2020           0 1    
│ │ │ │ -00062670: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00062570: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062580: 0a7c 2033 2020 2020 2020 2020 2020 2020  .| 3            
│ │ │ │ +00062590: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ +000625a0: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ +000625b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000625c0: 2020 2032 2020 2020 2020 2020 2020 207c     2           |
│ │ │ │ +000625d0: 0a7c 7820 202d 2039 3634 3037 3530 3133  .|x  - 964075013
│ │ │ │ +000625e0: 3531 3631 3630 7820 202d 2033 3136 3732  516160x  - 31672
│ │ │ │ +000625f0: 3134 3531 3435 3331 3230 3078 2078 2020  14514531200x x  
│ │ │ │ +00062600: 2b20 3836 3936 3437 3334 3733 3039 3436  + 86964734730946
│ │ │ │ +00062610: 3830 7820 7820 7820 2020 2020 2020 207c  80x x x        |
│ │ │ │ +00062620: 0a7c 2031 2020 2020 2020 2020 2020 2020  .| 1            
│ │ │ │ +00062630: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ +00062640: 2020 2020 2020 2020 2020 2020 3020 3220              0 2 
│ │ │ │ +00062650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062660: 2020 2030 2031 2020 2020 2020 2020 207c     0 1         |
│ │ │ │ +00062670: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00062680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000626a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000626b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000626c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000626d0: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ -000626e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000626f0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -00062700: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -00062710: 2020 2020 207c 0a7c 202d 2034 3231 3338       |.| - 42138
│ │ │ │ -00062720: 3432 3836 3833 3638 3030 7820 202b 2032  4286836800x  + 2
│ │ │ │ -00062730: 3334 3531 3837 3534 3139 3339 3230 3078  345187541939200x
│ │ │ │ -00062740: 2078 2020 2b20 3139 3234 3836 3330 3639   x  + 1924863069
│ │ │ │ -00062750: 3135 3030 3030 7820 7820 7820 2020 2020  150000x x x     
│ │ │ │ -00062760: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00062770: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -00062780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062790: 3020 3220 2020 2020 2020 2020 2020 2020  0 2             
│ │ │ │ -000627a0: 2020 2020 2020 2030 2031 2032 2020 2020         0 1 2    
│ │ │ │ -000627b0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +000626b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000626c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000626d0: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │ +000626e0: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ +000626f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062700: 2032 2020 2020 2020 2020 2020 2020 207c   2             |
│ │ │ │ +00062710: 0a7c 202d 2034 3231 3338 3432 3836 3833  .| - 42138428683
│ │ │ │ +00062720: 3638 3030 7820 202b 2032 3334 3531 3837  6800x  + 2345187
│ │ │ │ +00062730: 3534 3139 3339 3230 3078 2078 2020 2b20  541939200x x  + 
│ │ │ │ +00062740: 3139 3234 3836 3330 3639 3135 3030 3030  1924863069150000
│ │ │ │ +00062750: 7820 7820 7820 2020 2020 2020 2020 207c  x x x          |
│ │ │ │ +00062760: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062770: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +00062780: 2020 2020 2020 2020 2020 3020 3220 2020            0 2   
│ │ │ │ +00062790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000627a0: 2030 2031 2032 2020 2020 2020 2020 207c   0 1 2         |
│ │ │ │ +000627b0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  000627c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000627d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000627e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000627f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062800: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -00062810: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00062820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062830: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -00062840: 2020 2020 2020 2020 3220 3220 2020 2020          2 2     
│ │ │ │ -00062850: 2020 2020 207c 0a7c 3337 3330 3131 3035       |.|37301105
│ │ │ │ -00062860: 3534 3938 3332 3630 7820 7820 7820 202d  54983260x x x  -
│ │ │ │ -00062870: 2034 3535 3830 3939 3631 3335 3038 3030   455809961350800
│ │ │ │ -00062880: 7820 7820 202b 2031 3135 3533 3931 3639  x x  + 115539169
│ │ │ │ -00062890: 3734 3636 3732 3078 2078 2020 2d20 2020  7466720x x  -   
│ │ │ │ -000628a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000628b0: 2020 2020 2020 2020 2030 2031 2032 2020           0 1 2  
│ │ │ │ -000628c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000628d0: 2031 2032 2020 2020 2020 2020 2020 2020   1 2            
│ │ │ │ -000628e0: 2020 2020 2020 2020 3020 3220 2020 2020          0 2     
│ │ │ │ -000628f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000627f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00062800: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062810: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00062820: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ +00062830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062840: 2020 3220 3220 2020 2020 2020 2020 207c    2 2          |
│ │ │ │ +00062850: 0a7c 3337 3330 3131 3035 3534 3938 3332  .|37301105549832
│ │ │ │ +00062860: 3630 7820 7820 7820 202d 2034 3535 3830  60x x x  - 45580
│ │ │ │ +00062870: 3939 3631 3335 3038 3030 7820 7820 202b  9961350800x x  +
│ │ │ │ +00062880: 2031 3135 3533 3931 3639 3734 3636 3732   115539169746672
│ │ │ │ +00062890: 3078 2078 2020 2d20 2020 2020 2020 207c  0x x  -        |
│ │ │ │ +000628a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000628b0: 2020 2030 2031 2032 2020 2020 2020 2020     0 1 2        
│ │ │ │ +000628c0: 2020 2020 2020 2020 2020 2031 2032 2020             1 2  
│ │ │ │ +000628d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000628e0: 2020 3020 3220 2020 2020 2020 2020 207c    0 2          |
│ │ │ │ +000628f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00062900: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00062930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062940: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00062950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062960: 3220 2020 2020 2020 2020 2020 2020 2020  2               
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│ │ │ │ -00062980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062990: 2020 2020 207c 0a7c 2020 2b20 3135 3838       |.|  + 1588
│ │ │ │ -000629a0: 3733 3638 3336 3734 3336 3330 3078 2078  7368367436300x x
│ │ │ │ -000629b0: 2078 2020 2d20 3336 3936 3237 3233 3634   x  - 3696272364
│ │ │ │ -000629c0: 3838 3434 3030 7820 7820 202b 2032 3336  884400x x  + 236
│ │ │ │ -000629d0: 3039 3539 3231 3939 3336 3830 3078 2020  0959219936800x  
│ │ │ │ -000629e0: 2020 2020 207c 0a7c 3220 2020 2020 2020       |.|2       
│ │ │ │ -000629f0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00062a00: 3120 3220 2020 2020 2020 2020 2020 2020  1 2             
│ │ │ │ -00062a10: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ -00062a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062a30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00062930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062940: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062950: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00062960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062970: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +00062980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062990: 0a7c 2020 2b20 3135 3838 3733 3638 3336  .|  + 1588736836
│ │ │ │ +000629a0: 3734 3336 3330 3078 2078 2078 2020 2d20  7436300x x x  - 
│ │ │ │ +000629b0: 3336 3936 3237 3233 3634 3838 3434 3030  3696272364884400
│ │ │ │ +000629c0: 7820 7820 202b 2032 3336 3039 3539 3231  x x  + 236095921
│ │ │ │ +000629d0: 3939 3336 3830 3078 2020 2020 2020 207c  9936800x       |
│ │ │ │ +000629e0: 0a7c 3220 2020 2020 2020 2020 2020 2020  .|2             
│ │ │ │ +000629f0: 2020 2020 2020 2020 3020 3120 3220 2020          0 1 2   
│ │ │ │ +00062a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062a10: 2031 2032 2020 2020 2020 2020 2020 2020   1 2            
│ │ │ │ +00062a20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062a30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00062a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062a80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00062a90: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00062aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062ab0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
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│ │ │ │ -00062ad0: 2020 2020 207c 0a7c 3638 3934 3635 3036       |.|68946506
│ │ │ │ -00062ae0: 3734 3632 3639 3630 7820 7820 7820 202b  74626960x x x  +
│ │ │ │ -00062af0: 2035 3539 3037 3033 3530 3139 3139 3930   559070350191990
│ │ │ │ -00062b00: 7820 7820 202d 2035 3131 3935 3433 3531  x x  - 511954351
│ │ │ │ -00062b10: 3036 3233 3336 3078 2078 2020 2b20 2020  0623360x x  +   
│ │ │ │ -00062b20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00062b30: 2020 2020 2020 2020 2030 2031 2032 2020           0 1 2  
│ │ │ │ -00062b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00062b60: 2020 2020 2020 2020 3020 3220 2020 2020          0 2     
│ │ │ │ -00062b70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00062a70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062a80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062a90: 2020 2020 2032 2020 2020 2020 2020 2020       2          
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│ │ │ │ +00062ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062ac0: 2020 3220 3220 2020 2020 2020 2020 207c    2 2          |
│ │ │ │ +00062ad0: 0a7c 3638 3934 3635 3036 3734 3632 3639  .|68946506746269
│ │ │ │ +00062ae0: 3630 7820 7820 7820 202b 2035 3539 3037  60x x x  + 55907
│ │ │ │ +00062af0: 3033 3530 3139 3139 3930 7820 7820 202d  0350191990x x  -
│ │ │ │ +00062b00: 2035 3131 3935 3433 3531 3036 3233 3336   511954351062336
│ │ │ │ +00062b10: 3078 2078 2020 2b20 2020 2020 2020 207c  0x x  +        |
│ │ │ │ +00062b20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062b30: 2020 2030 2031 2032 2020 2020 2020 2020     0 1 2        
│ │ │ │ +00062b40: 2020 2020 2020 2020 2020 2031 2032 2020             1 2  
│ │ │ │ +00062b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062b60: 2020 3020 3220 2020 2020 2020 2020 207c    0 2          |
│ │ │ │ +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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00062bd0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00062bb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00062bc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00062bd0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  00062be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062bf0: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ -00062c00: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -00062c10: 2020 2020 207c 0a7c 2020 2d20 3638 3731       |.|  - 6871
│ │ │ │ -00062c20: 3639 3832 3036 3333 3738 3935 7820 7820  698206337895x x 
│ │ │ │ -00062c30: 7820 202b 2031 3837 3839 3730 3837 3434  x  + 18789708744
│ │ │ │ -00062c40: 3437 3532 3078 2078 2020 2b20 3932 3932  47520x x  + 9292
│ │ │ │ -00062c50: 3433 3535 3531 3633 3030 3078 2078 2020  43555163000x x  
│ │ │ │ -00062c60: 2020 2020 207c 0a7c 3220 2020 2020 2020       |.|2       
│ │ │ │ -00062c70: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
│ │ │ │ -00062c80: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00062c90: 2020 2020 2020 3120 3220 2020 2020 2020        1 2       
│ │ │ │ -00062ca0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ -00062cb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00062bf0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +00062c00: 2020 2020 2020 3220 2020 2020 2020 207c        2        |
│ │ │ │ +00062c10: 0a7c 2020 2d20 3638 3731 3639 3832 3036  .|  - 6871698206
│ │ │ │ +00062c20: 3333 3738 3935 7820 7820 7820 202b 2031  337895x x x  + 1
│ │ │ │ +00062c30: 3837 3839 3730 3837 3434 3437 3532 3078  878970874447520x
│ │ │ │ +00062c40: 2078 2020 2b20 3932 3932 3433 3535 3531   x  + 9292435551
│ │ │ │ +00062c50: 3633 3030 3078 2078 2020 2020 2020 207c  63000x x       |
│ │ │ │ +00062c60: 0a7c 3220 2020 2020 2020 2020 2020 2020  .|2             
│ │ │ │ +00062c70: 2020 2020 2020 2030 2031 2032 2020 2020         0 1 2    
│ │ │ │ +00062c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062c90: 3120 3220 2020 2020 2020 2020 2020 2020  1 2             
│ │ │ │ +00062ca0: 2020 2020 2020 3020 2020 2020 2020 207c        0        |
│ │ │ │ +00062cb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00062cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062d00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00062d10: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00062d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062d30: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ -00062d40: 2020 2020 2020 2020 2020 2032 2032 2020             2 2  
│ │ │ │ -00062d50: 2020 2020 207c 0a7c 2d20 3730 3930 3338       |.|- 709038
│ │ │ │ -00062d60: 3632 3630 3832 3639 3630 7820 7820 7820  6260826960x x x 
│ │ │ │ -00062d70: 202b 2032 3232 3233 3632 3933 3137 3438   + 2222362931748
│ │ │ │ -00062d80: 3132 3078 2078 2020 2b20 3333 3333 3533  120x x  + 333353
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│ │ │ │ -00062da0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00062db0: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
│ │ │ │ -00062dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00062d50: 0a7c 2d20 3730 3930 3338 3632 3630 3832  .|- 709038626082
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│ │ │ │ -00062fe0: 3238 3537 3239 3834 3935 3339 3430 7820  28572984953940x 
│ │ │ │ -00062ff0: 7820 7820 202d 2038 3035 3438 3236 3832  x x  - 805482682
│ │ │ │ -00063000: 3732 3136 3430 7820 7820 202b 2033 3631  721640x x  + 361
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│ │ │ │ +00062fd0: 0a7c 2078 2020 2d20 3231 3238 3537 3239  .| x  - 21285729
│ │ │ │ +00062fe0: 3834 3935 3339 3430 7820 7820 7820 202d  84953940x x x  -
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│ │ │ │ +00063000: 7820 7820 202b 2033 3631 3631 3437 3937  x x  + 361614797
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│ │ │ │ -00063140: 3078 2078 2020 2d20 3839 3532 3138 3831  0x x  - 89521881
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│ │ │ │ +00063140: 2d20 3839 3532 3138 3831 3930 3237 3132  - 89521881902712
│ │ │ │ +00063150: 3078 2078 2020 2b20 2020 2020 2020 207c  0x x  +        |
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│ │ │ │ +00063260: 3337 3236 3539 7820 7820 7820 202b 2032  372659x x x  + 2
│ │ │ │ +00063270: 3830 3330 3830 3431 3237 3436 3039 3678  803080412746096x
│ │ │ │ +00063280: 2078 2020 2b20 3932 3738 3234 3036 3430   x  + 9278240640
│ │ │ │ +00063290: 3130 3230 3078 2078 2020 2020 2020 207c  10200x x       |
│ │ │ │ +000632a0: 0a7c 3220 2020 2020 2020 2020 2020 2020  .|2             
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│ │ │ │ -000633b0: 2020 2b20 3438 3536 3430 3838 3231 3138    + 485640882118
│ │ │ │ -000633c0: 3232 3430 7820 7820 202b 2031 3433 3739  2240x x  + 14379
│ │ │ │ -000633d0: 3138 3336 3033 3033 3630 3078 2078 2020  18360303600x x  
│ │ │ │ -000633e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ +000633a0: 3137 3230 3078 2078 2078 2020 2b20 3438  17200x x x  + 48
│ │ │ │ +000633b0: 3536 3430 3838 3231 3138 3232 3430 7820  56408821182240x 
│ │ │ │ +000633c0: 7820 202b 2031 3433 3739 3138 3336 3033  x  + 14379183603
│ │ │ │ +000633d0: 3033 3630 3078 2078 2020 2020 2020 207c  03600x x       |
│ │ │ │ +000633e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000633f0: 2020 2020 2020 3020 3120 3220 2020 2020        0 1 2     
│ │ │ │ +00063400: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00063410: 2032 2020 2020 2020 2020 2020 2020 2020   2              
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│ │ │ │  00063450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000634a0: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ -000634b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000634c0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
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│ │ │ │ -000634e0: 3434 3732 3834 3078 2078 2020 2d20 3536  4472840x x  - 56
│ │ │ │ -000634f0: 3936 3431 3536 3133 3732 3830 7820 202d  964156137280x  -
│ │ │ │ -00063500: 2034 3132 3332 3833 3539 3435 3732 3830   412328359457280
│ │ │ │ -00063510: 7820 7820 202b 2032 3637 3639 3530 3635  x x  + 267695065
│ │ │ │ -00063520: 3330 3433 357c 0a7c 2020 2020 2020 2020  30435|.|        
│ │ │ │ -00063530: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ -00063540: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00063550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063560: 2030 2033 2020 2020 2020 2020 2020 2020   0 3            
│ │ │ │ -00063570: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00063470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
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│ │ │ │ +000634a0: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ +000634b0: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ +000634c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000634d0: 0a7c 3135 3336 3535 3930 3434 3732 3834  .|15365590447284
│ │ │ │ +000634e0: 3078 2078 2020 2d20 3536 3936 3431 3536  0x x  - 56964156
│ │ │ │ +000634f0: 3133 3732 3830 7820 202d 2034 3132 3332  137280x  - 41232
│ │ │ │ +00063500: 3833 3539 3435 3732 3830 7820 7820 202b  8359457280x x  +
│ │ │ │ +00063510: 2032 3637 3639 3530 3635 3330 3433 357c   26769506530435|
│ │ │ │ +00063520: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00063530: 2020 3120 3220 2020 2020 2020 2020 2020    1 2           
│ │ │ │ +00063540: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00063550: 2020 2020 2020 2020 2020 2030 2033 2020             0 3  
│ │ │ │ +00063560: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063570: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00063580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000635a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000635b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000635c0: 2020 2020 207c 0a7c 2033 2020 2020 2020       |.| 3      
│ │ │ │ -000635d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000635e0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -000635f0: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ -00063600: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ -00063610: 2020 2020 207c 0a7c 7820 202b 2032 3933       |.|x  + 293
│ │ │ │ -00063620: 3037 3830 3230 3238 3534 3631 3678 2078  0780202854616x x
│ │ │ │ -00063630: 2020 2d20 3133 3735 3835 3130 3330 3432    - 137585103042
│ │ │ │ -00063640: 3234 3078 2020 2d20 3131 3832 3534 3234  240x  - 11825424
│ │ │ │ -00063650: 3435 3530 3430 3030 7820 7820 202b 2033  45504000x x  + 3
│ │ │ │ -00063660: 3032 3230 357c 0a7c 2032 2020 2020 2020  02205|.| 2      
│ │ │ │ -00063670: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00063680: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00063690: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -000636a0: 2020 2020 2020 2020 2030 2033 2020 2020           0 3    
│ │ │ │ -000636b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000635b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000635c0: 0a7c 2033 2020 2020 2020 2020 2020 2020  .| 3            
│ │ │ │ +000635d0: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ +000635e0: 2020 2020 2020 2020 2020 2020 2020 3420                4 
│ │ │ │ +000635f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063600: 2020 2033 2020 2020 2020 2020 2020 207c     3           |
│ │ │ │ +00063610: 0a7c 7820 202b 2032 3933 3037 3830 3230  .|x  + 293078020
│ │ │ │ +00063620: 3238 3534 3631 3678 2078 2020 2d20 3133  2854616x x  - 13
│ │ │ │ +00063630: 3735 3835 3130 3330 3432 3234 3078 2020  7585103042240x  
│ │ │ │ +00063640: 2d20 3131 3832 3534 3234 3435 3530 3430  - 11825424455040
│ │ │ │ +00063650: 3030 7820 7820 202b 2033 3032 3230 357c  00x x  + 302205|
│ │ │ │ +00063660: 0a7c 2032 2020 2020 2020 2020 2020 2020  .| 2            
│ │ │ │ +00063670: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ +00063680: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00063690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000636a0: 2020 2030 2033 2020 2020 2020 2020 207c     0 3         |
│ │ │ │ +000636b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000636c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000636d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000636e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000636f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063700: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00063710: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ -00063720: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │ -00063730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063740: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -00063750: 2020 2020 207c 0a7c 3332 3733 3530 3033       |.|32735003
│ │ │ │ -00063760: 3933 3136 3836 3730 7820 7820 202d 2031  93168670x x  - 1
│ │ │ │ -00063770: 3635 3633 3831 3637 3435 3536 3030 7820  65638167455600x 
│ │ │ │ -00063780: 202b 2036 3131 3535 3734 3036 3230 3830   + 6115574062080
│ │ │ │ -00063790: 3078 2078 2020 2d20 3138 3530 3639 3136  0x x  - 18506916
│ │ │ │ -000637a0: 3731 3536 337c 0a7c 2020 2020 2020 2020  71563|.|        
│ │ │ │ -000637b0: 2020 2020 2020 2020 2031 2032 2020 2020           1 2    
│ │ │ │ -000637c0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -000637d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000637e0: 2020 3020 3320 2020 2020 2020 2020 2020    0 3           
│ │ │ │ -000637f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000636f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063700: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00063710: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ +00063720: 2020 2020 2020 2020 2034 2020 2020 2020           4      
│ │ │ │ +00063730: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ +00063740: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063750: 0a7c 3332 3733 3530 3033 3933 3136 3836  .|32735003931686
│ │ │ │ +00063760: 3730 7820 7820 202d 2031 3635 3633 3831  70x x  - 1656381
│ │ │ │ +00063770: 3637 3435 3536 3030 7820 202b 2036 3131  67455600x  + 611
│ │ │ │ +00063780: 3535 3734 3036 3230 3830 3078 2078 2020  55740620800x x  
│ │ │ │ +00063790: 2d20 3138 3530 3639 3136 3731 3536 337c  - 1850691671563|
│ │ │ │ +000637a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000637b0: 2020 2031 2032 2020 2020 2020 2020 2020     1 2          
│ │ │ │ +000637c0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +000637d0: 2020 2020 2020 2020 2020 2020 3020 3320              0 3 
│ │ │ │ +000637e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000637f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00063800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063840: 2020 2020 207c 0a7c 3320 2020 2020 2020       |.|3       
│ │ │ │ -00063850: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ -00063860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063870: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ -00063880: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ -00063890: 2020 2020 207c 0a7c 2020 2d20 3233 3832       |.|  - 2382
│ │ │ │ -000638a0: 3230 3137 3637 3337 3036 3030 7820 7820  201767370600x x 
│ │ │ │ -000638b0: 202b 2031 3333 3133 3433 3038 3335 3438   + 1331343083548
│ │ │ │ -000638c0: 3830 7820 202d 2031 3530 3239 3532 3530  80x  - 150295250
│ │ │ │ -000638d0: 3638 3939 3230 3078 2078 2020 2d20 3237  6899200x x  - 27
│ │ │ │ -000638e0: 3437 3032 337c 0a7c 3220 2020 2020 2020  47023|.|2       
│ │ │ │ -000638f0: 2020 2020 2020 2020 2020 2020 2031 2032               1 2
│ │ │ │ -00063900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063910: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00063920: 2020 2020 2020 2020 3020 3320 2020 2020          0 3     
│ │ │ │ -00063930: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00063830: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063840: 0a7c 3320 2020 2020 2020 2020 2020 2020  .|3             
│ │ │ │ +00063850: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ +00063860: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ +00063870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063880: 2020 3320 2020 2020 2020 2020 2020 207c    3            |
│ │ │ │ +00063890: 0a7c 2020 2d20 3233 3832 3230 3137 3637  .|  - 2382201767
│ │ │ │ +000638a0: 3337 3036 3030 7820 7820 202b 2031 3333  370600x x  + 133
│ │ │ │ +000638b0: 3133 3433 3038 3335 3438 3830 7820 202d  134308354880x  -
│ │ │ │ +000638c0: 2031 3530 3239 3532 3530 3638 3939 3230   150295250689920
│ │ │ │ +000638d0: 3078 2078 2020 2d20 3237 3437 3032 337c  0x x  - 2747023|
│ │ │ │ +000638e0: 0a7c 3220 2020 2020 2020 2020 2020 2020  .|2             
│ │ │ │ +000638f0: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ +00063900: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00063910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063920: 2020 3020 3320 2020 2020 2020 2020 207c    0 3          |
│ │ │ │ +00063930: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00063940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063980: 2020 2020 207c 0a7c 3320 2020 2020 2020       |.|3       
│ │ │ │ -00063990: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ -000639a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000639b0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ -000639c0: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ -000639d0: 2020 2020 207c 0a7c 2020 2d20 3535 3832       |.|  - 5582
│ │ │ │ -000639e0: 3933 3737 3031 3135 3637 3630 7820 7820  937701156760x x 
│ │ │ │ -000639f0: 202b 2033 3032 3631 3834 3134 3930 3136   + 3026184149016
│ │ │ │ -00063a00: 3030 7820 202b 2036 3730 3338 3431 3830  00x  + 670384180
│ │ │ │ -00063a10: 3831 3135 3230 3078 2078 2020 2b20 3837  8115200x x  + 87
│ │ │ │ -00063a20: 3138 3737 357c 0a7c 3220 2020 2020 2020  18775|.|2       
│ │ │ │ -00063a30: 2020 2020 2020 2020 2020 2020 2031 2032               1 2
│ │ │ │ -00063a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063a50: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00063a60: 2020 2020 2020 2020 3020 3320 2020 2020          0 3     
│ │ │ │ -00063a70: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00063970: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063980: 0a7c 3320 2020 2020 2020 2020 2020 2020  .|3             
│ │ │ │ +00063990: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ +000639a0: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ +000639b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000639c0: 2020 3320 2020 2020 2020 2020 2020 207c    3            |
│ │ │ │ +000639d0: 0a7c 2020 2d20 3535 3832 3933 3737 3031  .|  - 5582937701
│ │ │ │ +000639e0: 3135 3637 3630 7820 7820 202b 2033 3032  156760x x  + 302
│ │ │ │ +000639f0: 3631 3834 3134 3930 3136 3030 7820 202b  618414901600x  +
│ │ │ │ +00063a00: 2036 3730 3338 3431 3830 3831 3135 3230   670384180811520
│ │ │ │ +00063a10: 3078 2078 2020 2b20 3837 3138 3737 357c  0x x  + 8718775|
│ │ │ │ +00063a20: 0a7c 3220 2020 2020 2020 2020 2020 2020  .|2             
│ │ │ │ +00063a30: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ +00063a40: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00063a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063a60: 2020 3020 3320 2020 2020 2020 2020 207c    0 3          |
│ │ │ │ +00063a70: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00063a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063ac0: 2d2d 2d2d 2d7c 0a7c 2020 2032 2020 2020  -----|.|   2    
│ │ │ │ +00063ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00063ac0: 0a7c 2020 2032 2020 2020 2020 2020 2020  .|   2          
│ │ │ │  00063ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063ae0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -00063af0: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ -00063b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063b10: 2020 2020 207c 0a7c 3230 7820 7820 7820       |.|20x x x 
│ │ │ │ -00063b20: 202b 2031 3837 3739 3837 3332 3438 3438   + 1877987324848
│ │ │ │ -00063b30: 3332 3078 2078 2078 2020 2d20 3133 3338  320x x x  - 1338
│ │ │ │ -00063b40: 3536 3836 3332 3733 3732 3830 7820 7820  568632737280x x 
│ │ │ │ -00063b50: 202d 2031 3533 3831 3031 3139 3239 3735   - 1538101192975
│ │ │ │ -00063b60: 3230 3030 787c 0a7c 2020 2030 2031 2033  2000x|.|   0 1 3
│ │ │ │ -00063b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063b80: 2020 2020 3020 3120 3320 2020 2020 2020      0 1 3       
│ │ │ │ -00063b90: 2020 2020 2020 2020 2020 2020 2031 2033               1 3
│ │ │ │ -00063ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063bb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00063ae0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00063af0: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +00063b00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063b10: 0a7c 3230 7820 7820 7820 202b 2031 3837  .|20x x x  + 187
│ │ │ │ +00063b20: 3739 3837 3332 3438 3438 3332 3078 2078  7987324848320x x
│ │ │ │ +00063b30: 2078 2020 2d20 3133 3338 3536 3836 3332   x  - 1338568632
│ │ │ │ +00063b40: 3733 3732 3830 7820 7820 202d 2031 3533  737280x x  - 153
│ │ │ │ +00063b50: 3831 3031 3139 3239 3735 3230 3030 787c  81011929752000x|
│ │ │ │ +00063b60: 0a7c 2020 2030 2031 2033 2020 2020 2020  .|   0 1 3      
│ │ │ │ +00063b70: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00063b80: 3120 3320 2020 2020 2020 2020 2020 2020  1 3             
│ │ │ │ +00063b90: 2020 2020 2020 2031 2033 2020 2020 2020         1 3      
│ │ │ │ +00063ba0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063bb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00063bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063c00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00063c10: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -00063c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063c30: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00063c40: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ -00063c50: 2020 2020 207c 0a7c 3830 3133 3133 3738       |.|80131378
│ │ │ │ -00063c60: 3536 3078 2078 2078 2020 2d20 3432 3535  560x x x  - 4255
│ │ │ │ -00063c70: 3031 3734 3431 3237 3539 3136 3078 2078  0174412759160x x
│ │ │ │ -00063c80: 2078 2020 2b20 3132 3535 3230 3635 3731   x  + 1255206571
│ │ │ │ -00063c90: 3434 3632 3430 3078 2078 2020 2b20 3130  4462400x x  + 10
│ │ │ │ -00063ca0: 3834 3530 347c 0a7c 2020 2020 2020 2020  84504|.|        
│ │ │ │ -00063cb0: 2020 2020 3020 3120 3320 2020 2020 2020      0 1 3       
│ │ │ │ -00063cc0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00063cd0: 3120 3320 2020 2020 2020 2020 2020 2020  1 3             
│ │ │ │ -00063ce0: 2020 2020 2020 2020 3120 3320 2020 2020          1 3     
│ │ │ │ -00063cf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00063bf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063c00: 0a7c 2020 2020 2020 2020 2020 2020 3220  .|            2 
│ │ │ │ +00063c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063c20: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00063c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063c40: 2020 3320 2020 2020 2020 2020 2020 207c    3            |
│ │ │ │ +00063c50: 0a7c 3830 3133 3133 3738 3536 3078 2078  .|80131378560x x
│ │ │ │ +00063c60: 2078 2020 2d20 3432 3535 3031 3734 3431   x  - 4255017441
│ │ │ │ +00063c70: 3237 3539 3136 3078 2078 2078 2020 2b20  2759160x x x  + 
│ │ │ │ +00063c80: 3132 3535 3230 3635 3731 3434 3632 3430  1255206571446240
│ │ │ │ +00063c90: 3078 2078 2020 2b20 3130 3834 3530 347c  0x x  + 1084504|
│ │ │ │ +00063ca0: 0a7c 2020 2020 2020 2020 2020 2020 3020  .|            0 
│ │ │ │ +00063cb0: 3120 3320 2020 2020 2020 2020 2020 2020  1 3             
│ │ │ │ +00063cc0: 2020 2020 2020 2020 3020 3120 3320 2020          0 1 3   
│ │ │ │ +00063cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063ce0: 2020 3120 3320 2020 2020 2020 2020 207c    1 3          |
│ │ │ │ +00063cf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00063d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063d40: 2020 2020 207c 0a7c 2020 2020 2032 2020       |.|     2  
│ │ │ │ +00063d30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063d40: 0a7c 2020 2020 2032 2020 2020 2020 2020  .|     2        
│ │ │ │  00063d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063d60: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -00063d70: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ -00063d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063d90: 2020 2020 207c 0a7c 3339 3230 7820 7820       |.|3920x x 
│ │ │ │ -00063da0: 7820 202b 2032 3839 3031 3530 3939 3537  x  + 28901509957
│ │ │ │ -00063db0: 3435 3434 3078 2078 2078 2020 2b20 3235  45440x x x  + 25
│ │ │ │ -00063dc0: 3531 3635 3737 3435 3736 3732 3030 7820  51657745767200x 
│ │ │ │ -00063dd0: 7820 202b 2031 3535 3537 3837 3134 3635  x  + 15557871465
│ │ │ │ -00063de0: 3135 3834 307c 0a7c 2020 2020 2030 2031  15840|.|     0 1
│ │ │ │ -00063df0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -00063e00: 2020 2020 2020 3020 3120 3320 2020 2020        0 1 3     
│ │ │ │ -00063e10: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00063e20: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -00063e30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00063d60: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00063d70: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ +00063d80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063d90: 0a7c 3339 3230 7820 7820 7820 202b 2032  .|3920x x x  + 2
│ │ │ │ +00063da0: 3839 3031 3530 3939 3537 3435 3434 3078  890150995745440x
│ │ │ │ +00063db0: 2078 2078 2020 2b20 3235 3531 3635 3737   x x  + 25516577
│ │ │ │ +00063dc0: 3435 3736 3732 3030 7820 7820 202b 2031  45767200x x  + 1
│ │ │ │ +00063dd0: 3535 3537 3837 3134 3635 3135 3834 307c  555787146515840|
│ │ │ │ +00063de0: 0a7c 2020 2020 2030 2031 2033 2020 2020  .|     0 1 3    
│ │ │ │ +00063df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063e00: 3020 3120 3320 2020 2020 2020 2020 2020  0 1 3           
│ │ │ │ +00063e10: 2020 2020 2020 2020 2031 2033 2020 2020           1 3    
│ │ │ │ +00063e20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063e30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00063e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063e80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00063e90: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00063ea0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00063e70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063e80: 0a7c 2020 2020 2020 2020 2020 2032 2020  .|           2  
│ │ │ │ +00063e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063ea0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  00063eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063ec0: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ -00063ed0: 2020 2020 207c 0a7c 3835 3137 3733 3731       |.|85177371
│ │ │ │ -00063ee0: 3230 7820 7820 7820 202b 2033 3631 3933  20x x x  + 36193
│ │ │ │ -00063ef0: 3134 3339 3939 3637 3339 3230 7820 7820  143999673920x x 
│ │ │ │ -00063f00: 7820 202d 2031 3035 3039 3039 3437 3039  x  - 10509094709
│ │ │ │ -00063f10: 3437 3035 3630 7820 7820 202d 2038 3439  470560x x  - 849
│ │ │ │ -00063f20: 3837 3035 327c 0a7c 2020 2020 2020 2020  87052|.|        
│ │ │ │ -00063f30: 2020 2030 2031 2033 2020 2020 2020 2020     0 1 3        
│ │ │ │ -00063f40: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
│ │ │ │ -00063f50: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -00063f60: 2020 2020 2020 2031 2033 2020 2020 2020         1 3      
│ │ │ │ -00063f70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00063ec0: 2033 2020 2020 2020 2020 2020 2020 207c   3             |
│ │ │ │ +00063ed0: 0a7c 3835 3137 3733 3731 3230 7820 7820  .|8517737120x x 
│ │ │ │ +00063ee0: 7820 202b 2033 3631 3933 3134 3339 3939  x  + 36193143999
│ │ │ │ +00063ef0: 3637 3339 3230 7820 7820 7820 202d 2031  673920x x x  - 1
│ │ │ │ +00063f00: 3035 3039 3039 3437 3039 3437 3035 3630  0509094709470560
│ │ │ │ +00063f10: 7820 7820 202d 2038 3439 3837 3035 327c  x x  - 84987052|
│ │ │ │ +00063f20: 0a7c 2020 2020 2020 2020 2020 2030 2031  .|           0 1
│ │ │ │ +00063f30: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +00063f40: 2020 2020 2020 2030 2031 2033 2020 2020         0 1 3    
│ │ │ │ +00063f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063f60: 2031 2033 2020 2020 2020 2020 2020 207c   1 3           |
│ │ │ │ +00063f70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00063f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063fc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00063fd0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00063fe0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -00063ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064000: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -00064010: 2020 2020 207c 0a7c 3031 3737 3234 3830       |.|01772480
│ │ │ │ -00064020: 7820 7820 7820 202d 2031 3734 3930 3630  x x x  - 1749060
│ │ │ │ -00064030: 3832 3033 3638 3936 3078 2078 2078 2020  820368960x x x  
│ │ │ │ -00064040: 2d20 3132 3235 3239 3139 3536 3733 3932  - 12252919567392
│ │ │ │ -00064050: 3078 2078 2020 2d20 3333 3136 3531 3437  0x x  - 33165147
│ │ │ │ -00064060: 3237 3933 397c 0a7c 2020 2020 2020 2020  27939|.|        
│ │ │ │ -00064070: 2030 2031 2033 2020 2020 2020 2020 2020   0 1 3          
│ │ │ │ -00064080: 2020 2020 2020 2020 2020 3020 3120 3320            0 1 3 
│ │ │ │ -00064090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000640a0: 2020 3120 3320 2020 2020 2020 2020 2020    1 3           
│ │ │ │ -000640b0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00063fb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00063fc0: 0a7c 2020 2020 2020 2020 2032 2020 2020  .|         2    
│ │ │ │ +00063fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063fe0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00063ff0: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ +00064000: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00064010: 0a7c 3031 3737 3234 3830 7820 7820 7820  .|01772480x x x 
│ │ │ │ +00064020: 202d 2031 3734 3930 3630 3832 3033 3638   - 1749060820368
│ │ │ │ +00064030: 3936 3078 2078 2078 2020 2d20 3132 3235  960x x x  - 1225
│ │ │ │ +00064040: 3239 3139 3536 3733 3932 3078 2078 2020  29195673920x x  
│ │ │ │ +00064050: 2d20 3333 3136 3531 3437 3237 3933 397c  - 3316514727939|
│ │ │ │ +00064060: 0a7c 2020 2020 2020 2020 2030 2031 2033  .|         0 1 3
│ │ │ │ +00064070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064080: 2020 2020 3020 3120 3320 2020 2020 2020      0 1 3       
│ │ │ │ +00064090: 2020 2020 2020 2020 2020 2020 3120 3320              1 3 
│ │ │ │ +000640a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000640b0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  000640c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000640d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000640e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000640f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064100: 2d2d 2d2d 2d7c 0a7c 3220 2020 2020 2020  -----|.|2       
│ │ │ │ +000640f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00064100: 0a7c 3220 2020 2020 2020 2020 2020 2020  .|2             
│ │ │ │  00064110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064150: 2020 2020 207c 0a7c 2078 2078 2020 2b20       |.| x x  + 
│ │ │ │ -00064160: 3132 3734 3138 3730 3836 3834 3434 3533  1274187086844453
│ │ │ │ -00064170: 3678 2078 2078 2078 2020 2d20 2020 2020  6x x x x  -     
│ │ │ │ +00064140: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00064150: 0a7c 2078 2078 2020 2b20 3132 3734 3138  .| x x  + 127418
│ │ │ │ +00064160: 3730 3836 3834 3434 3533 3678 2078 2078  70868444536x x x
│ │ │ │ +00064170: 2078 2020 2d20 2020 2020 2020 2020 2020   x  -           
│ │ │ │  00064180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000641a0: 2020 2020 207c 0a7c 3020 3220 3320 2020       |.|0 2 3   
│ │ │ │ -000641b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000641c0: 2020 3020 3120 3220 3320 2020 2020 2020    0 1 2 3       
│ │ │ │ +00064190: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000641a0: 0a7c 3020 3220 3320 2020 2020 2020 2020  .|0 2 3         
│ │ │ │ +000641b0: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ +000641c0: 3220 3320 2020 2020 2020 2020 2020 2020  2 3             
│ │ │ │  000641d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000641e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000641f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000641e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000641f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00064200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00064250: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00064230: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00064240: 0a7c 2020 2020 2020 2020 2020 2032 2020  .|           2  
│ │ │ │ +00064250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064260: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00064280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064290: 2020 2020 207c 0a7c 3734 3933 3330 3234       |.|74933024
│ │ │ │ -000642a0: 3030 7820 7820 7820 202b 2037 3337 3531  00x x x  + 73751
│ │ │ │ -000642b0: 3632 3638 3935 3130 3030 3078 2020 2020  62689510000x    
│ │ │ │ +00064280: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00064290: 0a7c 3734 3933 3330 3234 3030 7820 7820  .|7493302400x x 
│ │ │ │ +000642a0: 7820 202b 2037 3337 3531 3632 3638 3935  x  + 73751626895
│ │ │ │ +000642b0: 3130 3030 3078 2020 2020 2020 2020 2020  10000x          
│ │ │ │  000642c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000642d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000642e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000642f0: 2020 2030 2032 2033 2020 2020 2020 2020     0 2 3        
│ │ │ │ +000642d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000642e0: 0a7c 2020 2020 2020 2020 2020 2030 2032  .|           0 2
│ │ │ │ +000642f0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  00064300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064310: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00064330: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00064320: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00064330: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00064340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064380: 2020 2020 207c 0a7c 2020 3220 2020 2020       |.|  2     
│ │ │ │ +00064370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00064380: 0a7c 2020 3220 2020 2020 2020 2020 2020  .|  2           
│ │ │ │  00064390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000643a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000643b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000643c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000643d0: 2020 2020 207c 0a7c 3078 2078 2078 2020       |.|0x x x  
│ │ │ │ -000643e0: 2d20 3131 3735 3139 3337 3134 3035 3039  - 11751937140509
│ │ │ │ -000643f0: 3636 3478 2078 2078 2078 2020 2020 2020  664x x x x      
│ │ │ │ +000643c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000643d0: 0a7c 3078 2078 2078 2020 2d20 3131 3735  .|0x x x  - 1175
│ │ │ │ +000643e0: 3139 3337 3134 3035 3039 3636 3478 2078  1937140509664x x
│ │ │ │ +000643f0: 2078 2078 2020 2020 2020 2020 2020 2020   x x            
│ │ │ │  00064400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064420: 2020 2020 207c 0a7c 2020 3020 3220 3320       |.|  0 2 3 
│ │ │ │ -00064430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064440: 2020 2020 3020 3120 3220 3320 2020 2020      0 1 2 3     
│ │ │ │ +00064410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00064420: 0a7c 2020 3020 3220 3320 2020 2020 2020  .|  0 2 3       
│ │ │ │ +00064430: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00064440: 3120 3220 3320 2020 2020 2020 2020 2020  1 2 3           
│ │ │ │  00064450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064470: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00064460: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00064470: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00064480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000644a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000644b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000644c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000644d0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000644b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000644c0: 0a7c 2020 2020 2020 2020 2032 2020 2020  .|         2    
│ │ │ │ +000644d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000644e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000644f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064510: 2020 2020 207c 0a7c 3138 3036 3939 3230       |.|18069920
│ │ │ │ -00064520: 7820 7820 7820 202b 2031 3434 3538 3537  x x x  + 1445857
│ │ │ │ -00064530: 3139 3830 3634 3737 3630 7820 2020 2020  1980647760x     
│ │ │ │ +00064500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00064510: 0a7c 3138 3036 3939 3230 7820 7820 7820  .|18069920x x x 
│ │ │ │ +00064520: 202b 2031 3434 3538 3537 3139 3830 3634   + 1445857198064
│ │ │ │ +00064530: 3737 3630 7820 2020 2020 2020 2020 2020  7760x           
│ │ │ │  00064540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064560: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00064570: 2030 2032 2033 2020 2020 2020 2020 2020   0 2 3          
│ │ │ │ -00064580: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ +00064550: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00064560: 0a7c 2020 2020 2020 2020 2030 2032 2033  .|         0 2 3
│ │ │ │ +00064570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064580: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │  00064590: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000645b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000645a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000645b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000645c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000645d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000645e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000645f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064600: 2020 2020 207c 0a7c 2020 2020 3220 2020       |.|    2   
│ │ │ │ +000645f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00064600: 0a7c 2020 2020 3220 2020 2020 2020 2020  .|    2         
│ │ │ │  00064610: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00064630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064650: 2020 2020 207c 0a7c 3834 3078 2078 2078       |.|840x x x
│ │ │ │ -00064660: 2020 2d20 3530 3137 3932 3836 3837 3830    - 501792868780
│ │ │ │ -00064670: 3938 3030 7820 7820 7820 7820 2020 2020  9800x x x x     
│ │ │ │ +00064640: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00064650: 0a7c 3834 3078 2078 2078 2020 2d20 3530  .|840x x x  - 50
│ │ │ │ +00064660: 3137 3932 3836 3837 3830 3938 3030 7820  17928687809800x 
│ │ │ │ +00064670: 7820 7820 7820 2020 2020 2020 2020 2020  x x x           
│ │ │ │  00064680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000646a0: 2020 2020 207c 0a7c 2020 2020 3020 3220       |.|    0 2 
│ │ │ │ -000646b0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -000646c0: 2020 2020 2030 2031 2032 2033 2020 2020       0 1 2 3    
│ │ │ │ +00064690: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000646a0: 0a7c 2020 2020 3020 3220 3320 2020 2020  .|    0 2 3     
│ │ │ │ +000646b0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +000646c0: 2031 2032 2033 2020 2020 2020 2020 2020   1 2 3          
│ │ │ │  000646d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000646e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000646f0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +000646e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000646f0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00064700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00064710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00064720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064740: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
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│ │ │ │ -00064760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064770: 2020 2032 2020 2020 2020 2020 2020 2020     2            
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│ │ │ │ -000647a0: 3238 3637 3432 3536 7820 7820 7820 202d  28674256x x x  -
│ │ │ │ -000647b0: 2033 3935 3537 3436 3237 3432 3730 3536   395574627427056
│ │ │ │ -000647c0: 7820 7820 7820 202b 2033 3633 3933 3636  x x x  + 3639366
│ │ │ │ -000647d0: 3630 3435 3535 3932 3078 2078 2078 2020  604555920x x x  
│ │ │ │ -000647e0: 2d20 3633 367c 0a7c 2020 2020 2020 2020  - 636|.|        
│ │ │ │ -000647f0: 2020 2020 2020 2020 2031 2032 2033 2020           1 2 3  
│ │ │ │ -00064800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064810: 2030 2032 2033 2020 2020 2020 2020 2020   0 2 3          
│ │ │ │ -00064820: 2020 2020 2020 2020 2020 3120 3220 3320            1 2 3 
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│ │ │ │ +00064730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
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│ │ │ │ +00064780: 2020 2020 2020 3220 2020 2020 2020 207c        2        |
│ │ │ │ +00064790: 0a7c 3431 3135 3231 3435 3238 3637 3432  .|41152145286742
│ │ │ │ +000647a0: 3536 7820 7820 7820 202d 2033 3935 3537  56x x x  - 39557
│ │ │ │ +000647b0: 3436 3237 3432 3730 3536 7820 7820 7820  4627427056x x x 
│ │ │ │ +000647c0: 202b 2033 3633 3933 3636 3630 3435 3535   + 3639366604555
│ │ │ │ +000647d0: 3932 3078 2078 2078 2020 2d20 3633 367c  920x x x  - 636|
│ │ │ │ +000647e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000647f0: 2020 2031 2032 2033 2020 2020 2020 2020     1 2 3        
│ │ │ │ +00064800: 2020 2020 2020 2020 2020 2030 2032 2033             0 2 3
│ │ │ │ +00064810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064820: 2020 2020 3120 3220 3320 2020 2020 207c      1 2 3      |
│ │ │ │ +00064830: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00064840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064880: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00064890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000648a0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -000648b0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -000648c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000648d0: 2020 2020 207c 0a7c 2078 2078 2078 2020       |.| x x x  
│ │ │ │ -000648e0: 2b20 3539 3236 3839 3733 3338 3631 3638  + 59268973386168
│ │ │ │ -000648f0: 3078 2078 2078 2020 2b20 3432 3331 3430  0x x x  + 423140
│ │ │ │ -00064900: 3838 3035 3935 3630 3830 7820 7820 7820  8805956080x x x 
│ │ │ │ -00064910: 202d 2031 3334 3837 3838 3338 3937 3230   - 1348788389720
│ │ │ │ -00064920: 3338 3536 787c 0a7c 3020 3120 3220 3320  3856x|.|0 1 2 3 
│ │ │ │ -00064930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064940: 2020 3120 3220 3320 2020 2020 2020 2020    1 2 3         
│ │ │ │ -00064950: 2020 2020 2020 2020 2020 2030 2032 2033             0 2 3
│ │ │ │ -00064960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064970: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00064870: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00064880: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00064890: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +000648a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000648b0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +000648c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000648d0: 0a7c 2078 2078 2078 2020 2b20 3539 3236  .| x x x  + 5926
│ │ │ │ +000648e0: 3839 3733 3338 3631 3638 3078 2078 2078  89733861680x x x
│ │ │ │ +000648f0: 2020 2b20 3432 3331 3430 3838 3035 3935    + 423140880595
│ │ │ │ +00064900: 3630 3830 7820 7820 7820 202d 2031 3334  6080x x x  - 134
│ │ │ │ +00064910: 3837 3838 3338 3937 3230 3338 3536 787c  87883897203856x|
│ │ │ │ +00064920: 0a7c 3020 3120 3220 3320 2020 2020 2020  .|0 1 2 3       
│ │ │ │ +00064930: 2020 2020 2020 2020 2020 2020 3120 3220              1 2 
│ │ │ │ +00064940: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +00064950: 2020 2020 2030 2032 2033 2020 2020 2020       0 2 3      
│ │ │ │ +00064960: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00064970: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00064980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000649d0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +000649b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000649c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000649d0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  000649e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000649f0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -00064a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064a10: 3220 2020 207c 0a7c 2b20 3732 3238 3032  2    |.|+ 722802
│ │ │ │ -00064a20: 3836 3938 3730 3433 3234 7820 7820 7820  8698704324x x x 
│ │ │ │ -00064a30: 202b 2032 3630 3235 3332 3634 3439 3831   + 2602532644981
│ │ │ │ -00064a40: 3932 3078 2078 2078 2020 2d20 3132 3334  920x x x  - 1234
│ │ │ │ -00064a50: 3933 3832 3635 3733 3138 3730 3078 2078  9382657318700x x
│ │ │ │ -00064a60: 2078 2020 2b7c 0a7c 2020 2020 2020 2020   x  +|.|        
│ │ │ │ -00064a70: 2020 2020 2020 2020 2020 2031 2032 2033             1 2 3
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│ │ │ │ -00064a90: 2020 2020 3020 3220 3320 2020 2020 2020      0 2 3       
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│ │ │ │ -00064ab0: 3220 3320 207c 0a7c 2020 2020 2020 2020  2 3  |.|        
│ │ │ │ +000649f0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00064a00: 2020 2020 2020 2020 2020 3220 2020 207c            2    |
│ │ │ │ +00064a10: 0a7c 2b20 3732 3238 3032 3836 3938 3730  .|+ 722802869870
│ │ │ │ +00064a20: 3433 3234 7820 7820 7820 202b 2032 3630  4324x x x  + 260
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│ │ │ │ +00064a40: 2078 2020 2d20 3132 3334 3933 3832 3635   x  - 1234938265
│ │ │ │ +00064a50: 3733 3138 3730 3078 2078 2078 2020 2b7c  7318700x x x  +|
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│ │ │ │ +00064a80: 2020 2020 2020 2020 2020 2020 2020 3020                0 
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│ │ │ │ -00064b80: 3338 3032 3635 3631 3234 7820 7820 7820  3802656124x x x 
│ │ │ │ -00064b90: 202b 2031 3032 3731 3435 3532 3130 3039   + 1027145521009
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│ │ │ │ -00064c90: 2020 2020 207c 0a7c 202b 2033 3236 3137       |.| + 32617
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│ │ │ │ -00064cd0: 3739 3235 3134 3935 3836 3430 7820 7820  792514958640x x 
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│ │ │ │ +00064cb0: 3437 3335 3037 3633 3938 3431 3230 7820  47350763984120x 
│ │ │ │ +00064cc0: 7820 7820 202b 2032 3436 3739 3235 3134  x x  + 246792514
│ │ │ │ +00064cd0: 3935 3836 3430 7820 7820 7820 202b 207c  958640x x x  + |
│ │ │ │ +00064ce0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ +00064e00: 3535 3639 3037 3532 3630 3830 7820 7820  556907526080x x 
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│ │ │ │ +00064e50: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
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│ │ │ │ -00064f50: 3232 3539 3636 3430 3832 3836 3430 7820  22596640828640x 
│ │ │ │ -00064f60: 7820 7820 207c 0a7c 3120 3220 3320 2020  x x  |.|1 2 3   
│ │ │ │ -00064f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00064f20: 3034 3031 3138 3838 3078 2078 2020 2d20  040118880x x  - 
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│ │ │ │ +00064f50: 3430 3832 3836 3430 7820 7820 7820 207c  40828640x x x  |
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│ │ │ │ +00065080: 3434 3330 3139 3934 3432 3339 3034 3078  443019944239040x
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│ │ │ │ +00065130: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ -000651b0: 7820 202b 2032 3237 3037 3732 3732 3436  x  + 22707727246
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│ │ │ │ -000651d0: 3335 3134 3938 3734 3434 3830 3078 2078  3514987444800x x
│ │ │ │ -000651e0: 2078 2020 2d7c 0a7c 3120 3220 3320 2020   x  -|.|1 2 3   
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│ │ │ │ +00065170: 2032 2032 2020 2020 2020 2020 2020 2020   2 2            
│ │ │ │ +00065180: 2020 2020 2020 2020 2020 2020 3220 207c              2  |
│ │ │ │ +00065190: 0a7c 2078 2078 2020 2d20 3436 3938 3131  .| x x  - 469811
│ │ │ │ +000651a0: 3430 3838 3935 3336 7820 7820 202b 2032  40889536x x  + 2
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│ │ │ │ +000651d0: 3734 3434 3830 3078 2078 2078 2020 2d7c  7444800x x x  -|
│ │ │ │ +000651e0: 0a7c 3120 3220 3320 2020 2020 2020 2020  .|1 2 3         
│ │ │ │ +000651f0: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
│ │ │ │ +00065200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065210: 2030 2033 2020 2020 2020 2020 2020 2020   0 3            
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│ │ │ │ -000652a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000652b0: 3220 3220 2020 2020 2020 2020 2020 2020  2 2             
│ │ │ │ -000652c0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
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│ │ │ │ -000652e0: 3630 3438 3537 3630 7820 7820 202d 2033  60485760x x  - 3
│ │ │ │ -000652f0: 3332 3937 3233 3335 3138 3038 3030 3078  329723351808000x
│ │ │ │ -00065300: 2078 2020 2b20 3636 3237 3131 3439 3733   x  + 6627114973
│ │ │ │ -00065310: 3438 3332 3078 2078 2078 2020 2b20 3132  48320x x x  + 12
│ │ │ │ -00065320: 3036 3035 377c 0a7c 2020 2020 2020 2020  06057|.|        
│ │ │ │ -00065330: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
│ │ │ │ -00065340: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000652a0: 2020 2020 2020 2020 2020 3220 3220 2020            2 2   
│ │ │ │ +000652b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000652d0: 0a7c 3131 3630 3839 3635 3630 3438 3537  .|11608965604857
│ │ │ │ +000652e0: 3630 7820 7820 202d 2033 3332 3937 3233  60x x  - 3329723
│ │ │ │ +000652f0: 3335 3138 3038 3030 3078 2078 2020 2b20  351808000x x  + 
│ │ │ │ +00065300: 3636 3237 3131 3439 3733 3438 3332 3078  662711497348320x
│ │ │ │ +00065310: 2078 2078 2020 2b20 3132 3036 3035 377c   x x  + 1206057|
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│ │ │ │ +00065340: 2020 2020 2020 2020 2020 3020 3320 2020            0 3   
│ │ │ │ +00065350: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00065370: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00065380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000653a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000653c0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2032  -----|.|       2
│ │ │ │ -000653d0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
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│ │ │ │ +000656c0: 3336 3238 3730 3839 3235 3678 2078 2078  36287089256x x x
│ │ │ │ +000656d0: 2020 2b20 3135 3632 3832 3031 3933 337c    + 15628201933|
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│ │ │ │ -000657d0: 2020 2020 207c 0a7c 2037 3330 3234 3735       |.| 7302475
│ │ │ │ -000657e0: 3939 3032 3631 3736 3078 2078 2020 2b20  990261760x x  + 
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│ │ │ │ +000657f0: 3837 3932 3932 3533 3630 7820 7820 7820  8792925360x x x 
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│ │ │ │ -00065a80: 3130 3533 3935 3230 7820 7820 202d 2039  10539520x x  - 9
│ │ │ │ -00065a90: 3133 3633 3833 3434 3230 3939 3278 2078  1363834420992x x
│ │ │ │ -00065aa0: 2020 2d20 317c 0a7c 2020 2032 2033 2020    - 1|.|   2 3  
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│ │ │ │ -00065bb0: 3138 3837 3430 3437 3732 3939 3230 7820  18874047729920x 
│ │ │ │ -00065bc0: 7820 202d 2032 3537 3139 3538 3237 3232  x  - 25719582722
│ │ │ │ -00065bd0: 3630 3136 3078 2078 2020 2b20 3236 3030  60160x x  + 2600
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│ │ │ │ -00065c10: 2033 2020 2020 2020 2020 2020 2020 2020   3              
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│ │ │ │ +00065bb0: 3437 3732 3939 3230 7820 7820 202d 2032  47729920x x  - 2
│ │ │ │ +00065bc0: 3537 3139 3538 3237 3232 3630 3136 3078  571958272260160x
│ │ │ │ +00065bd0: 2078 2020 2b20 3236 3030 3431 3838 347c   x  + 260041884|
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│ │ │ │ -00065c90: 3220 2020 2020 2020 2020 2020 2020 2020  2               
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│ │ │ │ -00065cd0: 2020 2033 207c 0a7c 3534 3839 3678 2078     3 |.|54896x x
│ │ │ │ -00065ce0: 2020 2d20 3232 3433 3730 3338 3338 3139    - 224370383819
│ │ │ │ -00065cf0: 3930 3430 7820 7820 202b 2038 3232 3238  9040x x  + 82228
│ │ │ │ -00065d00: 3737 3331 3638 3836 3430 7820 7820 202d  7731688640x x  -
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│ │ │ │ -00065d20: 7820 7820 207c 0a7c 2020 2020 2020 3220  x x  |.|      2 
│ │ │ │ -00065d30: 3320 2020 2020 2020 2020 2020 2020 2020  3               
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│ │ │ │ -00065d50: 2020 2020 2020 2020 2020 2031 2033 2020             1 3  
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│ │ │ │ +00065c70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +00065c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00065f90: 3433 3634 3838 3137 3736 3030 7820 7820  436488177600x x 
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│ │ │ │ +00065f90: 3137 3736 3030 7820 7820 202d 2031 317c  177600x x  - 11|
│ │ │ │ +00065fa0: 0a7c 2020 2032 2033 2020 2020 2020 2020  .|   2 3        
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│ │ │ │ +000660d0: 7820 7820 202d 2032 3433 3235 3737 367c  x x  - 24325776|
│ │ │ │ +000660e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ -00066330: 3339 3830 3834 3834 3335 3230 7820 7820  398084843520x x 
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│ │ │ │ +00066310: 0a7c 2d20 3830 3334 3236 3032 3039 3932  .|- 803426020992
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│ │ │ │ -00066480: 3532 3139 3039 3736 3078 2078 2020 2b20  521909760x x  + 
│ │ │ │ -00066490: 3137 3233 3136 3030 3337 3837 3935 3034  1723160037879504
│ │ │ │ -000664a0: 3078 2078 207c 0a7c 2020 2020 2020 2032  0x x |.|       2
│ │ │ │ -000664b0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
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│ │ │ │ +00066440: 2020 2020 2020 2020 2020 2020 3220 207c              2  |
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│ │ │ │ +00066460: 3136 3135 3136 3830 3638 3438 3030 7820  16151680684800x 
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│ │ │ │ +00066490: 3030 3337 3837 3935 3034 3078 2078 207c  00378795040x x |
│ │ │ │ +000664a0: 0a7c 2020 2020 2020 2032 2033 2020 2020  .|       2 3    
│ │ │ │ +000664b0: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ +000664c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000665b0: 3037 3334 3832 3234 3030 7820 7820 202d  0734822400x x  -
│ │ │ │ -000665c0: 2035 3736 3935 3231 3433 3835 3133 3238   576952143851328
│ │ │ │ -000665d0: 3030 7820 7820 7820 202b 2034 3134 3130  00x x x  + 41410
│ │ │ │ -000665e0: 3433 3730 377c 0a7c 2020 2020 2020 2020  43707|.|        
│ │ │ │ -000665f0: 2020 2020 2020 3320 2020 2020 2020 2020        3         
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│ │ │ │ +00066570: 2020 2020 2020 2020 2020 2020 2032 2020               2  
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│ │ │ │ +000665a0: 2020 2b20 3939 3037 3434 3037 3334 3832    + 990744073482
│ │ │ │ +000665b0: 3234 3030 7820 7820 202d 2035 3736 3935  2400x x  - 57695
│ │ │ │ +000665c0: 3231 3433 3835 3133 3238 3030 7820 7820  214385132800x x 
│ │ │ │ +000665d0: 7820 202b 2034 3134 3130 3433 3730 377c  x  + 4141043707|
│ │ │ │ +000665e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000665f0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +00066600: 2020 2020 2030 2034 2020 2020 2020 2020       0 4        
│ │ │ │ +00066610: 2020 2020 2020 2020 2020 2020 2030 2031               0 1
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│ │ │ │  00066660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00066d20: 2078 2020 2b20 3436 3433 3530 3030 3630   x  + 4643500060
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│ │ │ │ -00066e60: 3138 3435 3536 3933 3532 3078 2078 2078  18455693520x x x
│ │ │ │ -00066e70: 2078 2020 2b20 3834 3737 3739 3431 3732   x  + 8477794172
│ │ │ │ -00066e80: 3832 3932 3430 7820 7820 7820 202b 2031  829240x x x  + 1
│ │ │ │ -00066e90: 3736 3535 3239 3635 3930 3032 3936 3078  765529659002960x
│ │ │ │ -00066ea0: 2078 2078 207c 0a7c 2020 2020 2020 2020   x x |.|        
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│ │ │ │ +00066e60: 3933 3532 3078 2078 2078 2078 2020 2b20  93520x x x x  + 
│ │ │ │ +00066e70: 3834 3737 3739 3431 3732 3832 3932 3430  8477794172829240
│ │ │ │ +00066e80: 7820 7820 7820 202b 2031 3736 3535 3239  x x x  + 1765529
│ │ │ │ +00066e90: 3635 3930 3032 3936 3078 2078 2078 207c  659002960x x x |
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│ │ │ │ -00066fa0: 2078 2078 2078 2020 2d20 3239 3332 3038   x x x  - 293208
│ │ │ │ -00066fb0: 3533 3032 3036 3631 3739 3078 2078 2078  53020661790x x x
│ │ │ │ -00066fc0: 2020 2d20 3830 3637 3138 3834 3031 3530    - 806718840150
│ │ │ │ -00066fd0: 3038 3136 7820 7820 7820 202b 2032 3433  0816x x x  + 243
│ │ │ │ -00066fe0: 3535 3939 347c 0a7c 2020 2020 2020 2020  55994|.|        
│ │ │ │ -00066ff0: 3020 3120 3220 3420 2020 2020 2020 2020  0 1 2 4         
│ │ │ │ -00067000: 2020 2020 2020 2020 2020 2020 3120 3220              1 2 
│ │ │ │ -00067010: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00067020: 2020 2020 2030 2032 2034 2020 2020 2020       0 2 4      
│ │ │ │ -00067030: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00066f80: 2032 2020 2020 2020 2020 2020 2020 207c   2             |
│ │ │ │ +00066f90: 0a7c 3732 3336 3536 3078 2078 2078 2078  .|7236560x x x x
│ │ │ │ +00066fa0: 2020 2d20 3239 3332 3038 3533 3032 3036    - 293208530206
│ │ │ │ +00066fb0: 3631 3739 3078 2078 2078 2020 2d20 3830  61790x x x  - 80
│ │ │ │ +00066fc0: 3637 3138 3834 3031 3530 3038 3136 7820  67188401500816x 
│ │ │ │ +00066fd0: 7820 7820 202b 2032 3433 3535 3939 347c  x x  + 24355994|
│ │ │ │ +00066fe0: 0a7c 2020 2020 2020 2020 3020 3120 3220  .|        0 1 2 
│ │ │ │ +00066ff0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +00067000: 2020 2020 2020 3120 3220 3420 2020 2020        1 2 4     
│ │ │ │ +00067010: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00067020: 2032 2034 2020 2020 2020 2020 2020 207c   2 4           |
│ │ │ │ +00067030: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00067040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067070: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00067070: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ -000670a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000670b0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -000670c0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -000670d0: 2020 2020 207c 0a7c 3032 3732 3139 3336       |.|02721936
│ │ │ │ -000670e0: 3837 3832 3838 3030 7820 7820 7820 7820  87828800x x x x 
│ │ │ │ -000670f0: 202b 2036 3137 3133 3238 3138 3837 3437   + 6171328188747
│ │ │ │ -00067100: 3735 3078 2078 2078 2020 2b20 3433 3031  750x x x  + 4301
│ │ │ │ -00067110: 3138 3933 3236 3933 3438 3932 7820 7820  189326934892x x 
│ │ │ │ -00067120: 7820 202d 207c 0a7c 2020 2020 2020 2020  x  - |.|        
│ │ │ │ -00067130: 2020 2020 2020 2020 2030 2031 2032 2034           0 1 2 4
│ │ │ │ -00067140: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00067160: 2020 2020 2020 2020 2020 2020 2030 2032               0 2
│ │ │ │ -00067170: 2034 2020 207c 0a7c 2020 2020 2020 2020   4   |.|        
│ │ │ │ +000670a0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +000670b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000670c0: 2020 2020 2020 2020 2032 2020 2020 207c           2     |
│ │ │ │ +000670d0: 0a7c 3032 3732 3139 3336 3837 3832 3838  .|02721936878288
│ │ │ │ +000670e0: 3030 7820 7820 7820 7820 202b 2036 3137  00x x x x  + 617
│ │ │ │ +000670f0: 3133 3238 3138 3837 3437 3735 3078 2078  1328188747750x x
│ │ │ │ +00067100: 2078 2020 2b20 3433 3031 3138 3933 3236   x  + 4301189326
│ │ │ │ +00067110: 3933 3438 3932 7820 7820 7820 202d 207c  934892x x x  - |
│ │ │ │ +00067120: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00067130: 2020 2030 2031 2032 2034 2020 2020 2020     0 1 2 4      
│ │ │ │ +00067140: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +00067150: 3220 3420 2020 2020 2020 2020 2020 2020  2 4             
│ │ │ │ +00067160: 2020 2020 2020 2030 2032 2034 2020 207c         0 2 4   |
│ │ │ │ +00067170: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00067180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067190: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000671b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000671b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000671c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ -000671e0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -000671f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067200: 2020 3220 2020 2020 2020 2020 2020 2020    2             
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│ │ │ │ -00067220: 2078 2020 2b20 3132 3833 3832 3435 3537   x  + 1283824557
│ │ │ │ -00067230: 3731 3737 3836 3078 2078 2078 2020 2b20  7177860x x x  + 
│ │ │ │ -00067240: 3533 3833 3735 3134 3934 3236 3234 3078  538375149426240x
│ │ │ │ -00067250: 2078 2078 2020 2d20 3236 3538 3531 3735   x x  - 26585175
│ │ │ │ -00067260: 3339 3139 377c 0a7c 2020 2020 3020 3120  39197|.|    0 1 
│ │ │ │ -00067270: 3220 3420 2020 2020 2020 2020 2020 2020  2 4             
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│ │ │ │ -00067290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000672a0: 3020 3220 3420 2020 2020 2020 2020 2020  0 2 4           
│ │ │ │ -000672b0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
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│ │ │ │ +00067200: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00067210: 0a7c 3638 3078 2078 2078 2078 2020 2b20  .|680x x x x  + 
│ │ │ │ +00067220: 3132 3833 3832 3435 3537 3731 3737 3836  1283824557717786
│ │ │ │ +00067230: 3078 2078 2078 2020 2b20 3533 3833 3735  0x x x  + 538375
│ │ │ │ +00067240: 3134 3934 3236 3234 3078 2078 2078 2020  149426240x x x  
│ │ │ │ +00067250: 2d20 3236 3538 3531 3735 3339 3139 377c  - 2658517539197|
│ │ │ │ +00067260: 0a7c 2020 2020 3020 3120 3220 3420 2020  .|    0 1 2 4   
│ │ │ │ +00067270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067280: 2020 3120 3220 3420 2020 2020 2020 2020    1 2 4         
│ │ │ │ +00067290: 2020 2020 2020 2020 2020 3020 3220 3420            0 2 4 
│ │ │ │ +000672a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000672b0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  000672c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000672d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000672e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000672f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067300: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 3220  -----|.|      2 
│ │ │ │ -00067310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067320: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ -00067330: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -00067340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067350: 2020 2020 207c 0a7c 3838 3078 2078 2078       |.|880x x x
│ │ │ │ -00067360: 2020 2d20 3637 3432 3230 3037 3839 3637    - 674220078967
│ │ │ │ -00067370: 3638 7820 7820 202d 2033 3035 3432 3138  68x x  - 3054218
│ │ │ │ -00067380: 3238 3539 3332 3830 3078 2078 2078 2020  285932800x x x  
│ │ │ │ -00067390: 2b20 3332 3737 3233 3039 3132 3439 3037  + 32772309124907
│ │ │ │ -000673a0: 3532 3078 207c 0a7c 2020 2020 3120 3220  520x |.|    1 2 
│ │ │ │ -000673b0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -000673c0: 2020 2032 2034 2020 2020 2020 2020 2020     2 4          
│ │ │ │ -000673d0: 2020 2020 2020 2020 2020 3020 3320 3420            0 3 4 
│ │ │ │ -000673e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000673f0: 2020 2020 307c 0a7c 2020 2020 2020 2020      0|.|        
│ │ │ │ +000672f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
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│ │ │ │ +00067330: 2020 2020 3220 2020 2020 2020 2020 2020      2           
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│ │ │ │ +00067350: 0a7c 3838 3078 2078 2078 2020 2d20 3637  .|880x x x  - 67
│ │ │ │ +00067360: 3432 3230 3037 3839 3637 3638 7820 7820  422007896768x x 
│ │ │ │ +00067370: 202d 2033 3035 3432 3138 3238 3539 3332   - 3054218285932
│ │ │ │ +00067380: 3830 3078 2078 2078 2020 2b20 3332 3737  800x x x  + 3277
│ │ │ │ +00067390: 3233 3039 3132 3439 3037 3532 3078 207c  2309124907520x |
│ │ │ │ +000673a0: 0a7c 2020 2020 3120 3220 3420 2020 2020  .|    1 2 4     
│ │ │ │ +000673b0: 2020 2020 2020 2020 2020 2020 2032 2034               2 4
│ │ │ │ +000673c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000673d0: 2020 2020 3020 3320 3420 2020 2020 2020      0 3 4       
│ │ │ │ +000673e0: 2020 2020 2020 2020 2020 2020 2020 307c                0|
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│ │ │ │  00067400: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00067480: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00067490: 2020 2020 207c 0a7c 202b 2032 3239 3830       |.| + 22980
│ │ │ │ -000674a0: 3430 3136 3439 3237 3336 3078 2078 2078  40164927360x x x
│ │ │ │ -000674b0: 2020 2d20 3132 3133 3335 3038 3739 3539    - 121335087959
│ │ │ │ -000674c0: 3937 3132 7820 7820 202d 2032 3238 3734  9712x x  - 22874
│ │ │ │ -000674d0: 3731 3335 3631 3032 3931 3230 7820 7820  713561029120x x 
│ │ │ │ -000674e0: 7820 202b 207c 0a7c 2020 2020 2020 2020  x  + |.|        
│ │ │ │ -000674f0: 2020 2020 2020 2020 2020 2020 3120 3220              1 2 
│ │ │ │ -00067500: 3420 2020 2020 2020 2020 2020 2020 2020  4               
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│ │ │ │ -00067520: 2020 2020 2020 2020 2020 2020 2030 2033               0 3
│ │ │ │ -00067530: 2034 2020 207c 0a7c 2020 2020 2020 2020   4   |.|        
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│ │ │ │ +00067470: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00067490: 0a7c 202b 2032 3239 3830 3430 3136 3439  .| + 22980401649
│ │ │ │ +000674a0: 3237 3336 3078 2078 2078 2020 2d20 3132  27360x x x  - 12
│ │ │ │ +000674b0: 3133 3335 3038 3739 3539 3937 3132 7820  13350879599712x 
│ │ │ │ +000674c0: 7820 202d 2032 3238 3734 3731 3335 3631  x  - 22874713561
│ │ │ │ +000674d0: 3032 3931 3230 7820 7820 7820 202b 207c  029120x x x  + |
│ │ │ │ +000674e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000674f0: 2020 2020 2020 3120 3220 3420 2020 2020        1 2 4     
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│ │ │ │ -000675e0: 3078 2078 2078 2020 2d20 3233 3233 3432  0x x x  - 232342
│ │ │ │ -000675f0: 3637 3638 3231 3333 3230 7820 7820 202b  6768213320x x  +
│ │ │ │ -00067600: 2031 3234 3339 3433 3732 3835 3438 3836   124394372854886
│ │ │ │ -00067610: 3430 7820 7820 7820 202d 2031 3333 3932  40x x x  - 13392
│ │ │ │ -00067620: 3934 3533 347c 0a7c 2020 2020 2020 2020  94534|.|        
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│ │ │ │ +000675d0: 0a7c 3637 3030 3634 3839 3078 2078 2078  .|670064890x x x
│ │ │ │ +000675e0: 2020 2d20 3233 3233 3432 3637 3638 3231    - 232342676821
│ │ │ │ +000675f0: 3333 3230 7820 7820 202b 2031 3234 3339  3320x x  + 12439
│ │ │ │ +00067600: 3433 3732 3835 3438 3836 3430 7820 7820  437285488640x x 
│ │ │ │ +00067610: 7820 202d 2031 3333 3932 3934 3533 347c  x  - 1339294534|
│ │ │ │ +00067620: 0a7c 2020 2020 2020 2020 2020 3120 3220  .|          1 2 
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│ │ │ │ -00067730: 2b20 3135 3839 3638 3832 3334 3935 3432  + 15896882349542
│ │ │ │ -00067740: 3838 7820 7820 202b 2031 3430 3330 3032  88x x  + 1403002
│ │ │ │ -00067750: 3132 3036 3532 3136 3030 7820 7820 7820  1206521600x x x 
│ │ │ │ -00067760: 202d 2032 307c 0a7c 2020 2020 2020 2020   - 20|.|        
│ │ │ │ -00067770: 2020 2020 2020 2020 2020 3120 3220 3420            1 2 4 
│ │ │ │ -00067780: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000677a0: 2020 2020 2020 2020 2020 2030 2033 2034             0 3 4
│ │ │ │ -000677b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000676b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +00067720: 3430 3278 2078 2078 2020 2b20 3135 3839  402x x x  + 1589
│ │ │ │ +00067730: 3638 3832 3334 3935 3432 3838 7820 7820  688234954288x x 
│ │ │ │ +00067740: 202b 2031 3430 3330 3032 3132 3036 3532   + 1403002120652
│ │ │ │ +00067750: 3136 3030 7820 7820 7820 202d 2032 307c  1600x x x  - 20|
│ │ │ │ +00067760: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ +00067790: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000677f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ -00067840: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00067860: 7820 202b 2033 3232 3336 3834 3034 3130  x  + 32236840410
│ │ │ │ -00067870: 3337 3734 3078 2078 2020 2b20 3134 3234  37740x x  + 1424
│ │ │ │ -00067880: 3735 3038 3933 3638 3939 3834 3078 2078  7508936899840x x
│ │ │ │ -00067890: 2078 2020 2d20 3239 3431 3034 3838 3435   x  - 2941048845
│ │ │ │ -000678a0: 3139 3034 307c 0a7c 2020 2020 2031 2032  19040|.|     1 2
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│ │ │ │ +00067860: 3232 3336 3834 3034 3130 3337 3734 3078  223684041037740x
│ │ │ │ +00067870: 2078 2020 2b20 3134 3234 3735 3038 3933   x  + 1424750893
│ │ │ │ +00067880: 3638 3939 3834 3078 2078 2078 2020 2d20  6899840x x x  - 
│ │ │ │ +00067890: 3239 3431 3034 3838 3435 3139 3034 307c  294104884519040|
│ │ │ │ +000678a0: 0a7c 2020 2020 2031 2032 2034 2020 2020  .|     1 2 4    
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│ │ │ │ +00068280: 202b 2037 3738 3433 3739 3131 3436 3234   + 7784379114624
│ │ │ │ +00068290: 3030 7820 7820 7820 202b 2037 3431 377c  00x x x  + 7417|
│ │ │ │ +000682a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ +000682c0: 2020 2020 2020 2020 2020 2032 2033 2034             2 3 4
│ │ │ │ +000682d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000682f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ -000683c0: 3237 3238 3878 2078 2078 2020 2d20 3134  27288x x x  - 14
│ │ │ │ -000683d0: 3336 3339 3532 3332 3134 3732 3332 3078  363952321472320x
│ │ │ │ -000683e0: 2078 2078 207c 0a7c 2020 2020 2020 2020   x x |.|        
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│ │ │ │ +000683c0: 2078 2078 2020 2d20 3134 3336 3339 3532   x x  - 14363952
│ │ │ │ +000683d0: 3332 3134 3732 3332 3078 2078 2078 207c  321472320x x x |
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│ │ │ │ +00068400: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000684c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -000684d0: 2020 2020 207c 0a7c 3630 7820 7820 7820       |.|60x x x 
│ │ │ │ -000684e0: 7820 202b 2031 3937 3638 3333 3331 3836  x  + 19768333186
│ │ │ │ -000684f0: 3234 3932 3078 2078 2078 2020 2b20 3931  24920x x x  + 91
│ │ │ │ -00068500: 3239 3538 3937 3436 3435 3331 3230 7820  29589746453120x 
│ │ │ │ -00068510: 7820 7820 202b 2032 3534 3736 3334 3933  x x  + 254763493
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│ │ │ │ +000684d0: 0a7c 3630 7820 7820 7820 7820 202b 2031  .|60x x x x  + 1
│ │ │ │ +000684e0: 3937 3638 3333 3331 3836 3234 3932 3078  976833318624920x
│ │ │ │ +000684f0: 2078 2078 2020 2b20 3931 3239 3538 3937   x x  + 91295897
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│ │ │ │ +00068550: 2020 2020 2020 2020 2030 2033 2034 2020           0 3 4  
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│ │ │ │ -00068630: 3233 3832 3038 3078 2078 2078 2020 2d20  2382080x x x  - 
│ │ │ │ -00068640: 3235 3235 3630 3833 3637 3534 3638 3830  2525608367546880
│ │ │ │ -00068650: 7820 7820 202d 2031 3430 3833 3732 3939  x x  - 140837299
│ │ │ │ -00068660: 3239 3032 347c 0a7c 2020 2020 2020 3120  29024|.|      1 
│ │ │ │ -00068670: 3320 3420 2020 2020 2020 2020 2020 2020  3 4             
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│ │ │ │ +00068610: 0a7c 3432 3838 3078 2078 2078 2020 2b20  .|42880x x x  + 
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│ │ │ │ +00068630: 3078 2078 2078 2020 2d20 3235 3235 3630  0x x x  - 252560
│ │ │ │ +00068640: 3833 3637 3534 3638 3830 7820 7820 202d  8367546880x x  -
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│ │ │ │ -00068770: 7820 7820 202d 2031 3239 3836 3238 3931  x x  - 129862891
│ │ │ │ -00068780: 3930 3232 3430 3030 7820 7820 7820 202b  90224000x x x  +
│ │ │ │ -00068790: 2031 3430 3539 3336 3233 3637 3333 3035   140593623673305
│ │ │ │ -000687a0: 3630 7820 787c 0a7c 2034 2020 2020 2020  60x x|.| 4      
│ │ │ │ -000687b0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -000687c0: 2033 2034 2020 2020 2020 2020 2020 2020   3 4            
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│ │ │ │ +00068760: 3930 3431 3838 3830 7820 7820 7820 202d  90418880x x x  -
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│ │ │ │ +00068780: 3030 7820 7820 7820 202b 2031 3430 3539  00x x x  + 14059
│ │ │ │ +00068790: 3336 3233 3637 3333 3035 3630 7820 787c  362367330560x x|
│ │ │ │ +000687a0: 0a7c 2034 2020 2020 2020 2020 2020 2020  .| 4            
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│ │ │ │ -000688b0: 3938 3336 3239 3637 3035 3038 3878 2078  9836296705088x x
│ │ │ │ -000688c0: 2078 2020 2d20 3235 3930 3537 3132 3531   x  - 2590571251
│ │ │ │ -000688d0: 3537 3832 3430 7820 7820 202d 2032 3230  578240x x  - 220
│ │ │ │ -000688e0: 3036 3435 367c 0a7c 2020 2020 2020 2020  06456|.|        
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│ │ │ │ +000688a0: 7820 7820 202b 2035 3332 3938 3336 3239  x x  + 532983629
│ │ │ │ +000688b0: 3637 3035 3038 3878 2078 2078 2020 2d20  6705088x x x  - 
│ │ │ │ +000688c0: 3235 3930 3537 3132 3531 3537 3832 3430  2590571251578240
│ │ │ │ +000688d0: 7820 7820 202d 2032 3230 3036 3435 367c  x x  - 22006456|
│ │ │ │ +000688e0: 0a7c 2020 2020 2020 2020 2020 2020 2031  .|             1
│ │ │ │ +000688f0: 2033 2034 2020 2020 2020 2020 2020 2020   3 4            
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│ │ │ │ -000689d0: 3320 2020 207c 0a7c 202b 2033 3332 3035  3    |.| + 33205
│ │ │ │ -000689e0: 3031 3234 3434 3039 3434 3078 2078 2078  01244409440x x x
│ │ │ │ -000689f0: 2020 2d20 3133 3432 3434 3736 3330 3633    - 134244763063
│ │ │ │ -00068a00: 3437 3132 3078 2078 2078 2020 2b20 3139  47120x x x  + 19
│ │ │ │ -00068a10: 3139 3439 3635 3338 3138 3934 3430 3078  194965381894400x
│ │ │ │ -00068a20: 2078 2020 2d7c 0a7c 2020 2020 2020 2020   x  -|.|        
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│ │ │ │ +000689d0: 0a7c 202b 2033 3332 3035 3031 3234 3434  .| + 33205012444
│ │ │ │ +000689e0: 3039 3434 3078 2078 2078 2020 2d20 3133  09440x x x  - 13
│ │ │ │ +000689f0: 3432 3434 3736 3330 3633 3437 3132 3078  424476306347120x
│ │ │ │ +00068a00: 2078 2078 2020 2b20 3139 3139 3439 3635   x x  + 19194965
│ │ │ │ +00068a10: 3338 3138 3934 3430 3078 2078 2020 2d7c  381894400x x  -|
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│ │ │ │ +00068a60: 2020 2020 2020 2020 2020 3320 3420 207c            3 4  |
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│ │ │ │ -00068b20: 202b 2039 3031 3233 3133 3232 3631 3930   + 9012313226190
│ │ │ │ -00068b30: 3936 3078 2078 2078 2020 2d20 3230 3635  960x x x  - 2065
│ │ │ │ -00068b40: 3835 3539 3939 3737 3739 3230 7820 7820  855999777920x x 
│ │ │ │ -00068b50: 202b 2032 3038 3638 3733 3939 3936 3630   + 2086873999660
│ │ │ │ -00068b60: 3830 3030 787c 0a7c 2020 2031 2033 2034  8000x|.|   1 3 4
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│ │ │ │ -00068b90: 2020 2020 2020 2020 2020 2020 2033 2034               3 4
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│ │ │ │ +00068b10: 0a7c 3230 7820 7820 7820 202b 2039 3031  .|20x x x  + 901
│ │ │ │ +00068b20: 3233 3133 3232 3631 3930 3936 3078 2078  2313226190960x x
│ │ │ │ +00068b30: 2078 2020 2d20 3230 3635 3835 3539 3939   x  - 2065855999
│ │ │ │ +00068b40: 3737 3739 3230 7820 7820 202b 2032 3038  777920x x  + 208
│ │ │ │ +00068b50: 3638 3733 3939 3936 3630 3830 3030 787c  68739996608000x|
│ │ │ │ +00068b60: 0a7c 2020 2031 2033 2034 2020 2020 2020  .|   1 3 4      
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│ │ │ │ +00068b80: 3320 3420 2020 2020 2020 2020 2020 2020  3 4             
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│ │ │ │ +00068c00: 0a7c 2020 2032 2032 2020 2020 2020 2020  .|   2 2        
│ │ │ │  00068c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068c20: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -00068c30: 2020 2020 2020 2020 2020 2032 2032 2020             2 2  
│ │ │ │ -00068c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068c50: 2020 2020 207c 0a7c 3030 7820 7820 202d       |.|00x x  -
│ │ │ │ -00068c60: 2038 3939 3734 3437 3132 3635 3339 3230   899744712653920
│ │ │ │ -00068c70: 3078 2078 2078 2020 2b20 3637 3035 3431  0x x x  + 670541
│ │ │ │ -00068c80: 3734 3930 3137 3834 3030 7820 7820 202b  7490178400x x  +
│ │ │ │ -00068c90: 2036 3438 3239 3737 3834 3939 3838 3634   648297784998864
│ │ │ │ -00068ca0: 3078 2078 207c 0a7c 2020 2030 2034 2020  0x x |.|   0 4  
│ │ │ │ -00068cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00068cd0: 2020 2020 2020 2020 2020 2031 2034 2020             1 4  
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│ │ │ │ +00068c20: 3220 2020 2020 2020 2020 2020 2020 2020  2               
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│ │ │ │ +00068c40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00068c50: 0a7c 3030 7820 7820 202d 2038 3939 3734  .|00x x  - 89974
│ │ │ │ +00068c60: 3437 3132 3635 3339 3230 3078 2078 2078  47126539200x x x
│ │ │ │ +00068c70: 2020 2b20 3637 3035 3431 3734 3930 3137    + 670541749017
│ │ │ │ +00068c80: 3834 3030 7820 7820 202b 2036 3438 3239  8400x x  + 64829
│ │ │ │ +00068c90: 3737 3834 3939 3838 3634 3078 2078 207c  77849988640x x |
│ │ │ │ +00068ca0: 0a7c 2020 2030 2034 2020 2020 2020 2020  .|   0 4        
│ │ │ │ +00068cb0: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ +00068cc0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +00068cd0: 2020 2020 2031 2034 2020 2020 2020 2020       1 4        
│ │ │ │ +00068ce0: 2020 2020 2020 2020 2020 2020 3020 327c              0 2|
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│ │ │ │  00068d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00068d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068d40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00068d50: 2020 2020 2020 2020 2020 2020 2032 2032               2 2
│ │ │ │ +00068d30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ -00068d80: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00068d90: 3220 2020 207c 0a7c 2020 2b20 3435 3331  2    |.|  + 4531
│ │ │ │ -00068da0: 3133 3430 3637 3231 3932 3030 7820 7820  134067219200x x 
│ │ │ │ -00068db0: 202d 2031 3133 3736 3135 3934 3035 3535   - 1137615940555
│ │ │ │ -00068dc0: 3434 3030 7820 7820 7820 202b 2036 3435  4400x x x  + 645
│ │ │ │ -00068dd0: 3532 3935 3937 3335 3031 3230 3078 2078  5295973501200x x
│ │ │ │ -00068de0: 2020 2b20 377c 0a7c 3420 2020 2020 2020    + 7|.|4       
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│ │ │ │ +00068da0: 3231 3932 3030 7820 7820 202d 2031 3133  219200x x  - 113
│ │ │ │ +00068db0: 3736 3135 3934 3035 3535 3434 3030 7820  76159405554400x 
│ │ │ │ +00068dc0: 7820 7820 202b 2036 3435 3532 3935 3937  x x  + 645529597
│ │ │ │ +00068dd0: 3335 3031 3230 3078 2078 2020 2b20 377c  3501200x x  + 7|
│ │ │ │ +00068de0: 0a7c 3420 2020 2020 2020 2020 2020 2020  .|4             
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│ │ │ │ +00068e00: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00068e10: 2031 2034 2020 2020 2020 2020 2020 2020   1 4            
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│ │ │ │  00068e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00068e80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ -00068ea0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00068eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00068ef0: 3530 3036 3135 3336 3078 2078 2078 2020  500615360x x x  
│ │ │ │ -00068f00: 2d20 3138 3634 3139 3734 3538 3437 3536  - 18641974584756
│ │ │ │ -00068f10: 3630 3078 2078 2020 2d20 3332 3733 3933  600x x  - 327393
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│ │ │ │ -00068f30: 2020 3020 3420 2020 2020 2020 2020 2020    0 4           
│ │ │ │ -00068f40: 2020 2020 2020 2020 2020 3020 3120 3420            0 1 4 
│ │ │ │ -00068f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00068eb0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
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│ │ │ │ +00068ed0: 0a7c 3632 3230 3038 3332 3078 2078 2020  .|622008320x x  
│ │ │ │ +00068ee0: 2b20 3436 3737 3735 3737 3530 3036 3135  + 46777577500615
│ │ │ │ +00068ef0: 3336 3078 2078 2078 2020 2d20 3138 3634  360x x x  - 1864
│ │ │ │ +00068f00: 3139 3734 3538 3437 3536 3630 3078 2078  1974584756600x x
│ │ │ │ +00068f10: 2020 2d20 3332 3733 3933 3438 3534 357c    - 32739348545|
│ │ │ │ +00068f20: 0a7c 2020 2020 2020 2020 2020 3020 3420  .|          0 4 
│ │ │ │ +00068f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00068f50: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +00068f60: 3420 2020 2020 2020 2020 2020 2020 207c  4              |
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│ │ │ │ -00069000: 2020 2020 2020 2020 2020 2032 2032 2020             2 2  
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│ │ │ │ -00069020: 3134 3736 3339 3034 3078 2078 2020 2d20  147639040x x  - 
│ │ │ │ -00069030: 3130 3335 3835 3038 3533 3037 3339 3630  1035850853073960
│ │ │ │ -00069040: 3078 2078 2078 2020 2b20 3532 3539 3432  0x x x  + 525942
│ │ │ │ -00069050: 3739 3435 3730 3334 3030 7820 7820 202b  7945703400x x  +
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│ │ │ │ +00069010: 0a7c 2031 3633 3037 3933 3134 3736 3339  .| 1630793147639
│ │ │ │ +00069020: 3034 3078 2078 2020 2d20 3130 3335 3835  040x x  - 103585
│ │ │ │ +00069030: 3038 3533 3037 3339 3630 3078 2078 2078  08530739600x x x
│ │ │ │ +00069040: 2020 2b20 3532 3539 3432 3739 3435 3730    + 525942794570
│ │ │ │ +00069050: 3334 3030 7820 7820 202b 2038 3134 387c  3400x x  + 8148|
│ │ │ │ +00069060: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ -00069150: 2020 2020 207c 0a7c 2078 2020 2d20 3134       |.| x  - 14
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│ │ │ │ -00069170: 2078 2078 2020 2d20 3139 3830 3733 3531   x x  - 19807351
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│ │ │ │ +000691a0: 0a7c 3020 3420 2020 2020 2020 2020 2020  .|0 4           
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│ │ │ │  00069210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00069280: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000692a0: 3530 3535 3939 3831 3835 3131 3230 7820  50559981851120x 
│ │ │ │ -000692b0: 7820 7820 202d 2032 3931 3038 3438 3933  x x  - 291084893
│ │ │ │ -000692c0: 3833 3834 3830 7820 7820 202d 2031 3435  838480x x  - 145
│ │ │ │ -000692d0: 3434 3937 3732 3831 3032 3237 3230 7820  44977281022720x 
│ │ │ │ -000692e0: 7820 7820 207c 0a7c 2034 2020 2020 2020  x x  |.| 4      
│ │ │ │ -000692f0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00069300: 2032 2034 2020 2020 2020 2020 2020 2020   2 4            
│ │ │ │ -00069310: 2020 2020 2020 2032 2034 2020 2020 2020         2 4      
│ │ │ │ -00069320: 2020 2020 2020 2020 2020 2020 2020 2030                 0
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│ │ │ │ +00069270: 2032 2032 2020 2020 2020 2020 2020 2020   2 2            
│ │ │ │ +00069280: 2020 2020 2020 2020 2020 2020 2032 207c               2 |
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│ │ │ │ +000692a0: 3831 3835 3131 3230 7820 7820 7820 202d  81851120x x x  -
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│ │ │ │ +000692c0: 7820 7820 202d 2031 3435 3434 3937 3732  x x  - 145449772
│ │ │ │ +000692d0: 3831 3032 3237 3230 7820 7820 7820 207c  81022720x x x  |
│ │ │ │ +000692e0: 0a7c 2034 2020 2020 2020 2020 2020 2020  .| 4            
│ │ │ │ +000692f0: 2020 2020 2020 2020 2031 2032 2034 2020           1 2 4  
│ │ │ │ +00069300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069310: 2032 2034 2020 2020 2020 2020 2020 2020   2 4            
│ │ │ │ +00069320: 2020 2020 2020 2020 2030 2033 2034 207c           0 3 4 |
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│ │ │ │  00069340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069350: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00069370: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000693a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000693b0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -000693c0: 2020 2020 2020 2020 2032 2032 2020 2020           2 2    
│ │ │ │ -000693d0: 2020 2020 207c 0a7c 3035 3138 3030 3435       |.|05180045
│ │ │ │ -000693e0: 3630 3038 3936 3078 2078 2078 2020 2d20  6008960x x x  - 
│ │ │ │ -000693f0: 3833 3838 3731 3630 3034 3235 3438 3830  8388716004254880
│ │ │ │ -00069400: 7820 7820 7820 202d 2031 3633 3637 3232  x x x  - 1636722
│ │ │ │ -00069410: 3138 3132 3331 3230 7820 7820 202d 2032  18123120x x  - 2
│ │ │ │ -00069420: 3137 3735 367c 0a7c 2020 2020 2020 2020  17756|.|        
│ │ │ │ -00069430: 2020 2020 2020 2020 3020 3220 3420 2020          0 2 4   
│ │ │ │ -00069440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069450: 2031 2032 2034 2020 2020 2020 2020 2020   1 2 4          
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│ │ │ │ +000693b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000693d0: 0a7c 3035 3138 3030 3435 3630 3038 3936  .|05180045600896
│ │ │ │ +000693e0: 3078 2078 2078 2020 2d20 3833 3838 3731  0x x x  - 838871
│ │ │ │ +000693f0: 3630 3034 3235 3438 3830 7820 7820 7820  6004254880x x x 
│ │ │ │ +00069400: 202d 2031 3633 3637 3232 3138 3132 3331   - 1636722181231
│ │ │ │ +00069410: 3230 7820 7820 202d 2032 3137 3735 367c  20x x  - 217756|
│ │ │ │ +00069420: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ +00069440: 2020 2020 2020 2020 2020 2031 2032 2034             1 2 4
│ │ │ │ +00069450: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00069470: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │  00069490: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000694b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000694c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000694d0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -000694e0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -000694f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069500: 2020 2032 2032 2020 2020 2020 2020 2020     2 2          
│ │ │ │ -00069510: 2020 2020 207c 0a7c 3436 3732 3030 7820       |.|467200x 
│ │ │ │ -00069520: 7820 7820 202b 2034 3934 3639 3538 3939  x x  + 494695899
│ │ │ │ -00069530: 3635 3631 3739 3230 7820 7820 7820 202d  65617920x x x  -
│ │ │ │ -00069540: 2031 3034 3839 3133 3332 3831 3739 3535   104891332817955
│ │ │ │ -00069550: 3630 7820 7820 202b 2037 3735 3338 3037  60x x  + 7753807
│ │ │ │ -00069560: 3338 3738 387c 0a7c 2020 2020 2020 2030  38788|.|       0
│ │ │ │ -00069570: 2032 2034 2020 2020 2020 2020 2020 2020   2 4            
│ │ │ │ -00069580: 2020 2020 2020 2020 2031 2032 2034 2020           1 2 4  
│ │ │ │ -00069590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000695a0: 2020 2032 2034 2020 2020 2020 2020 2020     2 4          
│ │ │ │ -000695b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000694b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000694c0: 0a7c 2020 2020 2020 2020 2020 2032 2020  .|           2  
│ │ │ │ +000694d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000694e0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +000694f0: 2020 2020 2020 2020 2020 2020 2032 2032               2 2
│ │ │ │ +00069500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00069510: 0a7c 3436 3732 3030 7820 7820 7820 202b  .|467200x x x  +
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│ │ │ │ -00069ec0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -00069ed0: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ -00069ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069ef0: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ -00069f00: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ -00069f10: 2020 2020 207c 0a7c 2039 3533 3030 3631       |.| 9530061
│ │ │ │ -00069f20: 3032 3333 3437 3230 3078 2078 2020 2d20  023347200x x  - 
│ │ │ │ -00069f30: 3131 3838 3838 3435 3238 3132 3634 3030  1188884528126400
│ │ │ │ -00069f40: 3078 2078 2020 2b20 3234 3039 3639 3531  0x x  + 24096951
│ │ │ │ -00069f50: 3738 3633 3438 3030 7820 7820 202d 2031  78634800x x  - 1
│ │ │ │ -00069f60: 3130 3130 367c 0a7c 2020 2020 2020 2020  10106|.|        
│ │ │ │ -00069f70: 2020 2020 2020 2020 2020 3020 3420 2020            0 4   
│ │ │ │ -00069f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069f90: 2020 3120 3420 2020 2020 2020 2020 2020    1 4           
│ │ │ │ -00069fa0: 2020 2020 2020 2020 2032 2034 2020 2020           2 4    
│ │ │ │ -00069fb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00069eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00069ec0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00069ed0: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +00069ee0: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ +00069ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069f00: 2020 2020 2033 2020 2020 2020 2020 207c       3         |
│ │ │ │ +00069f10: 0a7c 2039 3533 3030 3631 3032 3333 3437  .| 9530061023347
│ │ │ │ +00069f20: 3230 3078 2078 2020 2d20 3131 3838 3838  200x x  - 118888
│ │ │ │ +00069f30: 3435 3238 3132 3634 3030 3078 2078 2020  45281264000x x  
│ │ │ │ +00069f40: 2b20 3234 3039 3639 3531 3738 3633 3438  + 24096951786348
│ │ │ │ +00069f50: 3030 7820 7820 202d 2031 3130 3130 367c  00x x  - 110106|
│ │ │ │ +00069f60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00069f70: 2020 2020 3020 3420 2020 2020 2020 2020      0 4         
│ │ │ │ +00069f80: 2020 2020 2020 2020 2020 2020 3120 3420              1 4 
│ │ │ │ +00069f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069fa0: 2020 2032 2034 2020 2020 2020 2020 207c     2 4         |
│ │ │ │ +00069fb0: 0a7c 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a000: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0006a010: 2020 3220 3220 2020 2020 2020 2020 2020    2 2           
│ │ │ │ -0006a020: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ -0006a030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a040: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -0006a050: 2020 2020 207c 0a7c 3537 3134 3531 3834       |.|57145184
│ │ │ │ -0006a060: 3078 2078 2020 2b20 3332 3431 3639 3336  0x x  + 32416936
│ │ │ │ -0006a070: 3436 3036 3038 3030 7820 7820 202d 2035  46060800x x  - 5
│ │ │ │ -0006a080: 3937 3933 3136 3339 3239 3537 3630 3078  979316392957600x
│ │ │ │ -0006a090: 2078 2020 2b20 3330 3030 3333 3036 3534   x  + 3000330654
│ │ │ │ -0006a0a0: 3337 3536 307c 0a7c 2020 2020 2020 2020  37560|.|        
│ │ │ │ -0006a0b0: 2020 3320 3420 2020 2020 2020 2020 2020    3 4           
│ │ │ │ -0006a0c0: 2020 2020 2020 2020 2030 2034 2020 2020           0 4    
│ │ │ │ -0006a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a0e0: 3120 3420 2020 2020 2020 2020 2020 2020  1 4             
│ │ │ │ -0006a0f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00069ff0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a000: 0a7c 2020 2020 2020 2020 2020 3220 3220  .|          2 2 
│ │ │ │ +0006a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a020: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ +0006a030: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ +0006a040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a050: 0a7c 3537 3134 3531 3834 3078 2078 2020  .|571451840x x  
│ │ │ │ +0006a060: 2b20 3332 3431 3639 3336 3436 3036 3038  + 32416936460608
│ │ │ │ +0006a070: 3030 7820 7820 202d 2035 3937 3933 3136  00x x  - 5979316
│ │ │ │ +0006a080: 3339 3239 3537 3630 3078 2078 2020 2b20  392957600x x  + 
│ │ │ │ +0006a090: 3330 3030 3333 3036 3534 3337 3536 307c  300033065437560|
│ │ │ │ +0006a0a0: 0a7c 2020 2020 2020 2020 2020 3320 3420  .|          3 4 
│ │ │ │ +0006a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a0c0: 2020 2030 2034 2020 2020 2020 2020 2020     0 4          
│ │ │ │ +0006a0d0: 2020 2020 2020 2020 2020 3120 3420 2020            1 4   
│ │ │ │ +0006a0e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a0f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0006a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a140: 2020 2020 207c 0a7c 2020 2020 2020 2032       |.|       2
│ │ │ │ -0006a150: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0006a160: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ -0006a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a180: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -0006a190: 2020 2020 207c 0a7c 3134 3430 3030 7820       |.|144000x 
│ │ │ │ -0006a1a0: 7820 202d 2031 3632 3130 3437 3731 3732  x  - 16210477172
│ │ │ │ -0006a1b0: 3733 3630 3030 7820 7820 202b 2032 3131  736000x x  + 211
│ │ │ │ -0006a1c0: 3333 3634 3438 3538 3439 3638 3030 7820  33644858496800x 
│ │ │ │ -0006a1d0: 7820 202d 2031 3534 3938 3133 3132 3536  x  - 15498131256
│ │ │ │ -0006a1e0: 3536 3838 307c 0a7c 2020 2020 2020 2033  56880|.|       3
│ │ │ │ -0006a1f0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -0006a200: 2020 2020 2020 2030 2034 2020 2020 2020         0 4      
│ │ │ │ -0006a210: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -0006a220: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -0006a230: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006a130: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a140: 0a7c 2020 2020 2020 2032 2032 2020 2020  .|       2 2    
│ │ │ │ +0006a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a160: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +0006a170: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ +0006a180: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a190: 0a7c 3134 3430 3030 7820 7820 202d 2031  .|144000x x  - 1
│ │ │ │ +0006a1a0: 3632 3130 3437 3731 3732 3733 3630 3030  6210477172736000
│ │ │ │ +0006a1b0: 7820 7820 202b 2032 3131 3333 3634 3438  x x  + 211336448
│ │ │ │ +0006a1c0: 3538 3439 3638 3030 7820 7820 202d 2031  58496800x x  - 1
│ │ │ │ +0006a1d0: 3534 3938 3133 3132 3536 3536 3838 307c  549813125656880|
│ │ │ │ +0006a1e0: 0a7c 2020 2020 2020 2033 2034 2020 2020  .|       3 4    
│ │ │ │ +0006a1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a200: 2030 2034 2020 2020 2020 2020 2020 2020   0 4            
│ │ │ │ +0006a210: 2020 2020 2020 2020 2031 2034 2020 2020           1 4    
│ │ │ │ +0006a220: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a230: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0006a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a280: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0006a290: 3220 3220 2020 2020 2020 2020 2020 2020  2 2             
│ │ │ │ -0006a2a0: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ -0006a2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a2c0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -0006a2d0: 2020 2020 207c 0a7c 3336 3839 3630 3078       |.|3689600x
│ │ │ │ -0006a2e0: 2078 2020 2b20 3538 3036 3535 3534 3434   x  + 5806555444
│ │ │ │ -0006a2f0: 3136 3634 3030 7820 7820 202d 2038 3639  166400x x  - 869
│ │ │ │ -0006a300: 3331 3734 3637 3833 3238 3030 3078 2078  3174678328000x x
│ │ │ │ -0006a310: 2020 2b20 3735 3435 3135 3436 3038 3838    + 754515460888
│ │ │ │ -0006a320: 3035 3630 787c 0a7c 2020 2020 2020 2020  0560x|.|        
│ │ │ │ -0006a330: 3320 3420 2020 2020 2020 2020 2020 2020  3 4             
│ │ │ │ -0006a340: 2020 2020 2020 2030 2034 2020 2020 2020         0 4      
│ │ │ │ -0006a350: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -0006a360: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -0006a370: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006a270: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a280: 0a7c 2020 2020 2020 2020 3220 3220 2020  .|        2 2   
│ │ │ │ +0006a290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a2a0: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +0006a2b0: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ +0006a2c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a2d0: 0a7c 3336 3839 3630 3078 2078 2020 2b20  .|3689600x x  + 
│ │ │ │ +0006a2e0: 3538 3036 3535 3534 3434 3136 3634 3030  5806555444166400
│ │ │ │ +0006a2f0: 7820 7820 202d 2038 3639 3331 3734 3637  x x  - 869317467
│ │ │ │ +0006a300: 3833 3238 3030 3078 2078 2020 2b20 3735  8328000x x  + 75
│ │ │ │ +0006a310: 3435 3135 3436 3038 3838 3035 3630 787c  45154608880560x|
│ │ │ │ +0006a320: 0a7c 2020 2020 2020 2020 3320 3420 2020  .|        3 4   
│ │ │ │ +0006a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a340: 2030 2034 2020 2020 2020 2020 2020 2020   0 4            
│ │ │ │ +0006a350: 2020 2020 2020 2020 3120 3420 2020 2020          1 4     
│ │ │ │ +0006a360: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a370: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0006a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a3c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0006a3d0: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ -0006a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a3f0: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ -0006a400: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ -0006a410: 2020 2020 207c 0a7c 2036 3636 3435 3137       |.| 6664517
│ │ │ │ -0006a420: 3837 3930 3430 3030 3078 2078 2020 2b20  879040000x x  + 
│ │ │ │ -0006a430: 3535 3637 3239 3034 3136 3231 3036 3030  5567290416210600
│ │ │ │ -0006a440: 3078 2078 2020 2d20 3139 3434 3936 3033  0x x  - 19449603
│ │ │ │ -0006a450: 3938 3531 3635 3230 3078 2078 2020 2b20  985165200x x  + 
│ │ │ │ -0006a460: 3632 3035 327c 0a7c 2020 2020 2020 2020  62052|.|        
│ │ │ │ -0006a470: 2020 2020 2020 2020 2020 3020 3420 2020            0 4   
│ │ │ │ -0006a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a490: 2020 3120 3420 2020 2020 2020 2020 2020    1 4           
│ │ │ │ -0006a4a0: 2020 2020 2020 2020 2020 3220 3420 2020            2 4   
│ │ │ │ -0006a4b0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +0006a3b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a3c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0006a3d0: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +0006a3e0: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ +0006a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a400: 2020 2020 2020 3320 2020 2020 2020 207c        3        |
│ │ │ │ +0006a410: 0a7c 2036 3636 3435 3137 3837 3930 3430  .| 6664517879040
│ │ │ │ +0006a420: 3030 3078 2078 2020 2b20 3535 3637 3239  000x x  + 556729
│ │ │ │ +0006a430: 3034 3136 3231 3036 3030 3078 2078 2020  04162106000x x  
│ │ │ │ +0006a440: 2d20 3139 3434 3936 3033 3938 3531 3635  - 19449603985165
│ │ │ │ +0006a450: 3230 3078 2078 2020 2b20 3632 3035 327c  200x x  + 62052|
│ │ │ │ +0006a460: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0006a470: 2020 2020 3020 3420 2020 2020 2020 2020      0 4         
│ │ │ │ +0006a480: 2020 2020 2020 2020 2020 2020 3120 3420              1 4 
│ │ │ │ +0006a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a4a0: 2020 2020 3220 3420 2020 2020 2020 207c      2 4        |
│ │ │ │ +0006a4b0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0006a4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a500: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -0006a510: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ -0006a520: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ +0006a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +0006a500: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0006a510: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0006a520: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │  0006a530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a550: 2020 2020 207c 0a7c 3831 3339 3739 3638       |.|81397968
│ │ │ │ -0006a560: 3030 3078 2078 2020 2b20 3236 3136 3833  000x x  + 261683
│ │ │ │ -0006a570: 3537 3439 3139 3230 3030 7820 2c20 2020  5749192000x ,   
│ │ │ │ +0006a540: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a550: 0a7c 3831 3339 3739 3638 3030 3078 2078  .|81397968000x x
│ │ │ │ +0006a560: 2020 2b20 3236 3136 3833 3537 3439 3139    + 261683574919
│ │ │ │ +0006a570: 3230 3030 7820 2c20 2020 2020 2020 2020  2000x ,         
│ │ │ │  0006a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a5a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0006a5b0: 2020 2020 3320 3420 2020 2020 2020 2020      3 4         
│ │ │ │ -0006a5c0: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ +0006a590: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a5a0: 0a7c 2020 2020 2020 2020 2020 2020 3320  .|            3 
│ │ │ │ +0006a5b0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +0006a5c0: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │  0006a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a5f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006a5e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a5f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0006a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a640: 2020 2020 207c 0a7c 2020 2020 3320 2020       |.|    3   
│ │ │ │ -0006a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a660: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ -0006a670: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ -0006a680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a690: 2020 2020 207c 0a7c 3078 2078 2020 2d20       |.|0x x  - 
│ │ │ │ -0006a6a0: 3938 3838 3331 3337 3531 3736 3936 3030  9888313751769600
│ │ │ │ -0006a6b0: 7820 7820 202b 2031 3738 3439 3736 3234  x x  + 178497624
│ │ │ │ -0006a6c0: 3936 3732 3030 3078 202c 2020 2020 2020  9672000x ,      
│ │ │ │ -0006a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a6e0: 2020 2020 207c 0a7c 2020 3220 3420 2020       |.|  2 4   
│ │ │ │ -0006a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a700: 2033 2034 2020 2020 2020 2020 2020 2020   3 4            
│ │ │ │ -0006a710: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ -0006a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a730: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006a630: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a640: 0a7c 2020 2020 3320 2020 2020 2020 2020  .|    3         
│ │ │ │ +0006a650: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ +0006a660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a670: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +0006a680: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a690: 0a7c 3078 2078 2020 2d20 3938 3838 3331  .|0x x  - 988831
│ │ │ │ +0006a6a0: 3337 3531 3736 3936 3030 7820 7820 202b  3751769600x x  +
│ │ │ │ +0006a6b0: 2031 3738 3439 3736 3234 3936 3732 3030   178497624967200
│ │ │ │ +0006a6c0: 3078 202c 2020 2020 2020 2020 2020 2020  0x ,            
│ │ │ │ +0006a6d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a6e0: 0a7c 2020 3220 3420 2020 2020 2020 2020  .|  2 4         
│ │ │ │ +0006a6f0: 2020 2020 2020 2020 2020 2033 2034 2020             3 4  
│ │ │ │ +0006a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a710: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +0006a720: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a730: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0006a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a780: 2020 2020 207c 0a7c 2020 2020 3320 2020       |.|    3   
│ │ │ │ -0006a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a7a0: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ -0006a7b0: 2020 2020 2020 2020 2034 2020 2020 2020           4      
│ │ │ │ -0006a7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a7d0: 2020 2020 207c 0a7c 3078 2078 2020 2b20       |.|0x x  + 
│ │ │ │ -0006a7e0: 3338 3938 3738 3535 3136 3930 3536 3634  3898785516905664
│ │ │ │ -0006a7f0: 3078 2078 2020 2d20 3538 3239 3431 3531  0x x  - 58294151
│ │ │ │ -0006a800: 3635 3332 3438 3030 7820 2c20 2020 2020  65324800x ,     
│ │ │ │ -0006a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a820: 2020 2020 207c 0a7c 2020 3220 3420 2020       |.|  2 4   
│ │ │ │ -0006a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a840: 2020 3320 3420 2020 2020 2020 2020 2020    3 4           
│ │ │ │ -0006a850: 2020 2020 2020 2020 2034 2020 2020 2020           4      
│ │ │ │ -0006a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a870: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006a770: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a780: 0a7c 2020 2020 3320 2020 2020 2020 2020  .|    3         
│ │ │ │ +0006a790: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ +0006a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a7b0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +0006a7c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a7d0: 0a7c 3078 2078 2020 2b20 3338 3938 3738  .|0x x  + 389878
│ │ │ │ +0006a7e0: 3535 3136 3930 3536 3634 3078 2078 2020  55169056640x x  
│ │ │ │ +0006a7f0: 2d20 3538 3239 3431 3531 3635 3332 3438  - 58294151653248
│ │ │ │ +0006a800: 3030 7820 2c20 2020 2020 2020 2020 2020  00x ,           
│ │ │ │ +0006a810: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a820: 0a7c 2020 3220 3420 2020 2020 2020 2020  .|  2 4         
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│ │ │ │ +0006a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a910: 2020 2020 207c 0a7c 2078 2020 2d20 3132       |.| x  - 12
│ │ │ │ -0006a920: 3235 3333 3539 3734 3939 3534 3430 3078  253359749954400x
│ │ │ │ -0006a930: 2078 2020 2b20 3237 3835 3633 3438 3538   x  + 2785634858
│ │ │ │ -0006a940: 3035 3932 3030 7820 2c20 2020 2020 2020  059200x ,       
│ │ │ │ -0006a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a960: 2020 2020 207c 0a7c 3220 3420 2020 2020       |.|2 4     
│ │ │ │ -0006a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006a990: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ -0006a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a9b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006a8b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006a8c0: 0a7c 2020 3320 2020 2020 2020 2020 2020  .|  3           
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│ │ │ │ +0006a910: 0a7c 2078 2020 2d20 3132 3235 3333 3539  .| x  - 12253359
│ │ │ │ +0006a920: 3734 3939 3534 3430 3078 2078 2020 2b20  749954400x x  + 
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│ │ │ │ +0006a940: 7820 2c20 2020 2020 2020 2020 2020 2020  x ,             
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│ │ │ │ +0006a9b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0006a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0006aa20: 2020 2020 2020 3420 2020 2020 2020 2020        4         
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│ │ │ │  0006aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0006aaf0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0006ab00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ab10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ab20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ab30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ab40: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f  -----+..See also
│ │ │ │ -0006ab50: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ -0006ab60: 6e6f 7465 2069 6e76 6572 7365 4d61 703a  note inverseMap:
│ │ │ │ -0006ab70: 2069 6e76 6572 7365 4d61 702c 202d 2d20   inverseMap, -- 
│ │ │ │ -0006ab80: 696e 7665 7273 6520 6f66 2061 2062 6972  inverse of a bir
│ │ │ │ -0006ab90: 6174 696f 6e61 6c20 6d61 700a 2020 2a20  ational map.  * 
│ │ │ │ -0006aba0: 2a6e 6f74 6520 5261 7469 6f6e 616c 4d61  *note RationalMa
│ │ │ │ -0006abb0: 7020 5e20 5a5a 3a20 5261 7469 6f6e 616c  p ^ ZZ: Rational
│ │ │ │ -0006abc0: 4d61 7020 5e20 5a5a 2c20 2d2d 2070 6f77  Map ^ ZZ, -- pow
│ │ │ │ -0006abd0: 6572 0a20 202a 202a 6e6f 7465 2069 7349  er.  * *note isI
│ │ │ │ -0006abe0: 6e76 6572 7365 4d61 7028 5261 7469 6f6e  nverseMap(Ration
│ │ │ │ -0006abf0: 616c 4d61 702c 5261 7469 6f6e 616c 4d61  alMap,RationalMa
│ │ │ │ -0006ac00: 7029 3a0a 2020 2020 6973 496e 7665 7273  p):.    isInvers
│ │ │ │ -0006ac10: 654d 6170 5f6c 7052 6174 696f 6e61 6c4d  eMap_lpRationalM
│ │ │ │ -0006ac20: 6170 5f63 6d52 6174 696f 6e61 6c4d 6170  ap_cmRationalMap
│ │ │ │ -0006ac30: 5f72 702c 202d 2d20 6368 6563 6b73 2077  _rp, -- checks w
│ │ │ │ -0006ac40: 6865 7468 6572 2074 776f 2072 6174 696f  hether two ratio
│ │ │ │ -0006ac50: 6e61 6c0a 2020 2020 6d61 7073 2061 7265  nal.    maps are
│ │ │ │ -0006ac60: 206f 6e65 2074 6865 2069 6e76 6572 7365   one the inverse
│ │ │ │ -0006ac70: 206f 6620 7468 6520 6f74 6865 720a 0a57   of the other..W
│ │ │ │ -0006ac80: 6179 7320 746f 2075 7365 2074 6869 7320  ays to use this 
│ │ │ │ -0006ac90: 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d  method:.========
│ │ │ │ -0006aca0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0006acb0: 0a0a 2020 2a20 2a6e 6f74 6520 696e 7665  ..  * *note inve
│ │ │ │ -0006acc0: 7273 6528 5261 7469 6f6e 616c 4d61 7029  rse(RationalMap)
│ │ │ │ -0006acd0: 3a20 696e 7665 7273 655f 6c70 5261 7469  : inverse_lpRati
│ │ │ │ -0006ace0: 6f6e 616c 4d61 705f 7270 2c20 2d2d 2069  onalMap_rp, -- i
│ │ │ │ -0006acf0: 6e76 6572 7365 206f 6620 610a 2020 2020  nverse of a.    
│ │ │ │ -0006ad00: 6269 7261 7469 6f6e 616c 206d 6170 0a20  birational map. 
│ │ │ │ -0006ad10: 202a 2022 696e 7665 7273 6528 5261 7469   * "inverse(Rati
│ │ │ │ -0006ad20: 6f6e 616c 4d61 702c 4f70 7469 6f6e 2922  onalMap,Option)"
│ │ │ │ -0006ad30: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │ +0006ab30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0006ab40: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ +0006ab50: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2069  ===..  * *note i
│ │ │ │ +0006ab60: 6e76 6572 7365 4d61 703a 2069 6e76 6572  nverseMap: inver
│ │ │ │ +0006ab70: 7365 4d61 702c 202d 2d20 696e 7665 7273  seMap, -- invers
│ │ │ │ +0006ab80: 6520 6f66 2061 2062 6972 6174 696f 6e61  e of a birationa
│ │ │ │ +0006ab90: 6c20 6d61 700a 2020 2a20 2a6e 6f74 6520  l map.  * *note 
│ │ │ │ +0006aba0: 5261 7469 6f6e 616c 4d61 7020 5e20 5a5a  RationalMap ^ ZZ
│ │ │ │ +0006abb0: 3a20 5261 7469 6f6e 616c 4d61 7020 5e20  : RationalMap ^ 
│ │ │ │ +0006abc0: 5a5a 2c20 2d2d 2070 6f77 6572 0a20 202a  ZZ, -- power.  *
│ │ │ │ +0006abd0: 202a 6e6f 7465 2069 7349 6e76 6572 7365   *note isInverse
│ │ │ │ +0006abe0: 4d61 7028 5261 7469 6f6e 616c 4d61 702c  Map(RationalMap,
│ │ │ │ +0006abf0: 5261 7469 6f6e 616c 4d61 7029 3a0a 2020  RationalMap):.  
│ │ │ │ +0006ac00: 2020 6973 496e 7665 7273 654d 6170 5f6c    isInverseMap_l
│ │ │ │ +0006ac10: 7052 6174 696f 6e61 6c4d 6170 5f63 6d52  pRationalMap_cmR
│ │ │ │ +0006ac20: 6174 696f 6e61 6c4d 6170 5f72 702c 202d  ationalMap_rp, -
│ │ │ │ +0006ac30: 2d20 6368 6563 6b73 2077 6865 7468 6572  - checks whether
│ │ │ │ +0006ac40: 2074 776f 2072 6174 696f 6e61 6c0a 2020   two rational.  
│ │ │ │ +0006ac50: 2020 6d61 7073 2061 7265 206f 6e65 2074    maps are one t
│ │ │ │ +0006ac60: 6865 2069 6e76 6572 7365 206f 6620 7468  he inverse of th
│ │ │ │ +0006ac70: 6520 6f74 6865 720a 0a57 6179 7320 746f  e other..Ways to
│ │ │ │ +0006ac80: 2075 7365 2074 6869 7320 6d65 7468 6f64   use this method
│ │ │ │ +0006ac90: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +0006aca0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +0006acb0: 2a6e 6f74 6520 696e 7665 7273 6528 5261  *note inverse(Ra
│ │ │ │ +0006acc0: 7469 6f6e 616c 4d61 7029 3a20 696e 7665  tionalMap): inve
│ │ │ │ +0006acd0: 7273 655f 6c70 5261 7469 6f6e 616c 4d61  rse_lpRationalMa
│ │ │ │ +0006ace0: 705f 7270 2c20 2d2d 2069 6e76 6572 7365  p_rp, -- inverse
│ │ │ │ +0006acf0: 206f 6620 610a 2020 2020 6269 7261 7469   of a.    birati
│ │ │ │ +0006ad00: 6f6e 616c 206d 6170 0a20 202a 2022 696e  onal map.  * "in
│ │ │ │ +0006ad10: 7665 7273 6528 5261 7469 6f6e 616c 4d61  verse(RationalMa
│ │ │ │ +0006ad20: 702c 4f70 7469 6f6e 2922 0a2d 2d2d 2d2d  p,Option)".-----
│ │ │ │ +0006ad30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ad40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ad50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ad60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ad80: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ -0006ad90: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
│ │ │ │ -0006ada0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f   in./build/repro
│ │ │ │ -0006adb0: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ -0006adc0: 6175 6c61 7932 2d31 2e32 352e 3131 2b64  aulay2-1.25.11+d
│ │ │ │ -0006add0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ -0006ade0: 6163 6b61 6765 732f 4372 656d 6f6e 612f  ackages/Cremona/
│ │ │ │ -0006adf0: 0a64 6f63 756d 656e 7461 7469 6f6e 2e6d  .documentation.m
│ │ │ │ -0006ae00: 323a 3431 323a 302e 0a1f 0a46 696c 653a  2:412:0....File:
│ │ │ │ -0006ae10: 2043 7265 6d6f 6e61 2e69 6e66 6f2c 204e   Cremona.info, N
│ │ │ │ -0006ae20: 6f64 653a 2069 6e76 6572 7365 4d61 702c  ode: inverseMap,
│ │ │ │ -0006ae30: 204e 6578 743a 2069 6e76 6572 7365 4d61   Next: inverseMa
│ │ │ │ -0006ae40: 705f 6c70 5f70 645f 7064 5f70 645f 636d  p_lp_pd_pd_pd_cm
│ │ │ │ -0006ae50: 5665 7262 6f73 653d 3e5f 7064 5f70 645f  Verbose=>_pd_pd_
│ │ │ │ -0006ae60: 7064 5f72 702c 2050 7265 763a 2069 6e76  pd_rp, Prev: inv
│ │ │ │ -0006ae70: 6572 7365 5f6c 7052 6174 696f 6e61 6c4d  erse_lpRationalM
│ │ │ │ -0006ae80: 6170 5f72 702c 2055 703a 2054 6f70 0a0a  ap_rp, Up: Top..
│ │ │ │ -0006ae90: 696e 7665 7273 654d 6170 202d 2d20 696e  inverseMap -- in
│ │ │ │ -0006aea0: 7665 7273 6520 6f66 2061 2062 6972 6174  verse of a birat
│ │ │ │ -0006aeb0: 696f 6e61 6c20 6d61 700a 2a2a 2a2a 2a2a  ional map.******
│ │ │ │ +0006ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
│ │ │ │ +0006ad80: 736f 7572 6365 206f 6620 7468 6973 2064  source of this d
│ │ │ │ +0006ad90: 6f63 756d 656e 7420 6973 2069 6e0a 2f62  ocument is in./b
│ │ │ │ +0006ada0: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ +0006adb0: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ +0006adc0: 2d31 2e32 352e 3131 2b64 732f 4d32 2f4d  -1.25.11+ds/M2/M
│ │ │ │ +0006add0: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ +0006ade0: 732f 4372 656d 6f6e 612f 0a64 6f63 756d  s/Cremona/.docum
│ │ │ │ +0006adf0: 656e 7461 7469 6f6e 2e6d 323a 3431 323a  entation.m2:412:
│ │ │ │ +0006ae00: 302e 0a1f 0a46 696c 653a 2043 7265 6d6f  0....File: Cremo
│ │ │ │ +0006ae10: 6e61 2e69 6e66 6f2c 204e 6f64 653a 2069  na.info, Node: i
│ │ │ │ +0006ae20: 6e76 6572 7365 4d61 702c 204e 6578 743a  nverseMap, Next:
│ │ │ │ +0006ae30: 2069 6e76 6572 7365 4d61 705f 6c70 5f70   inverseMap_lp_p
│ │ │ │ +0006ae40: 645f 7064 5f70 645f 636d 5665 7262 6f73  d_pd_pd_cmVerbos
│ │ │ │ +0006ae50: 653d 3e5f 7064 5f70 645f 7064 5f72 702c  e=>_pd_pd_pd_rp,
│ │ │ │ +0006ae60: 2050 7265 763a 2069 6e76 6572 7365 5f6c   Prev: inverse_l
│ │ │ │ +0006ae70: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ +0006ae80: 2055 703a 2054 6f70 0a0a 696e 7665 7273   Up: Top..invers
│ │ │ │ +0006ae90: 654d 6170 202d 2d20 696e 7665 7273 6520  eMap -- inverse 
│ │ │ │ +0006aea0: 6f66 2061 2062 6972 6174 696f 6e61 6c20  of a birational 
│ │ │ │ +0006aeb0: 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  map.************
│ │ │ │  0006aec0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006aed0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006aee0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ -0006aef0: 0a20 2020 2020 2020 2069 6e76 6572 7365  .        inverse
│ │ │ │ -0006af00: 4d61 7020 7068 690a 2020 2a20 496e 7075  Map phi.  * Inpu
│ │ │ │ -0006af10: 7473 3a0a 2020 2020 2020 2a20 7068 692c  ts:.      * phi,
│ │ │ │ -0006af20: 2061 202a 6e6f 7465 2072 6174 696f 6e61   a *note rationa
│ │ │ │ -0006af30: 6c20 6d61 703a 2052 6174 696f 6e61 6c4d  l map: RationalM
│ │ │ │ -0006af40: 6170 2c2c 2061 2062 6972 6174 696f 6e61  ap,, a birationa
│ │ │ │ -0006af50: 6c20 6d61 700a 2020 2a20 2a6e 6f74 6520  l map.  * *note 
│ │ │ │ -0006af60: 4f70 7469 6f6e 616c 2069 6e70 7574 733a  Optional inputs:
│ │ │ │ -0006af70: 2028 4d61 6361 756c 6179 3244 6f63 2975   (Macaulay2Doc)u
│ │ │ │ -0006af80: 7369 6e67 2066 756e 6374 696f 6e73 2077  sing functions w
│ │ │ │ -0006af90: 6974 6820 6f70 7469 6f6e 616c 2069 6e70  ith optional inp
│ │ │ │ -0006afa0: 7574 732c 3a0a 2020 2020 2020 2a20 2a6e  uts,:.      * *n
│ │ │ │ -0006afb0: 6f74 6520 426c 6f77 5570 5374 7261 7465  ote BlowUpStrate
│ │ │ │ -0006afc0: 6779 3a20 426c 6f77 5570 5374 7261 7465  gy: BlowUpStrate
│ │ │ │ -0006afd0: 6779 2c20 3d3e 202e 2e2e 2c20 6465 6661  gy, => ..., defa
│ │ │ │ -0006afe0: 756c 7420 7661 6c75 650a 2020 2020 2020  ult value.      
│ │ │ │ -0006aff0: 2020 2245 6c69 6d69 6e61 7465 222c 0a20    "Eliminate",. 
│ │ │ │ -0006b000: 2020 2020 202a 202a 6e6f 7465 2043 6572       * *note Cer
│ │ │ │ -0006b010: 7469 6679 3a20 4365 7274 6966 792c 203d  tify: Certify, =
│ │ │ │ -0006b020: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ -0006b030: 616c 7565 2066 616c 7365 2c20 7768 6574  alue false, whet
│ │ │ │ -0006b040: 6865 7220 746f 2065 6e73 7572 650a 2020  her to ensure.  
│ │ │ │ -0006b050: 2020 2020 2020 636f 7272 6563 746e 6573        correctnes
│ │ │ │ -0006b060: 7320 6f66 206f 7574 7075 740a 2020 2020  s of output.    
│ │ │ │ -0006b070: 2020 2a20 2a6e 6f74 6520 5665 7262 6f73    * *note Verbos
│ │ │ │ -0006b080: 653a 2069 6e76 6572 7365 4d61 705f 6c70  e: inverseMap_lp
│ │ │ │ -0006b090: 5f70 645f 7064 5f70 645f 636d 5665 7262  _pd_pd_pd_cmVerb
│ │ │ │ -0006b0a0: 6f73 653d 3e5f 7064 5f70 645f 7064 5f72  ose=>_pd_pd_pd_r
│ │ │ │ -0006b0b0: 702c 203d 3e20 2e2e 2e2c 0a20 2020 2020  p, => ...,.     
│ │ │ │ -0006b0c0: 2020 2064 6566 6175 6c74 2076 616c 7565     default value
│ │ │ │ -0006b0d0: 2074 7275 652c 0a20 202a 204f 7574 7075   true,.  * Outpu
│ │ │ │ -0006b0e0: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e  ts:.      * a *n
│ │ │ │ -0006b0f0: 6f74 6520 7261 7469 6f6e 616c 206d 6170  ote rational map
│ │ │ │ -0006b100: 3a20 5261 7469 6f6e 616c 4d61 702c 2c20  : RationalMap,, 
│ │ │ │ -0006b110: 7468 6520 696e 7665 7273 6520 6d61 7020  the inverse map 
│ │ │ │ -0006b120: 6f66 2070 6869 0a0a 4465 7363 7269 7074  of phi..Descript
│ │ │ │ -0006b130: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ -0006b140: 0a49 6620 7468 6520 736f 7572 6365 2076  .If the source v
│ │ │ │ -0006b150: 6172 6965 7479 2069 7320 6120 7072 6f6a  ariety is a proj
│ │ │ │ -0006b160: 6563 7469 7665 2073 7061 6365 2061 6e64  ective space and
│ │ │ │ -0006b170: 2069 6620 6120 6675 7274 6865 7220 7465   if a further te
│ │ │ │ -0006b180: 6368 6e69 6361 6c0a 636f 6e64 6974 696f  chnical.conditio
│ │ │ │ -0006b190: 6e20 6973 2073 6174 6973 6669 6564 2c20  n is satisfied, 
│ │ │ │ -0006b1a0: 7468 656e 2074 6865 2061 6c67 6f72 6974  then the algorit
│ │ │ │ -0006b1b0: 686d 2075 7365 6420 6973 2074 6861 7420  hm used is that 
│ │ │ │ -0006b1c0: 6465 7363 7269 6265 6420 696e 2074 6865  described in the
│ │ │ │ -0006b1d0: 2070 6170 6572 0a62 7920 5275 7373 6f20   paper.by Russo 
│ │ │ │ -0006b1e0: 616e 6420 5369 6d69 7320 2d20 4f6e 2062  and Simis - On b
│ │ │ │ -0006b1f0: 6972 6174 696f 6e61 6c20 6d61 7073 2061  irational maps a
│ │ │ │ -0006b200: 6e64 204a 6163 6f62 6961 6e20 6d61 7472  nd Jacobian matr
│ │ │ │ -0006b210: 6963 6573 202d 2043 6f6d 706f 732e 204d  ices - Compos. M
│ │ │ │ -0006b220: 6174 682e 0a31 3236 2028 3329 2c20 3333  ath..126 (3), 33
│ │ │ │ -0006b230: 352d 3335 382c 2032 3030 312e 2046 6f72  5-358, 2001. For
│ │ │ │ -0006b240: 2074 6865 2067 656e 6572 616c 2063 6173   the general cas
│ │ │ │ -0006b250: 652c 2074 6865 2061 6c67 6f72 6974 686d  e, the algorithm
│ │ │ │ -0006b260: 2075 7365 6420 6973 2074 6865 2073 616d   used is the sam
│ │ │ │ -0006b270: 6520 6173 0a66 6f72 202a 6e6f 7465 2069  e as.for *note i
│ │ │ │ -0006b280: 6e76 6572 7442 6972 6174 696f 6e61 6c4d  nvertBirationalM
│ │ │ │ -0006b290: 6170 3a20 2850 6172 616d 6574 7269 7a61  ap: (Parametriza
│ │ │ │ -0006b2a0: 7469 6f6e 2969 6e76 6572 7442 6972 6174  tion)invertBirat
│ │ │ │ -0006b2b0: 696f 6e61 6c4d 6170 2c20 696e 2074 6865  ionalMap, in the
│ │ │ │ -0006b2c0: 0a70 6163 6b61 6765 202a 6e6f 7465 2050  .package *note P
│ │ │ │ -0006b2d0: 6172 616d 6574 7269 7a61 7469 6f6e 3a20  arametrization: 
│ │ │ │ -0006b2e0: 2850 6172 616d 6574 7269 7a61 7469 6f6e  (Parametrization
│ │ │ │ -0006b2f0: 2954 6f70 2c2e 204e 6f74 6520 7468 6174  )Top,. Note that
│ │ │ │ -0006b300: 2069 6e20 7468 6973 2063 6173 652c 0a74   in this case,.t
│ │ │ │ -0006b310: 6865 2061 6e61 6c6f 676f 7573 206d 6574  he analogous met
│ │ │ │ -0006b320: 686f 6420 2a6e 6f74 6520 696e 7665 7273  hod *note invers
│ │ │ │ -0006b330: 654f 664d 6170 3a20 2852 6174 696f 6e61  eOfMap: (Rationa
│ │ │ │ -0006b340: 6c4d 6170 7329 696e 7665 7273 654f 664d  lMaps)inverseOfM
│ │ │ │ -0006b350: 6170 2c20 696e 2074 6865 0a70 6163 6b61  ap, in the.packa
│ │ │ │ -0006b360: 6765 2052 6174 696f 6e61 6c4d 6170 7320  ge RationalMaps 
│ │ │ │ -0006b370: 286d 6973 7369 6e67 2064 6f63 756d 656e  (missing documen
│ │ │ │ -0006b380: 7461 7469 6f6e 2920 6765 6e65 7261 6c6c  tation) generall
│ │ │ │ -0006b390: 7920 7475 726e 7320 6f75 7420 746f 2062  y turns out to b
│ │ │ │ -0006b3a0: 6520 6661 7374 6572 2e0a 0a2b 2d2d 2d2d  e faster...+----
│ │ │ │ +0006aed0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20  *************.. 
│ │ │ │ +0006aee0: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ +0006aef0: 2020 2069 6e76 6572 7365 4d61 7020 7068     inverseMap ph
│ │ │ │ +0006af00: 690a 2020 2a20 496e 7075 7473 3a0a 2020  i.  * Inputs:.  
│ │ │ │ +0006af10: 2020 2020 2a20 7068 692c 2061 202a 6e6f      * phi, a *no
│ │ │ │ +0006af20: 7465 2072 6174 696f 6e61 6c20 6d61 703a  te rational map:
│ │ │ │ +0006af30: 2052 6174 696f 6e61 6c4d 6170 2c2c 2061   RationalMap,, a
│ │ │ │ +0006af40: 2062 6972 6174 696f 6e61 6c20 6d61 700a   birational map.
│ │ │ │ +0006af50: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e    * *note Option
│ │ │ │ +0006af60: 616c 2069 6e70 7574 733a 2028 4d61 6361  al inputs: (Maca
│ │ │ │ +0006af70: 756c 6179 3244 6f63 2975 7369 6e67 2066  ulay2Doc)using f
│ │ │ │ +0006af80: 756e 6374 696f 6e73 2077 6974 6820 6f70  unctions with op
│ │ │ │ +0006af90: 7469 6f6e 616c 2069 6e70 7574 732c 3a0a  tional inputs,:.
│ │ │ │ +0006afa0: 2020 2020 2020 2a20 2a6e 6f74 6520 426c        * *note Bl
│ │ │ │ +0006afb0: 6f77 5570 5374 7261 7465 6779 3a20 426c  owUpStrategy: Bl
│ │ │ │ +0006afc0: 6f77 5570 5374 7261 7465 6779 2c20 3d3e  owUpStrategy, =>
│ │ │ │ +0006afd0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ +0006afe0: 6c75 650a 2020 2020 2020 2020 2245 6c69  lue.        "Eli
│ │ │ │ +0006aff0: 6d69 6e61 7465 222c 0a20 2020 2020 202a  minate",.      *
│ │ │ │ +0006b000: 202a 6e6f 7465 2043 6572 7469 6679 3a20   *note Certify: 
│ │ │ │ +0006b010: 4365 7274 6966 792c 203d 3e20 2e2e 2e2c  Certify, => ...,
│ │ │ │ +0006b020: 2064 6566 6175 6c74 2076 616c 7565 2066   default value f
│ │ │ │ +0006b030: 616c 7365 2c20 7768 6574 6865 7220 746f  alse, whether to
│ │ │ │ +0006b040: 2065 6e73 7572 650a 2020 2020 2020 2020   ensure.        
│ │ │ │ +0006b050: 636f 7272 6563 746e 6573 7320 6f66 206f  correctness of o
│ │ │ │ +0006b060: 7574 7075 740a 2020 2020 2020 2a20 2a6e  utput.      * *n
│ │ │ │ +0006b070: 6f74 6520 5665 7262 6f73 653a 2069 6e76  ote Verbose: inv
│ │ │ │ +0006b080: 6572 7365 4d61 705f 6c70 5f70 645f 7064  erseMap_lp_pd_pd
│ │ │ │ +0006b090: 5f70 645f 636d 5665 7262 6f73 653d 3e5f  _pd_cmVerbose=>_
│ │ │ │ +0006b0a0: 7064 5f70 645f 7064 5f72 702c 203d 3e20  pd_pd_pd_rp, => 
│ │ │ │ +0006b0b0: 2e2e 2e2c 0a20 2020 2020 2020 2064 6566  ...,.        def
│ │ │ │ +0006b0c0: 6175 6c74 2076 616c 7565 2074 7275 652c  ault value true,
│ │ │ │ +0006b0d0: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ +0006b0e0: 2020 2020 2a20 6120 2a6e 6f74 6520 7261      * a *note ra
│ │ │ │ +0006b0f0: 7469 6f6e 616c 206d 6170 3a20 5261 7469  tional map: Rati
│ │ │ │ +0006b100: 6f6e 616c 4d61 702c 2c20 7468 6520 696e  onalMap,, the in
│ │ │ │ +0006b110: 7665 7273 6520 6d61 7020 6f66 2070 6869  verse map of phi
│ │ │ │ +0006b120: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +0006b130: 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620 7468  =========..If th
│ │ │ │ +0006b140: 6520 736f 7572 6365 2076 6172 6965 7479  e source variety
│ │ │ │ +0006b150: 2069 7320 6120 7072 6f6a 6563 7469 7665   is a projective
│ │ │ │ +0006b160: 2073 7061 6365 2061 6e64 2069 6620 6120   space and if a 
│ │ │ │ +0006b170: 6675 7274 6865 7220 7465 6368 6e69 6361  further technica
│ │ │ │ +0006b180: 6c0a 636f 6e64 6974 696f 6e20 6973 2073  l.condition is s
│ │ │ │ +0006b190: 6174 6973 6669 6564 2c20 7468 656e 2074  atisfied, then t
│ │ │ │ +0006b1a0: 6865 2061 6c67 6f72 6974 686d 2075 7365  he algorithm use
│ │ │ │ +0006b1b0: 6420 6973 2074 6861 7420 6465 7363 7269  d is that descri
│ │ │ │ +0006b1c0: 6265 6420 696e 2074 6865 2070 6170 6572  bed in the paper
│ │ │ │ +0006b1d0: 0a62 7920 5275 7373 6f20 616e 6420 5369  .by Russo and Si
│ │ │ │ +0006b1e0: 6d69 7320 2d20 4f6e 2062 6972 6174 696f  mis - On biratio
│ │ │ │ +0006b1f0: 6e61 6c20 6d61 7073 2061 6e64 204a 6163  nal maps and Jac
│ │ │ │ +0006b200: 6f62 6961 6e20 6d61 7472 6963 6573 202d  obian matrices -
│ │ │ │ +0006b210: 2043 6f6d 706f 732e 204d 6174 682e 0a31   Compos. Math..1
│ │ │ │ +0006b220: 3236 2028 3329 2c20 3333 352d 3335 382c  26 (3), 335-358,
│ │ │ │ +0006b230: 2032 3030 312e 2046 6f72 2074 6865 2067   2001. For the g
│ │ │ │ +0006b240: 656e 6572 616c 2063 6173 652c 2074 6865  eneral case, the
│ │ │ │ +0006b250: 2061 6c67 6f72 6974 686d 2075 7365 6420   algorithm used 
│ │ │ │ +0006b260: 6973 2074 6865 2073 616d 6520 6173 0a66  is the same as.f
│ │ │ │ +0006b270: 6f72 202a 6e6f 7465 2069 6e76 6572 7442  or *note invertB
│ │ │ │ +0006b280: 6972 6174 696f 6e61 6c4d 6170 3a20 2850  irationalMap: (P
│ │ │ │ +0006b290: 6172 616d 6574 7269 7a61 7469 6f6e 2969  arametrization)i
│ │ │ │ +0006b2a0: 6e76 6572 7442 6972 6174 696f 6e61 6c4d  nvertBirationalM
│ │ │ │ +0006b2b0: 6170 2c20 696e 2074 6865 0a70 6163 6b61  ap, in the.packa
│ │ │ │ +0006b2c0: 6765 202a 6e6f 7465 2050 6172 616d 6574  ge *note Paramet
│ │ │ │ +0006b2d0: 7269 7a61 7469 6f6e 3a20 2850 6172 616d  rization: (Param
│ │ │ │ +0006b2e0: 6574 7269 7a61 7469 6f6e 2954 6f70 2c2e  etrization)Top,.
│ │ │ │ +0006b2f0: 204e 6f74 6520 7468 6174 2069 6e20 7468   Note that in th
│ │ │ │ +0006b300: 6973 2063 6173 652c 0a74 6865 2061 6e61  is case,.the ana
│ │ │ │ +0006b310: 6c6f 676f 7573 206d 6574 686f 6420 2a6e  logous method *n
│ │ │ │ +0006b320: 6f74 6520 696e 7665 7273 654f 664d 6170  ote inverseOfMap
│ │ │ │ +0006b330: 3a20 2852 6174 696f 6e61 6c4d 6170 7329  : (RationalMaps)
│ │ │ │ +0006b340: 696e 7665 7273 654f 664d 6170 2c20 696e  inverseOfMap, in
│ │ │ │ +0006b350: 2074 6865 0a70 6163 6b61 6765 2052 6174   the.package Rat
│ │ │ │ +0006b360: 696f 6e61 6c4d 6170 7320 286d 6973 7369  ionalMaps (missi
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│ │ │ │ +0006b3a0: 6572 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  er...+----------
│ │ │ │  0006b3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b3f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ -0006b400: 202d 2d20 4120 4372 656d 6f6e 6120 7472   -- A Cremona tr
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│ │ │ │ -0006b420: 505e 3230 2020 2020 2020 2020 2020 2020  P^20            
│ │ │ │ +0006b3f0: 2d2d 2d2b 0a7c 6931 203a 202d 2d20 4120  ---+.|i1 : -- A 
│ │ │ │ +0006b400: 4372 656d 6f6e 6120 7472 616e 7366 6f72  Cremona transfor
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│ │ │ │ +0006b420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b440: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006b450: 2070 6869 203d 2072 6174 696f 6e61 6c4d   phi = rationalM
│ │ │ │ -0006b460: 6170 206d 6170 2071 7561 6472 6f51 7561  ap map quadroQua
│ │ │ │ -0006b470: 6472 6963 4372 656d 6f6e 6154 7261 6e73  dricCremonaTrans
│ │ │ │ -0006b480: 666f 726d 6174 696f 6e28 3230 2c31 2920  formation(20,1) 
│ │ │ │ -0006b490: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006b440: 2020 207c 0a7c 2020 2020 2070 6869 203d     |.|     phi =
│ │ │ │ +0006b450: 2072 6174 696f 6e61 6c4d 6170 206d 6170   rationalMap map
│ │ │ │ +0006b460: 2071 7561 6472 6f51 7561 6472 6963 4372   quadroQuadricCr
│ │ │ │ +0006b470: 656d 6f6e 6154 7261 6e73 666f 726d 6174  emonaTransformat
│ │ │ │ +0006b480: 696f 6e28 3230 2c31 2920 2020 2020 2020  ion(20,1)       
│ │ │ │ +0006b490: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b4e0: 2020 2020 2020 2020 207c 0a7c 6f31 203d           |.|o1 =
│ │ │ │ -0006b4f0: 202d 2d20 7261 7469 6f6e 616c 206d 6170   -- rational map
│ │ │ │ -0006b500: 202d 2d20 2020 2020 2020 2020 2020 2020   --             
│ │ │ │ +0006b4e0: 2020 207c 0a7c 6f31 203d 202d 2d20 7261     |.|o1 = -- ra
│ │ │ │ +0006b4f0: 7469 6f6e 616c 206d 6170 202d 2d20 2020  tional map --   
│ │ │ │ +0006b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b530: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006b540: 2073 6f75 7263 653a 2050 726f 6a28 5151   source: Proj(QQ
│ │ │ │ -0006b550: 5b77 202c 2077 202c 2077 202c 2077 202c  [w , w , w , w ,
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│ │ │ │ -0006b580: 202c 2077 2020 2c20 207c 0a7c 2020 2020   , w  ,  |.|    
│ │ │ │ -0006b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b5a0: 2020 3020 2020 3120 2020 3220 2020 3320    0   1   2   3 
│ │ │ │ -0006b5b0: 2020 3420 2020 3520 2020 3620 2020 3720    4   5   6   7 
│ │ │ │ -0006b5c0: 2020 3820 2020 3920 2020 3130 2020 2031    8   9   10   1
│ │ │ │ -0006b5d0: 3120 2020 3132 2020 207c 0a7c 2020 2020  1   12   |.|    
│ │ │ │ -0006b5e0: 2074 6172 6765 743a 2050 726f 6a28 5151   target: Proj(QQ
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│ │ │ │ -0006b600: 2077 202c 2077 202c 2077 202c 2077 202c   w , w , w , w ,
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│ │ │ │ -0006b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006b650: 2020 3420 2020 3520 2020 3620 2020 3720    4   5   6   7 
│ │ │ │ -0006b660: 2020 3820 2020 3920 2020 3130 2020 2031    8   9   10   1
│ │ │ │ -0006b670: 3120 2020 3132 2020 207c 0a7c 2020 2020  1   12   |.|    
│ │ │ │ -0006b680: 2064 6566 696e 696e 6720 666f 726d 733a   defining forms:
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│ │ │ │ +0006b5a0: 3120 2020 3220 2020 3320 2020 3420 2020  1   2   3   4   
│ │ │ │ +0006b5b0: 3520 2020 3620 2020 3720 2020 3820 2020  5   6   7   8   
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│ │ │ │ +0006b5e0: 743a 2050 726f 6a28 5151 5b77 202c 2077  t: Proj(QQ[w , w
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│ │ │ │ +0006b600: 202c 2077 202c 2077 202c 2077 202c 2077   , w , w , w , w
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│ │ │ │ +0006b640: 3120 2020 3220 2020 3320 2020 3420 2020  1   2   3   4   
│ │ │ │ +0006b650: 3520 2020 3620 2020 3720 2020 3820 2020  5   6   7   8   
│ │ │ │ +0006b660: 3920 2020 3130 2020 2031 3120 2020 3132  9   10   11   12
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│ │ │ │ +0006b6d0: 2020 2020 2020 2020 2020 2020 7720 2077              w  w
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│ │ │ │ -0006b7b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006b7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b7d0: 2020 7720 2077 2020 202d 2077 2077 2020    w  w   - w w  
│ │ │ │ -0006b7e0: 202b 2077 2077 2020 2c20 2020 2020 2020   + w w  ,       
│ │ │ │ +0006b7b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006b7c0: 2020 2020 2020 2020 2020 2020 7720 2077              w  w
│ │ │ │ +0006b7d0: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ +0006b7e0: 2020 2c20 2020 2020 2020 2020 2020 2020    ,             
│ │ │ │  0006b7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006b820: 2020 2031 3020 3134 2020 2020 3820 3136     10 14    8 16
│ │ │ │ -0006b830: 2020 2020 3520 3230 2020 2020 2020 2020      5 20        
│ │ │ │ +0006b800: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006b810: 2020 2020 2020 2020 2020 2020 2031 3020               10 
│ │ │ │ +0006b820: 3134 2020 2020 3820 3136 2020 2020 3520  14    8 16    5 
│ │ │ │ +0006b830: 3230 2020 2020 2020 2020 2020 2020 2020  20              
│ │ │ │  0006b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b850: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006b850: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006b860: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006b8c0: 2020 7720 7720 2020 2d20 7720 7720 2020    w w   - w w   
│ │ │ │ -0006b8d0: 2b20 7720 7720 202c 2020 2020 2020 2020  + w w  ,        
│ │ │ │ +0006b8a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ +0006b8d0: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
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│ │ │ │ -0006b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006b920: 2020 2034 2032 3020 2020 2020 2020 2020     4 20         
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│ │ │ │ +0006b920: 3020 2020 2020 2020 2020 2020 2020 2020  0               
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│ │ │ │ -0006b9b0: 2020 7720 7720 2020 2d20 7720 7720 2020    w w   - w w   
│ │ │ │ -0006b9c0: 2b20 7720 7720 202c 2020 2020 2020 2020  + w w  ,        
│ │ │ │ +0006b990: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006b9a0: 2020 2020 2020 2020 2020 2020 7720 7720              w w 
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│ │ │ │ +0006b9c0: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
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│ │ │ │ -0006b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ba00: 2020 2036 2031 3420 2020 2035 2031 3520     6 14    5 15 
│ │ │ │ -0006ba10: 2020 2034 2031 3620 2020 2020 2020 2020     4 16         
│ │ │ │ +0006b9e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006b9f0: 2020 2020 2020 2020 2020 2020 2036 2031               6 1
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│ │ │ │ -0006baa0: 2020 7720 2077 2020 202d 2077 2020 7720    w  w   - w  w 
│ │ │ │ -0006bab0: 2020 2b20 7720 2077 2020 202d 2077 2020    + w  w   - w  
│ │ │ │ -0006bac0: 7720 2020 2b20 7720 2077 2020 2c20 2020  w   + w  w  ,   
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│ │ │ │ -0006bb00: 3720 2020 2031 3520 3138 2020 2020 3134  7    15 18    14
│ │ │ │ -0006bb10: 2031 3920 2020 2031 3220 3230 2020 2020   19    12 20    
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│ │ │ │ +0006ba90: 2020 2020 2020 2020 2020 2020 7720 2077              w  w
│ │ │ │ +0006baa0: 2020 202d 2077 2020 7720 2020 2b20 7720     - w  w   + w 
│ │ │ │ +0006bab0: 2077 2020 202d 2077 2020 7720 2020 2b20   w   - w  w   + 
│ │ │ │ +0006bac0: 7720 2077 2020 2c20 2020 2020 2020 2020  w  w  ,         
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│ │ │ │ +0006baf0: 3133 2020 2020 3136 2031 3720 2020 2031  13    16 17    1
│ │ │ │ +0006bb00: 3520 3138 2020 2020 3134 2031 3920 2020  5 18    14 19   
│ │ │ │ +0006bb10: 2031 3220 3230 2020 2020 2020 2020 2020   12 20          
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│ │ │ │ -0006bb90: 2020 7720 7720 2020 2d20 7720 2077 2020    w w   - w  w  
│ │ │ │ -0006bba0: 202b 2077 2077 2020 202d 2077 2077 2020   + w w   - w w  
│ │ │ │ -0006bbb0: 202b 2077 2077 2020 2c20 2020 2020 2020   + w w  ,       
│ │ │ │ -0006bbc0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bbe0: 2020 2033 2031 3320 2020 2031 3020 3137     3 13    10 17
│ │ │ │ -0006bbf0: 2020 2020 3920 3138 2020 2020 3820 3139      9 18    8 19
│ │ │ │ -0006bc00: 2020 2020 3720 3230 2020 2020 2020 2020      7 20        
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│ │ │ │ +0006bb80: 2020 2020 2020 2020 2020 2020 7720 7720              w w 
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│ │ │ │ +0006bba0: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ +0006bbb0: 2020 2c20 2020 2020 2020 2020 2020 2020    ,             
│ │ │ │ +0006bbc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006bbd0: 2020 2020 2020 2020 2020 2020 2033 2031               3 1
│ │ │ │ +0006bbe0: 3320 2020 2031 3020 3137 2020 2020 3920  3    10 17    9 
│ │ │ │ +0006bbf0: 3138 2020 2020 3820 3139 2020 2020 3720  18    8 19    7 
│ │ │ │ +0006bc00: 3230 2020 2020 2020 2020 2020 2020 2020  20              
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│ │ │ │ -0006bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bc80: 2020 7720 2077 2020 202d 2077 2077 2020    w  w   - w w  
│ │ │ │ -0006bc90: 202d 2077 2077 2020 202d 2077 2077 2020   - w w   - w w  
│ │ │ │ -0006bca0: 202b 2077 2077 2020 2c20 2020 2020 2020   + w w  ,       
│ │ │ │ -0006bcb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bcd0: 2020 2031 3020 3132 2020 2020 3220 3133     10 12    2 13
│ │ │ │ -0006bce0: 2020 2020 3720 3136 2020 2020 3620 3138      7 16    6 18
│ │ │ │ -0006bcf0: 2020 2020 3520 3139 2020 2020 2020 2020      5 19        
│ │ │ │ -0006bd00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
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│ │ │ │ +0006bc80: 2020 202d 2077 2077 2020 202d 2077 2077     - w w   - w w
│ │ │ │ +0006bc90: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ +0006bca0: 2020 2c20 2020 2020 2020 2020 2020 2020    ,             
│ │ │ │ +0006bcb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006bcc0: 2020 2020 2020 2020 2020 2020 2031 3020               10 
│ │ │ │ +0006bcd0: 3132 2020 2020 3220 3133 2020 2020 3720  12    2 13    7 
│ │ │ │ +0006bce0: 3136 2020 2020 3620 3138 2020 2020 3520  16    6 18    5 
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│ │ │ │  0006bd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bd70: 2020 7720 7720 2020 2d20 7720 7720 2020    w w   - w w   
│ │ │ │ -0006bd80: 2d20 7720 7720 2020 2d20 7720 7720 2020  - w w   - w w   
│ │ │ │ -0006bd90: 2b20 7720 7720 202c 2020 2020 2020 2020  + w w  ,        
│ │ │ │ -0006bda0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bdc0: 2020 2039 2031 3220 2020 2031 2031 3320     9 12    1 13 
│ │ │ │ -0006bdd0: 2020 2037 2031 3520 2020 2036 2031 3720     7 15    6 17 
│ │ │ │ -0006bde0: 2020 2034 2031 3920 2020 2020 2020 2020     4 19         
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│ │ │ │ +0006bd80: 2020 2d20 7720 7720 2020 2b20 7720 7720    - w w   + w w 
│ │ │ │ +0006bd90: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
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│ │ │ │ +0006bdb0: 2020 2020 2020 2020 2020 2020 2039 2031               9 1
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│ │ │ │ +0006bdd0: 3520 2020 2036 2031 3720 2020 2034 2031  5    6 17    4 1
│ │ │ │ +0006bde0: 3920 2020 2020 2020 2020 2020 2020 2020  9               
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│ │ │ │ -0006be60: 2020 7720 7720 2020 2d20 7720 7720 2020    w w   - w w   
│ │ │ │ -0006be70: 2d20 7720 7720 2020 2d20 7720 7720 2020  - w w   - w w   
│ │ │ │ -0006be80: 2b20 7720 7720 202c 2020 2020 2020 2020  + w w  ,        
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│ │ │ │ -0006bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006beb0: 2020 2038 2031 3220 2020 2030 2031 3320     8 12    0 13 
│ │ │ │ -0006bec0: 2020 2037 2031 3420 2020 2035 2031 3720     7 14    5 17 
│ │ │ │ -0006bed0: 2020 2034 2031 3820 2020 2020 2020 2020     4 18         
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│ │ │ │ +0006be50: 2020 2020 2020 2020 2020 2020 7720 7720              w w 
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│ │ │ │ +0006bea0: 2020 2020 2020 2020 2020 2020 2038 2031               8 1
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│ │ │ │ +0006bed0: 3820 2020 2020 2020 2020 2020 2020 2020  8               
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│ │ │ │ -0006bf50: 2020 7720 2077 2020 202d 2077 2077 2020    w  w   - w w  
│ │ │ │ -0006bf60: 202b 2077 2077 2020 2c20 2020 2020 2020   + w w  ,       
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│ │ │ │ +0006bf40: 2020 2020 2020 2020 2020 2020 7720 2077              w  w
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│ │ │ │ +0006bf60: 2020 2c20 2020 2020 2020 2020 2020 2020    ,             
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│ │ │ │ -0006bfa0: 2020 2031 3020 3131 2020 2020 3320 3136     10 11    3 16
│ │ │ │ -0006bfb0: 2020 2020 3220 3230 2020 2020 2020 2020      2 20        
│ │ │ │ +0006bf80: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ +0006bfa0: 3131 2020 2020 3320 3136 2020 2020 3220  11    3 16    2 
│ │ │ │ +0006bfb0: 3230 2020 2020 2020 2020 2020 2020 2020  20              
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│ │ │ │ -0006bfd0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006bfd0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ -0006c040: 2020 7720 7720 2020 2d20 7720 7720 2020    w w   - w w   
│ │ │ │ -0006c050: 2b20 7720 7720 202c 2020 2020 2020 2020  + w w  ,        
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│ │ │ │ +0006c030: 2020 2020 2020 2020 2020 2020 7720 7720              w w 
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│ │ │ │ +0006c050: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
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│ │ │ │ -0006c070: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c090: 2020 2039 2031 3120 2020 2033 2031 3520     9 11    3 15 
│ │ │ │ -0006c0a0: 2020 2031 2032 3020 2020 2020 2020 2020     1 20         
│ │ │ │ +0006c070: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c080: 2020 2020 2020 2020 2020 2020 2039 2031               9 1
│ │ │ │ +0006c090: 3120 2020 2033 2031 3520 2020 2031 2032  1    3 15    1 2
│ │ │ │ +0006c0a0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
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│ │ │ │ -0006c0c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006c0c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0006c100: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006c130: 2020 7720 7720 2020 2d20 7720 7720 2020    w w   - w w   
│ │ │ │ -0006c140: 2b20 7720 7720 202c 2020 2020 2020 2020  + w w  ,        
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│ │ │ │ +0006c120: 2020 2020 2020 2020 2020 2020 7720 7720              w w 
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│ │ │ │ -0006c160: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c180: 2020 2038 2031 3120 2020 2033 2031 3420     8 11    3 14 
│ │ │ │ -0006c190: 2020 2030 2032 3020 2020 2020 2020 2020     0 20         
│ │ │ │ +0006c160: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c170: 2020 2020 2020 2020 2020 2020 2038 2031               8 1
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│ │ │ │ +0006c190: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │  0006c1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c1b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006c1b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c200: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006c230: 2b20 7720 7720 2020 2d20 7720 7720 2020  + w w   - w w   
│ │ │ │ -0006c240: 2b20 7720 7720 202c 2020 2020 2020 2020  + w w  ,        
│ │ │ │ -0006c250: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
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│ │ │ │ -0006c270: 2020 2037 2031 3120 2020 2033 2031 3220     7 11    3 12 
│ │ │ │ -0006c280: 2020 2032 2031 3720 2020 2031 2031 3820     2 17    1 18 
│ │ │ │ -0006c290: 2020 2030 2031 3920 2020 2020 2020 2020     0 19         
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│ │ │ │ +0006c200: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c210: 2020 2020 2020 2020 2020 2020 7720 7720              w w 
│ │ │ │ +0006c220: 2020 2d20 7720 7720 2020 2b20 7720 7720    - w w   + w w 
│ │ │ │ +0006c230: 2020 2d20 7720 7720 2020 2b20 7720 7720    - w w   + w w 
│ │ │ │ +0006c240: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │ +0006c250: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c260: 2020 2020 2020 2020 2020 2020 2037 2031               7 1
│ │ │ │ +0006c270: 3120 2020 2033 2031 3220 2020 2032 2031  1    3 12    2 1
│ │ │ │ +0006c280: 3720 2020 2031 2031 3820 2020 2030 2031  7    1 18    0 1
│ │ │ │ +0006c290: 3920 2020 2020 2020 2020 2020 2020 2020  9               
│ │ │ │ +0006c2a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c2f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c310: 2020 7720 7720 2020 2d20 7720 7720 2020    w w   - w w   
│ │ │ │ -0006c320: 2b20 7720 7720 202c 2020 2020 2020 2020  + w w  ,        
│ │ │ │ +0006c2f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c300: 2020 2020 2020 2020 2020 2020 7720 7720              w w 
│ │ │ │ +0006c310: 2020 2d20 7720 7720 2020 2b20 7720 7720    - w w   + w w 
│ │ │ │ +0006c320: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  0006c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c340: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c360: 2020 2036 2031 3120 2020 2032 2031 3520     6 11    2 15 
│ │ │ │ -0006c370: 2020 2031 2031 3620 2020 2020 2020 2020     1 16         
│ │ │ │ +0006c340: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c350: 2020 2020 2020 2020 2020 2020 2036 2031               6 1
│ │ │ │ +0006c360: 3120 2020 2032 2031 3520 2020 2031 2031  1    2 15    1 1
│ │ │ │ +0006c370: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │  0006c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c390: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006c390: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006c3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c3e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c400: 2020 7720 7720 2020 2d20 7720 7720 2020    w w   - w w   
│ │ │ │ -0006c410: 2b20 7720 7720 202c 2020 2020 2020 2020  + w w  ,        
│ │ │ │ +0006c3e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c3f0: 2020 2020 2020 2020 2020 2020 7720 7720              w w 
│ │ │ │ +0006c400: 2020 2d20 7720 7720 2020 2b20 7720 7720    - w w   + w w 
│ │ │ │ +0006c410: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  0006c420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c430: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c450: 2020 2035 2031 3120 2020 2032 2031 3420     5 11    2 14 
│ │ │ │ -0006c460: 2020 2030 2031 3620 2020 2020 2020 2020     0 16         
│ │ │ │ +0006c430: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c440: 2020 2020 2020 2020 2020 2020 2035 2031               5 1
│ │ │ │ +0006c450: 3120 2020 2032 2031 3420 2020 2030 2031  1    2 14    0 1
│ │ │ │ +0006c460: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │  0006c470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c480: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006c480: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006c490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c4d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c4f0: 2020 7720 7720 2020 2d20 7720 7720 2020    w w   - w w   
│ │ │ │ -0006c500: 2b20 7720 7720 202c 2020 2020 2020 2020  + w w  ,        
│ │ │ │ +0006c4d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c4e0: 2020 2020 2020 2020 2020 2020 7720 7720              w w 
│ │ │ │ +0006c4f0: 2020 2d20 7720 7720 2020 2b20 7720 7720    - w w   + w w 
│ │ │ │ +0006c500: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  0006c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c520: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c540: 2020 2034 2031 3120 2020 2031 2031 3420     4 11    1 14 
│ │ │ │ -0006c550: 2020 2030 2031 3520 2020 2020 2020 2020     0 15         
│ │ │ │ +0006c520: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c530: 2020 2020 2020 2020 2020 2020 2034 2031               4 1
│ │ │ │ +0006c540: 3120 2020 2031 2031 3420 2020 2030 2031  1    1 14    0 1
│ │ │ │ +0006c550: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │  0006c560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c570: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006c570: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006c580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c5c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c5e0: 2020 7720 7720 202d 2077 2077 2020 2b20    w w  - w w  + 
│ │ │ │ -0006c5f0: 7720 7720 202c 2020 2020 2020 2020 2020  w w  ,          
│ │ │ │ +0006c5c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c5d0: 2020 2020 2020 2020 2020 2020 7720 7720              w w 
│ │ │ │ +0006c5e0: 202d 2077 2077 2020 2b20 7720 7720 202c   - w w  + w w  ,
│ │ │ │ +0006c5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c610: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c630: 2020 2036 2038 2020 2020 3520 3920 2020     6 8    5 9   
│ │ │ │ -0006c640: 2034 2031 3020 2020 2020 2020 2020 2020   4 10           
│ │ │ │ +0006c610: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c620: 2020 2020 2020 2020 2020 2020 2036 2038               6 8
│ │ │ │ +0006c630: 2020 2020 3520 3920 2020 2034 2031 3020      5 9    4 10 
│ │ │ │ +0006c640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c660: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006c660: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006c670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c6b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c6d0: 2020 7720 7720 202d 2077 2077 2020 2b20    w w  - w w  + 
│ │ │ │ -0006c6e0: 7720 7720 202c 2020 2020 2020 2020 2020  w w  ,          
│ │ │ │ +0006c6b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c6c0: 2020 2020 2020 2020 2020 2020 7720 7720              w w 
│ │ │ │ +0006c6d0: 202d 2077 2077 2020 2b20 7720 7720 202c   - w w  + w w  ,
│ │ │ │ +0006c6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c700: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c720: 2020 2033 2036 2020 2020 3220 3920 2020     3 6    2 9   
│ │ │ │ -0006c730: 2031 2031 3020 2020 2020 2020 2020 2020   1 10           
│ │ │ │ +0006c700: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c710: 2020 2020 2020 2020 2020 2020 2033 2036               3 6
│ │ │ │ +0006c720: 2020 2020 3220 3920 2020 2031 2031 3020      2 9    1 10 
│ │ │ │ +0006c730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c750: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006c750: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c7a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0006c7a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ +0006c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c7f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c810: 2020 2033 2035 2020 2020 3220 3820 2020     3 5    2 8   
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│ │ │ │ +0006c7f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c800: 2020 2020 2020 2020 2020 2020 2033 2035               3 5
│ │ │ │ +0006c810: 2020 2020 3220 3820 2020 2030 2031 3020      2 8    0 10 
│ │ │ │ +0006c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c830: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0006c840: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006c850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c880: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006c8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0006c890: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c8a0: 2020 2020 2020 2020 2020 2020 7720 7720              w w 
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│ │ │ │ +0006c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c8e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c900: 2020 2033 2034 2020 2020 3120 3820 2020     3 4    1 8   
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│ │ │ │ +0006c8e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006c8f0: 2020 2020 2020 2020 2020 2020 2033 2034               3 4
│ │ │ │ +0006c900: 2020 2020 3120 3820 2020 2030 2039 2020      1 8    0 9  
│ │ │ │ +0006c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c930: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006c930: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006c940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c980: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c9a0: 2020 7720 7720 202d 2077 2077 2020 2b20    w w  - w w  + 
│ │ │ │ -0006c9b0: 7720 7720 2020 2020 2020 2020 2020 2020  w w             
│ │ │ │ +0006c980: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │ -0006c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0006ca10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ca20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ca40: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │ +0006ca20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
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│ │ │ │  0006ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ca60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ca70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006ca70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cac0: 2020 2020 2020 2020 207c 0a7c 6f31 203a           |.|o1 :
│ │ │ │ -0006cad0: 2052 6174 696f 6e61 6c4d 6170 2028 7175   RationalMap (qu
│ │ │ │ -0006cae0: 6164 7261 7469 6320 4372 656d 6f6e 6120  adratic Cremona 
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│ │ │ │ -0006cb00: 6620 5050 5e32 3029 2020 2020 2020 2020  f PP^20)        
│ │ │ │ -0006cb10: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │ +0006cac0: 2020 207c 0a7c 6f31 203a 2052 6174 696f     |.|o1 : Ratio
│ │ │ │ +0006cad0: 6e61 6c4d 6170 2028 7175 6164 7261 7469  nalMap (quadrati
│ │ │ │ +0006cae0: 6320 4372 656d 6f6e 6120 7472 616e 7366  c Cremona transf
│ │ │ │ +0006caf0: 6f72 6d61 7469 6f6e 206f 6620 5050 5e32  ormation of PP^2
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│ │ │ │ +0006cb10: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │  0006cb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006cb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006cb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006cb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0006cb80: 7720 202c 2077 2020 2c20 7720 202c 2077  w  , w  , w  , w
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│ │ │ │ +0006cb70: 7720 202c 2077 2020 2c20 7720 202c 2077  w  , w  , w  , w
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│ │ │ │ +0006cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006cbb0: 2020 2020 2020 2020 207c 0a7c 2031 3320           |.| 13 
│ │ │ │ -0006cbc0: 2020 3134 2020 2031 3520 2020 3136 2020    14   15   16  
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│ │ │ │ +0006cbb0: 2020 207c 0a7c 2031 3320 2020 3134 2020     |.| 13   14  
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│ │ │ │ +0006cc00: 2020 207c 0a7c 7720 202c 2077 2020 2c20     |.|w  , w  , 
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│ │ │ │ +0006cc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006cc60: 2020 3134 2020 2031 3520 2020 3136 2020    14   15   16  
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│ │ │ │ +0006cc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  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
│ │ │ │ -0006ce00: 202d 2d20 2020 2020 2020 2020 2020 2020   --             
│ │ │ │ +0006cde0: 2020 207c 0a7c 6f32 203d 202d 2d20 7261     |.|o2 = -- ra
│ │ │ │ +0006cdf0: 7469 6f6e 616c 206d 6170 202d 2d20 2020  tional map --   
│ │ │ │ +0006ce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ce10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ce20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ce30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006ce40: 2073 6f75 7263 653a 2050 726f 6a28 5151   source: Proj(QQ
│ │ │ │ -0006ce50: 5b77 202c 2077 202c 2077 202c 2077 202c  [w , w , w , w ,
│ │ │ │ -0006ce60: 2077 202c 2077 202c 2077 202c 2077 202c   w , w , w , w ,
│ │ │ │ -0006ce70: 2077 202c 2077 202c 2077 2020 2c20 7720   w , w , w  , w 
│ │ │ │ -0006ce80: 202c 2077 2020 2c20 207c 0a7c 2020 2020   , w  ,  |.|    
│ │ │ │ -0006ce90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cea0: 2020 3020 2020 3120 2020 3220 2020 3320    0   1   2   3 
│ │ │ │ -0006ceb0: 2020 3420 2020 3520 2020 3620 2020 3720    4   5   6   7 
│ │ │ │ -0006cec0: 2020 3820 2020 3920 2020 3130 2020 2031    8   9   10   1
│ │ │ │ -0006ced0: 3120 2020 3132 2020 207c 0a7c 2020 2020  1   12   |.|    
│ │ │ │ -0006cee0: 2074 6172 6765 743a 2050 726f 6a28 5151   target: Proj(QQ
│ │ │ │ -0006cef0: 5b77 202c 2077 202c 2077 202c 2077 202c  [w , w , w , w ,
│ │ │ │ -0006cf00: 2077 202c 2077 202c 2077 202c 2077 202c   w , w , w , w ,
│ │ │ │ -0006cf10: 2077 202c 2077 202c 2077 2020 2c20 7720   w , w , w  , w 
│ │ │ │ -0006cf20: 202c 2077 2020 2c20 207c 0a7c 2020 2020   , w  ,  |.|    
│ │ │ │ -0006cf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cf40: 2020 3020 2020 3120 2020 3220 2020 3320    0   1   2   3 
│ │ │ │ -0006cf50: 2020 3420 2020 3520 2020 3620 2020 3720    4   5   6   7 
│ │ │ │ -0006cf60: 2020 3820 2020 3920 2020 3130 2020 2031    8   9   10   1
│ │ │ │ -0006cf70: 3120 2020 3132 2020 207c 0a7c 2020 2020  1   12   |.|    
│ │ │ │ -0006cf80: 2064 6566 696e 696e 6720 666f 726d 733a   defining forms:
│ │ │ │ -0006cf90: 207b 2020 2020 2020 2020 2020 2020 2020   {              
│ │ │ │ +0006ce30: 2020 207c 0a7c 2020 2020 2073 6f75 7263     |.|     sourc
│ │ │ │ +0006ce40: 653a 2050 726f 6a28 5151 5b77 202c 2077  e: Proj(QQ[w , w
│ │ │ │ +0006ce50: 202c 2077 202c 2077 202c 2077 202c 2077   , w , w , w , w
│ │ │ │ +0006ce60: 202c 2077 202c 2077 202c 2077 202c 2077   , w , w , w , w
│ │ │ │ +0006ce70: 202c 2077 2020 2c20 7720 202c 2077 2020   , w  , w  , w  
│ │ │ │ +0006ce80: 2c20 207c 0a7c 2020 2020 2020 2020 2020  ,  |.|          
│ │ │ │ +0006ce90: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +0006cea0: 3120 2020 3220 2020 3320 2020 3420 2020  1   2   3   4   
│ │ │ │ +0006ceb0: 3520 2020 3620 2020 3720 2020 3820 2020  5   6   7   8   
│ │ │ │ +0006cec0: 3920 2020 3130 2020 2031 3120 2020 3132  9   10   11   12
│ │ │ │ +0006ced0: 2020 207c 0a7c 2020 2020 2074 6172 6765     |.|     targe
│ │ │ │ +0006cee0: 743a 2050 726f 6a28 5151 5b77 202c 2077  t: Proj(QQ[w , w
│ │ │ │ +0006cef0: 202c 2077 202c 2077 202c 2077 202c 2077   , w , w , w , w
│ │ │ │ +0006cf00: 202c 2077 202c 2077 202c 2077 202c 2077   , w , w , w , w
│ │ │ │ +0006cf10: 202c 2077 2020 2c20 7720 202c 2077 2020   , w  , w  , w  
│ │ │ │ +0006cf20: 2c20 207c 0a7c 2020 2020 2020 2020 2020  ,  |.|          
│ │ │ │ +0006cf30: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +0006cf40: 3120 2020 3220 2020 3320 2020 3420 2020  1   2   3   4   
│ │ │ │ +0006cf50: 3520 2020 3620 2020 3720 2020 3820 2020  5   6   7   8   
│ │ │ │ +0006cf60: 3920 2020 3130 2020 2031 3120 2020 3132  9   10   11   12
│ │ │ │ +0006cf70: 2020 207c 0a7c 2020 2020 2064 6566 696e     |.|     defin
│ │ │ │ +0006cf80: 696e 6720 666f 726d 733a 207b 2020 2020  ing forms: {    
│ │ │ │ +0006cf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cfc0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006cfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cfe0: 2020 2d20 7720 2077 2020 202b 2077 2020    - w  w   + w  
│ │ │ │ -0006cff0: 7720 2020 2d20 7720 2077 2020 2c20 2020  w   - w  w  ,   
│ │ │ │ +0006cfc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006cfd0: 2020 2020 2020 2020 2020 2020 2d20 7720              - w 
│ │ │ │ +0006cfe0: 2077 2020 202b 2077 2020 7720 2020 2d20   w   + w  w   - 
│ │ │ │ +0006cff0: 7720 2077 2020 2c20 2020 2020 2020 2020  w  w  ,         
│ │ │ │  0006d000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d010: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d030: 2020 2020 2031 3520 3138 2020 2020 3134       15 18    14
│ │ │ │ -0006d040: 2031 3920 2020 2031 3120 3230 2020 2020   19    11 20    
│ │ │ │ +0006d010: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d020: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +0006d030: 3520 3138 2020 2020 3134 2031 3920 2020  5 18    14 19   
│ │ │ │ +0006d040: 2031 3120 3230 2020 2020 2020 2020 2020   11 20          
│ │ │ │  0006d050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d060: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006d060: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006d070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d0b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d0d0: 2020 2d20 7720 2077 2020 202b 2077 2020    - w  w   + w  
│ │ │ │ -0006d0e0: 7720 2020 2d20 7720 2077 2020 2c20 2020  w   - w  w  ,   
│ │ │ │ +0006d0b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d0c0: 2020 2020 2020 2020 2020 2020 2d20 7720              - w 
│ │ │ │ +0006d0d0: 2077 2020 202b 2077 2020 7720 2020 2d20   w   + w  w   - 
│ │ │ │ +0006d0e0: 7720 2077 2020 2c20 2020 2020 2020 2020  w  w  ,         
│ │ │ │  0006d0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d100: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d120: 2020 2020 2031 3520 3137 2020 2020 3133       15 17    13
│ │ │ │ -0006d130: 2031 3920 2020 2031 3020 3230 2020 2020   19    10 20    
│ │ │ │ +0006d100: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d110: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +0006d120: 3520 3137 2020 2020 3133 2031 3920 2020  5 17    13 19   
│ │ │ │ +0006d130: 2031 3020 3230 2020 2020 2020 2020 2020   10 20          
│ │ │ │  0006d140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d150: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006d150: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006d160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d1a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d1c0: 2020 2d20 7720 2077 2020 202b 2077 2020    - w  w   + w  
│ │ │ │ -0006d1d0: 7720 2020 2d20 7720 7720 202c 2020 2020  w   - w w  ,    
│ │ │ │ +0006d1a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d1b0: 2020 2020 2020 2020 2020 2020 2d20 7720              - w 
│ │ │ │ +0006d1c0: 2077 2020 202b 2077 2020 7720 2020 2d20   w   + w  w   - 
│ │ │ │ +0006d1d0: 7720 7720 202c 2020 2020 2020 2020 2020  w w  ,          
│ │ │ │  0006d1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d1f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d210: 2020 2020 2031 3420 3137 2020 2020 3133       14 17    13
│ │ │ │ -0006d220: 2031 3820 2020 2039 2032 3020 2020 2020   18    9 20     
│ │ │ │ +0006d1f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d200: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +0006d210: 3420 3137 2020 2020 3133 2031 3820 2020  4 17    13 18   
│ │ │ │ +0006d220: 2039 2032 3020 2020 2020 2020 2020 2020   9 20           
│ │ │ │  0006d230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d240: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006d240: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006d250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d290: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d2b0: 2020 2d20 7720 2077 2020 202b 2077 2020    - w  w   + w  
│ │ │ │ -0006d2c0: 7720 2020 2d20 7720 7720 202c 2020 2020  w   - w w  ,    
│ │ │ │ +0006d290: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d2a0: 2020 2020 2020 2020 2020 2020 2d20 7720              - w 
│ │ │ │ +0006d2b0: 2077 2020 202b 2077 2020 7720 2020 2d20   w   + w  w   - 
│ │ │ │ +0006d2c0: 7720 7720 202c 2020 2020 2020 2020 2020  w w  ,          
│ │ │ │  0006d2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d2e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d300: 2020 2020 2031 3120 3137 2020 2020 3130       11 17    10
│ │ │ │ -0006d310: 2031 3820 2020 2039 2031 3920 2020 2020   18    9 19     
│ │ │ │ +0006d2e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d2f0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +0006d300: 3120 3137 2020 2020 3130 2031 3820 2020  1 17    10 18   
│ │ │ │ +0006d310: 2039 2031 3920 2020 2020 2020 2020 2020   9 19           
│ │ │ │  0006d320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d330: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006d330: 2020 207c 0a7c 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 2020 2020                  
│ │ │ │  0006d370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d380: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d3a0: 2020 2d20 7720 2077 2020 202b 2077 2077    - w  w   + w w
│ │ │ │ -0006d3b0: 2020 202d 2077 2077 2020 2c20 2020 2020     - w w  ,     
│ │ │ │ +0006d380: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d390: 2020 2020 2020 2020 2020 2020 2d20 7720              - w 
│ │ │ │ +0006d3a0: 2077 2020 202b 2077 2077 2020 202d 2077   w   + w w   - w
│ │ │ │ +0006d3b0: 2077 2020 2c20 2020 2020 2020 2020 2020   w  ,           
│ │ │ │  0006d3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d3d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d3f0: 2020 2020 2031 3520 3136 2020 2020 3320       15 16    3 
│ │ │ │ -0006d400: 3139 2020 2020 3220 3230 2020 2020 2020  19    2 20      
│ │ │ │ +0006d3d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d3e0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +0006d3f0: 3520 3136 2020 2020 3320 3139 2020 2020  5 16    3 19    
│ │ │ │ +0006d400: 3220 3230 2020 2020 2020 2020 2020 2020  2 20            
│ │ │ │  0006d410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d420: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006d420: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006d430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d470: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d490: 2020 2d20 7720 2077 2020 202b 2077 2077    - w  w   + w w
│ │ │ │ -0006d4a0: 2020 202d 2077 2077 2020 2c20 2020 2020     - w w  ,     
│ │ │ │ +0006d470: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d480: 2020 2020 2020 2020 2020 2020 2d20 7720              - w 
│ │ │ │ +0006d490: 2077 2020 202b 2077 2077 2020 202d 2077   w   + w w   - w
│ │ │ │ +0006d4a0: 2077 2020 2c20 2020 2020 2020 2020 2020   w  ,           
│ │ │ │  0006d4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d4c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d4e0: 2020 2020 2031 3420 3136 2020 2020 3320       14 16    3 
│ │ │ │ -0006d4f0: 3138 2020 2020 3120 3230 2020 2020 2020  18    1 20      
│ │ │ │ +0006d4c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d4d0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +0006d4e0: 3420 3136 2020 2020 3320 3138 2020 2020  4 16    3 18    
│ │ │ │ +0006d4f0: 3120 3230 2020 2020 2020 2020 2020 2020  1 20            
│ │ │ │  0006d500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d510: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006d510: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006d520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d560: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d580: 2020 2d20 7720 2077 2020 202b 2077 2077    - w  w   + w w
│ │ │ │ -0006d590: 2020 202d 2077 2077 2020 2c20 2020 2020     - w w  ,     
│ │ │ │ +0006d560: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d570: 2020 2020 2020 2020 2020 2020 2d20 7720              - w 
│ │ │ │ +0006d580: 2077 2020 202b 2077 2077 2020 202d 2077   w   + w w   - w
│ │ │ │ +0006d590: 2077 2020 2c20 2020 2020 2020 2020 2020   w  ,           
│ │ │ │  0006d5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d5b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d5d0: 2020 2020 2031 3320 3136 2020 2020 3320       13 16    3 
│ │ │ │ -0006d5e0: 3137 2020 2020 3020 3230 2020 2020 2020  17    0 20      
│ │ │ │ +0006d5b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d5c0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +0006d5d0: 3320 3136 2020 2020 3320 3137 2020 2020  3 16    3 17    
│ │ │ │ +0006d5e0: 3020 3230 2020 2020 2020 2020 2020 2020  0 20            
│ │ │ │  0006d5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d600: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006d600: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006d610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d650: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d670: 2020 2d20 7720 2077 2020 202d 2077 2077    - w  w   - w w
│ │ │ │ -0006d680: 2020 202b 2077 2077 2020 202d 2077 2077     + w w   - w w
│ │ │ │ -0006d690: 2020 202d 2077 2077 2020 2c20 2020 2020     - w w  ,     
│ │ │ │ -0006d6a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d6c0: 2020 2020 2031 3220 3136 2020 2020 3820       12 16    8 
│ │ │ │ -0006d6d0: 3137 2020 2020 3720 3138 2020 2020 3620  17    7 18    6 
│ │ │ │ -0006d6e0: 3139 2020 2020 3520 3230 2020 2020 2020  19    5 20      
│ │ │ │ -0006d6f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006d650: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d660: 2020 2020 2020 2020 2020 2020 2d20 7720              - w 
│ │ │ │ +0006d670: 2077 2020 202d 2077 2077 2020 202b 2077   w   - w w   + w
│ │ │ │ +0006d680: 2077 2020 202d 2077 2077 2020 202d 2077   w   - w w   - w
│ │ │ │ +0006d690: 2077 2020 2c20 2020 2020 2020 2020 2020   w  ,           
│ │ │ │ +0006d6a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d6b0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +0006d6c0: 3220 3136 2020 2020 3820 3137 2020 2020  2 16    8 17    
│ │ │ │ +0006d6d0: 3720 3138 2020 2020 3620 3139 2020 2020  7 18    6 19    
│ │ │ │ +0006d6e0: 3520 3230 2020 2020 2020 2020 2020 2020  5 20            
│ │ │ │ +0006d6f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006d700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d740: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d760: 2020 2d20 7720 2077 2020 202b 2077 2077    - w  w   + w w
│ │ │ │ -0006d770: 2020 202d 2077 2077 2020 2c20 2020 2020     - w w  ,     
│ │ │ │ +0006d740: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d750: 2020 2020 2020 2020 2020 2020 2d20 7720              - w 
│ │ │ │ +0006d760: 2077 2020 202b 2077 2077 2020 202d 2077   w   + w w   - w
│ │ │ │ +0006d770: 2077 2020 2c20 2020 2020 2020 2020 2020   w  ,           
│ │ │ │  0006d780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d790: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d7b0: 2020 2020 2031 3120 3136 2020 2020 3220       11 16    2 
│ │ │ │ -0006d7c0: 3138 2020 2020 3120 3139 2020 2020 2020  18    1 19      
│ │ │ │ +0006d790: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d7a0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +0006d7b0: 3120 3136 2020 2020 3220 3138 2020 2020  1 16    2 18    
│ │ │ │ +0006d7c0: 3120 3139 2020 2020 2020 2020 2020 2020  1 19            
│ │ │ │  0006d7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d7e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006d7e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006d7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d830: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d850: 2020 2d20 7720 2077 2020 202b 2077 2077    - w  w   + w w
│ │ │ │ -0006d860: 2020 202d 2077 2077 2020 2c20 2020 2020     - w w  ,     
│ │ │ │ +0006d830: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0006d840: 2020 2020 2020 2020 2020 2020 2d20 7720              - w 
│ │ │ │ +0006d850: 2077 2020 202b 2077 2077 2020 202d 2077   w   + w w   - w
│ │ │ │ +0006d860: 2077 2020 2c20 2020 2020 2020 2020 2020   w  ,           
│ │ │ │  0006d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d880: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006d890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d8a0: 2020 2020 2031 3020 3136 2020 2020 3220       10 16    2 
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│ │ │ │ -0006e3c0: 2020 2020 2020 2020 207c 0a7c 6f32 203a           |.|o2 :
│ │ │ │ -0006e3d0: 2052 6174 696f 6e61 6c4d 6170 2028 7175   RationalMap (qu
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│ │ │ │ -0006e400: 746f 2050 505e 3230 2920 2020 2020 2020  to PP^20)       
│ │ │ │ -0006e410: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │ +0006e3c0: 2020 207c 0a7c 6f32 203a 2052 6174 696f     |.|o2 : Ratio
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│ │ │ │ +0006e3f0: 726f 6d20 5050 5e32 3020 746f 2050 505e  rom PP^20 to PP^
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│ │ │ │  0006e450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0006e480: 7720 202c 2077 2020 2c20 7720 202c 2077  w  , w  , w  , w
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│ │ │ │  0006e5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0006e600: 7428 7068 6920 2a20 7073 6920 3d3d 2031  t(phi * psi == 1
│ │ │ │ +0006e610: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  0006e620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e640: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0006e640: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  0006e650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e690: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468 6520  ---------+..The 
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│ │ │ │ -0006e760: 6465 6669 6e69 6e67 2024 5c50 6869 5e7b  defining $\Phi^{
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│ │ │ │ +0006e690: 2d2d 2d2b 0a0a 5468 6520 6d65 7468 6f64  ---+..The method
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│ │ │ │ +0006e6b0: 2069 6e70 7574 2061 202a 6e6f 7465 2072   input a *note r
│ │ │ │ +0006e6c0: 696e 6720 6d61 703a 2028 4d61 6361 756c  ing map: (Macaul
│ │ │ │ +0006e6d0: 6179 3244 6f63 2952 696e 674d 6170 2c0a  ay2Doc)RingMap,.
│ │ │ │ +0006e6e0: 7265 7072 6573 656e 7469 6e67 2061 2072  representing a r
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│ │ │ │ +0006e720: 2049 6e20 7468 6973 2063 6173 652c 0a74   In this case,.t
│ │ │ │ +0006e730: 6865 202a 6e6f 7465 2072 696e 6720 6d61  he *note ring ma
│ │ │ │ +0006e740: 703a 2028 4d61 6361 756c 6179 3244 6f63  p: (Macaulay2Doc
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│ │ │ │ +0006e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e7d0: 2d2b 0a7c 6934 203a 202d 2d20 4120 4372  -+.|i4 : -- A Cr
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│ │ │ │ -0006e7f0: 7469 6f6e 206f 6620 505e 3236 2020 2020  tion of P^26    
│ │ │ │ +0006e7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +0006e7d0: 203a 202d 2d20 4120 4372 656d 6f6e 6120   : -- A Cremona 
│ │ │ │ +0006e7e0: 7472 616e 7366 6f72 6d61 7469 6f6e 206f  transformation o
│ │ │ │ +0006e7f0: 6620 505e 3236 2020 2020 2020 2020 2020  f P^26          
│ │ │ │  0006e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e820: 207c 0a7c 2020 2020 2070 6869 203d 206d   |.|     phi = m
│ │ │ │ -0006e830: 6170 2071 7561 6472 6f51 7561 6472 6963  ap quadroQuadric
│ │ │ │ -0006e840: 4372 656d 6f6e 6154 7261 6e73 666f 726d  CremonaTransform
│ │ │ │ -0006e850: 6174 696f 6e28 3236 2c31 2920 2020 2020  ation(26,1)     
│ │ │ │ -0006e860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e870: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0006e810: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006e820: 2020 2070 6869 203d 206d 6170 2071 7561     phi = map qua
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│ │ │ │ +0006e840: 6154 7261 6e73 666f 726d 6174 696f 6e28  aTransformation(
│ │ │ │ +0006e850: 3236 2c31 2920 2020 2020 2020 2020 2020  26,1)           
│ │ │ │ +0006e860: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e8c0: 207c 0a7c 6f34 203d 206d 6170 2028 5151   |.|o4 = map (QQ
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│ │ │ │ -0006e8e0: 2e2e 7720 205d 2c20 7b77 2020 7720 2020  ..w  ], {w  w   
│ │ │ │ -0006e8f0: 2d20 7720 2077 2020 202d 2077 2020 7720  - w  w   - w  w 
│ │ │ │ -0006e900: 2020 2d20 7720 2077 2020 202d 2020 2020    - w  w   -    
│ │ │ │ -0006e910: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0006e920: 2020 3020 2020 3236 2020 2020 2020 2030    0   26       0
│ │ │ │ -0006e930: 2020 2032 3620 2020 2020 3231 2032 3220     26     21 22 
│ │ │ │ -0006e940: 2020 2032 3020 3233 2020 2020 3135 2032     20 23    15 2
│ │ │ │ -0006e950: 3420 2020 2031 3020 3235 2020 2020 2020  4    10 25      
│ │ │ │ -0006e960: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0006e8b0: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
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│ │ │ │ +0006e910: 2020 2020 2020 2020 2020 2020 3020 2020              0   
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│ │ │ │ +0006e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e980: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0006e9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e9b0: 207c 0a7c 6f34 203a 2052 696e 674d 6170   |.|o4 : RingMap
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│ │ │ │ -0006ead0: 2077 2020 2c20 7720 2077 2020 202d 2077   w  , w  w   - w
│ │ │ │ -0006eae0: 2020 7720 2020 2d20 7720 2077 2020 202d    w   - w  w   -
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│ │ │ │ +0006eb10: 2031 3120 3235 2020 2020 3120 3236 2020   11 25    1 26  
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│ │ │ │  0006eb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006eb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0006ebf0: 3236 2020 2031 3520 3139 2020 2020 3136  26   15 19    16
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│ │ │ │ -0006f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f8b0: 207c 0a7c 6f35 203d 206d 6170 2028 5151   |.|o5 = map (QQ
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│ │ │ │ +0006f8f0: 2077 2020 202d 2020 2020 207c 0a7c 2020   w   -     |.|  
│ │ │ │ +0006f900: 2020 2020 2020 2020 2020 2020 3020 2020              0   
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│ │ │ │ +0006f920: 2020 2020 2020 3520 3232 2020 2020 3820        5 22    8 
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│ │ │ │ +0006f940: 3320 3235 2020 2020 2020 207c 0a7c 2020  3 25       |.|  
│ │ │ │ +0006f950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f9a0: 207c 0a7c 6f35 203a 2052 696e 674d 6170   |.|o5 : RingMap
│ │ │ │ -0006f9b0: 2051 515b 7720 2e2e 7720 205d 203c 2d2d   QQ[w ..w  ] <--
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│ │ │ │ -0006fac0: 2077 2020 2c20 2d20 7720 7720 2020 2b20   w  , - w w   + 
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│ │ │ │ +0006faf0: 3820 3139 2020 2020 3134 2032 3020 2020  8 19    14 20   
│ │ │ │ +0006fb00: 2031 3020 3235 2020 2020 3120 3236 2020   10 25    1 26  
│ │ │ │ +0006fb10: 2020 2035 2031 3620 2020 2038 2031 3720     5 16    8 17 
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│ │ │ │ +00070280: 3320 3232 2020 2020 3620 3233 2020 2031  3 22    6 23   1
│ │ │ │ +00070290: 3020 3132 2020 2020 3920 3134 2020 2020  0 12    9 14    
│ │ │ │ +000702a0: 3120 3135 2020 2020 2020 207c 0a7c 2d2d  1 15       |.|--
│ │ │ │ +000702b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000702c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000702d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000702e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000702f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070300: 2d7c 0a7c 7720 7720 2020 2b20 7720 7720  -|.|w w   + w w 
│ │ │ │ -00070310: 202c 2077 2020 7720 2020 2d20 7720 7720   , w  w   - w w 
│ │ │ │ -00070320: 2020 2b20 7720 7720 2020 2d20 7720 7720    + w w   - w w 
│ │ │ │ -00070330: 2020 2b20 7720 7720 202c 2077 2077 2020    + w w  , w w  
│ │ │ │ -00070340: 2d20 7720 7720 2020 2b20 7720 7720 2020  - w w   + w w   
│ │ │ │ -00070350: 2d7c 0a7c 2033 2031 3820 2020 2036 2031  -|.| 3 18    6 1
│ │ │ │ -00070360: 3920 2020 3130 2031 3120 2020 2039 2031  9   10 11    9 1
│ │ │ │ -00070370: 3320 2020 2032 2031 3520 2020 2033 2031  3    2 15    3 1
│ │ │ │ -00070380: 3620 2020 2036 2031 3720 2020 3820 3920  6    6 17   8 9 
│ │ │ │ -00070390: 2020 2037 2031 3020 2020 2031 2031 3620     7 10    1 16 
│ │ │ │ -000703a0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ +000702f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7720  -----------|.|w 
│ │ │ │ +00070300: 7720 2020 2b20 7720 7720 202c 2077 2020  w   + w w  , w  
│ │ │ │ +00070310: 7720 2020 2d20 7720 7720 2020 2b20 7720  w   - w w   + w 
│ │ │ │ +00070320: 7720 2020 2d20 7720 7720 2020 2b20 7720  w   - w w   + w 
│ │ │ │ +00070330: 7720 202c 2077 2077 2020 2d20 7720 7720  w  , w w  - w w 
│ │ │ │ +00070340: 2020 2b20 7720 7720 2020 2d7c 0a7c 2033    + w w   -|.| 3
│ │ │ │ +00070350: 2031 3820 2020 2036 2031 3920 2020 3130   18    6 19   10
│ │ │ │ +00070360: 2031 3120 2020 2039 2031 3320 2020 2032   11    9 13    2
│ │ │ │ +00070370: 2031 3520 2020 2033 2031 3620 2020 2036   15    3 16    6
│ │ │ │ +00070380: 2031 3720 2020 3820 3920 2020 2037 2031   17   8 9    7 1
│ │ │ │ +00070390: 3020 2020 2031 2031 3620 207c 0a7c 2d2d  0    1 16  |.|--
│ │ │ │ +000703a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000703b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000703c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000703d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000703e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000703f0: 2d7c 0a7c 7720 7720 2020 2b20 7720 7720  -|.|w w   + w w 
│ │ │ │ -00070400: 202c 2077 2077 2020 2d20 7720 7720 2020   , w w  - w w   
│ │ │ │ -00070410: 2b20 7720 7720 2020 2d20 7720 7720 2020  + w w   - w w   
│ │ │ │ -00070420: 2b20 7720 7720 202c 2077 2077 2020 202d  + w w  , w w   -
│ │ │ │ -00070430: 2077 2077 2020 202b 2077 2077 2020 202d   w w   + w w   -
│ │ │ │ -00070440: 207c 0a7c 2032 2031 3820 2020 2036 2032   |.| 2 18    6 2
│ │ │ │ -00070450: 3020 2020 3520 3920 2020 2034 2031 3020  0   5 9    4 10 
│ │ │ │ -00070460: 2020 2031 2031 3720 2020 2032 2031 3920     1 17    2 19 
│ │ │ │ -00070470: 2020 2033 2032 3020 2020 3820 3131 2020     3 20   8 11  
│ │ │ │ -00070480: 2020 3720 3133 2020 2020 3020 3136 2020    7 13    0 16  
│ │ │ │ -00070490: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ +000703e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7720  -----------|.|w 
│ │ │ │ +000703f0: 7720 2020 2b20 7720 7720 202c 2077 2077  w   + w w  , w w
│ │ │ │ +00070400: 2020 2d20 7720 7720 2020 2b20 7720 7720    - w w   + w w 
│ │ │ │ +00070410: 2020 2d20 7720 7720 2020 2b20 7720 7720    - w w   + w w 
│ │ │ │ +00070420: 202c 2077 2077 2020 202d 2077 2077 2020   , w w   - w w  
│ │ │ │ +00070430: 202b 2077 2077 2020 202d 207c 0a7c 2032   + w w   - |.| 2
│ │ │ │ +00070440: 2031 3820 2020 2036 2032 3020 2020 3520   18    6 20   5 
│ │ │ │ +00070450: 3920 2020 2034 2031 3020 2020 2031 2031  9    4 10    1 1
│ │ │ │ +00070460: 3720 2020 2032 2031 3920 2020 2033 2032  7    2 19    3 2
│ │ │ │ +00070470: 3020 2020 3820 3131 2020 2020 3720 3133  0   8 11    7 13
│ │ │ │ +00070480: 2020 2020 3020 3136 2020 207c 0a7c 2d2d      0 16   |.|--
│ │ │ │ +00070490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000704a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000704b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000704c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000704d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000704e0: 2d7c 0a7c 7720 7720 2020 2b20 7720 7720  -|.|w w   + w w 
│ │ │ │ -000704f0: 202c 2077 2077 2020 202d 2077 2077 2020   , w w   - w w  
│ │ │ │ -00070500: 202b 2077 2077 2020 202d 2077 2077 2020   + w w   - w w  
│ │ │ │ -00070510: 202b 2077 2077 2020 2c20 7720 7720 2020   + w w  , w w   
│ │ │ │ -00070520: 2d20 7720 7720 2020 2b20 7720 7720 2020  - w w   + w w   
│ │ │ │ -00070530: 2d7c 0a7c 2032 2032 3220 2020 2036 2032  -|.| 2 22    6 2
│ │ │ │ -00070540: 3420 2020 3520 3131 2020 2020 3420 3133  4   5 11    4 13
│ │ │ │ -00070550: 2020 2020 3020 3137 2020 2020 3220 3233      0 17    2 23
│ │ │ │ -00070560: 2020 2020 3320 3234 2020 2038 2031 3220      3 24   8 12 
│ │ │ │ -00070570: 2020 2037 2031 3420 2020 2030 2031 3820     7 14    0 18 
│ │ │ │ -00070580: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ +000704d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7720  -----------|.|w 
│ │ │ │ +000704e0: 7720 2020 2b20 7720 7720 202c 2077 2077  w   + w w  , w w
│ │ │ │ +000704f0: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ +00070500: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ +00070510: 2020 2c20 7720 7720 2020 2d20 7720 7720    , w w   - w w 
│ │ │ │ +00070520: 2020 2b20 7720 7720 2020 2d7c 0a7c 2032    + w w   -|.| 2
│ │ │ │ +00070530: 2032 3220 2020 2036 2032 3420 2020 3520   22    6 24   5 
│ │ │ │ +00070540: 3131 2020 2020 3420 3133 2020 2020 3020  11    4 13    0 
│ │ │ │ +00070550: 3137 2020 2020 3220 3233 2020 2020 3320  17    2 23    3 
│ │ │ │ +00070560: 3234 2020 2038 2031 3220 2020 2037 2031  24   8 12    7 1
│ │ │ │ +00070570: 3420 2020 2030 2031 3820 207c 0a7c 2d2d  4    0 18  |.|--
│ │ │ │ +00070580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000705a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000705b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000705c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000705d0: 2d7c 0a7c 7720 7720 2020 2b20 7720 7720  -|.|w w   + w w 
│ │ │ │ -000705e0: 202c 2077 2077 2020 202d 2077 2077 2020   , w w   - w w  
│ │ │ │ -000705f0: 202b 2077 2077 2020 202d 2077 2077 2020   + w w   - w w  
│ │ │ │ -00070600: 202b 2077 2077 2020 2c20 7720 7720 202d   + w w  , w w  -
│ │ │ │ -00070610: 2077 2077 2020 2d20 7720 7720 2020 2b20   w w  - w w   + 
│ │ │ │ -00070620: 207c 0a7c 2031 2032 3220 2020 2036 2032   |.| 1 22    6 2
│ │ │ │ -00070630: 3520 2020 3520 3132 2020 2020 3420 3134  5   5 12    4 14
│ │ │ │ -00070640: 2020 2020 3020 3139 2020 2020 3120 3233      0 19    1 23
│ │ │ │ -00070650: 2020 2020 3320 3235 2020 2035 2037 2020      3 25   5 7  
│ │ │ │ -00070660: 2020 3420 3820 2020 2030 2032 3020 2020    4 8    0 20   
│ │ │ │ -00070670: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ +000705c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7720  -----------|.|w 
│ │ │ │ +000705d0: 7720 2020 2b20 7720 7720 202c 2077 2077  w   + w w  , w w
│ │ │ │ +000705e0: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ +000705f0: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ +00070600: 2020 2c20 7720 7720 202d 2077 2077 2020    , w w  - w w  
│ │ │ │ +00070610: 2d20 7720 7720 2020 2b20 207c 0a7c 2031  - w w   +  |.| 1
│ │ │ │ +00070620: 2032 3220 2020 2036 2032 3520 2020 3520   22    6 25   5 
│ │ │ │ +00070630: 3132 2020 2020 3420 3134 2020 2020 3020  12    4 14    0 
│ │ │ │ +00070640: 3139 2020 2020 3120 3233 2020 2020 3320  19    1 23    3 
│ │ │ │ +00070650: 3235 2020 2035 2037 2020 2020 3420 3820  25   5 7    4 8 
│ │ │ │ +00070660: 2020 2030 2032 3020 2020 207c 0a7c 2d2d     0 20    |.|--
│ │ │ │ +00070670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000706a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000706b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000706c0: 2d7c 0a7c 7720 7720 2020 2d20 7720 7720  -|.|w w   - w w 
│ │ │ │ -000706d0: 202c 202d 2077 2077 2020 2b20 7720 7720   , - w w  + w w 
│ │ │ │ -000706e0: 202b 2077 2077 2020 202d 2077 2077 2020   + w w   - w w  
│ │ │ │ -000706f0: 202b 2077 2077 2020 2c20 7720 7720 202d   + w w  , w w  -
│ │ │ │ -00070700: 2077 2077 2020 2d20 7720 7720 202b 2020   w w  - w w  +  
│ │ │ │ -00070710: 207c 0a7c 2031 2032 3420 2020 2032 2032   |.| 1 24    2 2
│ │ │ │ -00070720: 3520 2020 2020 3520 3620 2020 2033 2038  5     5 6    3 8
│ │ │ │ -00070730: 2020 2020 3020 3130 2020 2020 3120 3133      0 10    1 13
│ │ │ │ -00070740: 2020 2020 3220 3134 2020 2034 2036 2020      2 14   4 6  
│ │ │ │ -00070750: 2020 3320 3720 2020 2030 2039 2020 2020    3 7    0 9    
│ │ │ │ -00070760: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ +000706b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7720  -----------|.|w 
│ │ │ │ +000706c0: 7720 2020 2d20 7720 7720 202c 202d 2077  w   - w w  , - w
│ │ │ │ +000706d0: 2077 2020 2b20 7720 7720 202b 2077 2077   w  + w w  + w w
│ │ │ │ +000706e0: 2020 202d 2077 2077 2020 202b 2077 2077     - w w   + w w
│ │ │ │ +000706f0: 2020 2c20 7720 7720 202d 2077 2077 2020    , w w  - w w  
│ │ │ │ +00070700: 2d20 7720 7720 202b 2020 207c 0a7c 2031  - w w  +   |.| 1
│ │ │ │ +00070710: 2032 3420 2020 2032 2032 3520 2020 2020   24    2 25     
│ │ │ │ +00070720: 3520 3620 2020 2033 2038 2020 2020 3020  5 6    3 8    0 
│ │ │ │ +00070730: 3130 2020 2020 3120 3133 2020 2020 3220  10    1 13    2 
│ │ │ │ +00070740: 3134 2020 2034 2036 2020 2020 3320 3720  14   4 6    3 7 
│ │ │ │ +00070750: 2020 2030 2039 2020 2020 207c 0a7c 2d2d     0 9     |.|--
│ │ │ │ +00070760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000707a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000707b0: 2d7c 0a7c 7720 7720 2020 2d20 7720 7720  -|.|w w   - w w 
│ │ │ │ -000707c0: 207d 2920 2020 2020 2020 2020 2020 2020   })             
│ │ │ │ +000707a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7720  -----------|.|w 
│ │ │ │ +000707b0: 7720 2020 2d20 7720 7720 207d 2920 2020  w   - w w  })   
│ │ │ │ +000707c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000707d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000707e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000707f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070800: 207c 0a7c 2031 2031 3120 2020 2032 2031   |.| 1 11    2 1
│ │ │ │ -00070810: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000707f0: 2020 2020 2020 2020 2020 207c 0a7c 2031             |.| 1
│ │ │ │ +00070800: 2031 3120 2020 2032 2031 3220 2020 2020   11    2 12     
│ │ │ │ +00070810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070850: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00070840: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00070850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000708a0: 2d2b 0a7c 6936 203a 2061 7373 6572 7420  -+.|i6 : assert 
│ │ │ │ -000708b0: 6973 496e 7665 7273 654d 6170 2870 6869  isInverseMap(phi
│ │ │ │ -000708c0: 2c70 7369 2920 2020 2020 2020 2020 2020  ,psi)           
│ │ │ │ +00070890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ +000708a0: 203a 2061 7373 6572 7420 6973 496e 7665   : assert isInve
│ │ │ │ +000708b0: 7273 654d 6170 2870 6869 2c70 7369 2920  rseMap(phi,psi) 
│ │ │ │ +000708c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000708d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000708e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000708f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000708e0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000708f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070940: 2d2b 0a0a 4361 7665 6174 0a3d 3d3d 3d3d  -+..Caveat.=====
│ │ │ │ -00070950: 3d0a 0a49 6620 7468 6520 6d61 7020 7061  =..If the map pa
│ │ │ │ -00070960: 7373 6564 2069 7320 6e6f 7420 6269 7261  ssed is not bira
│ │ │ │ -00070970: 7469 6f6e 616c 2061 6e64 2074 6865 206f  tional and the o
│ │ │ │ -00070980: 7074 696f 6e20 2a6e 6f74 6520 4365 7274  ption *note Cert
│ │ │ │ -00070990: 6966 793a 2043 6572 7469 6679 2c20 6973  ify: Certify, is
│ │ │ │ -000709a0: 0a73 6574 2074 6f20 6661 6c73 652c 2079  .set to false, y
│ │ │ │ -000709b0: 6f75 206d 6967 6874 206e 6f74 2067 6574  ou might not get
│ │ │ │ -000709c0: 2061 6e79 2065 7272 6f72 206d 6573 7361   any error messa
│ │ │ │ -000709d0: 6765 2e0a 0a53 6565 2061 6c73 6f0a 3d3d  ge...See also.==
│ │ │ │ -000709e0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ -000709f0: 6520 6170 7072 6f78 696d 6174 6549 6e76  e approximateInv
│ │ │ │ -00070a00: 6572 7365 4d61 703a 2061 7070 726f 7869  erseMap: approxi
│ │ │ │ -00070a10: 6d61 7465 496e 7665 7273 654d 6170 2c20  mateInverseMap, 
│ │ │ │ -00070a20: 2d2d 2072 616e 646f 6d20 6d61 7020 7265  -- random map re
│ │ │ │ -00070a30: 6c61 7465 640a 2020 2020 746f 2074 6865  lated.    to the
│ │ │ │ -00070a40: 2069 6e76 6572 7365 206f 6620 6120 6269   inverse of a bi
│ │ │ │ -00070a50: 7261 7469 6f6e 616c 206d 6170 0a20 202a  rational map.  *
│ │ │ │ -00070a60: 202a 6e6f 7465 2069 6e76 6572 7365 2852   *note inverse(R
│ │ │ │ -00070a70: 6174 696f 6e61 6c4d 6170 293a 2069 6e76  ationalMap): inv
│ │ │ │ -00070a80: 6572 7365 5f6c 7052 6174 696f 6e61 6c4d  erse_lpRationalM
│ │ │ │ -00070a90: 6170 5f72 702c 202d 2d20 696e 7665 7273  ap_rp, -- invers
│ │ │ │ -00070aa0: 6520 6f66 2061 0a20 2020 2062 6972 6174  e of a.    birat
│ │ │ │ -00070ab0: 696f 6e61 6c20 6d61 700a 0a57 6179 7320  ional map..Ways 
│ │ │ │ -00070ac0: 746f 2075 7365 2069 6e76 6572 7365 4d61  to use inverseMa
│ │ │ │ -00070ad0: 703a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  p:.=============
│ │ │ │ -00070ae0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00070af0: 2269 6e76 6572 7365 4d61 7028 5261 7469  "inverseMap(Rati
│ │ │ │ -00070b00: 6f6e 616c 4d61 7029 220a 2020 2a20 2269  onalMap)".  * "i
│ │ │ │ -00070b10: 6e76 6572 7365 4d61 7028 5269 6e67 4d61  nverseMap(RingMa
│ │ │ │ -00070b20: 7029 220a 0a46 6f72 2074 6865 2070 726f  p)"..For the pro
│ │ │ │ -00070b30: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
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│ │ │ │ +00071140: 616d 6564 2056 6572 626f 7365 3a0a 3d3d  amed Verbose:.==
│ │ │ │ +00071150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00071160: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00071170: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00071180: 3d3d 3d0a 0a20 202a 2022 6368 6563 6b28  ===..  * "check(
│ │ │ │ -00071190: 2e2e 2e2c 5665 7262 6f73 653d 3e2e 2e2e  ...,Verbose=>...
│ │ │ │ -000711a0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -000711b0: 6368 6563 6b3a 2028 4d61 6361 756c 6179  check: (Macaulay
│ │ │ │ -000711c0: 3244 6f63 2963 6865 636b 2c20 2d2d 0a20  2Doc)check, --. 
│ │ │ │ -000711d0: 2020 2070 6572 666f 726d 2074 6573 7473     perform tests
│ │ │ │ -000711e0: 206f 6620 6120 7061 636b 6167 650a 2020   of a package.  
│ │ │ │ -000711f0: 2a20 2263 6f70 7944 6972 6563 746f 7279  * "copyDirectory
│ │ │ │ -00071200: 282e 2e2e 2c56 6572 626f 7365 3d3e 2e2e  (...,Verbose=>..
│ │ │ │ -00071210: 2e29 2220 2d2d 2073 6565 202a 6e6f 7465  .)" -- see *note
│ │ │ │ -00071220: 0a20 2020 2063 6f70 7944 6972 6563 746f  .    copyDirecto
│ │ │ │ -00071230: 7279 2853 7472 696e 672c 5374 7269 6e67  ry(String,String
│ │ │ │ -00071240: 293a 0a20 2020 2028 4d61 6361 756c 6179  ):.    (Macaulay
│ │ │ │ -00071250: 3244 6f63 2963 6f70 7944 6972 6563 746f  2Doc)copyDirecto
│ │ │ │ -00071260: 7279 5f6c 7053 7472 696e 675f 636d 5374  ry_lpString_cmSt
│ │ │ │ -00071270: 7269 6e67 5f72 702c 0a20 202a 2022 636f  ring_rp,.  * "co
│ │ │ │ -00071280: 7079 4669 6c65 282e 2e2e 2c56 6572 626f  pyFile(...,Verbo
│ │ │ │ -00071290: 7365 3d3e 2e2e 2e29 2220 2d2d 2073 6565  se=>...)" -- see
│ │ │ │ -000712a0: 202a 6e6f 7465 2063 6f70 7946 696c 6528   *note copyFile(
│ │ │ │ -000712b0: 5374 7269 6e67 2c53 7472 696e 6729 3a0a  String,String):.
│ │ │ │ -000712c0: 2020 2020 284d 6163 6175 6c61 7932 446f      (Macaulay2Do
│ │ │ │ -000712d0: 6329 636f 7079 4669 6c65 5f6c 7053 7472  c)copyFile_lpStr
│ │ │ │ -000712e0: 696e 675f 636d 5374 7269 6e67 5f72 702c  ing_cmString_rp,
│ │ │ │ -000712f0: 0a20 202a 2022 6669 6e64 5072 6f67 7261  .  * "findProgra
│ │ │ │ -00071300: 6d28 2e2e 2e2c 5665 7262 6f73 653d 3e2e  m(...,Verbose=>.
│ │ │ │ -00071310: 2e2e 2922 202d 2d20 7365 6520 2a6e 6f74  ..)" -- see *not
│ │ │ │ -00071320: 6520 6669 6e64 5072 6f67 7261 6d3a 0a20  e findProgram:. 
│ │ │ │ -00071330: 2020 2028 4d61 6361 756c 6179 3244 6f63     (Macaulay2Doc
│ │ │ │ -00071340: 2966 696e 6450 726f 6772 616d 2c20 2d2d  )findProgram, --
│ │ │ │ -00071350: 206c 6f61 6420 6578 7465 726e 616c 2070   load external p
│ │ │ │ -00071360: 726f 6772 616d 0a20 202a 2022 696e 7374  rogram.  * "inst
│ │ │ │ -00071370: 616c 6c50 6163 6b61 6765 282e 2e2e 2c56  allPackage(...,V
│ │ │ │ -00071380: 6572 626f 7365 3d3e 2e2e 2e29 2220 2d2d  erbose=>...)" --
│ │ │ │ -00071390: 2073 6565 202a 6e6f 7465 2069 6e73 7461   see *note insta
│ │ │ │ -000713a0: 6c6c 5061 636b 6167 653a 0a20 2020 2028  llPackage:.    (
│ │ │ │ -000713b0: 4d61 6361 756c 6179 3244 6f63 2969 6e73  Macaulay2Doc)ins
│ │ │ │ -000713c0: 7461 6c6c 5061 636b 6167 652c 202d 2d20  tallPackage, -- 
│ │ │ │ -000713d0: 6c6f 6164 2061 6e64 2069 6e73 7461 6c6c  load and install
│ │ │ │ -000713e0: 2061 2070 6163 6b61 6765 2061 6e64 2069   a package and i
│ │ │ │ -000713f0: 7473 0a20 2020 2064 6f63 756d 656e 7461  ts.    documenta
│ │ │ │ -00071400: 7469 6f6e 0a20 202a 2022 6170 7072 6f78  tion.  * "approx
│ │ │ │ -00071410: 696d 6174 6549 6e76 6572 7365 4d61 7028  imateInverseMap(
│ │ │ │ -00071420: 2e2e 2e2c 5665 7262 6f73 653d 3e2e 2e2e  ...,Verbose=>...
│ │ │ │ -00071430: 2922 0a20 202a 2022 4368 6572 6e53 6368  )".  * "ChernSch
│ │ │ │ -00071440: 7761 7274 7a4d 6163 5068 6572 736f 6e28  wartzMacPherson(
│ │ │ │ -00071450: 2e2e 2e2c 5665 7262 6f73 653d 3e2e 2e2e  ...,Verbose=>...
│ │ │ │ -00071460: 2922 0a20 202a 2022 6465 6772 6565 4d61  )".  * "degreeMa
│ │ │ │ -00071470: 7028 2e2e 2e2c 5665 7262 6f73 653d 3e2e  p(...,Verbose=>.
│ │ │ │ -00071480: 2e2e 2922 0a20 202a 2022 4575 6c65 7243  ..)".  * "EulerC
│ │ │ │ -00071490: 6861 7261 6374 6572 6973 7469 6328 2e2e  haracteristic(..
│ │ │ │ -000714a0: 2e2c 5665 7262 6f73 653d 3e2e 2e2e 2922  .,Verbose=>...)"
│ │ │ │ -000714b0: 0a20 202a 202a 6e6f 7465 2069 6e76 6572  .  * *note inver
│ │ │ │ -000714c0: 7365 4d61 7028 2e2e 2e2c 5665 7262 6f73  seMap(...,Verbos
│ │ │ │ -000714d0: 653d 3e2e 2e2e 293a 0a20 2020 2069 6e76  e=>...):.    inv
│ │ │ │ -000714e0: 6572 7365 4d61 705f 6c70 5f70 645f 7064  erseMap_lp_pd_pd
│ │ │ │ -000714f0: 5f70 645f 636d 5665 7262 6f73 653d 3e5f  _pd_cmVerbose=>_
│ │ │ │ -00071500: 7064 5f70 645f 7064 5f72 702c 0a20 202a  pd_pd_pd_rp,.  *
│ │ │ │ -00071510: 2022 6973 4269 7261 7469 6f6e 616c 282e   "isBirational(.
│ │ │ │ -00071520: 2e2e 2c56 6572 626f 7365 3d3e 2e2e 2e29  ..,Verbose=>...)
│ │ │ │ -00071530: 220a 2020 2a20 2269 7344 6f6d 696e 616e  ".  * "isDominan
│ │ │ │ -00071540: 7428 2e2e 2e2c 5665 7262 6f73 653d 3e2e  t(...,Verbose=>.
│ │ │ │ -00071550: 2e2e 2922 0a20 202a 2022 7072 6f6a 6563  ..)".  * "projec
│ │ │ │ -00071560: 7469 7665 4465 6772 6565 7328 2e2e 2e2c  tiveDegrees(...,
│ │ │ │ -00071570: 5665 7262 6f73 653d 3e2e 2e2e 2922 0a20  Verbose=>...)". 
│ │ │ │ -00071580: 202a 2022 5365 6772 6543 6c61 7373 282e   * "SegreClass(.
│ │ │ │ -00071590: 2e2e 2c56 6572 626f 7365 3d3e 2e2e 2e29  ..,Verbose=>...)
│ │ │ │ -000715a0: 220a 2020 2a20 226d 6f76 6546 696c 6528  ".  * "moveFile(
│ │ │ │ -000715b0: 2e2e 2e2c 5665 7262 6f73 653d 3e2e 2e2e  ...,Verbose=>...
│ │ │ │ -000715c0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -000715d0: 6d6f 7665 4669 6c65 2853 7472 696e 672c  moveFile(String,
│ │ │ │ -000715e0: 5374 7269 6e67 293a 0a20 2020 2028 4d61  String):.    (Ma
│ │ │ │ -000715f0: 6361 756c 6179 3244 6f63 296d 6f76 6546  caulay2Doc)moveF
│ │ │ │ -00071600: 696c 655f 6c70 5374 7269 6e67 5f63 6d53  ile_lpString_cmS
│ │ │ │ -00071610: 7472 696e 675f 7270 2c0a 2020 2a20 2272  tring_rp,.  * "r
│ │ │ │ -00071620: 756e 5072 6f67 7261 6d28 2e2e 2e2c 5665  unProgram(...,Ve
│ │ │ │ -00071630: 7262 6f73 653d 3e2e 2e2e 2922 202d 2d20  rbose=>...)" -- 
│ │ │ │ -00071640: 7365 6520 2a6e 6f74 6520 7275 6e50 726f  see *note runPro
│ │ │ │ -00071650: 6772 616d 3a0a 2020 2020 284d 6163 6175  gram:.    (Macau
│ │ │ │ -00071660: 6c61 7932 446f 6329 7275 6e50 726f 6772  lay2Doc)runProgr
│ │ │ │ -00071670: 616d 2c20 2d2d 2072 756e 2061 6e20 6578  am, -- run an ex
│ │ │ │ -00071680: 7465 726e 616c 2070 726f 6772 616d 0a20  ternal program. 
│ │ │ │ -00071690: 202a 2022 7379 6d6c 696e 6b44 6972 6563   * "symlinkDirec
│ │ │ │ -000716a0: 746f 7279 282e 2e2e 2c56 6572 626f 7365  tory(...,Verbose
│ │ │ │ -000716b0: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a  =>...)" -- see *
│ │ │ │ -000716c0: 6e6f 7465 0a20 2020 2073 796d 6c69 6e6b  note.    symlink
│ │ │ │ -000716d0: 4469 7265 6374 6f72 7928 5374 7269 6e67  Directory(String
│ │ │ │ -000716e0: 2c53 7472 696e 6729 3a0a 2020 2020 284d  ,String):.    (M
│ │ │ │ -000716f0: 6163 6175 6c61 7932 446f 6329 7379 6d6c  acaulay2Doc)syml
│ │ │ │ -00071700: 696e 6b44 6972 6563 746f 7279 5f6c 7053  inkDirectory_lpS
│ │ │ │ -00071710: 7472 696e 675f 636d 5374 7269 6e67 5f72  tring_cmString_r
│ │ │ │ -00071720: 702c 202d 2d20 6d61 6b65 2073 796d 626f  p, -- make symbo
│ │ │ │ -00071730: 6c69 6320 6c69 6e6b 730a 2020 2020 666f  lic links.    fo
│ │ │ │ -00071740: 7220 616c 6c20 6669 6c65 7320 696e 2061  r all files in a
│ │ │ │ -00071750: 2064 6972 6563 746f 7279 2074 7265 650a   directory tree.
│ │ │ │ -00071760: 0a46 7572 7468 6572 2069 6e66 6f72 6d61  .Further informa
│ │ │ │ -00071770: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -00071780: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 4465  ========..  * De
│ │ │ │ -00071790: 6661 756c 7420 7661 6c75 653a 202a 6e6f  fault value: *no
│ │ │ │ -000717a0: 7465 2074 7275 653a 2028 4d61 6361 756c  te true: (Macaul
│ │ │ │ -000717b0: 6179 3244 6f63 2974 7275 652c 0a20 202a  ay2Doc)true,.  *
│ │ │ │ -000717c0: 2046 756e 6374 696f 6e3a 202a 6e6f 7465   Function: *note
│ │ │ │ -000717d0: 2069 6e76 6572 7365 4d61 703a 2069 6e76   inverseMap: inv
│ │ │ │ -000717e0: 6572 7365 4d61 702c 202d 2d20 696e 7665  erseMap, -- inve
│ │ │ │ -000717f0: 7273 6520 6f66 2061 2062 6972 6174 696f  rse of a biratio
│ │ │ │ -00071800: 6e61 6c20 6d61 700a 2020 2a20 4f70 7469  nal map.  * Opti
│ │ │ │ -00071810: 6f6e 206b 6579 3a20 2a6e 6f74 6520 5665  on key: *note Ve
│ │ │ │ -00071820: 7262 6f73 653a 2028 4d61 6361 756c 6179  rbose: (Macaulay
│ │ │ │ -00071830: 3244 6f63 2956 6572 626f 7365 2c20 2d2d  2Doc)Verbose, --
│ │ │ │ -00071840: 2061 6e20 6f70 7469 6f6e 616c 2061 7267   an optional arg
│ │ │ │ -00071850: 756d 656e 740a 2d2d 2d2d 2d2d 2d2d 2d2d  ument.----------
│ │ │ │ +00071170: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +00071180: 202a 2022 6368 6563 6b28 2e2e 2e2c 5665   * "check(...,Ve
│ │ │ │ +00071190: 7262 6f73 653d 3e2e 2e2e 2922 202d 2d20  rbose=>...)" -- 
│ │ │ │ +000711a0: 7365 6520 2a6e 6f74 6520 6368 6563 6b3a  see *note check:
│ │ │ │ +000711b0: 2028 4d61 6361 756c 6179 3244 6f63 2963   (Macaulay2Doc)c
│ │ │ │ +000711c0: 6865 636b 2c20 2d2d 0a20 2020 2070 6572  heck, --.    per
│ │ │ │ +000711d0: 666f 726d 2074 6573 7473 206f 6620 6120  form tests of a 
│ │ │ │ +000711e0: 7061 636b 6167 650a 2020 2a20 2263 6f70  package.  * "cop
│ │ │ │ +000711f0: 7944 6972 6563 746f 7279 282e 2e2e 2c56  yDirectory(...,V
│ │ │ │ +00071200: 6572 626f 7365 3d3e 2e2e 2e29 2220 2d2d  erbose=>...)" --
│ │ │ │ +00071210: 2073 6565 202a 6e6f 7465 0a20 2020 2063   see *note.    c
│ │ │ │ +00071220: 6f70 7944 6972 6563 746f 7279 2853 7472  opyDirectory(Str
│ │ │ │ +00071230: 696e 672c 5374 7269 6e67 293a 0a20 2020  ing,String):.   
│ │ │ │ +00071240: 2028 4d61 6361 756c 6179 3244 6f63 2963   (Macaulay2Doc)c
│ │ │ │ +00071250: 6f70 7944 6972 6563 746f 7279 5f6c 7053  opyDirectory_lpS
│ │ │ │ +00071260: 7472 696e 675f 636d 5374 7269 6e67 5f72  tring_cmString_r
│ │ │ │ +00071270: 702c 0a20 202a 2022 636f 7079 4669 6c65  p,.  * "copyFile
│ │ │ │ +00071280: 282e 2e2e 2c56 6572 626f 7365 3d3e 2e2e  (...,Verbose=>..
│ │ │ │ +00071290: 2e29 2220 2d2d 2073 6565 202a 6e6f 7465  .)" -- see *note
│ │ │ │ +000712a0: 2063 6f70 7946 696c 6528 5374 7269 6e67   copyFile(String
│ │ │ │ +000712b0: 2c53 7472 696e 6729 3a0a 2020 2020 284d  ,String):.    (M
│ │ │ │ +000712c0: 6163 6175 6c61 7932 446f 6329 636f 7079  acaulay2Doc)copy
│ │ │ │ +000712d0: 4669 6c65 5f6c 7053 7472 696e 675f 636d  File_lpString_cm
│ │ │ │ +000712e0: 5374 7269 6e67 5f72 702c 0a20 202a 2022  String_rp,.  * "
│ │ │ │ +000712f0: 6669 6e64 5072 6f67 7261 6d28 2e2e 2e2c  findProgram(...,
│ │ │ │ +00071300: 5665 7262 6f73 653d 3e2e 2e2e 2922 202d  Verbose=>...)" -
│ │ │ │ +00071310: 2d20 7365 6520 2a6e 6f74 6520 6669 6e64  - see *note find
│ │ │ │ +00071320: 5072 6f67 7261 6d3a 0a20 2020 2028 4d61  Program:.    (Ma
│ │ │ │ +00071330: 6361 756c 6179 3244 6f63 2966 696e 6450  caulay2Doc)findP
│ │ │ │ +00071340: 726f 6772 616d 2c20 2d2d 206c 6f61 6420  rogram, -- load 
│ │ │ │ +00071350: 6578 7465 726e 616c 2070 726f 6772 616d  external program
│ │ │ │ +00071360: 0a20 202a 2022 696e 7374 616c 6c50 6163  .  * "installPac
│ │ │ │ +00071370: 6b61 6765 282e 2e2e 2c56 6572 626f 7365  kage(...,Verbose
│ │ │ │ +00071380: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a  =>...)" -- see *
│ │ │ │ +00071390: 6e6f 7465 2069 6e73 7461 6c6c 5061 636b  note installPack
│ │ │ │ +000713a0: 6167 653a 0a20 2020 2028 4d61 6361 756c  age:.    (Macaul
│ │ │ │ +000713b0: 6179 3244 6f63 2969 6e73 7461 6c6c 5061  ay2Doc)installPa
│ │ │ │ +000713c0: 636b 6167 652c 202d 2d20 6c6f 6164 2061  ckage, -- load a
│ │ │ │ +000713d0: 6e64 2069 6e73 7461 6c6c 2061 2070 6163  nd install a pac
│ │ │ │ +000713e0: 6b61 6765 2061 6e64 2069 7473 0a20 2020  kage and its.   
│ │ │ │ +000713f0: 2064 6f63 756d 656e 7461 7469 6f6e 0a20   documentation. 
│ │ │ │ +00071400: 202a 2022 6170 7072 6f78 696d 6174 6549   * "approximateI
│ │ │ │ +00071410: 6e76 6572 7365 4d61 7028 2e2e 2e2c 5665  nverseMap(...,Ve
│ │ │ │ +00071420: 7262 6f73 653d 3e2e 2e2e 2922 0a20 202a  rbose=>...)".  *
│ │ │ │ +00071430: 2022 4368 6572 6e53 6368 7761 7274 7a4d   "ChernSchwartzM
│ │ │ │ +00071440: 6163 5068 6572 736f 6e28 2e2e 2e2c 5665  acPherson(...,Ve
│ │ │ │ +00071450: 7262 6f73 653d 3e2e 2e2e 2922 0a20 202a  rbose=>...)".  *
│ │ │ │ +00071460: 2022 6465 6772 6565 4d61 7028 2e2e 2e2c   "degreeMap(...,
│ │ │ │ +00071470: 5665 7262 6f73 653d 3e2e 2e2e 2922 0a20  Verbose=>...)". 
│ │ │ │ +00071480: 202a 2022 4575 6c65 7243 6861 7261 6374   * "EulerCharact
│ │ │ │ +00071490: 6572 6973 7469 6328 2e2e 2e2c 5665 7262  eristic(...,Verb
│ │ │ │ +000714a0: 6f73 653d 3e2e 2e2e 2922 0a20 202a 202a  ose=>...)".  * *
│ │ │ │ +000714b0: 6e6f 7465 2069 6e76 6572 7365 4d61 7028  note inverseMap(
│ │ │ │ +000714c0: 2e2e 2e2c 5665 7262 6f73 653d 3e2e 2e2e  ...,Verbose=>...
│ │ │ │ +000714d0: 293a 0a20 2020 2069 6e76 6572 7365 4d61  ):.    inverseMa
│ │ │ │ +000714e0: 705f 6c70 5f70 645f 7064 5f70 645f 636d  p_lp_pd_pd_pd_cm
│ │ │ │ +000714f0: 5665 7262 6f73 653d 3e5f 7064 5f70 645f  Verbose=>_pd_pd_
│ │ │ │ +00071500: 7064 5f72 702c 0a20 202a 2022 6973 4269  pd_rp,.  * "isBi
│ │ │ │ +00071510: 7261 7469 6f6e 616c 282e 2e2e 2c56 6572  rational(...,Ver
│ │ │ │ +00071520: 626f 7365 3d3e 2e2e 2e29 220a 2020 2a20  bose=>...)".  * 
│ │ │ │ +00071530: 2269 7344 6f6d 696e 616e 7428 2e2e 2e2c  "isDominant(...,
│ │ │ │ +00071540: 5665 7262 6f73 653d 3e2e 2e2e 2922 0a20  Verbose=>...)". 
│ │ │ │ +00071550: 202a 2022 7072 6f6a 6563 7469 7665 4465   * "projectiveDe
│ │ │ │ +00071560: 6772 6565 7328 2e2e 2e2c 5665 7262 6f73  grees(...,Verbos
│ │ │ │ +00071570: 653d 3e2e 2e2e 2922 0a20 202a 2022 5365  e=>...)".  * "Se
│ │ │ │ +00071580: 6772 6543 6c61 7373 282e 2e2e 2c56 6572  greClass(...,Ver
│ │ │ │ +00071590: 626f 7365 3d3e 2e2e 2e29 220a 2020 2a20  bose=>...)".  * 
│ │ │ │ +000715a0: 226d 6f76 6546 696c 6528 2e2e 2e2c 5665  "moveFile(...,Ve
│ │ │ │ +000715b0: 7262 6f73 653d 3e2e 2e2e 2922 202d 2d20  rbose=>...)" -- 
│ │ │ │ +000715c0: 7365 6520 2a6e 6f74 6520 6d6f 7665 4669  see *note moveFi
│ │ │ │ +000715d0: 6c65 2853 7472 696e 672c 5374 7269 6e67  le(String,String
│ │ │ │ +000715e0: 293a 0a20 2020 2028 4d61 6361 756c 6179  ):.    (Macaulay
│ │ │ │ +000715f0: 3244 6f63 296d 6f76 6546 696c 655f 6c70  2Doc)moveFile_lp
│ │ │ │ +00071600: 5374 7269 6e67 5f63 6d53 7472 696e 675f  String_cmString_
│ │ │ │ +00071610: 7270 2c0a 2020 2a20 2272 756e 5072 6f67  rp,.  * "runProg
│ │ │ │ +00071620: 7261 6d28 2e2e 2e2c 5665 7262 6f73 653d  ram(...,Verbose=
│ │ │ │ +00071630: 3e2e 2e2e 2922 202d 2d20 7365 6520 2a6e  >...)" -- see *n
│ │ │ │ +00071640: 6f74 6520 7275 6e50 726f 6772 616d 3a0a  ote runProgram:.
│ │ │ │ +00071650: 2020 2020 284d 6163 6175 6c61 7932 446f      (Macaulay2Do
│ │ │ │ +00071660: 6329 7275 6e50 726f 6772 616d 2c20 2d2d  c)runProgram, --
│ │ │ │ +00071670: 2072 756e 2061 6e20 6578 7465 726e 616c   run an external
│ │ │ │ +00071680: 2070 726f 6772 616d 0a20 202a 2022 7379   program.  * "sy
│ │ │ │ +00071690: 6d6c 696e 6b44 6972 6563 746f 7279 282e  mlinkDirectory(.
│ │ │ │ +000716a0: 2e2e 2c56 6572 626f 7365 3d3e 2e2e 2e29  ..,Verbose=>...)
│ │ │ │ +000716b0: 2220 2d2d 2073 6565 202a 6e6f 7465 0a20  " -- see *note. 
│ │ │ │ +000716c0: 2020 2073 796d 6c69 6e6b 4469 7265 6374     symlinkDirect
│ │ │ │ +000716d0: 6f72 7928 5374 7269 6e67 2c53 7472 696e  ory(String,Strin
│ │ │ │ +000716e0: 6729 3a0a 2020 2020 284d 6163 6175 6c61  g):.    (Macaula
│ │ │ │ +000716f0: 7932 446f 6329 7379 6d6c 696e 6b44 6972  y2Doc)symlinkDir
│ │ │ │ +00071700: 6563 746f 7279 5f6c 7053 7472 696e 675f  ectory_lpString_
│ │ │ │ +00071710: 636d 5374 7269 6e67 5f72 702c 202d 2d20  cmString_rp, -- 
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│ │ │ │  000719f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00071a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00071c50: 2062 6972 6174 696f 6e61 6c0a 0a44 6573   birational..Des
│ │ │ │ -00071c60: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
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│ │ │ │ -00071c90: 2074 6865 206d 6574 686f 6473 202a 6e6f   the methods *no
│ │ │ │ -00071ca0: 7465 2070 726f 6a65 6374 6976 6544 6567  te projectiveDeg
│ │ │ │ -00071cb0: 7265 6573 3a0a 7072 6f6a 6563 7469 7665  rees:.projective
│ │ │ │ -00071cc0: 4465 6772 6565 732c 2c20 2a6e 6f74 6520  Degrees,, *note 
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│ │ │ │ +00071a10: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ +00071a20: 2020 6973 4269 7261 7469 6f6e 616c 2070    isBirational p
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│ │ │ │ +00071a60: 3a20 5261 7469 6f6e 616c 4d61 702c 0a20  : RationalMap,. 
│ │ │ │ +00071a70: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ +00071a80: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ +00071a90: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
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│ │ │ │ +00071ac0: 2020 2020 202a 202a 6e6f 7465 2042 6c6f       * *note Blo
│ │ │ │ +00071ad0: 7755 7053 7472 6174 6567 793a 2042 6c6f  wUpStrategy: Blo
│ │ │ │ +00071ae0: 7755 7053 7472 6174 6567 792c 203d 3e20  wUpStrategy, => 
│ │ │ │ +00071af0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
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│ │ │ │ +00071bf0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
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│ │ │ │  00071e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00071e50: 2020 2020 2030 2020 2034 2020 2020 2020       0   4      
│ │ │ │  00071e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00071e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00071e90: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00071ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00071eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00071ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00071ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00071f30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00071f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00071f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00071f80: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ -00071f90: 7068 6920 3d20 7261 7469 6f6e 616c 4d61  phi = rationalMa
│ │ │ │ -00071fa0: 7028 6d69 6e6f 7273 2832 2c6d 6174 7269  p(minors(2,matri
│ │ │ │ -00071fb0: 787b 7b74 5f30 2e2e 745f 337d 2c7b 745f  x{{t_0..t_3},{t_
│ │ │ │ -00071fc0: 312e 2e74 5f34 7d7d 292c 2020 2020 2020  1..t_4}}),      
│ │ │ │ -00071fd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00071f80: 2d2d 2b0a 7c69 3220 3a20 7068 6920 3d20  --+.|i2 : phi = 
│ │ │ │ +00071f90: 7261 7469 6f6e 616c 4d61 7028 6d69 6e6f  rationalMap(mino
│ │ │ │ +00071fa0: 7273 2832 2c6d 6174 7269 787b 7b74 5f30  rs(2,matrix{{t_0
│ │ │ │ +00071fb0: 2e2e 745f 337d 2c7b 745f 312e 2e74 5f34  ..t_3},{t_1..t_4
│ │ │ │ +00071fc0: 7d7d 292c 2020 2020 2020 2020 2020 2020  }}),            
│ │ │ │ +00071fd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00071fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00071ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072020: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ -00072030: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ -00072040: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +00072020: 2020 7c0a 7c6f 3220 3d20 2d2d 2072 6174    |.|o2 = -- rat
│ │ │ │ +00072030: 696f 6e61 6c20 6d61 7020 2d2d 2020 2020  ional map --    
│ │ │ │ +00072040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072070: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00072080: 736f 7572 6365 3a20 5072 6f6a 2847 4620  source: Proj(GF 
│ │ │ │ -00072090: 3130 3935 3631 5b74 202c 2074 202c 2074  109561[t , t , t
│ │ │ │ -000720a0: 202c 2074 202c 2074 205d 2920 2020 2020   , t , t ])     
│ │ │ │ +00072070: 2020 7c0a 7c20 2020 2020 736f 7572 6365    |.|     source
│ │ │ │ +00072080: 3a20 5072 6f6a 2847 4620 3130 3935 3631  : Proj(GF 109561
│ │ │ │ +00072090: 5b74 202c 2074 202c 2074 202c 2074 202c  [t , t , t , t ,
│ │ │ │ +000720a0: 2074 205d 2920 2020 2020 2020 2020 2020   t ])           
│ │ │ │  000720b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000720c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000720c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000720d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000720e0: 2020 2020 2020 2020 3020 2020 3120 2020          0   1   
│ │ │ │ -000720f0: 3220 2020 3320 2020 3420 2020 2020 2020  2   3   4       
│ │ │ │ +000720e0: 2020 3020 2020 3120 2020 3220 2020 3320    0   1   2   3 
│ │ │ │ +000720f0: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │  00072100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072110: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00072120: 7461 7267 6574 3a20 7375 6276 6172 6965  target: subvarie
│ │ │ │ -00072130: 7479 206f 6620 5072 6f6a 2847 4620 3130  ty of Proj(GF 10
│ │ │ │ -00072140: 3935 3631 5b78 202c 2078 202c 2078 202c  9561[x , x , x ,
│ │ │ │ -00072150: 2078 202c 2078 202c 2078 2020 2020 2020   x , x , x      
│ │ │ │ -00072160: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00072110: 2020 7c0a 7c20 2020 2020 7461 7267 6574    |.|     target
│ │ │ │ +00072120: 3a20 7375 6276 6172 6965 7479 206f 6620  : subvariety of 
│ │ │ │ +00072130: 5072 6f6a 2847 4620 3130 3935 3631 5b78  Proj(GF 109561[x
│ │ │ │ +00072140: 202c 2078 202c 2078 202c 2078 202c 2078   , x , x , x , x
│ │ │ │ +00072150: 202c 2078 2020 2020 2020 2020 2020 2020   , x            
│ │ │ │ +00072160: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00072170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072190: 2020 2020 2020 3020 2020 3120 2020 3220        0   1   2 
│ │ │ │ -000721a0: 2020 3320 2020 3420 2020 2020 2020 2020    3   4         
│ │ │ │ -000721b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000721c0: 2020 2020 2020 2020 7b20 2020 2020 2020          {       
│ │ │ │ +00072190: 3020 2020 3120 2020 3220 2020 3320 2020  0   1   2   3   
│ │ │ │ +000721a0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +000721b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000721c0: 2020 7b20 2020 2020 2020 2020 2020 2020    {             
│ │ │ │  000721d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000721e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000721f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072200: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00072210: 2020 2020 2020 2020 2078 2078 2020 2d20           x x  - 
│ │ │ │ -00072220: 7820 7820 202b 2078 2078 2020 2020 2020  x x  + x x      
│ │ │ │ +00072200: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00072210: 2020 2078 2078 2020 2d20 7820 7820 202b     x x  - x x  +
│ │ │ │ +00072220: 2078 2078 2020 2020 2020 2020 2020 2020   x x            
│ │ │ │  00072230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072250: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00072260: 2020 2020 2020 2020 2020 3220 3320 2020            2 3   
│ │ │ │ -00072270: 2031 2034 2020 2020 3020 3520 2020 2020   1 4    0 5     
│ │ │ │ +00072250: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00072260: 2020 2020 3220 3320 2020 2031 2034 2020      2 3    1 4  
│ │ │ │ +00072270: 2020 3020 3520 2020 2020 2020 2020 2020    0 5           
│ │ │ │  00072280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000722a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000722b0: 2020 2020 2020 2020 7d20 2020 2020 2020          }       
│ │ │ │ +000722a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000722b0: 2020 7d20 2020 2020 2020 2020 2020 2020    }             
│ │ │ │  000722c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000722d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000722e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000722f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00072300: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20  defining forms: 
│ │ │ │ -00072310: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │ +000722f0: 2020 7c0a 7c20 2020 2020 6465 6669 6e69    |.|     defini
│ │ │ │ +00072300: 6e67 2066 6f72 6d73 3a20 7b20 2020 2020  ng forms: {     
│ │ │ │ +00072310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072340: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00072350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072360: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00072340: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00072350: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00072360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072390: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000723a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000723b0: 202d 2074 2020 2b20 7420 7420 2c20 2020   - t  + t t ,   
│ │ │ │ +00072390: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000723a0: 2020 2020 2020 2020 2020 202d 2074 2020             - t  
│ │ │ │ +000723b0: 2b20 7420 7420 2c20 2020 2020 2020 2020  + t t ,         
│ │ │ │  000723c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000723d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000723e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000723f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072400: 2020 2020 3120 2020 2030 2032 2020 2020      1    0 2    
│ │ │ │ +000723e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000723f0: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +00072400: 2020 2030 2032 2020 2020 2020 2020 2020     0 2          
│ │ │ │  00072410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072430: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00072430: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00072440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072480: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00072490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000724a0: 202d 2074 2074 2020 2b20 7420 7420 2c20   - t t  + t t , 
│ │ │ │ +00072480: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00072490: 2020 2020 2020 2020 2020 202d 2074 2074             - t t
│ │ │ │ +000724a0: 2020 2b20 7420 7420 2c20 2020 2020 2020    + t t ,       
│ │ │ │  000724b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000724c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000724d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000724e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000724f0: 2020 2020 3120 3220 2020 2030 2033 2020      1 2    0 3  
│ │ │ │ +000724d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000724e0: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +000724f0: 3220 2020 2030 2033 2020 2020 2020 2020  2    0 3        
│ │ │ │  00072500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072520: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00072520: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00072530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072570: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00072580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072590: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00072570: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00072580: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00072590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000725a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000725b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000725c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000725d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000725e0: 202d 2074 2020 2b20 7420 7420 2c20 2020   - t  + t t ,   
│ │ │ │ +000725c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000725d0: 2020 2020 2020 2020 2020 202d 2074 2020             - t  
│ │ │ │ +000725e0: 2b20 7420 7420 2c20 2020 2020 2020 2020  + t t ,         
│ │ │ │  000725f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072610: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00072620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072630: 2020 2020 3220 2020 2031 2033 2020 2020      2    1 3    
│ │ │ │ +00072610: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00072620: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00072630: 2020 2031 2033 2020 2020 2020 2020 2020     1 3          
│ │ │ │  00072640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072660: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00072660: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00072670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000726a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000726b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000726c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000726d0: 202d 2074 2074 2020 2b20 7420 7420 2c20   - t t  + t t , 
│ │ │ │ +000726b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000726c0: 2020 2020 2020 2020 2020 202d 2074 2074             - t t
│ │ │ │ +000726d0: 2020 2b20 7420 7420 2c20 2020 2020 2020    + t t ,       
│ │ │ │  000726e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000726f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072700: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00072710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072720: 2020 2020 3120 3320 2020 2030 2034 2020      1 3    0 4  
│ │ │ │ +00072700: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00072710: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +00072720: 3320 2020 2030 2034 2020 2020 2020 2020  3    0 4        
│ │ │ │  00072730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072750: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00072750: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00072760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000727a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000727b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000727c0: 202d 2074 2074 2020 2b20 7420 7420 2c20   - t t  + t t , 
│ │ │ │ +000727a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000727b0: 2020 2020 2020 2020 2020 202d 2074 2074             - t t
│ │ │ │ +000727c0: 2020 2b20 7420 7420 2c20 2020 2020 2020    + t t ,       
│ │ │ │  000727d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000727e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000727f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00072800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072810: 2020 2020 3220 3320 2020 2031 2034 2020      2 3    1 4  
│ │ │ │ +000727f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00072800: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00072810: 3320 2020 2031 2034 2020 2020 2020 2020  3    1 4        
│ │ │ │  00072820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072840: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00072840: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00072850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072890: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000728a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000728b0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00072890: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000728a0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +000728b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000728c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000728d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000728e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000728f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072900: 202d 2074 2020 2b20 7420 7420 2020 2020   - t  + t t     
│ │ │ │ +000728e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000728f0: 2020 2020 2020 2020 2020 202d 2074 2020             - t  
│ │ │ │ +00072900: 2b20 7420 7420 2020 2020 2020 2020 2020  + t t           
│ │ │ │  00072910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072930: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00072940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072950: 2020 2020 3320 2020 2032 2034 2020 2020      3    2 4    
│ │ │ │ +00072930: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00072940: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ +00072950: 2020 2032 2034 2020 2020 2020 2020 2020     2 4          
│ │ │ │  00072960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072980: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00072990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000729a0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00072980: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00072990: 2020 2020 2020 2020 2020 7d20 2020 2020            }     
│ │ │ │ +000729a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000729b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000729c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000729d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000729d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000729e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000729f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00072a20: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ -00072a30: 5261 7469 6f6e 616c 4d61 7020 2871 7561  RationalMap (qua
│ │ │ │ -00072a40: 6472 6174 6963 2064 6f6d 696e 616e 7420  dratic dominant 
│ │ │ │ -00072a50: 7261 7469 6f6e 616c 206d 6170 2066 726f  rational map fro
│ │ │ │ -00072a60: 6d20 5050 5e34 2074 6f20 2020 2020 2020  m PP^4 to       
│ │ │ │ -00072a70: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d          |.|-----
│ │ │ │ +00072a20: 2020 7c0a 7c6f 3220 3a20 5261 7469 6f6e    |.|o2 : Ration
│ │ │ │ +00072a30: 616c 4d61 7020 2871 7561 6472 6174 6963  alMap (quadratic
│ │ │ │ +00072a40: 2064 6f6d 696e 616e 7420 7261 7469 6f6e   dominant ration
│ │ │ │ +00072a50: 616c 206d 6170 2066 726f 6d20 5050 5e34  al map from PP^4
│ │ │ │ +00072a60: 2074 6f20 2020 2020 2020 2020 2020 2020   to             
│ │ │ │ +00072a70: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │  00072a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00072ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00072ac0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c44 6f6d 696e  --------|.|Domin
│ │ │ │ -00072ad0: 616e 743d 3e69 6e66 696e 6974 7929 2020  ant=>infinity)  
│ │ │ │ +00072ac0: 2d2d 7c0a 7c44 6f6d 696e 616e 743d 3e69  --|.|Dominant=>i
│ │ │ │ +00072ad0: 6e66 696e 6974 7929 2020 2020 2020 2020  nfinity)        
│ │ │ │  00072ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00072af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00072c60: 2062 7920 2020 2020 2020 2020 2020 2020   by             
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│ │ │ │  00072cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00072fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00073000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073010: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │  00073020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073040: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00073090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000730a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000730b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │  000730c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000730d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000730e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000730f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00073180: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00073360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00073380: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │  000733a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000733b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000733c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000737a0: 7469 6d65 2069 7342 6972 6174 696f 6e61  time isBirationa
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│ │ │ │ -00073970: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ -00073980: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
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│ │ │ │ +00073c50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00073c60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00073c70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00073c90: 0a20 2020 2020 2020 2069 7344 6f6d 696e  .        isDomin
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│ │ │ │ +00073da0: 202a 202a 6e6f 7465 2056 6572 626f 7365   * *note Verbose
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│ │ │ │ +00073ff0: 2070 6869 203d 2072 6174 696f 6e61 6c4d   phi = rationalM
│ │ │ │ +00074000: 6170 2069 6465 616c 206a 6163 6f62 6961  ap ideal jacobia
│ │ │ │ +00074010: 6e20 6964 6561 6c20 2020 2020 2020 2020  n ideal         
│ │ │ │  00074020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074030: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00074040: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00074030: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00074040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00074090: 0a7c 6f32 203a 2052 6174 696f 6e61 6c4d  .|o2 : RationalM
│ │ │ │ -000740a0: 6170 2028 7261 7469 6f6e 616c 206d 6170  ap (rational map
│ │ │ │ -000740b0: 2066 726f 6d20 5050 5e38 2074 6f20 2020   from PP^8 to   
│ │ │ │ +00074080: 2020 2020 2020 2020 207c 0a7c 6f32 203a           |.|o2 :
│ │ │ │ +00074090: 2052 6174 696f 6e61 6c4d 6170 2028 7261   RationalMap (ra
│ │ │ │ +000740a0: 7469 6f6e 616c 206d 6170 2066 726f 6d20  tional map from 
│ │ │ │ +000740b0: 5050 5e38 2074 6f20 2020 2020 2020 2020  PP^8 to         
│ │ │ │  000740c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000740d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000740e0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000740d0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │ +000740e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000740f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00074130: 0a7c 6465 7420 6d61 7472 6978 7b7b 785f  .|det matrix{{x_
│ │ │ │ -00074140: 302e 2e78 5f34 7d2c 7b78 5f31 2e2e 785f  0..x_4},{x_1..x_
│ │ │ │ -00074150: 357d 2c7b 785f 322e 2e78 5f36 7d2c 7b78  5},{x_2..x_6},{x
│ │ │ │ -00074160: 5f33 2e2e 785f 377d 2c7b 785f 342e 2e78  _3..x_7},{x_4..x
│ │ │ │ -00074170: 5f38 7d7d 3b20 2020 2020 2020 2020 207c  _8}};          |
│ │ │ │ -00074180: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00074120: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 6465 7420  ---------|.|det 
│ │ │ │ +00074130: 6d61 7472 6978 7b7b 785f 302e 2e78 5f34  matrix{{x_0..x_4
│ │ │ │ +00074140: 7d2c 7b78 5f31 2e2e 785f 357d 2c7b 785f  },{x_1..x_5},{x_
│ │ │ │ +00074150: 322e 2e78 5f36 7d2c 7b78 5f33 2e2e 785f  2..x_6},{x_3..x_
│ │ │ │ +00074160: 377d 2c7b 785f 342e 2e78 5f38 7d7d 3b20  7},{x_4..x_8}}; 
│ │ │ │ +00074170: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00074180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000741a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000741b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000741c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000741d0: 0a7c 5050 5e38 2920 2020 2020 2020 2020  .|PP^8)         
│ │ │ │ +000741c0: 2020 2020 2020 2020 207c 0a7c 5050 5e38           |.|PP^8
│ │ │ │ +000741d0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  000741e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000741f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00074220: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00074210: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00074220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00074270: 0a7c 6933 203a 2074 696d 6520 6973 446f  .|i3 : time isDo
│ │ │ │ -00074280: 6d69 6e61 6e74 2870 6869 2c43 6572 7469  minant(phi,Certi
│ │ │ │ -00074290: 6679 3d3e 7472 7565 2920 2020 2020 2020  fy=>true)       
│ │ │ │ +00074260: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ +00074270: 2074 696d 6520 6973 446f 6d69 6e61 6e74   time isDominant
│ │ │ │ +00074280: 2870 6869 2c43 6572 7469 6679 3d3e 7472  (phi,Certify=>tr
│ │ │ │ +00074290: 7565 2920 2020 2020 2020 2020 2020 2020  ue)             
│ │ │ │  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!    
│ │ │ │ +000742b0: 2020 2020 2020 2020 207c 0a7c 4365 7274           |.|Cert
│ │ │ │ +000742c0: 6966 793a 206f 7574 7075 7420 6365 7274  ify: output cert
│ │ │ │ +000742d0: 6966 6965 6421 2020 2020 2020 2020 2020  ified!          
│ │ │ │  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 2e35 3833  .| -- used 2.583
│ │ │ │ -00074320: 3435 7320 2863 7075 293b 2032 2e30 3131  45s (cpu); 2.011
│ │ │ │ -00074330: 3334 7320 2874 6872 6561 6429 3b20 3073  34s (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  .|              
│ │ │ │ +00074300: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ +00074310: 7573 6564 2032 2e36 3534 3733 7320 2863  used 2.65473s (c
│ │ │ │ +00074320: 7075 293b 2032 2e33 3139 3936 7320 2874  pu); 2.31996s (t
│ │ │ │ +00074330: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +00074340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00074350: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00074360: 2020 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                 |
│ │ │ │ -000743b0: 0a7c 6f33 203d 2074 7275 6520 2020 2020  .|o3 = true     
│ │ │ │ +000743a0: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ +000743b0: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │  000743c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000743d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000743e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000743f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00074400: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000743f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00074400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00074450: 0a7c 6934 203a 2050 3720 3d20 5a5a 2f31  .|i4 : P7 = ZZ/1
│ │ │ │ -00074460: 3031 5b78 5f30 2e2e 785f 375d 3b20 2020  01[x_0..x_7];   
│ │ │ │ +00074440: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ +00074450: 2050 3720 3d20 5a5a 2f31 3031 5b78 5f30   P7 = ZZ/101[x_0
│ │ │ │ +00074460: 2e2e 785f 375d 3b20 2020 2020 2020 2020  ..x_7];         
│ │ │ │  00074470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074490: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000744a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00074490: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000744a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000744b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000744c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000744d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000744e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000744f0: 0a7c 6935 203a 202d 2d20 6879 7065 7265  .|i5 : -- hypere
│ │ │ │ -00074500: 6c6c 6970 7469 6320 6375 7276 6520 6f66  lliptic curve of
│ │ │ │ -00074510: 2067 656e 7573 2033 2020 2020 2020 2020   genus 3        
│ │ │ │ +000744e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ +000744f0: 202d 2d20 6879 7065 7265 6c6c 6970 7469   -- hyperellipti
│ │ │ │ +00074500: 6320 6375 7276 6520 6f66 2067 656e 7573  c curve of genus
│ │ │ │ +00074510: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  00074520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074530: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00074540: 0a7c 2020 2020 2043 203d 2069 6465 616c  .|     C = ideal
│ │ │ │ -00074550: 2878 5f34 2a78 5f35 2b32 332a 785f 355e  (x_4*x_5+23*x_5^
│ │ │ │ -00074560: 322d 3233 2a78 5f30 2a78 5f36 2d31 382a  2-23*x_0*x_6-18*
│ │ │ │ -00074570: 785f 312a 785f 362b 362a 785f 322a 785f  x_1*x_6+6*x_2*x_
│ │ │ │ -00074580: 362b 3337 2a78 5f33 2a78 5f36 2b32 337c  6+37*x_3*x_6+23|
│ │ │ │ -00074590: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00074530: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00074540: 2043 203d 2069 6465 616c 2878 5f34 2a78   C = ideal(x_4*x
│ │ │ │ +00074550: 5f35 2b32 332a 785f 355e 322d 3233 2a78  _5+23*x_5^2-23*x
│ │ │ │ +00074560: 5f30 2a78 5f36 2d31 382a 785f 312a 785f  _0*x_6-18*x_1*x_
│ │ │ │ +00074570: 362b 362a 785f 322a 785f 362b 3337 2a78  6+6*x_2*x_6+37*x
│ │ │ │ +00074580: 5f33 2a78 5f36 2b32 337c 0a7c 2020 2020  _3*x_6+23|.|    
│ │ │ │ +00074590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000745a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000745b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000745c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000745d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000745e0: 0a7c 6f35 203a 2049 6465 616c 206f 6620  .|o5 : Ideal of 
│ │ │ │ -000745f0: 5037 2020 2020 2020 2020 2020 2020 2020  P7              
│ │ │ │ +000745d0: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ +000745e0: 2049 6465 616c 206f 6620 5037 2020 2020   Ideal of P7    
│ │ │ │ +000745f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00074620: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00074630: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00074620: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │ +00074630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00074680: 0a7c 2a78 5f34 2a78 5f36 2d32 362a 785f  .|*x_4*x_6-26*x_
│ │ │ │ -00074690: 352a 785f 362b 322a 785f 365e 322d 3235  5*x_6+2*x_6^2-25
│ │ │ │ -000746a0: 2a78 5f30 2a78 5f37 2b34 352a 785f 312a  *x_0*x_7+45*x_1*
│ │ │ │ -000746b0: 785f 372b 3330 2a78 5f32 2a78 5f37 2d34  x_7+30*x_2*x_7-4
│ │ │ │ -000746c0: 392a 785f 332a 785f 372d 3439 2a78 5f7c  9*x_3*x_7-49*x_|
│ │ │ │ -000746d0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00074670: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2a78 5f34  ---------|.|*x_4
│ │ │ │ +00074680: 2a78 5f36 2d32 362a 785f 352a 785f 362b  *x_6-26*x_5*x_6+
│ │ │ │ +00074690: 322a 785f 365e 322d 3235 2a78 5f30 2a78  2*x_6^2-25*x_0*x
│ │ │ │ +000746a0: 5f37 2b34 352a 785f 312a 785f 372b 3330  _7+45*x_1*x_7+30
│ │ │ │ +000746b0: 2a78 5f32 2a78 5f37 2d34 392a 785f 332a  *x_2*x_7-49*x_3*
│ │ │ │ +000746c0: 785f 372d 3439 2a78 5f7c 0a7c 2d2d 2d2d  x_7-49*x_|.|----
│ │ │ │ +000746d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000746e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000746f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00074720: 0a7c 342a 785f 372b 3530 2a78 5f35 2a78  .|4*x_7+50*x_5*x
│ │ │ │ -00074730: 5f37 2c78 5f33 2a78 5f35 2d32 342a 785f  _7,x_3*x_5-24*x_
│ │ │ │ -00074740: 355e 322b 3231 2a78 5f30 2a78 5f36 2b78  5^2+21*x_0*x_6+x
│ │ │ │ -00074750: 5f31 2a78 5f36 2b34 362a 785f 332a 785f  _1*x_6+46*x_3*x_
│ │ │ │ -00074760: 362b 3237 2a78 5f34 2a78 5f36 2b35 2a7c  6+27*x_4*x_6+5*|
│ │ │ │ -00074770: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00074710: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 342a 785f  ---------|.|4*x_
│ │ │ │ +00074720: 372b 3530 2a78 5f35 2a78 5f37 2c78 5f33  7+50*x_5*x_7,x_3
│ │ │ │ +00074730: 2a78 5f35 2d32 342a 785f 355e 322b 3231  *x_5-24*x_5^2+21
│ │ │ │ +00074740: 2a78 5f30 2a78 5f36 2b78 5f31 2a78 5f36  *x_0*x_6+x_1*x_6
│ │ │ │ +00074750: 2b34 362a 785f 332a 785f 362b 3237 2a78  +46*x_3*x_6+27*x
│ │ │ │ +00074760: 5f34 2a78 5f36 2b35 2a7c 0a7c 2d2d 2d2d  _4*x_6+5*|.|----
│ │ │ │ +00074770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000747a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000747b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000747c0: 0a7c 785f 352a 785f 362b 3335 2a78 5f36  .|x_5*x_6+35*x_6
│ │ │ │ -000747d0: 5e32 2b32 302a 785f 302a 785f 372d 3233  ^2+20*x_0*x_7-23
│ │ │ │ -000747e0: 2a78 5f31 2a78 5f37 2b38 2a78 5f32 2a78  *x_1*x_7+8*x_2*x
│ │ │ │ -000747f0: 5f37 2d32 322a 785f 332a 785f 372b 3230  _7-22*x_3*x_7+20
│ │ │ │ -00074800: 2a78 5f34 2a78 5f37 2d31 352a 785f 357c  *x_4*x_7-15*x_5|
│ │ │ │ -00074810: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000747b0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 785f 352a  ---------|.|x_5*
│ │ │ │ +000747c0: 785f 362b 3335 2a78 5f36 5e32 2b32 302a  x_6+35*x_6^2+20*
│ │ │ │ +000747d0: 785f 302a 785f 372d 3233 2a78 5f31 2a78  x_0*x_7-23*x_1*x
│ │ │ │ +000747e0: 5f37 2b38 2a78 5f32 2a78 5f37 2d32 322a  _7+8*x_2*x_7-22*
│ │ │ │ +000747f0: 785f 332a 785f 372b 3230 2a78 5f34 2a78  x_3*x_7+20*x_4*x
│ │ │ │ +00074800: 5f37 2d31 352a 785f 357c 0a7c 2d2d 2d2d  _7-15*x_5|.|----
│ │ │ │ +00074810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00074860: 0a7c 2a78 5f37 2c78 5f32 2a78 5f35 2b34  .|*x_7,x_2*x_5+4
│ │ │ │ -00074870: 372a 785f 355e 322d 3430 2a78 5f30 2a78  7*x_5^2-40*x_0*x
│ │ │ │ -00074880: 5f36 2b33 372a 785f 312a 785f 362d 3235  _6+37*x_1*x_6-25
│ │ │ │ -00074890: 2a78 5f32 2a78 5f36 2d32 322a 785f 332a  *x_2*x_6-22*x_3*
│ │ │ │ -000748a0: 785f 362d 382a 785f 342a 785f 362b 207c  x_6-8*x_4*x_6+ |
│ │ │ │ -000748b0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00074850: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2a78 5f37  ---------|.|*x_7
│ │ │ │ +00074860: 2c78 5f32 2a78 5f35 2b34 372a 785f 355e  ,x_2*x_5+47*x_5^
│ │ │ │ +00074870: 322d 3430 2a78 5f30 2a78 5f36 2b33 372a  2-40*x_0*x_6+37*
│ │ │ │ +00074880: 785f 312a 785f 362d 3235 2a78 5f32 2a78  x_1*x_6-25*x_2*x
│ │ │ │ +00074890: 5f36 2d32 322a 785f 332a 785f 362d 382a  _6-22*x_3*x_6-8*
│ │ │ │ +000748a0: 785f 342a 785f 362b 207c 0a7c 2d2d 2d2d  x_4*x_6+ |.|----
│ │ │ │ +000748b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000748c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000748d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000748e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000748f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00074900: 0a7c 3237 2a78 5f35 2a78 5f36 2b31 352a  .|27*x_5*x_6+15*
│ │ │ │ -00074910: 785f 365e 322d 3233 2a78 5f30 2a78 5f37  x_6^2-23*x_0*x_7
│ │ │ │ -00074920: 2d34 322a 785f 312a 785f 372b 3237 2a78  -42*x_1*x_7+27*x
│ │ │ │ -00074930: 5f32 2a78 5f37 2b33 352a 785f 332a 785f  _2*x_7+35*x_3*x_
│ │ │ │ -00074940: 372b 3339 2a78 5f34 2a78 5f37 2b32 347c  7+39*x_4*x_7+24|
│ │ │ │ -00074950: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000748f0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3237 2a78  ---------|.|27*x
│ │ │ │ +00074900: 5f35 2a78 5f36 2b31 352a 785f 365e 322d  _5*x_6+15*x_6^2-
│ │ │ │ +00074910: 3233 2a78 5f30 2a78 5f37 2d34 322a 785f  23*x_0*x_7-42*x_
│ │ │ │ +00074920: 312a 785f 372b 3237 2a78 5f32 2a78 5f37  1*x_7+27*x_2*x_7
│ │ │ │ +00074930: 2b33 352a 785f 332a 785f 372b 3339 2a78  +35*x_3*x_7+39*x
│ │ │ │ +00074940: 5f34 2a78 5f37 2b32 347c 0a7c 2d2d 2d2d  _4*x_7+24|.|----
│ │ │ │ +00074950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000749a0: 0a7c 2a78 5f35 2a78 5f37 2c78 5f31 2a78  .|*x_5*x_7,x_1*x
│ │ │ │ -000749b0: 5f35 2b31 352a 785f 355e 322b 3439 2a78  _5+15*x_5^2+49*x
│ │ │ │ -000749c0: 5f30 2a78 5f36 2b38 2a78 5f31 2a78 5f36  _0*x_6+8*x_1*x_6
│ │ │ │ -000749d0: 2d33 312a 785f 322a 785f 362b 392a 785f  -31*x_2*x_6+9*x_
│ │ │ │ -000749e0: 332a 785f 362b 3338 2a78 5f34 2a78 5f7c  3*x_6+38*x_4*x_|
│ │ │ │ -000749f0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00074990: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2a78 5f35  ---------|.|*x_5
│ │ │ │ +000749a0: 2a78 5f37 2c78 5f31 2a78 5f35 2b31 352a  *x_7,x_1*x_5+15*
│ │ │ │ +000749b0: 785f 355e 322b 3439 2a78 5f30 2a78 5f36  x_5^2+49*x_0*x_6
│ │ │ │ +000749c0: 2b38 2a78 5f31 2a78 5f36 2d33 312a 785f  +8*x_1*x_6-31*x_
│ │ │ │ +000749d0: 322a 785f 362b 392a 785f 332a 785f 362b  2*x_6+9*x_3*x_6+
│ │ │ │ +000749e0: 3338 2a78 5f34 2a78 5f7c 0a7c 2d2d 2d2d  38*x_4*x_|.|----
│ │ │ │ +000749f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00074a40: 0a7c 362d 3336 2a78 5f35 2a78 5f36 2d33  .|6-36*x_5*x_6-3
│ │ │ │ -00074a50: 302a 785f 365e 322d 3333 2a78 5f30 2a78  0*x_6^2-33*x_0*x
│ │ │ │ -00074a60: 5f37 2b32 362a 785f 312a 785f 372b 3332  _7+26*x_1*x_7+32
│ │ │ │ -00074a70: 2a78 5f32 2a78 5f37 2b32 372a 785f 332a  *x_2*x_7+27*x_3*
│ │ │ │ -00074a80: 785f 372b 362a 785f 342a 785f 372b 207c  x_7+6*x_4*x_7+ |
│ │ │ │ -00074a90: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00074a30: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 362d 3336  ---------|.|6-36
│ │ │ │ +00074a40: 2a78 5f35 2a78 5f36 2d33 302a 785f 365e  *x_5*x_6-30*x_6^
│ │ │ │ +00074a50: 322d 3333 2a78 5f30 2a78 5f37 2b32 362a  2-33*x_0*x_7+26*
│ │ │ │ +00074a60: 785f 312a 785f 372b 3332 2a78 5f32 2a78  x_1*x_7+32*x_2*x
│ │ │ │ +00074a70: 5f37 2b32 372a 785f 332a 785f 372b 362a  _7+27*x_3*x_7+6*
│ │ │ │ +00074a80: 785f 342a 785f 372b 207c 0a7c 2d2d 2d2d  x_4*x_7+ |.|----
│ │ │ │ +00074a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00074ae0: 0a7c 3336 2a78 5f35 2a78 5f37 2c78 5f30  .|36*x_5*x_7,x_0
│ │ │ │ -00074af0: 2a78 5f35 2b33 302a 785f 355e 322d 3131  *x_5+30*x_5^2-11
│ │ │ │ -00074b00: 2a78 5f30 2a78 5f36 2d33 382a 785f 312a  *x_0*x_6-38*x_1*
│ │ │ │ -00074b10: 785f 362b 3133 2a78 5f32 2a78 5f36 2d33  x_6+13*x_2*x_6-3
│ │ │ │ -00074b20: 322a 785f 332a 785f 362d 3330 2a78 5f7c  2*x_3*x_6-30*x_|
│ │ │ │ -00074b30: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00074ad0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3336 2a78  ---------|.|36*x
│ │ │ │ +00074ae0: 5f35 2a78 5f37 2c78 5f30 2a78 5f35 2b33  _5*x_7,x_0*x_5+3
│ │ │ │ +00074af0: 302a 785f 355e 322d 3131 2a78 5f30 2a78  0*x_5^2-11*x_0*x
│ │ │ │ +00074b00: 5f36 2d33 382a 785f 312a 785f 362b 3133  _6-38*x_1*x_6+13
│ │ │ │ +00074b10: 2a78 5f32 2a78 5f36 2d33 322a 785f 332a  *x_2*x_6-32*x_3*
│ │ │ │ +00074b20: 785f 362d 3330 2a78 5f7c 0a7c 2d2d 2d2d  x_6-30*x_|.|----
│ │ │ │ +00074b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00074b80: 0a7c 342a 785f 362b 342a 785f 352a 785f  .|4*x_6+4*x_5*x_
│ │ │ │ -00074b90: 362d 3238 2a78 5f36 5e32 2d33 302a 785f  6-28*x_6^2-30*x_
│ │ │ │ -00074ba0: 302a 785f 372d 362a 785f 312a 785f 372d  0*x_7-6*x_1*x_7-
│ │ │ │ -00074bb0: 3435 2a78 5f32 2a78 5f37 2b33 342a 785f  45*x_2*x_7+34*x_
│ │ │ │ -00074bc0: 332a 785f 372b 3230 2a78 5f34 2a78 5f7c  3*x_7+20*x_4*x_|
│ │ │ │ -00074bd0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00074b70: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 342a 785f  ---------|.|4*x_
│ │ │ │ +00074b80: 362b 342a 785f 352a 785f 362d 3238 2a78  6+4*x_5*x_6-28*x
│ │ │ │ +00074b90: 5f36 5e32 2d33 302a 785f 302a 785f 372d  _6^2-30*x_0*x_7-
│ │ │ │ +00074ba0: 362a 785f 312a 785f 372d 3435 2a78 5f32  6*x_1*x_7-45*x_2
│ │ │ │ +00074bb0: 2a78 5f37 2b33 342a 785f 332a 785f 372b  *x_7+34*x_3*x_7+
│ │ │ │ +00074bc0: 3230 2a78 5f34 2a78 5f7c 0a7c 2d2d 2d2d  20*x_4*x_|.|----
│ │ │ │ +00074bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00074c20: 0a7c 372b 3438 2a78 5f35 2a78 5f37 2c78  .|7+48*x_5*x_7,x
│ │ │ │ -00074c30: 5f33 2a78 5f34 2b34 362a 785f 355e 322d  _3*x_4+46*x_5^2-
│ │ │ │ -00074c40: 3337 2a78 5f30 2a78 5f36 2b32 372a 785f  37*x_0*x_6+27*x_
│ │ │ │ -00074c50: 312a 785f 362b 3333 2a78 5f32 2a78 5f36  1*x_6+33*x_2*x_6
│ │ │ │ -00074c60: 2b38 2a78 5f33 2a78 5f36 2d33 322a 787c  +8*x_3*x_6-32*x|
│ │ │ │ -00074c70: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00074c10: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 372b 3438  ---------|.|7+48
│ │ │ │ +00074c20: 2a78 5f35 2a78 5f37 2c78 5f33 2a78 5f34  *x_5*x_7,x_3*x_4
│ │ │ │ +00074c30: 2b34 362a 785f 355e 322d 3337 2a78 5f30  +46*x_5^2-37*x_0
│ │ │ │ +00074c40: 2a78 5f36 2b32 372a 785f 312a 785f 362b  *x_6+27*x_1*x_6+
│ │ │ │ +00074c50: 3333 2a78 5f32 2a78 5f36 2b38 2a78 5f33  33*x_2*x_6+8*x_3
│ │ │ │ +00074c60: 2a78 5f36 2d33 322a 787c 0a7c 2d2d 2d2d  *x_6-32*x|.|----
│ │ │ │ +00074c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00074cc0: 0a7c 5f34 2a78 5f36 2b34 322a 785f 352a  .|_4*x_6+42*x_5*
│ │ │ │ -00074cd0: 785f 362d 3334 2a78 5f36 5e32 2d33 372a  x_6-34*x_6^2-37*
│ │ │ │ -00074ce0: 785f 302a 785f 372d 3238 2a78 5f31 2a78  x_0*x_7-28*x_1*x
│ │ │ │ -00074cf0: 5f37 2b31 302a 785f 322a 785f 372d 3237  _7+10*x_2*x_7-27
│ │ │ │ -00074d00: 2a78 5f33 2a78 5f37 2d34 322a 785f 347c  *x_3*x_7-42*x_4|
│ │ │ │ -00074d10: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00074cb0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 5f34 2a78  ---------|.|_4*x
│ │ │ │ +00074cc0: 5f36 2b34 322a 785f 352a 785f 362d 3334  _6+42*x_5*x_6-34
│ │ │ │ +00074cd0: 2a78 5f36 5e32 2d33 372a 785f 302a 785f  *x_6^2-37*x_0*x_
│ │ │ │ +00074ce0: 372d 3238 2a78 5f31 2a78 5f37 2b31 302a  7-28*x_1*x_7+10*
│ │ │ │ +00074cf0: 785f 322a 785f 372d 3237 2a78 5f33 2a78  x_2*x_7-27*x_3*x
│ │ │ │ +00074d00: 5f37 2d34 322a 785f 347c 0a7c 2d2d 2d2d  _7-42*x_4|.|----
│ │ │ │ +00074d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00074d60: 0a7c 2a78 5f37 2d38 2a78 5f35 2a78 5f37  .|*x_7-8*x_5*x_7
│ │ │ │ -00074d70: 2c78 5f32 2a78 5f34 2d32 352a 785f 355e  ,x_2*x_4-25*x_5^
│ │ │ │ -00074d80: 322d 342a 785f 302a 785f 362b 322a 785f  2-4*x_0*x_6+2*x_
│ │ │ │ -00074d90: 312a 785f 362d 3331 2a78 5f32 2a78 5f36  1*x_6-31*x_2*x_6
│ │ │ │ -00074da0: 2d35 2a78 5f33 2a78 5f36 2b31 362a 787c  -5*x_3*x_6+16*x|
│ │ │ │ -00074db0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00074d50: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2a78 5f37  ---------|.|*x_7
│ │ │ │ +00074d60: 2d38 2a78 5f35 2a78 5f37 2c78 5f32 2a78  -8*x_5*x_7,x_2*x
│ │ │ │ +00074d70: 5f34 2d32 352a 785f 355e 322d 342a 785f  _4-25*x_5^2-4*x_
│ │ │ │ +00074d80: 302a 785f 362b 322a 785f 312a 785f 362d  0*x_6+2*x_1*x_6-
│ │ │ │ +00074d90: 3331 2a78 5f32 2a78 5f36 2d35 2a78 5f33  31*x_2*x_6-5*x_3
│ │ │ │ +00074da0: 2a78 5f36 2b31 362a 787c 0a7c 2d2d 2d2d  *x_6+16*x|.|----
│ │ │ │ +00074db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00074e00: 0a7c 5f34 2a78 5f36 2d32 342a 785f 352a  .|_4*x_6-24*x_5*
│ │ │ │ -00074e10: 785f 362b 3331 2a78 5f36 5e32 2d33 302a  x_6+31*x_6^2-30*
│ │ │ │ -00074e20: 785f 302a 785f 372b 3332 2a78 5f31 2a78  x_0*x_7+32*x_1*x
│ │ │ │ -00074e30: 5f37 2b31 322a 785f 322a 785f 372d 3430  _7+12*x_2*x_7-40
│ │ │ │ -00074e40: 2a78 5f33 2a78 5f37 2b33 2a78 5f34 2a7c  *x_3*x_7+3*x_4*|
│ │ │ │ -00074e50: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00074df0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 5f34 2a78  ---------|.|_4*x
│ │ │ │ +00074e00: 5f36 2d32 342a 785f 352a 785f 362b 3331  _6-24*x_5*x_6+31
│ │ │ │ +00074e10: 2a78 5f36 5e32 2d33 302a 785f 302a 785f  *x_6^2-30*x_0*x_
│ │ │ │ +00074e20: 372b 3332 2a78 5f31 2a78 5f37 2b31 322a  7+32*x_1*x_7+12*
│ │ │ │ +00074e30: 785f 322a 785f 372d 3430 2a78 5f33 2a78  x_2*x_7-40*x_3*x
│ │ │ │ +00074e40: 5f37 2b33 2a78 5f34 2a7c 0a7c 2d2d 2d2d  _7+3*x_4*|.|----
│ │ │ │ +00074e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00074ea0: 0a7c 785f 372d 3238 2a78 5f35 2a78 5f37  .|x_7-28*x_5*x_7
│ │ │ │ -00074eb0: 2c78 5f30 2a78 5f34 2b31 352a 785f 355e  ,x_0*x_4+15*x_5^
│ │ │ │ -00074ec0: 322b 3438 2a78 5f30 2a78 5f36 2d35 302a  2+48*x_0*x_6-50*
│ │ │ │ -00074ed0: 785f 312a 785f 362b 3436 2a78 5f32 2a78  x_1*x_6+46*x_2*x
│ │ │ │ -00074ee0: 5f36 2d34 382a 785f 332a 785f 362d 207c  _6-48*x_3*x_6- |
│ │ │ │ -00074ef0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00074e90: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 785f 372d  ---------|.|x_7-
│ │ │ │ +00074ea0: 3238 2a78 5f35 2a78 5f37 2c78 5f30 2a78  28*x_5*x_7,x_0*x
│ │ │ │ +00074eb0: 5f34 2b31 352a 785f 355e 322b 3438 2a78  _4+15*x_5^2+48*x
│ │ │ │ +00074ec0: 5f30 2a78 5f36 2d35 302a 785f 312a 785f  _0*x_6-50*x_1*x_
│ │ │ │ +00074ed0: 362b 3436 2a78 5f32 2a78 5f36 2d34 382a  6+46*x_2*x_6-48*
│ │ │ │ +00074ee0: 785f 332a 785f 362d 207c 0a7c 2d2d 2d2d  x_3*x_6- |.|----
│ │ │ │ +00074ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00074f40: 0a7c 3233 2a78 5f34 2a78 5f36 2d32 382a  .|23*x_4*x_6-28*
│ │ │ │ -00074f50: 785f 352a 785f 362b 3339 2a78 5f36 5e32  x_5*x_6+39*x_6^2
│ │ │ │ -00074f60: 2b33 382a 785f 312a 785f 372d 352a 785f  +38*x_1*x_7-5*x_
│ │ │ │ -00074f70: 332a 785f 372b 352a 785f 342a 785f 372d  3*x_7+5*x_4*x_7-
│ │ │ │ -00074f80: 3334 2a78 5f35 2a78 5f37 2c78 5f33 5e7c  34*x_5*x_7,x_3^|
│ │ │ │ -00074f90: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00074f30: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3233 2a78  ---------|.|23*x
│ │ │ │ +00074f40: 5f34 2a78 5f36 2d32 382a 785f 352a 785f  _4*x_6-28*x_5*x_
│ │ │ │ +00074f50: 362b 3339 2a78 5f36 5e32 2b33 382a 785f  6+39*x_6^2+38*x_
│ │ │ │ +00074f60: 312a 785f 372d 352a 785f 332a 785f 372b  1*x_7-5*x_3*x_7+
│ │ │ │ +00074f70: 352a 785f 342a 785f 372d 3334 2a78 5f35  5*x_4*x_7-34*x_5
│ │ │ │ +00074f80: 2a78 5f37 2c78 5f33 5e7c 0a7c 2d2d 2d2d  *x_7,x_3^|.|----
│ │ │ │ +00074f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00074fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00074fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00074fe0: 0a7c 322d 3331 2a78 5f35 5e32 2b34 312a  .|2-31*x_5^2+41*
│ │ │ │ -00074ff0: 785f 302a 785f 362d 3330 2a78 5f31 2a78  x_0*x_6-30*x_1*x
│ │ │ │ -00075000: 5f36 2d34 2a78 5f32 2a78 5f36 2b34 332a  _6-4*x_2*x_6+43*
│ │ │ │ -00075010: 785f 332a 785f 362b 3233 2a78 5f34 2a78  x_3*x_6+23*x_4*x
│ │ │ │ -00075020: 5f36 2b37 2a78 5f35 2a78 5f36 2b33 317c  _6+7*x_5*x_6+31|
│ │ │ │ -00075030: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00074fd0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 322d 3331  ---------|.|2-31
│ │ │ │ +00074fe0: 2a78 5f35 5e32 2b34 312a 785f 302a 785f  *x_5^2+41*x_0*x_
│ │ │ │ +00074ff0: 362d 3330 2a78 5f31 2a78 5f36 2d34 2a78  6-30*x_1*x_6-4*x
│ │ │ │ +00075000: 5f32 2a78 5f36 2b34 332a 785f 332a 785f  _2*x_6+43*x_3*x_
│ │ │ │ +00075010: 362b 3233 2a78 5f34 2a78 5f36 2b37 2a78  6+23*x_4*x_6+7*x
│ │ │ │ +00075020: 5f35 2a78 5f36 2b33 317c 0a7c 2d2d 2d2d  _5*x_6+31|.|----
│ │ │ │ +00075030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00075070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00075080: 0a7c 2a78 5f36 5e32 2d31 392a 785f 302a  .|*x_6^2-19*x_0*
│ │ │ │ -00075090: 785f 372b 3235 2a78 5f31 2a78 5f37 2d34  x_7+25*x_1*x_7-4
│ │ │ │ -000750a0: 392a 785f 322a 785f 372d 3136 2a78 5f33  9*x_2*x_7-16*x_3
│ │ │ │ -000750b0: 2a78 5f37 2d34 352a 785f 342a 785f 372b  *x_7-45*x_4*x_7+
│ │ │ │ -000750c0: 3235 2a78 5f35 2a78 5f37 2c78 5f32 2a7c  25*x_5*x_7,x_2*|
│ │ │ │ -000750d0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00075070: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2a78 5f36  ---------|.|*x_6
│ │ │ │ +00075080: 5e32 2d31 392a 785f 302a 785f 372b 3235  ^2-19*x_0*x_7+25
│ │ │ │ +00075090: 2a78 5f31 2a78 5f37 2d34 392a 785f 322a  *x_1*x_7-49*x_2*
│ │ │ │ +000750a0: 785f 372d 3136 2a78 5f33 2a78 5f37 2d34  x_7-16*x_3*x_7-4
│ │ │ │ +000750b0: 352a 785f 342a 785f 372b 3235 2a78 5f35  5*x_4*x_7+25*x_5
│ │ │ │ +000750c0: 2a78 5f37 2c78 5f32 2a7c 0a7c 2d2d 2d2d  *x_7,x_2*|.|----
│ │ │ │ +000750d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000750e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000750f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00075110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00075120: 0a7c 785f 332b 3133 2a78 5f35 5e32 2d34  .|x_3+13*x_5^2-4
│ │ │ │ -00075130: 352a 785f 302a 785f 362d 3232 2a78 5f31  5*x_0*x_6-22*x_1
│ │ │ │ -00075140: 2a78 5f36 2b33 332a 785f 322a 785f 362d  *x_6+33*x_2*x_6-
│ │ │ │ -00075150: 3236 2a78 5f33 2a78 5f36 2d32 312a 785f  26*x_3*x_6-21*x_
│ │ │ │ -00075160: 342a 785f 362b 3334 2a78 5f35 2a78 5f7c  4*x_6+34*x_5*x_|
│ │ │ │ -00075170: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00075110: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 785f 332b  ---------|.|x_3+
│ │ │ │ +00075120: 3133 2a78 5f35 5e32 2d34 352a 785f 302a  13*x_5^2-45*x_0*
│ │ │ │ +00075130: 785f 362d 3232 2a78 5f31 2a78 5f36 2b33  x_6-22*x_1*x_6+3
│ │ │ │ +00075140: 332a 785f 322a 785f 362d 3236 2a78 5f33  3*x_2*x_6-26*x_3
│ │ │ │ +00075150: 2a78 5f36 2d32 312a 785f 342a 785f 362b  *x_6-21*x_4*x_6+
│ │ │ │ +00075160: 3334 2a78 5f35 2a78 5f7c 0a7c 2d2d 2d2d  34*x_5*x_|.|----
│ │ │ │ +00075170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000751a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000751b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000751c0: 0a7c 362d 3231 2a78 5f36 5e32 2d34 372a  .|6-21*x_6^2-47*
│ │ │ │ -000751d0: 785f 302a 785f 372d 3130 2a78 5f31 2a78  x_0*x_7-10*x_1*x
│ │ │ │ -000751e0: 5f37 2b32 392a 785f 322a 785f 372d 3436  _7+29*x_2*x_7-46
│ │ │ │ -000751f0: 2a78 5f33 2a78 5f37 2d78 5f34 2a78 5f37  *x_3*x_7-x_4*x_7
│ │ │ │ -00075200: 2b32 302a 785f 352a 785f 372c 785f 317c  +20*x_5*x_7,x_1|
│ │ │ │ -00075210: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000751b0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 362d 3231  ---------|.|6-21
│ │ │ │ +000751c0: 2a78 5f36 5e32 2d34 372a 785f 302a 785f  *x_6^2-47*x_0*x_
│ │ │ │ +000751d0: 372d 3130 2a78 5f31 2a78 5f37 2b32 392a  7-10*x_1*x_7+29*
│ │ │ │ +000751e0: 785f 322a 785f 372d 3436 2a78 5f33 2a78  x_2*x_7-46*x_3*x
│ │ │ │ +000751f0: 5f37 2d78 5f34 2a78 5f37 2b32 302a 785f  _7-x_4*x_7+20*x_
│ │ │ │ +00075200: 352a 785f 372c 785f 317c 0a7c 2d2d 2d2d  5*x_7,x_1|.|----
│ │ │ │ +00075210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00075250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00075260: 0a7c 2a78 5f33 2b32 322a 785f 355e 322b  .|*x_3+22*x_5^2+
│ │ │ │ -00075270: 342a 785f 302a 785f 362b 332a 785f 312a  4*x_0*x_6+3*x_1*
│ │ │ │ -00075280: 785f 362b 3435 2a78 5f32 2a78 5f36 2b33  x_6+45*x_2*x_6+3
│ │ │ │ -00075290: 372a 785f 332a 785f 362b 3137 2a78 5f34  7*x_3*x_6+17*x_4
│ │ │ │ -000752a0: 2a78 5f36 2b33 362a 785f 352a 785f 367c  *x_6+36*x_5*x_6|
│ │ │ │ -000752b0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00075250: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2a78 5f33  ---------|.|*x_3
│ │ │ │ +00075260: 2b32 322a 785f 355e 322b 342a 785f 302a  +22*x_5^2+4*x_0*
│ │ │ │ +00075270: 785f 362b 332a 785f 312a 785f 362b 3435  x_6+3*x_1*x_6+45
│ │ │ │ +00075280: 2a78 5f32 2a78 5f36 2b33 372a 785f 332a  *x_2*x_6+37*x_3*
│ │ │ │ +00075290: 785f 362b 3137 2a78 5f34 2a78 5f36 2b33  x_6+17*x_4*x_6+3
│ │ │ │ +000752a0: 362a 785f 352a 785f 367c 0a7c 2d2d 2d2d  6*x_5*x_6|.|----
│ │ │ │ +000752b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000752c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000752d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000752e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000752f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00075300: 0a7c 2d32 2a78 5f36 5e32 2d33 312a 785f  .|-2*x_6^2-31*x_
│ │ │ │ -00075310: 302a 785f 372b 332a 785f 312a 785f 372d  0*x_7+3*x_1*x_7-
│ │ │ │ -00075320: 3132 2a78 5f32 2a78 5f37 2b31 392a 785f  12*x_2*x_7+19*x_
│ │ │ │ -00075330: 332a 785f 372b 3238 2a78 5f34 2a78 5f37  3*x_7+28*x_4*x_7
│ │ │ │ -00075340: 2b33 302a 785f 352a 785f 372c 785f 307c  +30*x_5*x_7,x_0|
│ │ │ │ -00075350: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000752f0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2d32 2a78  ---------|.|-2*x
│ │ │ │ +00075300: 5f36 5e32 2d33 312a 785f 302a 785f 372b  _6^2-31*x_0*x_7+
│ │ │ │ +00075310: 332a 785f 312a 785f 372d 3132 2a78 5f32  3*x_1*x_7-12*x_2
│ │ │ │ +00075320: 2a78 5f37 2b31 392a 785f 332a 785f 372b  *x_7+19*x_3*x_7+
│ │ │ │ +00075330: 3238 2a78 5f34 2a78 5f37 2b33 302a 785f  28*x_4*x_7+30*x_
│ │ │ │ +00075340: 352a 785f 372c 785f 307c 0a7c 2d2d 2d2d  5*x_7,x_0|.|----
│ │ │ │ +00075350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00075390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000753a0: 0a7c 2a78 5f33 2d34 372a 785f 355e 322d  .|*x_3-47*x_5^2-
│ │ │ │ -000753b0: 3433 2a78 5f30 2a78 5f36 2b36 2a78 5f31  43*x_0*x_6+6*x_1
│ │ │ │ -000753c0: 2a78 5f36 2d34 302a 785f 322a 785f 362b  *x_6-40*x_2*x_6+
│ │ │ │ -000753d0: 3231 2a78 5f33 2a78 5f36 2b32 362a 785f  21*x_3*x_6+26*x_
│ │ │ │ -000753e0: 342a 785f 362d 352a 785f 352a 785f 367c  4*x_6-5*x_5*x_6|
│ │ │ │ -000753f0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00075390: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2a78 5f33  ---------|.|*x_3
│ │ │ │ +000753a0: 2d34 372a 785f 355e 322d 3433 2a78 5f30  -47*x_5^2-43*x_0
│ │ │ │ +000753b0: 2a78 5f36 2b36 2a78 5f31 2a78 5f36 2d34  *x_6+6*x_1*x_6-4
│ │ │ │ +000753c0: 302a 785f 322a 785f 362b 3231 2a78 5f33  0*x_2*x_6+21*x_3
│ │ │ │ +000753d0: 2a78 5f36 2b32 362a 785f 342a 785f 362d  *x_6+26*x_4*x_6-
│ │ │ │ +000753e0: 352a 785f 352a 785f 367c 0a7c 2d2d 2d2d  5*x_5*x_6|.|----
│ │ │ │ +000753f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00075430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00075440: 0a7c 2d35 2a78 5f36 5e32 2b34 2a78 5f30  .|-5*x_6^2+4*x_0
│ │ │ │ -00075450: 2a78 5f37 2d31 352a 785f 312a 785f 372b  *x_7-15*x_1*x_7+
│ │ │ │ -00075460: 3138 2a78 5f32 2a78 5f37 2d33 312a 785f  18*x_2*x_7-31*x_
│ │ │ │ -00075470: 332a 785f 372b 3530 2a78 5f34 2a78 5f37  3*x_7+50*x_4*x_7
│ │ │ │ -00075480: 2d34 362a 785f 352a 785f 372c 785f 327c  -46*x_5*x_7,x_2|
│ │ │ │ -00075490: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00075430: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2d35 2a78  ---------|.|-5*x
│ │ │ │ +00075440: 5f36 5e32 2b34 2a78 5f30 2a78 5f37 2d31  _6^2+4*x_0*x_7-1
│ │ │ │ +00075450: 352a 785f 312a 785f 372b 3138 2a78 5f32  5*x_1*x_7+18*x_2
│ │ │ │ +00075460: 2a78 5f37 2d33 312a 785f 332a 785f 372b  *x_7-31*x_3*x_7+
│ │ │ │ +00075470: 3530 2a78 5f34 2a78 5f37 2d34 362a 785f  50*x_4*x_7-46*x_
│ │ │ │ +00075480: 352a 785f 372c 785f 327c 0a7c 2d2d 2d2d  5*x_7,x_2|.|----
│ │ │ │ +00075490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000754a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000754b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000754c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000754d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000754e0: 0a7c 5e32 2b34 2a78 5f35 5e32 2b33 312a  .|^2+4*x_5^2+31*
│ │ │ │ -000754f0: 785f 302a 785f 362b 3431 2a78 5f31 2a78  x_0*x_6+41*x_1*x
│ │ │ │ -00075500: 5f36 2b33 312a 785f 322a 785f 362b 3238  _6+31*x_2*x_6+28
│ │ │ │ -00075510: 2a78 5f33 2a78 5f36 2b34 322a 785f 342a  *x_3*x_6+42*x_4*
│ │ │ │ -00075520: 785f 362d 3238 2a78 5f35 2a78 5f36 2d7c  x_6-28*x_5*x_6-|
│ │ │ │ -00075530: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000754d0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 5e32 2b34  ---------|.|^2+4
│ │ │ │ +000754e0: 2a78 5f35 5e32 2b33 312a 785f 302a 785f  *x_5^2+31*x_0*x_
│ │ │ │ +000754f0: 362b 3431 2a78 5f31 2a78 5f36 2b33 312a  6+41*x_1*x_6+31*
│ │ │ │ +00075500: 785f 322a 785f 362b 3238 2a78 5f33 2a78  x_2*x_6+28*x_3*x
│ │ │ │ +00075510: 5f36 2b34 322a 785f 342a 785f 362d 3238  _6+42*x_4*x_6-28
│ │ │ │ +00075520: 2a78 5f35 2a78 5f36 2d7c 0a7c 2d2d 2d2d  *x_5*x_6-|.|----
│ │ │ │ +00075530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00075570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00075580: 0a7c 342a 785f 365e 322d 372a 785f 302a  .|4*x_6^2-7*x_0*
│ │ │ │ -00075590: 785f 372b 3135 2a78 5f31 2a78 5f37 2d39  x_7+15*x_1*x_7-9
│ │ │ │ -000755a0: 2a78 5f32 2a78 5f37 2b33 312a 785f 332a  *x_2*x_7+31*x_3*
│ │ │ │ -000755b0: 785f 372b 332a 785f 342a 785f 372b 372a  x_7+3*x_4*x_7+7*
│ │ │ │ -000755c0: 785f 352a 785f 372c 785f 312a 785f 327c  x_5*x_7,x_1*x_2|
│ │ │ │ -000755d0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00075570: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 342a 785f  ---------|.|4*x_
│ │ │ │ +00075580: 365e 322d 372a 785f 302a 785f 372b 3135  6^2-7*x_0*x_7+15
│ │ │ │ +00075590: 2a78 5f31 2a78 5f37 2d39 2a78 5f32 2a78  *x_1*x_7-9*x_2*x
│ │ │ │ +000755a0: 5f37 2b33 312a 785f 332a 785f 372b 332a  _7+31*x_3*x_7+3*
│ │ │ │ +000755b0: 785f 342a 785f 372b 372a 785f 352a 785f  x_4*x_7+7*x_5*x_
│ │ │ │ +000755c0: 372c 785f 312a 785f 327c 0a7c 2d2d 2d2d  7,x_1*x_2|.|----
│ │ │ │ +000755d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000755e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000755f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00075610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00075620: 0a7c 2d34 362a 785f 355e 322d 362a 785f  .|-46*x_5^2-6*x_
│ │ │ │ -00075630: 302a 785f 362d 3530 2a78 5f31 2a78 5f36  0*x_6-50*x_1*x_6
│ │ │ │ -00075640: 2b33 322a 785f 322a 785f 362d 3130 2a78  +32*x_2*x_6-10*x
│ │ │ │ -00075650: 5f33 2a78 5f36 2b34 322a 785f 342a 785f  _3*x_6+42*x_4*x_
│ │ │ │ -00075660: 362b 3333 2a78 5f35 2a78 5f36 2b31 387c  6+33*x_5*x_6+18|
│ │ │ │ -00075670: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00075610: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2d34 362a  ---------|.|-46*
│ │ │ │ +00075620: 785f 355e 322d 362a 785f 302a 785f 362d  x_5^2-6*x_0*x_6-
│ │ │ │ +00075630: 3530 2a78 5f31 2a78 5f36 2b33 322a 785f  50*x_1*x_6+32*x_
│ │ │ │ +00075640: 322a 785f 362d 3130 2a78 5f33 2a78 5f36  2*x_6-10*x_3*x_6
│ │ │ │ +00075650: 2b34 322a 785f 342a 785f 362b 3333 2a78  +42*x_4*x_6+33*x
│ │ │ │ +00075660: 5f35 2a78 5f36 2b31 387c 0a7c 2d2d 2d2d  _5*x_6+18|.|----
│ │ │ │ +00075670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000756a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000756b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000756c0: 0a7c 2a78 5f36 5e32 2d39 2a78 5f30 2a78  .|*x_6^2-9*x_0*x
│ │ │ │ -000756d0: 5f37 2d32 302a 785f 312a 785f 372b 3435  _7-20*x_1*x_7+45
│ │ │ │ -000756e0: 2a78 5f32 2a78 5f37 2d39 2a78 5f33 2a78  *x_2*x_7-9*x_3*x
│ │ │ │ -000756f0: 5f37 2b31 302a 785f 342a 785f 372d 382a  _7+10*x_4*x_7-8*
│ │ │ │ -00075700: 785f 352a 785f 372c 785f 302a 785f 327c  x_5*x_7,x_0*x_2|
│ │ │ │ -00075710: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000756b0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2a78 5f36  ---------|.|*x_6
│ │ │ │ +000756c0: 5e32 2d39 2a78 5f30 2a78 5f37 2d32 302a  ^2-9*x_0*x_7-20*
│ │ │ │ +000756d0: 785f 312a 785f 372b 3435 2a78 5f32 2a78  x_1*x_7+45*x_2*x
│ │ │ │ +000756e0: 5f37 2d39 2a78 5f33 2a78 5f37 2b31 302a  _7-9*x_3*x_7+10*
│ │ │ │ +000756f0: 785f 342a 785f 372d 382a 785f 352a 785f  x_4*x_7-8*x_5*x_
│ │ │ │ +00075700: 372c 785f 302a 785f 327c 0a7c 2d2d 2d2d  7,x_0*x_2|.|----
│ │ │ │ +00075710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00075750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00075760: 0a7c 2d39 2a78 5f35 5e32 2b33 342a 785f  .|-9*x_5^2+34*x_
│ │ │ │ -00075770: 302a 785f 362d 3435 2a78 5f31 2a78 5f36  0*x_6-45*x_1*x_6
│ │ │ │ -00075780: 2b31 392a 785f 322a 785f 362b 3234 2a78  +19*x_2*x_6+24*x
│ │ │ │ -00075790: 5f33 2a78 5f36 2b32 332a 785f 342a 785f  _3*x_6+23*x_4*x_
│ │ │ │ -000757a0: 362d 3337 2a78 5f35 2a78 5f36 2d34 347c  6-37*x_5*x_6-44|
│ │ │ │ -000757b0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00075750: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2d39 2a78  ---------|.|-9*x
│ │ │ │ +00075760: 5f35 5e32 2b33 342a 785f 302a 785f 362d  _5^2+34*x_0*x_6-
│ │ │ │ +00075770: 3435 2a78 5f31 2a78 5f36 2b31 392a 785f  45*x_1*x_6+19*x_
│ │ │ │ +00075780: 322a 785f 362b 3234 2a78 5f33 2a78 5f36  2*x_6+24*x_3*x_6
│ │ │ │ +00075790: 2b32 332a 785f 342a 785f 362d 3337 2a78  +23*x_4*x_6-37*x
│ │ │ │ +000757a0: 5f35 2a78 5f36 2d34 347c 0a7c 2d2d 2d2d  _5*x_6-44|.|----
│ │ │ │ +000757b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000757c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000757d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000757e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000757f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00075800: 0a7c 2a78 5f36 5e32 2b32 342a 785f 302a  .|*x_6^2+24*x_0*
│ │ │ │ -00075810: 785f 372d 3333 2a78 5f32 2a78 5f37 2b34  x_7-33*x_2*x_7+4
│ │ │ │ -00075820: 312a 785f 332a 785f 372d 3430 2a78 5f34  1*x_3*x_7-40*x_4
│ │ │ │ -00075830: 2a78 5f37 2b34 2a78 5f35 2a78 5f37 2c78  *x_7+4*x_5*x_7,x
│ │ │ │ -00075840: 5f31 5e32 2b78 5f31 2a78 5f34 2b78 5f7c  _1^2+x_1*x_4+x_|
│ │ │ │ -00075850: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000757f0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2a78 5f36  ---------|.|*x_6
│ │ │ │ +00075800: 5e32 2b32 342a 785f 302a 785f 372d 3333  ^2+24*x_0*x_7-33
│ │ │ │ +00075810: 2a78 5f32 2a78 5f37 2b34 312a 785f 332a  *x_2*x_7+41*x_3*
│ │ │ │ +00075820: 785f 372d 3430 2a78 5f34 2a78 5f37 2b34  x_7-40*x_4*x_7+4
│ │ │ │ +00075830: 2a78 5f35 2a78 5f37 2c78 5f31 5e32 2b78  *x_5*x_7,x_1^2+x
│ │ │ │ +00075840: 5f31 2a78 5f34 2b78 5f7c 0a7c 2d2d 2d2d  _1*x_4+x_|.|----
│ │ │ │ +00075850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00075890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000758a0: 0a7c 345e 322d 3238 2a78 5f35 5e32 2d33  .|4^2-28*x_5^2-3
│ │ │ │ -000758b0: 332a 785f 302a 785f 362d 3137 2a78 5f31  3*x_0*x_6-17*x_1
│ │ │ │ -000758c0: 2a78 5f36 2b31 312a 785f 332a 785f 362b  *x_6+11*x_3*x_6+
│ │ │ │ -000758d0: 3230 2a78 5f34 2a78 5f36 2b32 352a 785f  20*x_4*x_6+25*x_
│ │ │ │ -000758e0: 352a 785f 362d 3231 2a78 5f36 5e32 2d7c  5*x_6-21*x_6^2-|
│ │ │ │ -000758f0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00075890: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 345e 322d  ---------|.|4^2-
│ │ │ │ +000758a0: 3238 2a78 5f35 5e32 2d33 332a 785f 302a  28*x_5^2-33*x_0*
│ │ │ │ +000758b0: 785f 362d 3137 2a78 5f31 2a78 5f36 2b31  x_6-17*x_1*x_6+1
│ │ │ │ +000758c0: 312a 785f 332a 785f 362b 3230 2a78 5f34  1*x_3*x_6+20*x_4
│ │ │ │ +000758d0: 2a78 5f36 2b32 352a 785f 352a 785f 362d  *x_6+25*x_5*x_6-
│ │ │ │ +000758e0: 3231 2a78 5f36 5e32 2d7c 0a7c 2d2d 2d2d  21*x_6^2-|.|----
│ │ │ │ +000758f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00075930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00075940: 0a7c 3232 2a78 5f30 2a78 5f37 2b32 342a  .|22*x_0*x_7+24*
│ │ │ │ -00075950: 785f 312a 785f 372d 3134 2a78 5f32 2a78  x_1*x_7-14*x_2*x
│ │ │ │ -00075960: 5f37 2b35 2a78 5f33 2a78 5f37 2d33 392a  _7+5*x_3*x_7-39*
│ │ │ │ -00075970: 785f 342a 785f 372d 3138 2a78 5f35 2a78  x_4*x_7-18*x_5*x
│ │ │ │ -00075980: 5f37 2c78 5f30 2a78 5f31 2d34 372a 787c  _7,x_0*x_1-47*x|
│ │ │ │ -00075990: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00075930: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3232 2a78  ---------|.|22*x
│ │ │ │ +00075940: 5f30 2a78 5f37 2b32 342a 785f 312a 785f  _0*x_7+24*x_1*x_
│ │ │ │ +00075950: 372d 3134 2a78 5f32 2a78 5f37 2b35 2a78  7-14*x_2*x_7+5*x
│ │ │ │ +00075960: 5f33 2a78 5f37 2d33 392a 785f 342a 785f  _3*x_7-39*x_4*x_
│ │ │ │ +00075970: 372d 3138 2a78 5f35 2a78 5f37 2c78 5f30  7-18*x_5*x_7,x_0
│ │ │ │ +00075980: 2a78 5f31 2d34 372a 787c 0a7c 2d2d 2d2d  *x_1-47*x|.|----
│ │ │ │ +00075990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000759a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000759b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000759c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000759d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000759e0: 0a7c 5f35 5e32 2d35 2a78 5f30 2a78 5f36  .|_5^2-5*x_0*x_6
│ │ │ │ -000759f0: 2d39 2a78 5f31 2a78 5f36 2d34 352a 785f  -9*x_1*x_6-45*x_
│ │ │ │ -00075a00: 322a 785f 362b 3438 2a78 5f33 2a78 5f36  2*x_6+48*x_3*x_6
│ │ │ │ -00075a10: 2b34 352a 785f 342a 785f 362d 3239 2a78  +45*x_4*x_6-29*x
│ │ │ │ -00075a20: 5f35 2a78 5f36 2b33 2a78 5f36 5e32 2b7c  _5*x_6+3*x_6^2+|
│ │ │ │ -00075a30: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000759d0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 5f35 5e32  ---------|.|_5^2
│ │ │ │ +000759e0: 2d35 2a78 5f30 2a78 5f36 2d39 2a78 5f31  -5*x_0*x_6-9*x_1
│ │ │ │ +000759f0: 2a78 5f36 2d34 352a 785f 322a 785f 362b  *x_6-45*x_2*x_6+
│ │ │ │ +00075a00: 3438 2a78 5f33 2a78 5f36 2b34 352a 785f  48*x_3*x_6+45*x_
│ │ │ │ +00075a10: 342a 785f 362d 3239 2a78 5f35 2a78 5f36  4*x_6-29*x_5*x_6
│ │ │ │ +00075a20: 2b33 2a78 5f36 5e32 2b7c 0a7c 2d2d 2d2d  +3*x_6^2+|.|----
│ │ │ │ +00075a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00075a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00075a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00075a80: 0a7c 3239 2a78 5f30 2a78 5f37 2b34 302a  .|29*x_0*x_7+40*
│ │ │ │ -00075a90: 785f 312a 785f 372b 3436 2a78 5f32 2a78  x_1*x_7+46*x_2*x
│ │ │ │ -00075aa0: 5f37 2b32 372a 785f 332a 785f 372d 3336  _7+27*x_3*x_7-36
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│ │ │ │ +00075c60: 2070 6869 203d 2072 6174 696f 6e61 6c4d   phi = rationalM
│ │ │ │ +00075c70: 6170 2843 2c33 2c32 293b 2020 2020 2020  ap(C,3,2);      
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│ │ │ │  00075c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00075ca0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00075cb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00075ca0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00075cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00075d00: 0a7c 6f36 203a 2052 6174 696f 6e61 6c4d  .|o6 : RationalM
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│ │ │ │ +00075da0: 2074 696d 6520 6973 446f 6d69 6e61 6e74   time isDominant
│ │ │ │ +00075db0: 2870 6869 2c43 6572 7469 6679 3d3e 7472  (phi,Certify=>tr
│ │ │ │ +00075dc0: 7565 2920 2020 2020 2020 2020 2020 2020  ue)             
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│ │ │ │ -00075de0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00075df0: 0a7c 4365 7274 6966 793a 206f 7574 7075  .|Certify: outpu
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│ │ │ │ -00075e40: 0a7c 202d 2d20 7573 6564 2033 2e38 3838  .| -- used 3.888
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│ │ │ │ +000761f0: 7874 3a20 6973 496e 7665 7273 654d 6170  xt: isInverseMap
│ │ │ │ +00076200: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f63  _lpRationalMap_c
│ │ │ │ +00076210: 6d52 6174 696f 6e61 6c4d 6170 5f72 702c  mRationalMap_rp,
│ │ │ │ +00076220: 2050 7265 763a 2069 7344 6f6d 696e 616e   Prev: isDominan
│ │ │ │ +00076230: 742c 2055 703a 2054 6f70 0a0a 6973 496e  t, Up: Top..isIn
│ │ │ │ +00076240: 7665 7273 654d 6170 202d 2d20 6368 6563  verseMap -- chec
│ │ │ │ +00076250: 6b73 2077 6865 7468 6572 2061 2072 6174  ks whether a rat
│ │ │ │ +00076260: 696f 6e61 6c20 6d61 7020 6973 2074 6865  ional map is the
│ │ │ │ +00076270: 2069 6e76 6572 7365 206f 6620 616e 6f74   inverse of anot
│ │ │ │ +00076280: 6865 720a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  her.************
│ │ │ │  00076290: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000762a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000762b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000762c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000762d0: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20  *..  * Usage: . 
│ │ │ │ -000762e0: 2020 2020 2020 2069 7349 6e76 6572 7365         isInverse
│ │ │ │ -000762f0: 4d61 7028 7068 692c 7073 6929 0a20 202a  Map(phi,psi).  *
│ │ │ │ -00076300: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -00076310: 2070 6869 2c20 6120 2a6e 6f74 6520 7269   phi, a *note ri
│ │ │ │ -00076320: 6e67 206d 6170 3a20 284d 6163 6175 6c61  ng map: (Macaula
│ │ │ │ -00076330: 7932 446f 6329 5269 6e67 4d61 702c 2c20  y2Doc)RingMap,, 
│ │ │ │ -00076340: 7265 7072 6573 656e 7469 6e67 2061 2072  representing a r
│ │ │ │ -00076350: 6174 696f 6e61 6c0a 2020 2020 2020 2020  ational.        
│ │ │ │ -00076360: 6d61 7020 245c 5068 693a 5820 5c64 6173  map $\Phi:X \das
│ │ │ │ -00076370: 6872 6967 6874 6172 726f 7720 5924 0a20  hrightarrow Y$. 
│ │ │ │ -00076380: 2020 2020 202a 2070 7369 2c20 6120 2a6e       * psi, a *n
│ │ │ │ -00076390: 6f74 6520 7269 6e67 206d 6170 3a20 284d  ote ring map: (M
│ │ │ │ -000763a0: 6163 6175 6c61 7932 446f 6329 5269 6e67  acaulay2Doc)Ring
│ │ │ │ -000763b0: 4d61 702c 2c20 7265 7072 6573 656e 7469  Map,, representi
│ │ │ │ -000763c0: 6e67 2061 2072 6174 696f 6e61 6c0a 2020  ng a rational.  
│ │ │ │ -000763d0: 2020 2020 2020 6d61 7020 245c 5073 693a        map $\Psi:
│ │ │ │ -000763e0: 5920 5c64 6173 6872 6967 6874 6172 726f  Y \dashrightarro
│ │ │ │ -000763f0: 7720 5824 0a20 202a 204f 7574 7075 7473  w X$.  * Outputs
│ │ │ │ -00076400: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74  :.      * a *not
│ │ │ │ -00076410: 6520 426f 6f6c 6561 6e20 7661 6c75 653a  e Boolean value:
│ │ │ │ -00076420: 2028 4d61 6361 756c 6179 3244 6f63 2942   (Macaulay2Doc)B
│ │ │ │ -00076430: 6f6f 6c65 616e 2c2c 2061 6363 6f72 6469  oolean,, accordi
│ │ │ │ -00076440: 6e67 2074 6f20 7468 650a 2020 2020 2020  ng to the.      
│ │ │ │ -00076450: 2020 636f 6e64 6974 696f 6e20 7468 6174    condition that
│ │ │ │ -00076460: 2074 6865 2063 6f6d 706f 7369 7469 6f6e   the composition
│ │ │ │ -00076470: 2024 5c50 7369 5c2c 5c50 6869 3a58 205c   $\Psi\,\Phi:X \
│ │ │ │ -00076480: 6461 7368 7269 6768 7461 7272 6f77 2058  dashrightarrow X
│ │ │ │ -00076490: 240a 2020 2020 2020 2020 636f 696e 6369  $.        coinci
│ │ │ │ -000764a0: 6465 7320 7769 7468 2074 6865 2069 6465  des with the ide
│ │ │ │ -000764b0: 6e74 6974 7920 6f66 2024 5824 2028 6173  ntity of $X$ (as
│ │ │ │ -000764c0: 2061 2072 6174 696f 6e61 6c20 6d61 7029   a rational map)
│ │ │ │ -000764d0: 0a0a 5761 7973 2074 6f20 7573 6520 6973  ..Ways to use is
│ │ │ │ -000764e0: 496e 7665 7273 654d 6170 3a0a 3d3d 3d3d  InverseMap:.====
│ │ │ │ -000764f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00076500: 3d3d 3d3d 3d0a 0a20 202a 2022 6973 496e  =====..  * "isIn
│ │ │ │ -00076510: 7665 7273 654d 6170 2852 696e 674d 6170  verseMap(RingMap
│ │ │ │ -00076520: 2c52 696e 674d 6170 2922 0a20 202a 202a  ,RingMap)".  * *
│ │ │ │ -00076530: 6e6f 7465 2069 7349 6e76 6572 7365 4d61  note isInverseMa
│ │ │ │ -00076540: 7028 5261 7469 6f6e 616c 4d61 702c 5261  p(RationalMap,Ra
│ │ │ │ -00076550: 7469 6f6e 616c 4d61 7029 3a0a 2020 2020  tionalMap):.    
│ │ │ │ -00076560: 6973 496e 7665 7273 654d 6170 5f6c 7052  isInverseMap_lpR
│ │ │ │ -00076570: 6174 696f 6e61 6c4d 6170 5f63 6d52 6174  ationalMap_cmRat
│ │ │ │ -00076580: 696f 6e61 6c4d 6170 5f72 702c 202d 2d20  ionalMap_rp, -- 
│ │ │ │ -00076590: 6368 6563 6b73 2077 6865 7468 6572 2074  checks whether t
│ │ │ │ -000765a0: 776f 2072 6174 696f 6e61 6c0a 2020 2020  wo rational.    
│ │ │ │ -000765b0: 6d61 7073 2061 7265 206f 6e65 2074 6865  maps are one the
│ │ │ │ -000765c0: 2069 6e76 6572 7365 206f 6620 7468 6520   inverse of the 
│ │ │ │ -000765d0: 6f74 6865 720a 0a46 6f72 2074 6865 2070  other..For the p
│ │ │ │ -000765e0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
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│ │ │ │ -00076600: 6520 6f62 6a65 6374 202a 6e6f 7465 2069  e object *note i
│ │ │ │ -00076610: 7349 6e76 6572 7365 4d61 703a 2069 7349  sInverseMap: isI
│ │ │ │ -00076620: 6e76 6572 7365 4d61 702c 2069 7320 6120  nverseMap, is a 
│ │ │ │ -00076630: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e  *note method fun
│ │ │ │ -00076640: 6374 696f 6e3a 0a28 4d61 6361 756c 6179  ction:.(Macaulay
│ │ │ │ -00076650: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ -00076660: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d  ion,...---------
│ │ │ │ +000762c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ +000762d0: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +000762e0: 2069 7349 6e76 6572 7365 4d61 7028 7068   isInverseMap(ph
│ │ │ │ +000762f0: 692c 7073 6929 0a20 202a 2049 6e70 7574  i,psi).  * Input
│ │ │ │ +00076300: 733a 0a20 2020 2020 202a 2070 6869 2c20  s:.      * phi, 
│ │ │ │ +00076310: 6120 2a6e 6f74 6520 7269 6e67 206d 6170  a *note ring map
│ │ │ │ +00076320: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00076330: 5269 6e67 4d61 702c 2c20 7265 7072 6573  RingMap,, repres
│ │ │ │ +00076340: 656e 7469 6e67 2061 2072 6174 696f 6e61  enting a rationa
│ │ │ │ +00076350: 6c0a 2020 2020 2020 2020 6d61 7020 245c  l.        map $\
│ │ │ │ +00076360: 5068 693a 5820 5c64 6173 6872 6967 6874  Phi:X \dashright
│ │ │ │ +00076370: 6172 726f 7720 5924 0a20 2020 2020 202a  arrow Y$.      *
│ │ │ │ +00076380: 2070 7369 2c20 6120 2a6e 6f74 6520 7269   psi, a *note ri
│ │ │ │ +00076390: 6e67 206d 6170 3a20 284d 6163 6175 6c61  ng map: (Macaula
│ │ │ │ +000763a0: 7932 446f 6329 5269 6e67 4d61 702c 2c20  y2Doc)RingMap,, 
│ │ │ │ +000763b0: 7265 7072 6573 656e 7469 6e67 2061 2072  representing a r
│ │ │ │ +000763c0: 6174 696f 6e61 6c0a 2020 2020 2020 2020  ational.        
│ │ │ │ +000763d0: 6d61 7020 245c 5073 693a 5920 5c64 6173  map $\Psi:Y \das
│ │ │ │ +000763e0: 6872 6967 6874 6172 726f 7720 5824 0a20  hrightarrow X$. 
│ │ │ │ +000763f0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +00076400: 2020 2a20 6120 2a6e 6f74 6520 426f 6f6c    * a *note Bool
│ │ │ │ +00076410: 6561 6e20 7661 6c75 653a 2028 4d61 6361  ean value: (Maca
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│ │ │ │ +00076430: 2c2c 2061 6363 6f72 6469 6e67 2074 6f20  ,, according to 
│ │ │ │ +00076440: 7468 650a 2020 2020 2020 2020 636f 6e64  the.        cond
│ │ │ │ +00076450: 6974 696f 6e20 7468 6174 2074 6865 2063  ition that the c
│ │ │ │ +00076460: 6f6d 706f 7369 7469 6f6e 2024 5c50 7369  omposition $\Psi
│ │ │ │ +00076470: 5c2c 5c50 6869 3a58 205c 6461 7368 7269  \,\Phi:X \dashri
│ │ │ │ +00076480: 6768 7461 7272 6f77 2058 240a 2020 2020  ghtarrow X$.    
│ │ │ │ +00076490: 2020 2020 636f 696e 6369 6465 7320 7769      coincides wi
│ │ │ │ +000764a0: 7468 2074 6865 2069 6465 6e74 6974 7920  th the identity 
│ │ │ │ +000764b0: 6f66 2024 5824 2028 6173 2061 2072 6174  of $X$ (as a rat
│ │ │ │ +000764c0: 696f 6e61 6c20 6d61 7029 0a0a 5761 7973  ional map)..Ways
│ │ │ │ +000764d0: 2074 6f20 7573 6520 6973 496e 7665 7273   to use isInvers
│ │ │ │ +000764e0: 654d 6170 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  eMap:.==========
│ │ │ │ +000764f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00076500: 0a20 202a 2022 6973 496e 7665 7273 654d  .  * "isInverseM
│ │ │ │ +00076510: 6170 2852 696e 674d 6170 2c52 696e 674d  ap(RingMap,RingM
│ │ │ │ +00076520: 6170 2922 0a20 202a 202a 6e6f 7465 2069  ap)".  * *note i
│ │ │ │ +00076530: 7349 6e76 6572 7365 4d61 7028 5261 7469  sInverseMap(Rati
│ │ │ │ +00076540: 6f6e 616c 4d61 702c 5261 7469 6f6e 616c  onalMap,Rational
│ │ │ │ +00076550: 4d61 7029 3a0a 2020 2020 6973 496e 7665  Map):.    isInve
│ │ │ │ +00076560: 7273 654d 6170 5f6c 7052 6174 696f 6e61  rseMap_lpRationa
│ │ │ │ +00076570: 6c4d 6170 5f63 6d52 6174 696f 6e61 6c4d  lMap_cmRationalM
│ │ │ │ +00076580: 6170 5f72 702c 202d 2d20 6368 6563 6b73  ap_rp, -- checks
│ │ │ │ +00076590: 2077 6865 7468 6572 2074 776f 2072 6174   whether two rat
│ │ │ │ +000765a0: 696f 6e61 6c0a 2020 2020 6d61 7073 2061  ional.    maps a
│ │ │ │ +000765b0: 7265 206f 6e65 2074 6865 2069 6e76 6572  re one the inver
│ │ │ │ +000765c0: 7365 206f 6620 7468 6520 6f74 6865 720a  se of the other.
│ │ │ │ +000765d0: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ +000765e0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ +000765f0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ +00076600: 6374 202a 6e6f 7465 2069 7349 6e76 6572  ct *note isInver
│ │ │ │ +00076610: 7365 4d61 703a 2069 7349 6e76 6572 7365  seMap: isInverse
│ │ │ │ +00076620: 4d61 702c 2069 7320 6120 2a6e 6f74 6520  Map, is a *note 
│ │ │ │ +00076630: 6d65 7468 6f64 2066 756e 6374 696f 6e3a  method function:
│ │ │ │ +00076640: 0a28 4d61 6361 756c 6179 3244 6f63 294d  .(Macaulay2Doc)M
│ │ │ │ +00076650: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ +00076660: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │  00076670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00076680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  000766a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00076770: 616c 4d61 705f 636d 5261 7469 6f6e 616c  alMap_cmRational
│ │ │ │ -00076780: 4d61 705f 7270 2c20 4e65 7874 3a20 6973  Map_rp, Next: is
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│ │ │ │ -000767a0: 7469 6f6e 616c 4d61 705f 7270 2c20 5072  tionalMap_rp, Pr
│ │ │ │ -000767b0: 6576 3a20 6973 496e 7665 7273 654d 6170  ev: isInverseMap
│ │ │ │ -000767c0: 2c20 5570 3a20 546f 700a 0a69 7349 6e76  , Up: Top..isInv
│ │ │ │ -000767d0: 6572 7365 4d61 7028 5261 7469 6f6e 616c  erseMap(Rational
│ │ │ │ -000767e0: 4d61 702c 5261 7469 6f6e 616c 4d61 7029  Map,RationalMap)
│ │ │ │ -000767f0: 202d 2d20 6368 6563 6b73 2077 6865 7468   -- checks wheth
│ │ │ │ -00076800: 6572 2074 776f 2072 6174 696f 6e61 6c20  er two rational 
│ │ │ │ -00076810: 6d61 7073 2061 7265 206f 6e65 2074 6865  maps are one the
│ │ │ │ -00076820: 2069 6e76 6572 7365 206f 6620 7468 6520   inverse of the 
│ │ │ │ -00076830: 6f74 6865 720a 2a2a 2a2a 2a2a 2a2a 2a2a  other.**********
│ │ │ │ +000766b0: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ +000766c0: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
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│ │ │ │ +000766f0: 6175 6c61 7932 2d31 2e32 352e 3131 2b64  aulay2-1.25.11+d
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│ │ │ │ +00076740: 2043 7265 6d6f 6e61 2e69 6e66 6f2c 204e   Cremona.info, N
│ │ │ │ +00076750: 6f64 653a 2069 7349 6e76 6572 7365 4d61  ode: isInverseMa
│ │ │ │ +00076760: 705f 6c70 5261 7469 6f6e 616c 4d61 705f  p_lpRationalMap_
│ │ │ │ +00076770: 636d 5261 7469 6f6e 616c 4d61 705f 7270  cmRationalMap_rp
│ │ │ │ +00076780: 2c20 4e65 7874 3a20 6973 4973 6f6d 6f72  , Next: isIsomor
│ │ │ │ +00076790: 7068 6973 6d5f 6c70 5261 7469 6f6e 616c  phism_lpRational
│ │ │ │ +000767a0: 4d61 705f 7270 2c20 5072 6576 3a20 6973  Map_rp, Prev: is
│ │ │ │ +000767b0: 496e 7665 7273 654d 6170 2c20 5570 3a20  InverseMap, Up: 
│ │ │ │ +000767c0: 546f 700a 0a69 7349 6e76 6572 7365 4d61  Top..isInverseMa
│ │ │ │ +000767d0: 7028 5261 7469 6f6e 616c 4d61 702c 5261  p(RationalMap,Ra
│ │ │ │ +000767e0: 7469 6f6e 616c 4d61 7029 202d 2d20 6368  tionalMap) -- ch
│ │ │ │ +000767f0: 6563 6b73 2077 6865 7468 6572 2074 776f  ecks whether two
│ │ │ │ +00076800: 2072 6174 696f 6e61 6c20 6d61 7073 2061   rational maps a
│ │ │ │ +00076810: 7265 206f 6e65 2074 6865 2069 6e76 6572  re one the inver
│ │ │ │ +00076820: 7365 206f 6620 7468 6520 6f74 6865 720a  se of the other.
│ │ │ │ +00076830: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00076840: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00076850: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00076860: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00076870: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00076880: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00076890: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000768a0: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20  ..  * Function: 
│ │ │ │ -000768b0: 2a6e 6f74 6520 6973 496e 7665 7273 654d  *note isInverseM
│ │ │ │ -000768c0: 6170 3a20 6973 496e 7665 7273 654d 6170  ap: isInverseMap
│ │ │ │ -000768d0: 2c0a 2020 2a20 5573 6167 653a 200a 2020  ,.  * Usage: .  
│ │ │ │ -000768e0: 2020 2020 2020 6973 496e 7665 7273 654d        isInverseM
│ │ │ │ -000768f0: 6170 2870 6869 2c70 7369 290a 2020 2a20  ap(phi,psi).  * 
│ │ │ │ -00076900: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -00076910: 7068 692c 2061 202a 6e6f 7465 2072 6174  phi, a *note rat
│ │ │ │ -00076920: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ -00076930: 6e61 6c4d 6170 2c0a 2020 2020 2020 2a20  nalMap,.      * 
│ │ │ │ -00076940: 7073 692c 2061 202a 6e6f 7465 2072 6174  psi, a *note rat
│ │ │ │ -00076950: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ -00076960: 6e61 6c4d 6170 2c0a 2020 2a20 4f75 7470  nalMap,.  * Outp
│ │ │ │ -00076970: 7574 733a 0a20 2020 2020 202a 2061 202a  uts:.      * a *
│ │ │ │ -00076980: 6e6f 7465 2042 6f6f 6c65 616e 2076 616c  note Boolean val
│ │ │ │ -00076990: 7565 3a20 284d 6163 6175 6c61 7932 446f  ue: (Macaulay2Do
│ │ │ │ -000769a0: 6329 426f 6f6c 6561 6e2c 2c20 7768 6574  c)Boolean,, whet
│ │ │ │ -000769b0: 6865 7220 7068 6920 2a20 7073 6920 3d3d  her phi * psi ==
│ │ │ │ -000769c0: 2031 0a20 2020 2020 2020 2061 6e64 2070   1.        and p
│ │ │ │ -000769d0: 7369 202a 2070 6869 203d 3d20 310a 2020  si * phi == 1.  
│ │ │ │ -000769e0: 2a20 436f 6e73 6571 7565 6e63 6573 3a0a  * Consequences:.
│ │ │ │ -000769f0: 2020 2020 2020 2a20 4966 2074 6865 2061        * If the a
│ │ │ │ -00076a00: 6e73 7765 7220 6973 2061 6666 6972 6d61  nswer is affirma
│ │ │ │ -00076a10: 7469 7665 2c20 7468 656e 2074 6865 2073  tive, then the s
│ │ │ │ -00076a20: 7973 7465 6d20 7769 6c6c 2062 6520 696e  ystem will be in
│ │ │ │ -00076a30: 666f 726d 6564 2061 6e64 2073 6f0a 2020  formed and so.  
│ │ │ │ -00076a40: 2020 2020 2020 636f 6d6d 616e 6473 206c        commands l
│ │ │ │ -00076a50: 696b 6520 2769 6e76 6572 7365 2070 6869  ike 'inverse phi
│ │ │ │ -00076a60: 2720 7769 6c6c 2065 7865 6375 7465 2066  ' will execute f
│ │ │ │ -00076a70: 6173 742e 0a0a 5365 6520 616c 736f 0a3d  ast...See also.=
│ │ │ │ -00076a80: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │ -00076a90: 7465 2052 6174 696f 6e61 6c4d 6170 203d  te RationalMap =
│ │ │ │ -00076aa0: 3d20 5261 7469 6f6e 616c 4d61 703a 2052  = RationalMap: R
│ │ │ │ -00076ab0: 6174 696f 6e61 6c4d 6170 203d 3d20 5261  ationalMap == Ra
│ │ │ │ -00076ac0: 7469 6f6e 616c 4d61 702c 202d 2d20 6571  tionalMap, -- eq
│ │ │ │ -00076ad0: 7561 6c69 7479 0a20 2020 206f 6620 7261  uality.    of ra
│ │ │ │ -00076ae0: 7469 6f6e 616c 206d 6170 730a 2020 2a20  tional maps.  * 
│ │ │ │ -00076af0: 2a6e 6f74 6520 5261 7469 6f6e 616c 4d61  *note RationalMa
│ │ │ │ -00076b00: 7020 2a20 5261 7469 6f6e 616c 4d61 703a  p * RationalMap:
│ │ │ │ -00076b10: 2052 6174 696f 6e61 6c4d 6170 205f 7374   RationalMap _st
│ │ │ │ -00076b20: 2052 6174 696f 6e61 6c4d 6170 2c20 2d2d   RationalMap, --
│ │ │ │ -00076b30: 0a20 2020 2063 6f6d 706f 7369 7469 6f6e  .    composition
│ │ │ │ -00076b40: 206f 6620 7261 7469 6f6e 616c 206d 6170   of rational map
│ │ │ │ -00076b50: 730a 2020 2a20 2a6e 6f74 6520 6973 496e  s.  * *note isIn
│ │ │ │ -00076b60: 7665 7273 654d 6170 2852 696e 674d 6170  verseMap(RingMap
│ │ │ │ -00076b70: 2c52 696e 674d 6170 293a 2069 7349 6e76  ,RingMap): isInv
│ │ │ │ -00076b80: 6572 7365 4d61 702c 202d 2d20 6368 6563  erseMap, -- chec
│ │ │ │ -00076b90: 6b73 2077 6865 7468 6572 2061 0a20 2020  ks whether a.   
│ │ │ │ -00076ba0: 2072 6174 696f 6e61 6c20 6d61 7020 6973   rational map is
│ │ │ │ -00076bb0: 2074 6865 2069 6e76 6572 7365 206f 6620   the inverse of 
│ │ │ │ -00076bc0: 616e 6f74 6865 720a 2020 2a20 2a6e 6f74  another.  * *not
│ │ │ │ -00076bd0: 6520 696e 7665 7273 6528 5261 7469 6f6e  e inverse(Ration
│ │ │ │ -00076be0: 616c 4d61 7029 3a20 696e 7665 7273 655f  alMap): inverse_
│ │ │ │ -00076bf0: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ -00076c00: 2c20 2d2d 2069 6e76 6572 7365 206f 6620  , -- inverse of 
│ │ │ │ -00076c10: 610a 2020 2020 6269 7261 7469 6f6e 616c  a.    birational
│ │ │ │ -00076c20: 206d 6170 0a20 202a 202a 6e6f 7465 2066   map.  * *note f
│ │ │ │ -00076c30: 6f72 6365 496e 7665 7273 654d 6170 3a20  orceInverseMap: 
│ │ │ │ -00076c40: 666f 7263 6549 6e76 6572 7365 4d61 702c  forceInverseMap,
│ │ │ │ -00076c50: 202d 2d20 6465 636c 6172 6520 7468 6174   -- declare that
│ │ │ │ -00076c60: 2074 776f 2072 6174 696f 6e61 6c20 6d61   two rational ma
│ │ │ │ -00076c70: 7073 0a20 2020 2061 7265 206f 6e65 2074  ps.    are one t
│ │ │ │ -00076c80: 6865 2069 6e76 6572 7365 206f 6620 7468  he inverse of th
│ │ │ │ -00076c90: 6520 6f74 6865 720a 0a57 6179 7320 746f  e other..Ways to
│ │ │ │ -00076ca0: 2075 7365 2074 6869 7320 6d65 7468 6f64   use this method
│ │ │ │ -00076cb0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ -00076cc0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00076cd0: 2a6e 6f74 6520 6973 496e 7665 7273 654d  *note isInverseM
│ │ │ │ -00076ce0: 6170 2852 6174 696f 6e61 6c4d 6170 2c52  ap(RationalMap,R
│ │ │ │ -00076cf0: 6174 696f 6e61 6c4d 6170 293a 0a20 2020  ationalMap):.   
│ │ │ │ -00076d00: 2069 7349 6e76 6572 7365 4d61 705f 6c70   isInverseMap_lp
│ │ │ │ -00076d10: 5261 7469 6f6e 616c 4d61 705f 636d 5261  RationalMap_cmRa
│ │ │ │ -00076d20: 7469 6f6e 616c 4d61 705f 7270 2c20 2d2d  tionalMap_rp, --
│ │ │ │ -00076d30: 2063 6865 636b 7320 7768 6574 6865 7220   checks whether 
│ │ │ │ -00076d40: 7477 6f20 7261 7469 6f6e 616c 0a20 2020  two rational.   
│ │ │ │ -00076d50: 206d 6170 7320 6172 6520 6f6e 6520 7468   maps are one th
│ │ │ │ -00076d60: 6520 696e 7665 7273 6520 6f66 2074 6865  e inverse of the
│ │ │ │ -00076d70: 206f 7468 6572 0a2d 2d2d 2d2d 2d2d 2d2d   other.---------
│ │ │ │ +00076890: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +000768a0: 4675 6e63 7469 6f6e 3a20 2a6e 6f74 6520  Function: *note 
│ │ │ │ +000768b0: 6973 496e 7665 7273 654d 6170 3a20 6973  isInverseMap: is
│ │ │ │ +000768c0: 496e 7665 7273 654d 6170 2c0a 2020 2a20  InverseMap,.  * 
│ │ │ │ +000768d0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +000768e0: 6973 496e 7665 7273 654d 6170 2870 6869  isInverseMap(phi
│ │ │ │ +000768f0: 2c70 7369 290a 2020 2a20 496e 7075 7473  ,psi).  * Inputs
│ │ │ │ +00076900: 3a0a 2020 2020 2020 2a20 7068 692c 2061  :.      * phi, a
│ │ │ │ +00076910: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ +00076920: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ +00076930: 2c0a 2020 2020 2020 2a20 7073 692c 2061  ,.      * psi, a
│ │ │ │ +00076940: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ +00076950: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ +00076960: 2c0a 2020 2a20 4f75 7470 7574 733a 0a20  ,.  * Outputs:. 
│ │ │ │ +00076970: 2020 2020 202a 2061 202a 6e6f 7465 2042       * a *note B
│ │ │ │ +00076980: 6f6f 6c65 616e 2076 616c 7565 3a20 284d  oolean value: (M
│ │ │ │ +00076990: 6163 6175 6c61 7932 446f 6329 426f 6f6c  acaulay2Doc)Bool
│ │ │ │ +000769a0: 6561 6e2c 2c20 7768 6574 6865 7220 7068  ean,, whether ph
│ │ │ │ +000769b0: 6920 2a20 7073 6920 3d3d 2031 0a20 2020  i * psi == 1.   
│ │ │ │ +000769c0: 2020 2020 2061 6e64 2070 7369 202a 2070       and psi * p
│ │ │ │ +000769d0: 6869 203d 3d20 310a 2020 2a20 436f 6e73  hi == 1.  * Cons
│ │ │ │ +000769e0: 6571 7565 6e63 6573 3a0a 2020 2020 2020  equences:.      
│ │ │ │ +000769f0: 2a20 4966 2074 6865 2061 6e73 7765 7220  * If the answer 
│ │ │ │ +00076a00: 6973 2061 6666 6972 6d61 7469 7665 2c20  is affirmative, 
│ │ │ │ +00076a10: 7468 656e 2074 6865 2073 7973 7465 6d20  then the system 
│ │ │ │ +00076a20: 7769 6c6c 2062 6520 696e 666f 726d 6564  will be informed
│ │ │ │ +00076a30: 2061 6e64 2073 6f0a 2020 2020 2020 2020   and so.        
│ │ │ │ +00076a40: 636f 6d6d 616e 6473 206c 696b 6520 2769  commands like 'i
│ │ │ │ +00076a50: 6e76 6572 7365 2070 6869 2720 7769 6c6c  nverse phi' will
│ │ │ │ +00076a60: 2065 7865 6375 7465 2066 6173 742e 0a0a   execute fast...
│ │ │ │ +00076a70: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ +00076a80: 3d0a 0a20 202a 202a 6e6f 7465 2052 6174  =..  * *note Rat
│ │ │ │ +00076a90: 696f 6e61 6c4d 6170 203d 3d20 5261 7469  ionalMap == Rati
│ │ │ │ +00076aa0: 6f6e 616c 4d61 703a 2052 6174 696f 6e61  onalMap: Rationa
│ │ │ │ +00076ab0: 6c4d 6170 203d 3d20 5261 7469 6f6e 616c  lMap == Rational
│ │ │ │ +00076ac0: 4d61 702c 202d 2d20 6571 7561 6c69 7479  Map, -- equality
│ │ │ │ +00076ad0: 0a20 2020 206f 6620 7261 7469 6f6e 616c  .    of rational
│ │ │ │ +00076ae0: 206d 6170 730a 2020 2a20 2a6e 6f74 6520   maps.  * *note 
│ │ │ │ +00076af0: 5261 7469 6f6e 616c 4d61 7020 2a20 5261  RationalMap * Ra
│ │ │ │ +00076b00: 7469 6f6e 616c 4d61 703a 2052 6174 696f  tionalMap: Ratio
│ │ │ │ +00076b10: 6e61 6c4d 6170 205f 7374 2052 6174 696f  nalMap _st Ratio
│ │ │ │ +00076b20: 6e61 6c4d 6170 2c20 2d2d 0a20 2020 2063  nalMap, --.    c
│ │ │ │ +00076b30: 6f6d 706f 7369 7469 6f6e 206f 6620 7261  omposition of ra
│ │ │ │ +00076b40: 7469 6f6e 616c 206d 6170 730a 2020 2a20  tional maps.  * 
│ │ │ │ +00076b50: 2a6e 6f74 6520 6973 496e 7665 7273 654d  *note isInverseM
│ │ │ │ +00076b60: 6170 2852 696e 674d 6170 2c52 696e 674d  ap(RingMap,RingM
│ │ │ │ +00076b70: 6170 293a 2069 7349 6e76 6572 7365 4d61  ap): isInverseMa
│ │ │ │ +00076b80: 702c 202d 2d20 6368 6563 6b73 2077 6865  p, -- checks whe
│ │ │ │ +00076b90: 7468 6572 2061 0a20 2020 2072 6174 696f  ther a.    ratio
│ │ │ │ +00076ba0: 6e61 6c20 6d61 7020 6973 2074 6865 2069  nal map is the i
│ │ │ │ +00076bb0: 6e76 6572 7365 206f 6620 616e 6f74 6865  nverse of anothe
│ │ │ │ +00076bc0: 720a 2020 2a20 2a6e 6f74 6520 696e 7665  r.  * *note inve
│ │ │ │ +00076bd0: 7273 6528 5261 7469 6f6e 616c 4d61 7029  rse(RationalMap)
│ │ │ │ +00076be0: 3a20 696e 7665 7273 655f 6c70 5261 7469  : inverse_lpRati
│ │ │ │ +00076bf0: 6f6e 616c 4d61 705f 7270 2c20 2d2d 2069  onalMap_rp, -- i
│ │ │ │ +00076c00: 6e76 6572 7365 206f 6620 610a 2020 2020  nverse of a.    
│ │ │ │ +00076c10: 6269 7261 7469 6f6e 616c 206d 6170 0a20  birational map. 
│ │ │ │ +00076c20: 202a 202a 6e6f 7465 2066 6f72 6365 496e   * *note forceIn
│ │ │ │ +00076c30: 7665 7273 654d 6170 3a20 666f 7263 6549  verseMap: forceI
│ │ │ │ +00076c40: 6e76 6572 7365 4d61 702c 202d 2d20 6465  nverseMap, -- de
│ │ │ │ +00076c50: 636c 6172 6520 7468 6174 2074 776f 2072  clare that two r
│ │ │ │ +00076c60: 6174 696f 6e61 6c20 6d61 7073 0a20 2020  ational maps.   
│ │ │ │ +00076c70: 2061 7265 206f 6e65 2074 6865 2069 6e76   are one the inv
│ │ │ │ +00076c80: 6572 7365 206f 6620 7468 6520 6f74 6865  erse of the othe
│ │ │ │ +00076c90: 720a 0a57 6179 7320 746f 2075 7365 2074  r..Ways to use t
│ │ │ │ +00076ca0: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d  his method:.====
│ │ │ │ +00076cb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00076cc0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ +00076cd0: 6973 496e 7665 7273 654d 6170 2852 6174  isInverseMap(Rat
│ │ │ │ +00076ce0: 696f 6e61 6c4d 6170 2c52 6174 696f 6e61  ionalMap,Rationa
│ │ │ │ +00076cf0: 6c4d 6170 293a 0a20 2020 2069 7349 6e76  lMap):.    isInv
│ │ │ │ +00076d00: 6572 7365 4d61 705f 6c70 5261 7469 6f6e  erseMap_lpRation
│ │ │ │ +00076d10: 616c 4d61 705f 636d 5261 7469 6f6e 616c  alMap_cmRational
│ │ │ │ +00076d20: 4d61 705f 7270 2c20 2d2d 2063 6865 636b  Map_rp, -- check
│ │ │ │ +00076d30: 7320 7768 6574 6865 7220 7477 6f20 7261  s whether two ra
│ │ │ │ +00076d40: 7469 6f6e 616c 0a20 2020 206d 6170 7320  tional.    maps 
│ │ │ │ +00076d50: 6172 6520 6f6e 6520 7468 6520 696e 7665  are one the inve
│ │ │ │ +00076d60: 7273 6520 6f66 2074 6865 206f 7468 6572  rse of the other
│ │ │ │ +00076d70: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │  00076d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00076d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00076da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00076db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00076dc0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ -00076dd0: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
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│ │ │ │ -00076e00: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
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│ │ │ │ -00076e60: 6e66 6f2c 204e 6f64 653a 2069 7349 736f  nfo, Node: isIso
│ │ │ │ -00076e70: 6d6f 7270 6869 736d 5f6c 7052 6174 696f  morphism_lpRatio
│ │ │ │ -00076e80: 6e61 6c4d 6170 5f72 702c 204e 6578 743a  nalMap_rp, Next:
│ │ │ │ -00076e90: 2069 734d 6f72 7068 6973 6d2c 2050 7265   isMorphism, Pre
│ │ │ │ -00076ea0: 763a 2069 7349 6e76 6572 7365 4d61 705f  v: isInverseMap_
│ │ │ │ -00076eb0: 6c70 5261 7469 6f6e 616c 4d61 705f 636d  lpRationalMap_cm
│ │ │ │ -00076ec0: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ -00076ed0: 5570 3a20 546f 700a 0a69 7349 736f 6d6f  Up: Top..isIsomo
│ │ │ │ -00076ee0: 7270 6869 736d 2852 6174 696f 6e61 6c4d  rphism(RationalM
│ │ │ │ -00076ef0: 6170 2920 2d2d 2077 6865 7468 6572 2061  ap) -- whether a
│ │ │ │ -00076f00: 2062 6972 6174 696f 6e61 6c20 6d61 7020   birational map 
│ │ │ │ -00076f10: 6973 2061 6e20 6973 6f6d 6f72 7068 6973  is an isomorphis
│ │ │ │ -00076f20: 6d0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  m.**************
│ │ │ │ +00076dc0: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ +00076dd0: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
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│ │ │ │ +00076e10: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ +00076e20: 6163 6b61 6765 732f 4372 656d 6f6e 612f  ackages/Cremona/
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│ │ │ │ +00076e50: 2043 7265 6d6f 6e61 2e69 6e66 6f2c 204e   Cremona.info, N
│ │ │ │ +00076e60: 6f64 653a 2069 7349 736f 6d6f 7270 6869  ode: isIsomorphi
│ │ │ │ +00076e70: 736d 5f6c 7052 6174 696f 6e61 6c4d 6170  sm_lpRationalMap
│ │ │ │ +00076e80: 5f72 702c 204e 6578 743a 2069 734d 6f72  _rp, Next: isMor
│ │ │ │ +00076e90: 7068 6973 6d2c 2050 7265 763a 2069 7349  phism, Prev: isI
│ │ │ │ +00076ea0: 6e76 6572 7365 4d61 705f 6c70 5261 7469  nverseMap_lpRati
│ │ │ │ +00076eb0: 6f6e 616c 4d61 705f 636d 5261 7469 6f6e  onalMap_cmRation
│ │ │ │ +00076ec0: 616c 4d61 705f 7270 2c20 5570 3a20 546f  alMap_rp, Up: To
│ │ │ │ +00076ed0: 700a 0a69 7349 736f 6d6f 7270 6869 736d  p..isIsomorphism
│ │ │ │ +00076ee0: 2852 6174 696f 6e61 6c4d 6170 2920 2d2d  (RationalMap) --
│ │ │ │ +00076ef0: 2077 6865 7468 6572 2061 2062 6972 6174   whether a birat
│ │ │ │ +00076f00: 696f 6e61 6c20 6d61 7020 6973 2061 6e20  ional map is an 
│ │ │ │ +00076f10: 6973 6f6d 6f72 7068 6973 6d0a 2a2a 2a2a  isomorphism.****
│ │ │ │ +00076f20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00076f30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00076f40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00076f50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00076f60: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -00076f70: 4675 6e63 7469 6f6e 3a20 2a6e 6f74 6520  Function: *note 
│ │ │ │ -00076f80: 6973 4973 6f6d 6f72 7068 6973 6d3a 2028  isIsomorphism: (
│ │ │ │ -00076f90: 4d61 6361 756c 6179 3244 6f63 2969 7349  Macaulay2Doc)isI
│ │ │ │ -00076fa0: 736f 6d6f 7270 6869 736d 2c0a 2020 2a20  somorphism,.  * 
│ │ │ │ -00076fb0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -00076fc0: 6973 4973 6f6d 6f72 7068 6973 6d20 7068  isIsomorphism ph
│ │ │ │ -00076fd0: 690a 2020 2a20 496e 7075 7473 3a0a 2020  i.  * Inputs:.  
│ │ │ │ -00076fe0: 2020 2020 2a20 7068 692c 2061 202a 6e6f      * phi, a *no
│ │ │ │ -00076ff0: 7465 2072 6174 696f 6e61 6c20 6d61 703a  te rational map:
│ │ │ │ -00077000: 2052 6174 696f 6e61 6c4d 6170 2c0a 2020   RationalMap,.  
│ │ │ │ -00077010: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -00077020: 202a 2061 202a 6e6f 7465 2042 6f6f 6c65   * a *note Boole
│ │ │ │ -00077030: 616e 2076 616c 7565 3a20 284d 6163 6175  an value: (Macau
│ │ │ │ -00077040: 6c61 7932 446f 6329 426f 6f6c 6561 6e2c  lay2Doc)Boolean,
│ │ │ │ -00077050: 2c20 7768 6574 6865 7220 7068 6920 6973  , whether phi is
│ │ │ │ -00077060: 2061 6e0a 2020 2020 2020 2020 6973 6f6d   an.        isom
│ │ │ │ -00077070: 6f72 7068 6973 6d0a 0a44 6573 6372 6970  orphism..Descrip
│ │ │ │ -00077080: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -00077090: 0a0a 5468 6973 206d 6574 686f 6420 636f  ..This method co
│ │ │ │ -000770a0: 6d70 7574 6573 2074 6865 2069 6e76 6572  mputes the inver
│ │ │ │ -000770b0: 7365 2072 6174 696f 6e61 6c20 6d61 7020  se rational map 
│ │ │ │ -000770c0: 7573 696e 6720 2a6e 6f74 6520 696e 7665  using *note inve
│ │ │ │ -000770d0: 7273 653a 0a69 6e76 6572 7365 5f6c 7052  rse:.inverse_lpR
│ │ │ │ -000770e0: 6174 696f 6e61 6c4d 6170 5f72 702c 2e0a  ationalMap_rp,..
│ │ │ │ -000770f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00076f60: 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 7469  ****..  * Functi
│ │ │ │ +00076f70: 6f6e 3a20 2a6e 6f74 6520 6973 4973 6f6d  on: *note isIsom
│ │ │ │ +00076f80: 6f72 7068 6973 6d3a 2028 4d61 6361 756c  orphism: (Macaul
│ │ │ │ +00076f90: 6179 3244 6f63 2969 7349 736f 6d6f 7270  ay2Doc)isIsomorp
│ │ │ │ +00076fa0: 6869 736d 2c0a 2020 2a20 5573 6167 653a  hism,.  * Usage:
│ │ │ │ +00076fb0: 200a 2020 2020 2020 2020 6973 4973 6f6d   .        isIsom
│ │ │ │ +00076fc0: 6f72 7068 6973 6d20 7068 690a 2020 2a20  orphism phi.  * 
│ │ │ │ +00076fd0: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +00076fe0: 7068 692c 2061 202a 6e6f 7465 2072 6174  phi, a *note rat
│ │ │ │ +00076ff0: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ +00077000: 6e61 6c4d 6170 2c0a 2020 2a20 4f75 7470  nalMap,.  * Outp
│ │ │ │ +00077010: 7574 733a 0a20 2020 2020 202a 2061 202a  uts:.      * a *
│ │ │ │ +00077020: 6e6f 7465 2042 6f6f 6c65 616e 2076 616c  note Boolean val
│ │ │ │ +00077030: 7565 3a20 284d 6163 6175 6c61 7932 446f  ue: (Macaulay2Do
│ │ │ │ +00077040: 6329 426f 6f6c 6561 6e2c 2c20 7768 6574  c)Boolean,, whet
│ │ │ │ +00077050: 6865 7220 7068 6920 6973 2061 6e0a 2020  her phi is an.  
│ │ │ │ +00077060: 2020 2020 2020 6973 6f6d 6f72 7068 6973        isomorphis
│ │ │ │ +00077070: 6d0a 0a44 6573 6372 6970 7469 6f6e 0a3d  m..Description.=
│ │ │ │ +00077080: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973  ==========..This
│ │ │ │ +00077090: 206d 6574 686f 6420 636f 6d70 7574 6573   method computes
│ │ │ │ +000770a0: 2074 6865 2069 6e76 6572 7365 2072 6174   the inverse rat
│ │ │ │ +000770b0: 696f 6e61 6c20 6d61 7020 7573 696e 6720  ional map using 
│ │ │ │ +000770c0: 2a6e 6f74 6520 696e 7665 7273 653a 0a69  *note inverse:.i
│ │ │ │ +000770d0: 6e76 6572 7365 5f6c 7052 6174 696f 6e61  nverse_lpRationa
│ │ │ │ +000770e0: 6c4d 6170 5f72 702c 2e0a 0a2b 2d2d 2d2d  lMap_rp,...+----
│ │ │ │ +000770f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00077130: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5031  ------+.|i1 : P1
│ │ │ │ -00077140: 203a 3d20 5151 5b61 2c62 5d3b 2050 3420   := QQ[a,b]; P4 
│ │ │ │ -00077150: 3a3d 2051 515b 782c 792c 7a2c 775d 3b20  := QQ[x,y,z,w]; 
│ │ │ │ +00077130: 2b0a 7c69 3120 3a20 5031 203a 3d20 5151  +.|i1 : P1 := QQ
│ │ │ │ +00077140: 5b61 2c62 5d3b 2050 3420 3a3d 2051 515b  [a,b]; P4 := QQ[
│ │ │ │ +00077150: 782c 792c 7a2c 775d 3b20 2020 2020 2020  x,y,z,w];       
│ │ │ │  00077160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077170: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00077170: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  00077180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000771a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000771b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000771c0: 2d2d 2d2d 2b0a 7c69 3320 3a20 7068 6920  ----+.|i3 : phi 
│ │ │ │ -000771d0: 3d20 7261 7469 6f6e 616c 4d61 7028 7b61  = rationalMap({a
│ │ │ │ -000771e0: 5e34 2c61 5e33 2a62 2c61 5e32 2a62 5e32  ^4,a^3*b,a^2*b^2
│ │ │ │ -000771f0: 2c61 2a62 5e33 2c62 5e34 7d2c 446f 6d69  ,a*b^3,b^4},Domi
│ │ │ │ -00077200: 6e61 6e74 3d3e 7472 7565 297c 0a7c 2020  nant=>true)|.|  
│ │ │ │ +000771b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000771c0: 7c69 3320 3a20 7068 6920 3d20 7261 7469  |i3 : phi = rati
│ │ │ │ +000771d0: 6f6e 616c 4d61 7028 7b61 5e34 2c61 5e33  onalMap({a^4,a^3
│ │ │ │ +000771e0: 2a62 2c61 5e32 2a62 5e32 2c61 2a62 5e33  *b,a^2*b^2,a*b^3
│ │ │ │ +000771f0: 2c62 5e34 7d2c 446f 6d69 6e61 6e74 3d3e  ,b^4},Dominant=>
│ │ │ │ +00077200: 7472 7565 297c 0a7c 2020 2020 2020 2020  true)|.|        
│ │ │ │  00077210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077250: 2020 7c0a 7c6f 3320 3d20 2d2d 2072 6174    |.|o3 = -- rat
│ │ │ │ -00077260: 696f 6e61 6c20 6d61 7020 2d2d 2020 2020  ional map --    
│ │ │ │ +00077240: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00077250: 3320 3d20 2d2d 2072 6174 696f 6e61 6c20  3 = -- rational 
│ │ │ │ +00077260: 6d61 7020 2d2d 2020 2020 2020 2020 2020  map --          
│ │ │ │  00077270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077290: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000772a0: 2073 6f75 7263 653a 2050 726f 6a28 5151   source: Proj(QQ
│ │ │ │ -000772b0: 5b61 2c20 625d 2920 2020 2020 2020 2020  [a, b])         
│ │ │ │ +00077290: 2020 207c 0a7c 2020 2020 2073 6f75 7263     |.|     sourc
│ │ │ │ +000772a0: 653a 2050 726f 6a28 5151 5b61 2c20 625d  e: Proj(QQ[a, b]
│ │ │ │ +000772b0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  000772c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000772d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000772e0: 7c0a 7c20 2020 2020 7461 7267 6574 3a20  |.|     target: 
│ │ │ │ -000772f0: 7375 6276 6172 6965 7479 206f 6620 5072  subvariety of Pr
│ │ │ │ -00077300: 6f6a 2851 515b 7420 2c20 7420 2c20 7420  oj(QQ[t , t , t 
│ │ │ │ -00077310: 2c20 7420 2c20 7420 5d29 2064 6566 696e  , t , t ]) defin
│ │ │ │ -00077320: 6564 2062 7920 207c 0a7c 2020 2020 2020  ed by  |.|      
│ │ │ │ +000772d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000772e0: 2020 7461 7267 6574 3a20 7375 6276 6172    target: subvar
│ │ │ │ +000772f0: 6965 7479 206f 6620 5072 6f6a 2851 515b  iety of Proj(QQ[
│ │ │ │ +00077300: 7420 2c20 7420 2c20 7420 2c20 7420 2c20  t , t , t , t , 
│ │ │ │ +00077310: 7420 5d29 2064 6566 696e 6564 2062 7920  t ]) defined by 
│ │ │ │ +00077320: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00077330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077340: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00077350: 2020 3120 2020 3220 2020 3320 2020 3420    1   2   3   4 
│ │ │ │ -00077360: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00077370: 7c20 2020 2020 2020 2020 2020 2020 7b20  |             { 
│ │ │ │ +00077340: 2020 2020 2020 2020 3020 2020 3120 2020          0   1   
│ │ │ │ +00077350: 3220 2020 3320 2020 3420 2020 2020 2020  2   3   4       
│ │ │ │ +00077360: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00077370: 2020 2020 2020 2020 7b20 2020 2020 2020          {       
│ │ │ │  00077380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000773a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000773b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000773c0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +000773a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000773b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000773c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  000773d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000773e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000773f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00077400: 2020 2020 2020 2020 2020 2020 2074 2020               t  
│ │ │ │ -00077410: 2d20 7420 7420 2c20 2020 2020 2020 2020  - t t ,         
│ │ │ │ +000773f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00077400: 2020 2020 2020 2074 2020 2d20 7420 7420         t  - t t 
│ │ │ │ +00077410: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  00077420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077440: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00077450: 2020 2020 2033 2020 2020 3220 3420 2020       3    2 4   
│ │ │ │ +00077430: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00077440: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ +00077450: 2020 2020 3220 3420 2020 2020 2020 2020      2 4         
│ │ │ │  00077460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077480: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00077480: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00077490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000774a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000774b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000774c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000774d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000774e0: 2020 7420 7420 202d 2074 2074 202c 2020    t t  - t t ,  
│ │ │ │ +000774c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000774d0: 2020 2020 2020 2020 2020 2020 7420 7420              t t 
│ │ │ │ +000774e0: 202d 2074 2074 202c 2020 2020 2020 2020   - t t ,        
│ │ │ │  000774f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077510: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00077520: 2020 2020 2020 2020 2020 3220 3320 2020            2 3   
│ │ │ │ -00077530: 2031 2034 2020 2020 2020 2020 2020 2020   1 4            
│ │ │ │ +00077510: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00077520: 2020 2020 3220 3320 2020 2031 2034 2020      2 3    1 4  
│ │ │ │ +00077530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077550: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00077560: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00077550: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00077560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000775a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000775b0: 2020 2020 2020 2074 2074 2020 2d20 7420         t t  - t 
│ │ │ │ -000775c0: 7420 2c20 2020 2020 2020 2020 2020 2020  t ,             
│ │ │ │ +000775a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000775b0: 2074 2074 2020 2d20 7420 7420 2c20 2020   t t  - t t ,   
│ │ │ │ +000775c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000775d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000775e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000775f0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00077600: 2033 2020 2020 3020 3420 2020 2020 2020   3    0 4       
│ │ │ │ +000775e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000775f0: 2020 2020 2020 2020 2031 2033 2020 2020           1 3    
│ │ │ │ +00077600: 3020 3420 2020 2020 2020 2020 2020 2020  0 4             
│ │ │ │  00077610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077630: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00077620: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00077630: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00077640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077670: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00077680: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00077670: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00077680: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  00077690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000776a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000776b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000776c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -000776d0: 2020 2074 2020 2d20 7420 7420 2c20 2020     t  - t t ,   
│ │ │ │ +000776b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000776c0: 2020 2020 2020 2020 2020 2020 2074 2020               t  
│ │ │ │ +000776d0: 2d20 7420 7420 2c20 2020 2020 2020 2020  - t t ,         
│ │ │ │  000776e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000776f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077700: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00077710: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00077720: 3020 3420 2020 2020 2020 2020 2020 2020  0 4             
│ │ │ │ +00077700: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00077710: 2020 2020 2032 2020 2020 3020 3420 2020       2    0 4   
│ │ │ │ +00077720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077750: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00077740: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00077750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077790: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000777a0: 2020 2020 2020 2020 7420 7420 202d 2074          t t  - t
│ │ │ │ -000777b0: 2074 202c 2020 2020 2020 2020 2020 2020   t ,            
│ │ │ │ +00077790: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000777a0: 2020 7420 7420 202d 2074 2074 202c 2020    t t  - t t ,  
│ │ │ │ +000777b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000777c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000777d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000777e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000777f0: 3120 3220 2020 2030 2033 2020 2020 2020  1 2    0 3      
│ │ │ │ +000777d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000777e0: 2020 2020 2020 2020 2020 3120 3220 2020            1 2   
│ │ │ │ +000777f0: 2030 2033 2020 2020 2020 2020 2020 2020   0 3            
│ │ │ │  00077800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077820: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00077810: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00077820: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00077830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077860: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00077870: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00077860: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00077870: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  00077880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000778a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000778b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000778c0: 2020 2020 7420 202d 2074 2074 2020 2020      t  - t t    
│ │ │ │ +000778a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000778b0: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ +000778c0: 202d 2074 2074 2020 2020 2020 2020 2020   - t t          
│ │ │ │  000778d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000778e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000778f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00077900: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ -00077910: 2030 2032 2020 2020 2020 2020 2020 2020   0 2            
│ │ │ │ +000778f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00077900: 2020 2020 2020 3120 2020 2030 2032 2020        1    0 2  
│ │ │ │ +00077910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077930: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00077950: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │ +00077930: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00077940: 2020 2020 2020 2020 2020 207d 2020 2020             }    
│ │ │ │ +00077950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077980: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00077990: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20  defining forms: 
│ │ │ │ -000779a0: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │ +00077980: 2020 7c0a 7c20 2020 2020 6465 6669 6e69    |.|     defini
│ │ │ │ +00077990: 6e67 2066 6f72 6d73 3a20 7b20 2020 2020  ng forms: {     
│ │ │ │ +000779a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000779b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000779c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000779d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000779e0: 2020 2020 2020 2020 2034 2020 2020 2020           4      
│ │ │ │ +000779c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000779d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000779e0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │  000779f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077a10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00077a20: 2020 2020 2020 2020 2020 2020 2020 2061                 a
│ │ │ │ -00077a30: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │ +00077a10: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00077a20: 2020 2020 2020 2020 2061 202c 2020 2020           a ,    
│ │ │ │ +00077a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077a50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00077a50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00077a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077aa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00077ab0: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ +00077a90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00077aa0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00077ab0: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │  00077ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077ae0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00077af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077b00: 2020 2020 6120 622c 2020 2020 2020 2020      a b,        
│ │ │ │ +00077ae0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00077af0: 2020 2020 2020 2020 2020 2020 2020 6120                a 
│ │ │ │ +00077b00: 622c 2020 2020 2020 2020 2020 2020 2020  b,              
│ │ │ │  00077b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077b30: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00077b20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00077b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077b70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00077b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077b90: 2020 2032 2032 2020 2020 2020 2020 2020     2 2          
│ │ │ │ +00077b70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00077b80: 2020 2020 2020 2020 2020 2020 2032 2032               2 2
│ │ │ │ +00077b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077bc0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00077bd0: 2020 2020 2020 2020 2061 2062 202c 2020           a b ,  
│ │ │ │ +00077bb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00077bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077bd0: 2020 2061 2062 202c 2020 2020 2020 2020     a b ,        
│ │ │ │  00077be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077c00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00077c00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00077c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077c40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00077c50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00077c60: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ +00077c40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00077c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077c60: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │  00077c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077c90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00077ca0: 2020 2020 2020 2020 2020 2020 2020 612a                a*
│ │ │ │ -00077cb0: 6220 2c20 2020 2020 2020 2020 2020 2020  b ,             
│ │ │ │ +00077c80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00077c90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00077ca0: 2020 2020 2020 2020 612a 6220 2c20 2020          a*b ,   
│ │ │ │ +00077cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077cd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00077cd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00077ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077d20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00077d30: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ +00077d10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00077d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077d30: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │  00077d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077d60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00077d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077d80: 2020 2062 2020 2020 2020 2020 2020 2020     b            
│ │ │ │ +00077d60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00077d70: 2020 2020 2020 2020 2020 2020 2062 2020               b  
│ │ │ │ +00077d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077db0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00077dc0: 2020 2020 2020 2020 207d 2020 2020 2020           }      
│ │ │ │ +00077da0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00077db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00077dc0: 2020 207d 2020 2020 2020 2020 2020 2020     }            
│ │ │ │  00077dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077df0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00077df0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00077e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077e30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00077e40: 0a7c 6f33 203a 2052 6174 696f 6e61 6c4d  .|o3 : RationalM
│ │ │ │ -00077e50: 6170 2028 646f 6d69 6e61 6e74 2072 6174  ap (dominant rat
│ │ │ │ -00077e60: 696f 6e61 6c20 6d61 7020 6672 6f6d 2050  ional map from P
│ │ │ │ -00077e70: 505e 3120 746f 2063 7572 7665 2069 6e20  P^1 to curve in 
│ │ │ │ -00077e80: 5050 5e34 2920 7c0a 2b2d 2d2d 2d2d 2d2d  PP^4) |.+-------
│ │ │ │ +00077e30: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ +00077e40: 2052 6174 696f 6e61 6c4d 6170 2028 646f   RationalMap (do
│ │ │ │ +00077e50: 6d69 6e61 6e74 2072 6174 696f 6e61 6c20  minant rational 
│ │ │ │ +00077e60: 6d61 7020 6672 6f6d 2050 505e 3120 746f  map from PP^1 to
│ │ │ │ +00077e70: 2063 7572 7665 2069 6e20 5050 5e34 2920   curve in PP^4) 
│ │ │ │ +00077e80: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00077e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00077ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00077ed0: 6934 203a 2069 7349 736f 6d6f 7270 6869  i4 : isIsomorphi
│ │ │ │ -00077ee0: 736d 2070 6869 2020 2020 2020 2020 2020  sm phi          
│ │ │ │ +00077ec0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2069  -------+.|i4 : i
│ │ │ │ +00077ed0: 7349 736f 6d6f 7270 6869 736d 2070 6869  sIsomorphism phi
│ │ │ │ +00077ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077f10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00077f00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00077f10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00077f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077f50: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ -00077f60: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ +00077f50: 2020 2020 207c 0a7c 6f34 203d 2074 7275       |.|o4 = tru
│ │ │ │ +00077f60: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │  00077f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00077f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00077fa0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00077f90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00077fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00077fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00077fe0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520  ---------+..See 
│ │ │ │ -00077ff0: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ -00078000: 202a 202a 6e6f 7465 2069 6465 616c 2852   * *note ideal(R
│ │ │ │ -00078010: 6174 696f 6e61 6c4d 6170 293a 2069 6465  ationalMap): ide
│ │ │ │ -00078020: 616c 5f6c 7052 6174 696f 6e61 6c4d 6170  al_lpRationalMap
│ │ │ │ -00078030: 5f72 702c 202d 2d20 6261 7365 206c 6f63  _rp, -- base loc
│ │ │ │ -00078040: 7573 206f 6620 610a 2020 2020 7261 7469  us of a.    rati
│ │ │ │ -00078050: 6f6e 616c 206d 6170 0a20 202a 202a 6e6f  onal map.  * *no
│ │ │ │ -00078060: 7465 2069 7342 6972 6174 696f 6e61 6c3a  te isBirational:
│ │ │ │ -00078070: 2069 7342 6972 6174 696f 6e61 6c2c 202d   isBirational, -
│ │ │ │ -00078080: 2d20 7768 6574 6865 7220 6120 7261 7469  - whether a rati
│ │ │ │ -00078090: 6f6e 616c 206d 6170 2069 7320 6269 7261  onal map is bira
│ │ │ │ -000780a0: 7469 6f6e 616c 0a20 202a 202a 6e6f 7465  tional.  * *note
│ │ │ │ -000780b0: 2069 734d 6f72 7068 6973 6d3a 2069 734d   isMorphism: isM
│ │ │ │ -000780c0: 6f72 7068 6973 6d2c 202d 2d20 7768 6574  orphism, -- whet
│ │ │ │ -000780d0: 6865 7220 6120 7261 7469 6f6e 616c 206d  her a rational m
│ │ │ │ -000780e0: 6170 2069 7320 6120 6d6f 7270 6869 736d  ap is a morphism
│ │ │ │ -000780f0: 0a0a 5761 7973 2074 6f20 7573 6520 7468  ..Ways to use th
│ │ │ │ -00078100: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d  is method:.=====
│ │ │ │ -00078110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00078120: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2069  ===..  * *note i
│ │ │ │ -00078130: 7349 736f 6d6f 7270 6869 736d 2852 6174  sIsomorphism(Rat
│ │ │ │ -00078140: 696f 6e61 6c4d 6170 293a 2069 7349 736f  ionalMap): isIso
│ │ │ │ -00078150: 6d6f 7270 6869 736d 5f6c 7052 6174 696f  morphism_lpRatio
│ │ │ │ -00078160: 6e61 6c4d 6170 5f72 702c 202d 2d0a 2020  nalMap_rp, --.  
│ │ │ │ -00078170: 2020 7768 6574 6865 7220 6120 6269 7261    whether a bira
│ │ │ │ -00078180: 7469 6f6e 616c 206d 6170 2069 7320 616e  tional map is an
│ │ │ │ -00078190: 2069 736f 6d6f 7270 6869 736d 0a2d 2d2d   isomorphism.---
│ │ │ │ +00077fe0: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d  ---+..See also.=
│ │ │ │ +00077ff0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │ +00078000: 7465 2069 6465 616c 2852 6174 696f 6e61  te ideal(Rationa
│ │ │ │ +00078010: 6c4d 6170 293a 2069 6465 616c 5f6c 7052  lMap): ideal_lpR
│ │ │ │ +00078020: 6174 696f 6e61 6c4d 6170 5f72 702c 202d  ationalMap_rp, -
│ │ │ │ +00078030: 2d20 6261 7365 206c 6f63 7573 206f 6620  - base locus of 
│ │ │ │ +00078040: 610a 2020 2020 7261 7469 6f6e 616c 206d  a.    rational m
│ │ │ │ +00078050: 6170 0a20 202a 202a 6e6f 7465 2069 7342  ap.  * *note isB
│ │ │ │ +00078060: 6972 6174 696f 6e61 6c3a 2069 7342 6972  irational: isBir
│ │ │ │ +00078070: 6174 696f 6e61 6c2c 202d 2d20 7768 6574  ational, -- whet
│ │ │ │ +00078080: 6865 7220 6120 7261 7469 6f6e 616c 206d  her a rational m
│ │ │ │ +00078090: 6170 2069 7320 6269 7261 7469 6f6e 616c  ap is birational
│ │ │ │ +000780a0: 0a20 202a 202a 6e6f 7465 2069 734d 6f72  .  * *note isMor
│ │ │ │ +000780b0: 7068 6973 6d3a 2069 734d 6f72 7068 6973  phism: isMorphis
│ │ │ │ +000780c0: 6d2c 202d 2d20 7768 6574 6865 7220 6120  m, -- whether a 
│ │ │ │ +000780d0: 7261 7469 6f6e 616c 206d 6170 2069 7320  rational map is 
│ │ │ │ +000780e0: 6120 6d6f 7270 6869 736d 0a0a 5761 7973  a morphism..Ways
│ │ │ │ +000780f0: 2074 6f20 7573 6520 7468 6973 206d 6574   to use this met
│ │ │ │ +00078100: 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  hod:.===========
│ │ │ │ +00078110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +00078120: 202a 202a 6e6f 7465 2069 7349 736f 6d6f   * *note isIsomo
│ │ │ │ +00078130: 7270 6869 736d 2852 6174 696f 6e61 6c4d  rphism(RationalM
│ │ │ │ +00078140: 6170 293a 2069 7349 736f 6d6f 7270 6869  ap): isIsomorphi
│ │ │ │ +00078150: 736d 5f6c 7052 6174 696f 6e61 6c4d 6170  sm_lpRationalMap
│ │ │ │ +00078160: 5f72 702c 202d 2d0a 2020 2020 7768 6574  _rp, --.    whet
│ │ │ │ +00078170: 6865 7220 6120 6269 7261 7469 6f6e 616c  her a birational
│ │ │ │ +00078180: 206d 6170 2069 7320 616e 2069 736f 6d6f   map is an isomo
│ │ │ │ +00078190: 7270 6869 736d 0a2d 2d2d 2d2d 2d2d 2d2d  rphism.---------
│ │ │ │  000781a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000781b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000781c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000781d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000781e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
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│ │ │ │ -00078230: 7932 2d31 2e32 352e 3131 2b64 732f 4d32  y2-1.25.11+ds/M2
│ │ │ │ -00078240: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ -00078250: 6765 732f 4372 656d 6f6e 612f 0a64 6f63  ges/Cremona/.doc
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│ │ │ │ -000782b0: 674d 6170 5f63 6d5a 5a5f 7270 2c20 5072  gMap_cmZZ_rp, Pr
│ │ │ │ -000782c0: 6576 3a20 6973 4973 6f6d 6f72 7068 6973  ev: isIsomorphis
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│ │ │ │ -000782e0: 7270 2c20 5570 3a20 546f 700a 0a69 734d  rp, Up: Top..isM
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│ │ │ │ +000781e0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
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│ │ │ │ +00078200: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ +00078210: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ +00078220: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ +00078230: 352e 3131 2b64 732f 4d32 2f4d 6163 6175  5.11+ds/M2/Macau
│ │ │ │ +00078240: 6c61 7932 2f70 6163 6b61 6765 732f 4372  lay2/packages/Cr
│ │ │ │ +00078250: 656d 6f6e 612f 0a64 6f63 756d 656e 7461  emona/.documenta
│ │ │ │ +00078260: 7469 6f6e 2e6d 323a 3131 3437 3a30 2e0a  tion.m2:1147:0..
│ │ │ │ +00078270: 1f0a 4669 6c65 3a20 4372 656d 6f6e 612e  ..File: Cremona.
│ │ │ │ +00078280: 696e 666f 2c20 4e6f 6465 3a20 6973 4d6f  info, Node: isMo
│ │ │ │ +00078290: 7270 6869 736d 2c20 4e65 7874 3a20 6b65  rphism, Next: ke
│ │ │ │ +000782a0: 726e 656c 5f6c 7052 696e 674d 6170 5f63  rnel_lpRingMap_c
│ │ │ │ +000782b0: 6d5a 5a5f 7270 2c20 5072 6576 3a20 6973  mZZ_rp, Prev: is
│ │ │ │ +000782c0: 4973 6f6d 6f72 7068 6973 6d5f 6c70 5261  Isomorphism_lpRa
│ │ │ │ +000782d0: 7469 6f6e 616c 4d61 705f 7270 2c20 5570  tionalMap_rp, Up
│ │ │ │ +000782e0: 3a20 546f 700a 0a69 734d 6f72 7068 6973  : Top..isMorphis
│ │ │ │ +000782f0: 6d20 2d2d 2077 6865 7468 6572 2061 2072  m -- whether a r
│ │ │ │ +00078300: 6174 696f 6e61 6c20 6d61 7020 6973 2061  ational map is a
│ │ │ │ +00078310: 206d 6f72 7068 6973 6d0a 2a2a 2a2a 2a2a   morphism.******
│ │ │ │  00078320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00078330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00078340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00078350: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a  **..  * Usage: .
│ │ │ │ -00078360: 2020 2020 2020 2020 6973 4d6f 7270 6869          isMorphi
│ │ │ │ -00078370: 736d 2070 6869 0a20 202a 2049 6e70 7574  sm phi.  * Input
│ │ │ │ -00078380: 733a 0a20 2020 2020 202a 2070 6869 2c20  s:.      * phi, 
│ │ │ │ -00078390: 6120 2a6e 6f74 6520 7261 7469 6f6e 616c  a *note rational
│ │ │ │ -000783a0: 206d 6170 3a20 5261 7469 6f6e 616c 4d61   map: RationalMa
│ │ │ │ -000783b0: 702c 0a20 202a 204f 7574 7075 7473 3a0a  p,.  * Outputs:.
│ │ │ │ -000783c0: 2020 2020 2020 2a20 6120 2a6e 6f74 6520        * a *note 
│ │ │ │ -000783d0: 426f 6f6c 6561 6e20 7661 6c75 653a 2028  Boolean value: (
│ │ │ │ -000783e0: 4d61 6361 756c 6179 3244 6f63 2942 6f6f  Macaulay2Doc)Boo
│ │ │ │ -000783f0: 6c65 616e 2c2c 2077 6865 7468 6572 2070  lean,, whether p
│ │ │ │ -00078400: 6869 2069 7320 610a 2020 2020 2020 2020  hi is a.        
│ │ │ │ -00078410: 6d6f 7270 6869 736d 2028 692e 652e 2c20  morphism (i.e., 
│ │ │ │ -00078420: 6576 6572 7977 6865 7265 2064 6566 696e  everywhere defin
│ │ │ │ -00078430: 6564 290a 0a44 6573 6372 6970 7469 6f6e  ed)..Description
│ │ │ │ -00078440: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d  .===========..+-
│ │ │ │ +00078340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020  ************..  
│ │ │ │ +00078350: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ +00078360: 2020 6973 4d6f 7270 6869 736d 2070 6869    isMorphism phi
│ │ │ │ +00078370: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +00078380: 2020 202a 2070 6869 2c20 6120 2a6e 6f74     * phi, a *not
│ │ │ │ +00078390: 6520 7261 7469 6f6e 616c 206d 6170 3a20  e rational map: 
│ │ │ │ +000783a0: 5261 7469 6f6e 616c 4d61 702c 0a20 202a  RationalMap,.  *
│ │ │ │ +000783b0: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +000783c0: 2a20 6120 2a6e 6f74 6520 426f 6f6c 6561  * a *note Boolea
│ │ │ │ +000783d0: 6e20 7661 6c75 653a 2028 4d61 6361 756c  n value: (Macaul
│ │ │ │ +000783e0: 6179 3244 6f63 2942 6f6f 6c65 616e 2c2c  ay2Doc)Boolean,,
│ │ │ │ +000783f0: 2077 6865 7468 6572 2070 6869 2069 7320   whether phi is 
│ │ │ │ +00078400: 610a 2020 2020 2020 2020 6d6f 7270 6869  a.        morphi
│ │ │ │ +00078410: 736d 2028 692e 652e 2c20 6576 6572 7977  sm (i.e., everyw
│ │ │ │ +00078420: 6865 7265 2064 6566 696e 6564 290a 0a44  here defined)..D
│ │ │ │ +00078430: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00078440: 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d  ======..+-------
│ │ │ │  00078450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00078460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00078470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00078480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00078490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000784a0: 3120 3a20 7068 6920 3d20 7175 6164 726f  1 : phi = quadro
│ │ │ │ -000784b0: 5175 6164 7269 6343 7265 6d6f 6e61 5472  QuadricCremonaTr
│ │ │ │ -000784c0: 616e 7366 6f72 6d61 7469 6f6e 2835 2c31  ansformation(5,1
│ │ │ │ -000784d0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -000784e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00078490: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7068  ------+.|i1 : ph
│ │ │ │ +000784a0: 6920 3d20 7175 6164 726f 5175 6164 7269  i = quadroQuadri
│ │ │ │ +000784b0: 6343 7265 6d6f 6e61 5472 616e 7366 6f72  cCremonaTransfor
│ │ │ │ +000784c0: 6d61 7469 6f6e 2835 2c31 2920 2020 2020  mation(5,1)     
│ │ │ │ +000784d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000784e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000784f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078530: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00078540: 3120 3d20 2d2d 2072 6174 696f 6e61 6c20  1 = -- rational 
│ │ │ │ -00078550: 6d61 7020 2d2d 2020 2020 2020 2020 2020  map --          
│ │ │ │ +00078530: 2020 2020 2020 7c0a 7c6f 3120 3d20 2d2d        |.|o1 = --
│ │ │ │ +00078540: 2072 6174 696f 6e61 6c20 6d61 7020 2d2d   rational map --
│ │ │ │ +00078550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078580: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00078590: 2020 2020 736f 7572 6365 3a20 5072 6f6a      source: Proj
│ │ │ │ -000785a0: 2851 515b 782c 2079 2c20 7a2c 2074 2c20  (QQ[x, y, z, t, 
│ │ │ │ -000785b0: 752c 2076 5d29 2020 2020 2020 2020 2020  u, v])          
│ │ │ │ +00078580: 2020 2020 2020 7c0a 7c20 2020 2020 736f        |.|     so
│ │ │ │ +00078590: 7572 6365 3a20 5072 6f6a 2851 515b 782c  urce: Proj(QQ[x,
│ │ │ │ +000785a0: 2079 2c20 7a2c 2074 2c20 752c 2076 5d29   y, z, t, u, v])
│ │ │ │ +000785b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000785c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000785d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000785e0: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
│ │ │ │ -000785f0: 2851 515b 782c 2079 2c20 7a2c 2074 2c20  (QQ[x, y, z, t, 
│ │ │ │ -00078600: 752c 2076 5d29 2020 2020 2020 2020 2020  u, v])          
│ │ │ │ +000785d0: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ +000785e0: 7267 6574 3a20 5072 6f6a 2851 515b 782c  rget: Proj(QQ[x,
│ │ │ │ +000785f0: 2079 2c20 7a2c 2074 2c20 752c 2076 5d29   y, z, t, u, v])
│ │ │ │ +00078600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078620: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00078630: 2020 2020 6465 6669 6e69 6e67 2066 6f72      defining for
│ │ │ │ -00078640: 6d73 3a20 7b20 2020 2020 2020 2020 2020  ms: {           
│ │ │ │ +00078620: 2020 2020 2020 7c0a 7c20 2020 2020 6465        |.|     de
│ │ │ │ +00078630: 6669 6e69 6e67 2066 6f72 6d73 3a20 7b20  fining forms: { 
│ │ │ │ +00078640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078670: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00078670: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00078680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078690: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00078690: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  000786a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000786b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000786c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000786d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000786e0: 2020 2020 2079 2a7a 202d 2076 202c 2020       y*z - v ,  
│ │ │ │ +000786c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000786d0: 2020 2020 2020 2020 2020 2020 2020 2079                 y
│ │ │ │ +000786e0: 2a7a 202d 2076 202c 2020 2020 2020 2020  *z - v ,        
│ │ │ │  000786f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078710: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00078710: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00078720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078760: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00078760: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00078770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078780: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00078780: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00078790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000787a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000787b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000787c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000787d0: 2020 2020 2078 2a7a 202d 2075 202c 2020       x*z - u ,  
│ │ │ │ +000787b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000787c0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +000787d0: 2a7a 202d 2075 202c 2020 2020 2020 2020  *z - u ,        
│ │ │ │  000787e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000787f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078800: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00078800: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00078810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078850: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00078850: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00078860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078870: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00078870: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00078880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000788a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000788b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000788c0: 2020 2020 2078 2a79 202d 2074 202c 2020       x*y - t ,  
│ │ │ │ +000788a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000788b0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +000788c0: 2a79 202d 2074 202c 2020 2020 2020 2020  *y - t ,        
│ │ │ │  000788d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000788e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000788f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000788f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00078900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078940: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00078950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078960: 2020 2020 202d 207a 2a74 202b 2075 2a76       - z*t + u*v
│ │ │ │ -00078970: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00078940: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00078950: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ +00078960: 207a 2a74 202b 2075 2a76 2c20 2020 2020   z*t + u*v,     
│ │ │ │ +00078970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078990: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00078990: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000789a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000789b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000789c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000789d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000789e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000789f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078a00: 2020 2020 202d 2079 2a75 202b 2074 2a76       - y*u + t*v
│ │ │ │ -00078a10: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +000789e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000789f0: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ +00078a00: 2079 2a75 202b 2074 2a76 2c20 2020 2020   y*u + t*v,     
│ │ │ │ +00078a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078a30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00078a30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00078a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078a80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00078a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078aa0: 2020 2020 2074 2a75 202d 2078 2a76 2020       t*u - x*v  
│ │ │ │ +00078a80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00078a90: 2020 2020 2020 2020 2020 2020 2020 2074                 t
│ │ │ │ +00078aa0: 2a75 202d 2078 2a76 2020 2020 2020 2020  *u - x*v        
│ │ │ │  00078ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078ad0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00078ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078af0: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │ +00078ad0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00078ae0: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
│ │ │ │ +00078af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078b20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00078b20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00078b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078b70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00078b80: 3120 3a20 5261 7469 6f6e 616c 4d61 7020  1 : RationalMap 
│ │ │ │ -00078b90: 2843 7265 6d6f 6e61 2074 7261 6e73 666f  (Cremona transfo
│ │ │ │ -00078ba0: 726d 6174 696f 6e20 6f66 2050 505e 3520  rmation of PP^5 
│ │ │ │ -00078bb0: 6f66 2074 7970 6520 2832 2c32 2929 2020  of type (2,2))  
│ │ │ │ -00078bc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00078b70: 2020 2020 2020 7c0a 7c6f 3120 3a20 5261        |.|o1 : Ra
│ │ │ │ +00078b80: 7469 6f6e 616c 4d61 7020 2843 7265 6d6f  tionalMap (Cremo
│ │ │ │ +00078b90: 6e61 2074 7261 6e73 666f 726d 6174 696f  na transformatio
│ │ │ │ +00078ba0: 6e20 6f66 2050 505e 3520 6f66 2074 7970  n of PP^5 of typ
│ │ │ │ +00078bb0: 6520 2832 2c32 2929 2020 2020 2020 2020  e (2,2))        
│ │ │ │ +00078bc0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  00078bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00078be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00078bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00078c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00078c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00078c20: 3220 3a20 6973 4d6f 7270 6869 736d 2070  2 : isMorphism p
│ │ │ │ -00078c30: 6869 2020 2020 2020 2020 2020 2020 2020  hi              
│ │ │ │ +00078c10: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 6973  ------+.|i2 : is
│ │ │ │ +00078c20: 4d6f 7270 6869 736d 2070 6869 2020 2020  Morphism phi    
│ │ │ │ +00078c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078c60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00078c60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00078c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078cb0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00078cc0: 3220 3d20 6661 6c73 6520 2020 2020 2020  2 = false       
│ │ │ │ +00078cb0: 2020 2020 2020 7c0a 7c6f 3220 3d20 6661        |.|o2 = fa
│ │ │ │ +00078cc0: 6c73 6520 2020 2020 2020 2020 2020 2020  lse             
│ │ │ │  00078cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078d00: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
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│ │ │ │ +00078d70: 7068 693b 2020 2020 2020 2020 2020 2020  phi;            
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│ │ │ │  00078d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  00078db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00078de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00078e00: 6c74 6968 6f6d 6f67 656e 656f 7573 5261  ltihomogeneousRa
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│ │ │ │  00078ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078ee0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
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│ │ │ │  00078ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00078f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00078f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00078f40: 3420 3a20 6973 4d6f 7270 6869 736d 2070  4 : isMorphism p
│ │ │ │ -00078f50: 6869 2720 2020 2020 2020 2020 2020 2020  hi'             
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│ │ │ │ +00078f40: 4d6f 7270 6869 736d 2070 6869 2720 2020  Morphism phi'   
│ │ │ │ +00078f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00078f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00078f80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │ -00078fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00078fe0: 3420 3d20 7472 7565 2020 2020 2020 2020  4 = true        
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│ │ │ │ +00078fe0: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
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│ │ │ │  00079000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00079010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00079020: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
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│ │ │ │  00079030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00079040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00079050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00079060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00079070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ -00079080: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ -00079090: 0a0a 2020 2a20 2a6e 6f74 6520 6964 6561  ..  * *note idea
│ │ │ │ -000790a0: 6c28 5261 7469 6f6e 616c 4d61 7029 3a20  l(RationalMap): 
│ │ │ │ -000790b0: 6964 6561 6c5f 6c70 5261 7469 6f6e 616c  ideal_lpRational
│ │ │ │ -000790c0: 4d61 705f 7270 2c20 2d2d 2062 6173 6520  Map_rp, -- base 
│ │ │ │ -000790d0: 6c6f 6375 7320 6f66 2061 0a20 2020 2072  locus of a.    r
│ │ │ │ -000790e0: 6174 696f 6e61 6c20 6d61 700a 2020 2a20  ational map.  * 
│ │ │ │ -000790f0: 2a6e 6f74 6520 6973 4973 6f6d 6f72 7068  *note isIsomorph
│ │ │ │ -00079100: 6973 6d28 5261 7469 6f6e 616c 4d61 7029  ism(RationalMap)
│ │ │ │ -00079110: 3a20 6973 4973 6f6d 6f72 7068 6973 6d5f  : isIsomorphism_
│ │ │ │ -00079120: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ -00079130: 2c20 2d2d 0a20 2020 2077 6865 7468 6572  , --.    whether
│ │ │ │ -00079140: 2061 2062 6972 6174 696f 6e61 6c20 6d61   a birational ma
│ │ │ │ -00079150: 7020 6973 2061 6e20 6973 6f6d 6f72 7068  p is an isomorph
│ │ │ │ -00079160: 6973 6d0a 0a57 6179 7320 746f 2075 7365  ism..Ways to use
│ │ │ │ -00079170: 2069 734d 6f72 7068 6973 6d3a 0a3d 3d3d   isMorphism:.===
│ │ │ │ -00079180: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00079190: 3d3d 3d3d 0a0a 2020 2a20 2269 734d 6f72  ====..  * "isMor
│ │ │ │ -000791a0: 7068 6973 6d28 5261 7469 6f6e 616c 4d61  phism(RationalMa
│ │ │ │ -000791b0: 7029 220a 0a46 6f72 2074 6865 2070 726f  p)"..For the pro
│ │ │ │ -000791c0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ -000791d0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ -000791e0: 6f62 6a65 6374 202a 6e6f 7465 2069 734d  object *note isM
│ │ │ │ -000791f0: 6f72 7068 6973 6d3a 2069 734d 6f72 7068  orphism: isMorph
│ │ │ │ -00079200: 6973 6d2c 2069 7320 6120 2a6e 6f74 6520  ism, is a *note 
│ │ │ │ -00079210: 6d65 7468 6f64 2066 756e 6374 696f 6e3a  method function:
│ │ │ │ -00079220: 0a28 4d61 6361 756c 6179 3244 6f63 294d  .(Macaulay2Doc)M
│ │ │ │ -00079230: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ -00079240: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │ +00079070: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73  ------+..See als
│ │ │ │ +00079080: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ +00079090: 2a6e 6f74 6520 6964 6561 6c28 5261 7469  *note ideal(Rati
│ │ │ │ +000790a0: 6f6e 616c 4d61 7029 3a20 6964 6561 6c5f  onalMap): ideal_
│ │ │ │ +000790b0: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ +000790c0: 2c20 2d2d 2062 6173 6520 6c6f 6375 7320  , -- base locus 
│ │ │ │ +000790d0: 6f66 2061 0a20 2020 2072 6174 696f 6e61  of a.    rationa
│ │ │ │ +000790e0: 6c20 6d61 700a 2020 2a20 2a6e 6f74 6520  l map.  * *note 
│ │ │ │ +000790f0: 6973 4973 6f6d 6f72 7068 6973 6d28 5261  isIsomorphism(Ra
│ │ │ │ +00079100: 7469 6f6e 616c 4d61 7029 3a20 6973 4973  tionalMap): isIs
│ │ │ │ +00079110: 6f6d 6f72 7068 6973 6d5f 6c70 5261 7469  omorphism_lpRati
│ │ │ │ +00079120: 6f6e 616c 4d61 705f 7270 2c20 2d2d 0a20  onalMap_rp, --. 
│ │ │ │ +00079130: 2020 2077 6865 7468 6572 2061 2062 6972     whether a bir
│ │ │ │ +00079140: 6174 696f 6e61 6c20 6d61 7020 6973 2061  ational map is a
│ │ │ │ +00079150: 6e20 6973 6f6d 6f72 7068 6973 6d0a 0a57  n isomorphism..W
│ │ │ │ +00079160: 6179 7320 746f 2075 7365 2069 734d 6f72  ays to use isMor
│ │ │ │ +00079170: 7068 6973 6d3a 0a3d 3d3d 3d3d 3d3d 3d3d  phism:.=========
│ │ │ │ +00079180: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00079190: 2020 2a20 2269 734d 6f72 7068 6973 6d28    * "isMorphism(
│ │ │ │ +000791a0: 5261 7469 6f6e 616c 4d61 7029 220a 0a46  RationalMap)"..F
│ │ │ │ +000791b0: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ +000791c0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ +000791d0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ +000791e0: 202a 6e6f 7465 2069 734d 6f72 7068 6973   *note isMorphis
│ │ │ │ +000791f0: 6d3a 2069 734d 6f72 7068 6973 6d2c 2069  m: isMorphism, i
│ │ │ │ +00079200: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64  s a *note method
│ │ │ │ +00079210: 2066 756e 6374 696f 6e3a 0a28 4d61 6361   function:.(Maca
│ │ │ │ +00079220: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
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│ │ │ │ -00079350: 4e65 7874 3a20 6d61 705f 6c70 5261 7469  Next: map_lpRati
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│ │ │ │ -00079370: 3a20 6973 4d6f 7270 6869 736d 2c20 5570  : isMorphism, Up
│ │ │ │ -00079380: 3a20 546f 700a 0a6b 6572 6e65 6c28 5269  : Top..kernel(Ri
│ │ │ │ -00079390: 6e67 4d61 702c 5a5a 2920 2d2d 2068 6f6d  ngMap,ZZ) -- hom
│ │ │ │ -000793a0: 6f67 656e 656f 7573 2063 6f6d 706f 6e65  ogeneous compone
│ │ │ │ -000793b0: 6e74 7320 6f66 2074 6865 206b 6572 6e65  nts of the kerne
│ │ │ │ -000793c0: 6c20 6f66 2061 2068 6f6d 6f67 656e 656f  l of a homogeneo
│ │ │ │ -000793d0: 7573 2072 696e 6720 6d61 700a 2a2a 2a2a  us ring map.****
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│ │ │ │ +00079300: 656e 7461 7469 6f6e 2e6d 323a 3131 3334  entation.m2:1134
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│ │ │ │ +00079320: 6f6e 612e 696e 666f 2c20 4e6f 6465 3a20  ona.info, Node: 
│ │ │ │ +00079330: 6b65 726e 656c 5f6c 7052 696e 674d 6170  kernel_lpRingMap
│ │ │ │ +00079340: 5f63 6d5a 5a5f 7270 2c20 4e65 7874 3a20  _cmZZ_rp, Next: 
│ │ │ │ +00079350: 6d61 705f 6c70 5261 7469 6f6e 616c 4d61  map_lpRationalMa
│ │ │ │ +00079360: 705f 7270 2c20 5072 6576 3a20 6973 4d6f  p_rp, Prev: isMo
│ │ │ │ +00079370: 7270 6869 736d 2c20 5570 3a20 546f 700a  rphism, Up: Top.
│ │ │ │ +00079380: 0a6b 6572 6e65 6c28 5269 6e67 4d61 702c  .kernel(RingMap,
│ │ │ │ +00079390: 5a5a 2920 2d2d 2068 6f6d 6f67 656e 656f  ZZ) -- homogeneo
│ │ │ │ +000793a0: 7573 2063 6f6d 706f 6e65 6e74 7320 6f66  us components of
│ │ │ │ +000793b0: 2074 6865 206b 6572 6e65 6c20 6f66 2061   the kernel of a
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│ │ │ │ +000793d0: 6720 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a  g map.**********
│ │ │ │  000793e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000793f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00079400: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00079410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00079420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00079430: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20  ..  * Function: 
│ │ │ │ -00079440: 2a6e 6f74 6520 6b65 726e 656c 3a20 284d  *note kernel: (M
│ │ │ │ -00079450: 6163 6175 6c61 7932 446f 6329 6b65 726e  acaulay2Doc)kern
│ │ │ │ -00079460: 656c 2c0a 2020 2a20 5573 6167 653a 200a  el,.  * Usage: .
│ │ │ │ -00079470: 2020 2020 2020 2020 6b65 726e 656c 2870          kernel(p
│ │ │ │ -00079480: 6869 2c64 290a 2020 2a20 496e 7075 7473  hi,d).  * Inputs
│ │ │ │ -00079490: 3a0a 2020 2020 2020 2a20 7068 692c 2061  :.      * phi, a
│ │ │ │ -000794a0: 202a 6e6f 7465 2072 696e 6720 6d61 703a   *note ring map:
│ │ │ │ -000794b0: 2028 4d61 6361 756c 6179 3244 6f63 2952   (Macaulay2Doc)R
│ │ │ │ -000794c0: 696e 674d 6170 2c2c 2024 4b5b 795f 302c  ingMap,, $K[y_0,
│ │ │ │ -000794d0: 5c6c 646f 7473 2c79 5f6d 5d2f 4a20 5c74  \ldots,y_m]/J \t
│ │ │ │ -000794e0: 6f0a 2020 2020 2020 2020 4b5b 785f 302c  o.        K[x_0,
│ │ │ │ -000794f0: 5c6c 646f 7473 2c78 5f6e 5d2f 4924 2c20  \ldots,x_n]/I$, 
│ │ │ │ -00079500: 6465 6669 6e65 6420 6279 2068 6f6d 6f67  defined by homog
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│ │ │ │ -00079520: 7468 6520 7361 6d65 2064 6567 7265 650a  the same degree.
│ │ │ │ -00079530: 2020 2020 2020 2020 616e 6420 7768 6572          and wher
│ │ │ │ -00079540: 6520 244a 2420 616e 6420 2449 2420 6172  e $J$ and $I$ ar
│ │ │ │ -00079550: 6520 686f 6d6f 6765 6e65 6f75 7320 6964  e homogeneous id
│ │ │ │ -00079560: 6561 6c73 0a20 2020 2020 202a 2064 2c20  eals.      * d, 
│ │ │ │ -00079570: 616e 202a 6e6f 7465 2069 6e74 6567 6572  an *note integer
│ │ │ │ -00079580: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00079590: 5a5a 2c0a 2020 2a20 2a6e 6f74 6520 4f70  ZZ,.  * *note Op
│ │ │ │ -000795a0: 7469 6f6e 616c 2069 6e70 7574 733a 2028  tional inputs: (
│ │ │ │ -000795b0: 4d61 6361 756c 6179 3244 6f63 2975 7369  Macaulay2Doc)usi
│ │ │ │ -000795c0: 6e67 2066 756e 6374 696f 6e73 2077 6974  ng functions wit
│ │ │ │ -000795d0: 6820 6f70 7469 6f6e 616c 2069 6e70 7574  h optional input
│ │ │ │ -000795e0: 732c 3a0a 2020 2020 2020 2a20 4465 6772  s,:.      * Degr
│ │ │ │ -000795f0: 6565 4c69 6d69 7420 286d 6973 7369 6e67  eeLimit (missing
│ │ │ │ -00079600: 2064 6f63 756d 656e 7461 7469 6f6e 2920   documentation) 
│ │ │ │ -00079610: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ -00079620: 7661 6c75 6520 7b7d 2c20 0a20 2020 2020  value {}, .     
│ │ │ │ -00079630: 202a 2053 7472 6174 6567 7920 286d 6973   * Strategy (mis
│ │ │ │ -00079640: 7369 6e67 2064 6f63 756d 656e 7461 7469  sing documentati
│ │ │ │ -00079650: 6f6e 2920 3d3e 202e 2e2e 2c20 6465 6661  on) => ..., defa
│ │ │ │ -00079660: 756c 7420 7661 6c75 6520 7b7d 2c20 0a20  ult value {}, . 
│ │ │ │ -00079670: 2020 2020 202a 202a 6e6f 7465 2053 7562       * *note Sub
│ │ │ │ -00079680: 7269 6e67 4c69 6d69 743a 2028 4d61 6361  ringLimit: (Maca
│ │ │ │ -00079690: 756c 6179 3244 6f63 296b 6572 6e65 6c5f  ulay2Doc)kernel_
│ │ │ │ -000796a0: 6c70 5f70 645f 7064 5f70 645f 636d 5375  lp_pd_pd_pd_cmSu
│ │ │ │ -000796b0: 6272 696e 674c 696d 6974 3d3e 5f0a 2020  bringLimit=>_.  
│ │ │ │ -000796c0: 2020 2020 2020 7064 5f70 645f 7064 5f72        pd_pd_pd_r
│ │ │ │ -000796d0: 702c 203d 3e20 2e2e 2e2c 2064 6566 6175  p, => ..., defau
│ │ │ │ -000796e0: 6c74 2076 616c 7565 2069 6e66 696e 6974  lt value infinit
│ │ │ │ -000796f0: 792c 0a20 202a 204f 7574 7075 7473 3a0a  y,.  * Outputs:.
│ │ │ │ -00079700: 2020 2020 2020 2a20 7468 6520 2a6e 6f74        * the *not
│ │ │ │ -00079710: 6520 6964 6561 6c3a 2028 4d61 6361 756c  e ideal: (Macaul
│ │ │ │ -00079720: 6179 3244 6f63 2949 6465 616c 2c20 6765  ay2Doc)Ideal, ge
│ │ │ │ -00079730: 6e65 7261 7465 6420 6279 2061 6c6c 2068  nerated by all h
│ │ │ │ -00079740: 6f6d 6f67 656e 656f 7573 0a20 2020 2020  omogeneous.     
│ │ │ │ -00079750: 2020 2065 6c65 6d65 6e74 7320 6f66 2064     elements of d
│ │ │ │ -00079760: 6567 7265 6520 6420 6265 6c6f 6e67 696e  egree d belongin
│ │ │ │ -00079770: 6720 746f 2074 6865 206b 6572 6e65 6c20  g to the kernel 
│ │ │ │ -00079780: 6f66 2070 6869 0a0a 4465 7363 7269 7074  of phi..Descript
│ │ │ │ -00079790: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ -000797a0: 0a54 6869 7320 6973 2065 7175 6976 616c  .This is equival
│ │ │ │ -000797b0: 656e 7420 746f 2069 6465 616c 2069 6d61  ent to ideal ima
│ │ │ │ -000797c0: 6765 2062 6173 6973 2864 2c6b 6572 6e65  ge basis(d,kerne
│ │ │ │ -000797d0: 6c20 7068 6929 2c20 6275 7420 7765 2075  l phi), but we u
│ │ │ │ -000797e0: 7365 2061 206d 6f72 6520 6469 7265 6374  se a more direct
│ │ │ │ -000797f0: 0a61 6c67 6f72 6974 686d 2e20 5765 2074  .algorithm. We t
│ │ │ │ -00079800: 616b 6520 6164 7661 6e74 6167 6520 6f66  ake advantage of
│ │ │ │ -00079810: 2074 6865 2068 6f6d 6f67 656e 6569 7479   the homogeneity
│ │ │ │ -00079820: 2061 6e64 2072 6564 7563 6520 7468 6520   and reduce the 
│ │ │ │ -00079830: 7072 6f62 6c65 6d20 746f 0a6c 696e 6561  problem to.linea
│ │ │ │ -00079840: 7220 616c 6765 6272 612e 2046 6f72 2073  r algebra. For s
│ │ │ │ -00079850: 6d61 6c6c 2076 616c 7565 7320 6f66 2064  mall values of d
│ │ │ │ -00079860: 2074 6869 7320 6d65 7468 6f64 2063 616e   this method can
│ │ │ │ -00079870: 2062 6520 7665 7279 2066 6173 742c 2061   be very fast, a
│ │ │ │ -00079880: 7320 7468 650a 666f 6c6c 6f77 696e 6720  s the.following 
│ │ │ │ -00079890: 6578 616d 706c 6520 7368 6f77 732e 0a0a  example shows...
│ │ │ │ -000798a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00079420: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +00079430: 4675 6e63 7469 6f6e 3a20 2a6e 6f74 6520  Function: *note 
│ │ │ │ +00079440: 6b65 726e 656c 3a20 284d 6163 6175 6c61  kernel: (Macaula
│ │ │ │ +00079450: 7932 446f 6329 6b65 726e 656c 2c0a 2020  y2Doc)kernel,.  
│ │ │ │ +00079460: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ +00079470: 2020 6b65 726e 656c 2870 6869 2c64 290a    kernel(phi,d).
│ │ │ │ +00079480: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +00079490: 2020 2a20 7068 692c 2061 202a 6e6f 7465    * phi, a *note
│ │ │ │ +000794a0: 2072 696e 6720 6d61 703a 2028 4d61 6361   ring map: (Maca
│ │ │ │ +000794b0: 756c 6179 3244 6f63 2952 696e 674d 6170  ulay2Doc)RingMap
│ │ │ │ +000794c0: 2c2c 2024 4b5b 795f 302c 5c6c 646f 7473  ,, $K[y_0,\ldots
│ │ │ │ +000794d0: 2c79 5f6d 5d2f 4a20 5c74 6f0a 2020 2020  ,y_m]/J \to.    
│ │ │ │ +000794e0: 2020 2020 4b5b 785f 302c 5c6c 646f 7473      K[x_0,\ldots
│ │ │ │ +000794f0: 2c78 5f6e 5d2f 4924 2c20 6465 6669 6e65  ,x_n]/I$, define
│ │ │ │ +00079500: 6420 6279 2068 6f6d 6f67 656e 656f 7573  d by homogeneous
│ │ │ │ +00079510: 2066 6f72 6d73 206f 6620 7468 6520 7361   forms of the sa
│ │ │ │ +00079520: 6d65 2064 6567 7265 650a 2020 2020 2020  me degree.      
│ │ │ │ +00079530: 2020 616e 6420 7768 6572 6520 244a 2420    and where $J$ 
│ │ │ │ +00079540: 616e 6420 2449 2420 6172 6520 686f 6d6f  and $I$ are homo
│ │ │ │ +00079550: 6765 6e65 6f75 7320 6964 6561 6c73 0a20  geneous ideals. 
│ │ │ │ +00079560: 2020 2020 202a 2064 2c20 616e 202a 6e6f       * d, an *no
│ │ │ │ +00079570: 7465 2069 6e74 6567 6572 3a20 284d 6163  te integer: (Mac
│ │ │ │ +00079580: 6175 6c61 7932 446f 6329 5a5a 2c0a 2020  aulay2Doc)ZZ,.  
│ │ │ │ +00079590: 2a20 2a6e 6f74 6520 4f70 7469 6f6e 616c  * *note Optional
│ │ │ │ +000795a0: 2069 6e70 7574 733a 2028 4d61 6361 756c   inputs: (Macaul
│ │ │ │ +000795b0: 6179 3244 6f63 2975 7369 6e67 2066 756e  ay2Doc)using fun
│ │ │ │ +000795c0: 6374 696f 6e73 2077 6974 6820 6f70 7469  ctions with opti
│ │ │ │ +000795d0: 6f6e 616c 2069 6e70 7574 732c 3a0a 2020  onal inputs,:.  
│ │ │ │ +000795e0: 2020 2020 2a20 4465 6772 6565 4c69 6d69      * DegreeLimi
│ │ │ │ +000795f0: 7420 286d 6973 7369 6e67 2064 6f63 756d  t (missing docum
│ │ │ │ +00079600: 656e 7461 7469 6f6e 2920 3d3e 202e 2e2e  entation) => ...
│ │ │ │ +00079610: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +00079620: 7b7d 2c20 0a20 2020 2020 202a 2053 7472  {}, .      * Str
│ │ │ │ +00079630: 6174 6567 7920 286d 6973 7369 6e67 2064  ategy (missing d
│ │ │ │ +00079640: 6f63 756d 656e 7461 7469 6f6e 2920 3d3e  ocumentation) =>
│ │ │ │ +00079650: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ +00079660: 6c75 6520 7b7d 2c20 0a20 2020 2020 202a  lue {}, .      *
│ │ │ │ +00079670: 202a 6e6f 7465 2053 7562 7269 6e67 4c69   *note SubringLi
│ │ │ │ +00079680: 6d69 743a 2028 4d61 6361 756c 6179 3244  mit: (Macaulay2D
│ │ │ │ +00079690: 6f63 296b 6572 6e65 6c5f 6c70 5f70 645f  oc)kernel_lp_pd_
│ │ │ │ +000796a0: 7064 5f70 645f 636d 5375 6272 696e 674c  pd_pd_cmSubringL
│ │ │ │ +000796b0: 696d 6974 3d3e 5f0a 2020 2020 2020 2020  imit=>_.        
│ │ │ │ +000796c0: 7064 5f70 645f 7064 5f72 702c 203d 3e20  pd_pd_pd_rp, => 
│ │ │ │ +000796d0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ +000796e0: 7565 2069 6e66 696e 6974 792c 0a20 202a  ue infinity,.  *
│ │ │ │ +000796f0: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +00079700: 2a20 7468 6520 2a6e 6f74 6520 6964 6561  * the *note idea
│ │ │ │ +00079710: 6c3a 2028 4d61 6361 756c 6179 3244 6f63  l: (Macaulay2Doc
│ │ │ │ +00079720: 2949 6465 616c 2c20 6765 6e65 7261 7465  )Ideal, generate
│ │ │ │ +00079730: 6420 6279 2061 6c6c 2068 6f6d 6f67 656e  d by all homogen
│ │ │ │ +00079740: 656f 7573 0a20 2020 2020 2020 2065 6c65  eous.        ele
│ │ │ │ +00079750: 6d65 6e74 7320 6f66 2064 6567 7265 6520  ments of degree 
│ │ │ │ +00079760: 6420 6265 6c6f 6e67 696e 6720 746f 2074  d belonging to t
│ │ │ │ +00079770: 6865 206b 6572 6e65 6c20 6f66 2070 6869  he kernel of phi
│ │ │ │ +00079780: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +00079790: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320  =========..This 
│ │ │ │ +000797a0: 6973 2065 7175 6976 616c 656e 7420 746f  is equivalent to
│ │ │ │ +000797b0: 2069 6465 616c 2069 6d61 6765 2062 6173   ideal image bas
│ │ │ │ +000797c0: 6973 2864 2c6b 6572 6e65 6c20 7068 6929  is(d,kernel phi)
│ │ │ │ +000797d0: 2c20 6275 7420 7765 2075 7365 2061 206d  , but we use a m
│ │ │ │ +000797e0: 6f72 6520 6469 7265 6374 0a61 6c67 6f72  ore direct.algor
│ │ │ │ +000797f0: 6974 686d 2e20 5765 2074 616b 6520 6164  ithm. We take ad
│ │ │ │ +00079800: 7661 6e74 6167 6520 6f66 2074 6865 2068  vantage of the h
│ │ │ │ +00079810: 6f6d 6f67 656e 6569 7479 2061 6e64 2072  omogeneity and r
│ │ │ │ +00079820: 6564 7563 6520 7468 6520 7072 6f62 6c65  educe the proble
│ │ │ │ +00079830: 6d20 746f 0a6c 696e 6561 7220 616c 6765  m to.linear alge
│ │ │ │ +00079840: 6272 612e 2046 6f72 2073 6d61 6c6c 2076  bra. For small v
│ │ │ │ +00079850: 616c 7565 7320 6f66 2064 2074 6869 7320  alues of d this 
│ │ │ │ +00079860: 6d65 7468 6f64 2063 616e 2062 6520 7665  method can be ve
│ │ │ │ +00079870: 7279 2066 6173 742c 2061 7320 7468 650a  ry fast, as the.
│ │ │ │ +00079880: 666f 6c6c 6f77 696e 6720 6578 616d 706c  following exampl
│ │ │ │ +00079890: 6520 7368 6f77 732e 0a0a 2b2d 2d2d 2d2d  e shows...+-----
│ │ │ │ +000798a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000798b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000798c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000798d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000798e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000798f0: 7c69 3120 3a20 7068 6920 3d20 746f 4d61  |i1 : phi = toMa
│ │ │ │ -00079900: 7020 6d61 7020 7370 6563 6961 6c51 7561  p map specialQua
│ │ │ │ -00079910: 6472 6174 6963 5472 616e 7366 6f72 6d61  draticTransforma
│ │ │ │ -00079920: 7469 6f6e 2038 2020 2020 2020 2020 2020  tion 8          
│ │ │ │ -00079930: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00079940: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000798e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +000798f0: 7068 6920 3d20 746f 4d61 7020 6d61 7020  phi = toMap map 
│ │ │ │ +00079900: 7370 6563 6961 6c51 7561 6472 6174 6963  specialQuadratic
│ │ │ │ +00079910: 5472 616e 7366 6f72 6d61 7469 6f6e 2038  Transformation 8
│ │ │ │ +00079920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00079930: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00079940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00079950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00079960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00079970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00079980: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00079990: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00079980: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00079990: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000799c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000799d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000799e0: 7c6f 3120 3d20 6d61 7020 2851 515b 7820  |o1 = map (QQ[x 
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│ │ │ │ -00079a50: 3120 2020 2020 2020 2030 2033 2020 2020  1        0 3    
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│ │ │ │ -00079ad0: 7c6f 3120 3a20 5269 6e67 4d61 7020 5151  |o1 : RingMap QQ
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│ │ │ │ -00079c60: 7c20 3320 3520 2020 2034 2035 2020 2020  | 3 5    4 5    
│ │ │ │ -00079c70: 2020 3520 2020 2020 3020 3620 2020 2020    5     0 6     
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│ │ │ │ -00079c90: 3520 3620 2020 2020 3620 2020 2034 2037  5 6     6    4 7
│ │ │ │ -00079ca0: 2020 2020 2035 2037 2020 2020 2020 7c0a       5 7      |.
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│ │ │ │ +00079c00: 2020 2020 2020 2020 7c0a 7c78 2078 2020          |.|x x  
│ │ │ │ +00079c10: 2d20 7820 7820 202d 2034 3978 2020 2d20  - x x  - 49x  - 
│ │ │ │ +00079c20: 3578 2078 2020 2b20 3278 2078 2020 2d20  5x x  + 2x x  - 
│ │ │ │ +00079c30: 7820 7820 202b 2032 3778 2078 2020 2d20  x x  + 27x x  - 
│ │ │ │ +00079c40: 3478 2020 2b20 7820 7820 202d 2037 7820  4x  + x x  - 7x 
│ │ │ │ +00079c50: 7820 202b 2020 2020 7c0a 7c20 3320 3520  x  +    |.| 3 5 
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│ │ │ │ +00079cf0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
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│ │ │ │ -00079d40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ -00079d90: 2b20 7820 7820 202b 2078 2078 2020 7c0a  + x x  + x x  |.
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│ │ │ │ -00079db0: 2020 2020 3520 3820 2020 2020 3620 3820      5 8     6 8 
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│ │ │ │ -00079dd0: 2020 2020 3020 3620 2020 2020 3120 3620      0 6     1 6 
│ │ │ │ -00079de0: 2020 2034 2036 2020 2020 3520 3620 7c0a     4 6    5 6 |.
│ │ │ │ -00079df0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00079d40: 2020 2020 2020 2020 7c0a 7c32 7820 7820          |.|2x x 
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│ │ │ │ +00079de0: 2020 2020 3520 3620 7c0a 7c2d 2d2d 2d2d      5 6 |.|-----
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│ │ │ │ -00079e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
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│ │ │ │ -00079e80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ -00079ed0: 2b20 3978 2078 2020 2b20 2020 2020 7c0a  + 9x x  +     |.
│ │ │ │ -00079ee0: 7c20 2020 2030 2037 2020 2020 3120 3720  |    0 7    1 7 
│ │ │ │ -00079ef0: 2020 2020 3220 3720 2020 2020 3520 3720      2 7     5 7 
│ │ │ │ -00079f00: 2020 2020 3620 3720 2020 2020 3120 3820      6 7     1 8 
│ │ │ │ -00079f10: 2020 2020 3720 3820 2020 2020 2020 3020      7 8       0 
│ │ │ │ -00079f20: 2020 2020 3020 3220 2020 2020 2020 7c0a      0 2       |.
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│ │ │ │ -00079f80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ -00079fd0: 7c31 3078 2078 2020 2d20 3278 2078 2020  |10x x  - 2x x  
│ │ │ │ -00079fe0: 2d20 7820 202b 2032 3978 2078 2020 2d20  - x  + 29x x  - 
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│ │ │ │ +00079fc0: 2020 2020 2020 2020 7c0a 7c31 3078 2078          |.|10x x
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│ │ │ │ +0007a010: 2078 2078 2020 2b20 7c0a 7c20 2020 3020   x x  + |.|   0 
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│ │ │ │ -0007a150: 7820 202b 2078 2020 2b20 2020 2020 7c0a  x  + x  +     |.
│ │ │ │ -0007a160: 7c20 2032 2037 2020 2020 3420 3720 2020  |  2 7    4 7   
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│ │ │ │ -0007a1a0: 2034 2020 2020 3420 2020 2020 2020 7c0a   4    4       |.
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│ │ │ │ +0007a100: 3220 2020 2020 2020 7c0a 7c32 7820 7820  2       |.|2x x 
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│ │ │ │ -0007a210: 2020 2020 2020 2020 2020 2020 3220 2020              2   
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│ │ │ │ -0007a240: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007a250: 7c37 7820 7820 202d 2039 7820 7820 202b  |7x x  - 9x x  +
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│ │ │ │ -0007a290: 2078 2078 2020 2d20 7820 7820 202d 7c0a   x x  - x x  -|.
│ │ │ │ -0007a2a0: 7c20 2032 2035 2020 2020 2034 2035 2020  |  2 5     4 5  
│ │ │ │ -0007a2b0: 2020 2020 3520 3620 2020 2020 3620 2020      5 6     6   
│ │ │ │ -0007a2c0: 2020 3320 3720 2020 2020 3420 3720 2020    3 7     4 7   
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│ │ │ │ -0007a2e0: 2020 3320 3820 2020 2034 2038 2020 7c0a    3 8    4 8  |.
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│ │ │ │ +0007a240: 2020 2020 2020 2020 7c0a 7c37 7820 7820          |.|7x x 
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│ │ │ │ +0007a280: 202b 2034 7820 7820 202b 2078 2078 2020   + 4x x  + x x  
│ │ │ │ +0007a290: 2d20 7820 7820 202d 7c0a 7c20 2032 2035  - x x  -|.|  2 5
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│ │ │ │ +0007a2b0: 3620 2020 2020 3620 2020 2020 3320 3720  6     6     3 7 
│ │ │ │ +0007a2c0: 2020 2020 3420 3720 2020 2020 2035 2037      4 7      5 7
│ │ │ │ +0007a2d0: 2020 2020 2036 2037 2020 2020 3320 3820       6 7    3 8 
│ │ │ │ +0007a2e0: 2020 2034 2038 2020 7c0a 7c2d 2d2d 2d2d     4 8  |.|-----
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│ │ │ │ -0007a330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007a340: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007a330: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0007a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007a380: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007a390: 7c31 3478 2078 2020 2b20 7820 7820 2c20  |14x x  + x x , 
│ │ │ │ -0007a3a0: 2d20 3578 2078 2020 2b20 7820 7820 202d  - 5x x  + x x  -
│ │ │ │ -0007a3b0: 2037 7820 7820 202b 2038 7820 7820 202d   7x x  + 8x x  -
│ │ │ │ -0007a3c0: 2035 7820 7820 202b 2032 7820 7820 202d   5x x  + 2x x  -
│ │ │ │ -0007a3d0: 2078 2078 2020 2b20 7820 7820 202d 7c0a   x x  + x x  -|.
│ │ │ │ -0007a3e0: 7c20 2020 3520 3820 2020 2036 2038 2020  |   5 8    6 8  
│ │ │ │ -0007a3f0: 2020 2020 3020 3420 2020 2032 2034 2020      0 4    2 4  
│ │ │ │ -0007a400: 2020 2032 2035 2020 2020 2034 2035 2020     2 5     4 5  
│ │ │ │ -0007a410: 2020 2030 2036 2020 2020 2032 2036 2020     0 6     2 6  
│ │ │ │ -0007a420: 2020 3420 3620 2020 2035 2036 2020 7c0a    4 6    5 6  |.
│ │ │ │ -0007a430: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +0007a380: 2020 2020 2020 2020 7c0a 7c31 3478 2078          |.|14x x
│ │ │ │ +0007a390: 2020 2b20 7820 7820 2c20 2d20 3578 2078    + x x , - 5x x
│ │ │ │ +0007a3a0: 2020 2b20 7820 7820 202d 2037 7820 7820    + x x  - 7x x 
│ │ │ │ +0007a3b0: 202b 2038 7820 7820 202d 2035 7820 7820   + 8x x  - 5x x 
│ │ │ │ +0007a3c0: 202b 2032 7820 7820 202d 2078 2078 2020   + 2x x  - x x  
│ │ │ │ +0007a3d0: 2b20 7820 7820 202d 7c0a 7c20 2020 3520  + x x  -|.|   5 
│ │ │ │ +0007a3e0: 3820 2020 2036 2038 2020 2020 2020 3020  8    6 8      0 
│ │ │ │ +0007a3f0: 3420 2020 2032 2034 2020 2020 2032 2035  4    2 4     2 5
│ │ │ │ +0007a400: 2020 2020 2034 2035 2020 2020 2030 2036       4 5     0 6
│ │ │ │ +0007a410: 2020 2020 2032 2036 2020 2020 3420 3620       2 6    4 6 
│ │ │ │ +0007a420: 2020 2035 2036 2020 7c0a 7c2d 2d2d 2d2d     5 6  |.|-----
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│ │ │ │  0007a460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007a470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007a480: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007a470: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007a4b0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -0007a4c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007a4d0: 7c78 2078 2020 2b20 3778 2078 2020 2d20  |x x  + 7x x  - 
│ │ │ │ -0007a4e0: 3278 2078 2020 2d20 7820 7820 202b 2037  2x x  - x x  + 7
│ │ │ │ -0007a4f0: 7820 7820 202d 2032 7820 7820 2c20 7820  x x  - 2x x , x 
│ │ │ │ -0007a500: 7820 202b 2078 2020 2d20 3778 2078 2020  x  + x  - 7x x  
│ │ │ │ -0007a510: 2d20 3878 2078 2020 2b20 7820 7820 7c0a  - 8x x  + x x |.
│ │ │ │ -0007a520: 7c20 3420 3720 2020 2020 3520 3720 2020  | 4 7     5 7   
│ │ │ │ -0007a530: 2020 3620 3720 2020 2034 2038 2020 2020    6 7    4 8    
│ │ │ │ -0007a540: 2035 2038 2020 2020 2036 2038 2020 2030   5 8     6 8   0
│ │ │ │ -0007a550: 2034 2020 2020 3420 2020 2020 3120 3520   4    4     1 5 
│ │ │ │ -0007a560: 2020 2020 3420 3520 2020 2030 2036 7c0a      4 5    0 6|.
│ │ │ │ -0007a570: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +0007a4b0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0007a4c0: 2020 2020 2020 2020 7c0a 7c78 2078 2020          |.|x x  
│ │ │ │ +0007a4d0: 2b20 3778 2078 2020 2d20 3278 2078 2020  + 7x x  - 2x x  
│ │ │ │ +0007a4e0: 2d20 7820 7820 202b 2037 7820 7820 202d  - x x  + 7x x  -
│ │ │ │ +0007a4f0: 2032 7820 7820 2c20 7820 7820 202b 2078   2x x , x x  + x
│ │ │ │ +0007a500: 2020 2d20 3778 2078 2020 2d20 3878 2078    - 7x x  - 8x x
│ │ │ │ +0007a510: 2020 2b20 7820 7820 7c0a 7c20 3420 3720    + x x |.| 4 7 
│ │ │ │ +0007a520: 2020 2020 3520 3720 2020 2020 3620 3720      5 7     6 7 
│ │ │ │ +0007a530: 2020 2034 2038 2020 2020 2035 2038 2020     4 8     5 8  
│ │ │ │ +0007a540: 2020 2036 2038 2020 2030 2034 2020 2020     6 8   0 4    
│ │ │ │ +0007a550: 3420 2020 2020 3120 3520 2020 2020 3420  4     1 5     4 
│ │ │ │ +0007a560: 3520 2020 2030 2036 7c0a 7c2d 2d2d 2d2d  5    0 6|.|-----
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│ │ │ │  0007a590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007a5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007a5c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007a5b0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007a600: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007a610: 7c2b 2078 2078 2020 2b20 3278 2078 2020  |+ x x  + 2x x  
│ │ │ │ -0007a620: 2d20 7820 7820 202b 2078 2078 2020 2d20  - x x  + x x  - 
│ │ │ │ -0007a630: 3778 2078 2020 2b20 3278 2078 2020 2b20  7x x  + 2x x  + 
│ │ │ │ -0007a640: 7820 7820 202d 2037 7820 7820 202b 2032  x x  - 7x x  + 2
│ │ │ │ -0007a650: 7820 7820 2c20 7820 7820 202b 2020 7c0a  x x , x x  +  |.
│ │ │ │ -0007a660: 7c20 2020 3120 3620 2020 2020 3420 3620  |   1 6     4 6 
│ │ │ │ -0007a670: 2020 2035 2036 2020 2020 3420 3720 2020     5 6    4 7   
│ │ │ │ -0007a680: 2020 3520 3720 2020 2020 3620 3720 2020    5 7     6 7   
│ │ │ │ -0007a690: 2034 2038 2020 2020 2035 2038 2020 2020   4 8     5 8    
│ │ │ │ -0007a6a0: 2036 2038 2020 2032 2033 2020 2020 7c0a   6 8   2 3    |.
│ │ │ │ -0007a6b0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +0007a600: 2020 2020 2020 2020 7c0a 7c2b 2078 2078          |.|+ x x
│ │ │ │ +0007a610: 2020 2b20 3278 2078 2020 2d20 7820 7820    + 2x x  - x x 
│ │ │ │ +0007a620: 202b 2078 2078 2020 2d20 3778 2078 2020   + x x  - 7x x  
│ │ │ │ +0007a630: 2b20 3278 2078 2020 2b20 7820 7820 202d  + 2x x  + x x  -
│ │ │ │ +0007a640: 2037 7820 7820 202b 2032 7820 7820 2c20   7x x  + 2x x , 
│ │ │ │ +0007a650: 7820 7820 202b 2020 7c0a 7c20 2020 3120  x x  +  |.|   1 
│ │ │ │ +0007a660: 3620 2020 2020 3420 3620 2020 2035 2036  6     4 6    5 6
│ │ │ │ +0007a670: 2020 2020 3420 3720 2020 2020 3520 3720      4 7     5 7 
│ │ │ │ +0007a680: 2020 2020 3620 3720 2020 2034 2038 2020      6 7    4 8  
│ │ │ │ +0007a690: 2020 2035 2038 2020 2020 2036 2038 2020     5 8     6 8  
│ │ │ │ +0007a6a0: 2032 2033 2020 2020 7c0a 7c2d 2d2d 2d2d   2 3    |.|-----
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│ │ │ │  0007a6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007a6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007a700: 7c20 3220 2020 2020 2020 2020 2020 2020  | 2             
│ │ │ │ -0007a710: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +0007a6f0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 3220 2020  --------|.| 2   
│ │ │ │ +0007a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007a710: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  0007a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007a740: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007a750: 7c78 2020 2d20 3878 2078 2020 2b20 7820  |x  - 8x x  + x 
│ │ │ │ -0007a760: 7820 202b 2036 7820 7820 202d 2032 7820  x  + 6x x  - 2x 
│ │ │ │ -0007a770: 202b 2078 2078 2020 2b20 7820 7820 202d   + x x  + x x  -
│ │ │ │ -0007a780: 2037 7820 7820 202b 2032 7820 7820 202b   7x x  + 2x x  +
│ │ │ │ -0007a790: 2078 2078 2020 2d20 3778 2078 2020 7c0a   x x  - 7x x  |.
│ │ │ │ -0007a7a0: 7c20 3420 2020 2020 3420 3520 2020 2034  | 4     4 5    4
│ │ │ │ -0007a7b0: 2036 2020 2020 2035 2036 2020 2020 2036   6     5 6     6
│ │ │ │ -0007a7c0: 2020 2020 3320 3720 2020 2034 2037 2020      3 7    4 7  
│ │ │ │ -0007a7d0: 2020 2035 2037 2020 2020 2036 2037 2020     5 7     6 7  
│ │ │ │ -0007a7e0: 2020 3420 3820 2020 2020 3520 3820 7c0a    4 8     5 8 |.
│ │ │ │ -0007a7f0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +0007a740: 2020 2020 2020 2020 7c0a 7c78 2020 2d20          |.|x  - 
│ │ │ │ +0007a750: 3878 2078 2020 2b20 7820 7820 202b 2036  8x x  + x x  + 6
│ │ │ │ +0007a760: 7820 7820 202d 2032 7820 202b 2078 2078  x x  - 2x  + x x
│ │ │ │ +0007a770: 2020 2b20 7820 7820 202d 2037 7820 7820    + x x  - 7x x 
│ │ │ │ +0007a780: 202b 2032 7820 7820 202b 2078 2078 2020   + 2x x  + x x  
│ │ │ │ +0007a790: 2d20 3778 2078 2020 7c0a 7c20 3420 2020  - 7x x  |.| 4   
│ │ │ │ +0007a7a0: 2020 3420 3520 2020 2034 2036 2020 2020    4 5    4 6    
│ │ │ │ +0007a7b0: 2035 2036 2020 2020 2036 2020 2020 3320   5 6     6    3 
│ │ │ │ +0007a7c0: 3720 2020 2034 2037 2020 2020 2035 2037  7    4 7     5 7
│ │ │ │ +0007a7d0: 2020 2020 2036 2037 2020 2020 3420 3820       6 7    4 8 
│ │ │ │ +0007a7e0: 2020 2020 3520 3820 7c0a 7c2d 2d2d 2d2d      5 8 |.|-----
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│ │ │ │  0007a800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007a830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007a840: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007a830: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007a870: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0007a880: 2020 2020 2020 2020 2020 2020 2032 7c0a               2|.
│ │ │ │ -0007a890: 7c2b 2032 7820 7820 2c20 7820 7820 202d  |+ 2x x , x x  -
│ │ │ │ -0007a8a0: 2037 7820 7820 202b 2078 2078 2020 2b20   7x x  + x x  + 
│ │ │ │ -0007a8b0: 7820 7820 202d 2037 7820 7820 202b 2032  x x  - 7x x  + 2
│ │ │ │ -0007a8c0: 7820 202d 2078 2078 202c 202d 2034 7820  x  - x x , - 4x 
│ │ │ │ -0007a8d0: 7820 202b 2078 2078 2020 2d20 7820 7c0a  x  + x x  - x |.
│ │ │ │ -0007a8e0: 7c20 2020 2036 2038 2020 2031 2033 2020  |    6 8   1 3  
│ │ │ │ -0007a8f0: 2020 2031 2035 2020 2020 3120 3620 2020     1 5    1 6   
│ │ │ │ -0007a900: 2034 2036 2020 2020 2035 2036 2020 2020   4 6     5 6    
│ │ │ │ -0007a910: 2036 2020 2020 3320 3720 2020 2020 2030   6    3 7      0
│ │ │ │ -0007a920: 2033 2020 2020 3320 3420 2020 2034 7c0a   3    3 4    4|.
│ │ │ │ -0007a930: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +0007a860: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0007a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007a880: 2020 2020 2020 2032 7c0a 7c2b 2032 7820         2|.|+ 2x 
│ │ │ │ +0007a890: 7820 2c20 7820 7820 202d 2037 7820 7820  x , x x  - 7x x 
│ │ │ │ +0007a8a0: 202b 2078 2078 2020 2b20 7820 7820 202d   + x x  + x x  -
│ │ │ │ +0007a8b0: 2037 7820 7820 202b 2032 7820 202d 2078   7x x  + 2x  - x
│ │ │ │ +0007a8c0: 2078 202c 202d 2034 7820 7820 202b 2078   x , - 4x x  + x
│ │ │ │ +0007a8d0: 2078 2020 2d20 7820 7c0a 7c20 2020 2036   x  - x |.|    6
│ │ │ │ +0007a8e0: 2038 2020 2031 2033 2020 2020 2031 2035   8   1 3     1 5
│ │ │ │ +0007a8f0: 2020 2020 3120 3620 2020 2034 2036 2020      1 6    4 6  
│ │ │ │ +0007a900: 2020 2035 2036 2020 2020 2036 2020 2020     5 6     6    
│ │ │ │ +0007a910: 3320 3720 2020 2020 2030 2033 2020 2020  3 7      0 3    
│ │ │ │ +0007a920: 3320 3420 2020 2034 7c0a 7c2d 2d2d 2d2d  3 4    4|.|-----
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│ │ │ │  0007a940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007a960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007a970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007a980: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007a970: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007a9a0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0007a9a0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  0007a9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007a9c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007a9d0: 7c2d 2037 7820 7820 202b 2038 7820 7820  |- 7x x  + 8x x 
│ │ │ │ -0007a9e0: 202b 2078 2078 2020 2d20 7820 7820 202d   + x x  - x x  -
│ │ │ │ -0007a9f0: 2036 7820 7820 202b 2032 7820 202d 2078   6x x  + 2x  - x
│ │ │ │ -0007aa00: 2078 2020 2d20 7820 7820 202b 2037 7820   x  - x x  + 7x 
│ │ │ │ -0007aa10: 7820 202d 2032 7820 7820 202d 2020 7c0a  x  - 2x x  -  |.
│ │ │ │ -0007aa20: 7c20 2020 2030 2035 2020 2020 2034 2035  |    0 5     4 5
│ │ │ │ -0007aa30: 2020 2020 3020 3620 2020 2034 2036 2020      0 6    4 6  
│ │ │ │ -0007aa40: 2020 2035 2036 2020 2020 2036 2020 2020     5 6     6    
│ │ │ │ -0007aa50: 3320 3720 2020 2034 2037 2020 2020 2035  3 7    4 7     5
│ │ │ │ -0007aa60: 2037 2020 2020 2036 2037 2020 2020 7c0a   7     6 7    |.
│ │ │ │ -0007aa70: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +0007a9c0: 2020 2020 2020 2020 7c0a 7c2d 2037 7820          |.|- 7x 
│ │ │ │ +0007a9d0: 7820 202b 2038 7820 7820 202b 2078 2078  x  + 8x x  + x x
│ │ │ │ +0007a9e0: 2020 2d20 7820 7820 202d 2036 7820 7820    - x x  - 6x x 
│ │ │ │ +0007a9f0: 202b 2032 7820 202d 2078 2078 2020 2d20   + 2x  - x x  - 
│ │ │ │ +0007aa00: 7820 7820 202b 2037 7820 7820 202d 2032  x x  + 7x x  - 2
│ │ │ │ +0007aa10: 7820 7820 202d 2020 7c0a 7c20 2020 2030  x x  -  |.|    0
│ │ │ │ +0007aa20: 2035 2020 2020 2034 2035 2020 2020 3020   5     4 5    0 
│ │ │ │ +0007aa30: 3620 2020 2034 2036 2020 2020 2035 2036  6    4 6     5 6
│ │ │ │ +0007aa40: 2020 2020 2036 2020 2020 3320 3720 2020       6    3 7   
│ │ │ │ +0007aa50: 2034 2037 2020 2020 2035 2037 2020 2020   4 7     5 7    
│ │ │ │ +0007aa60: 2036 2037 2020 2020 7c0a 7c2d 2d2d 2d2d   6 7    |.|-----
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│ │ │ │  0007aa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007aa90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007aaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007aab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007aac0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007aae0: 2020 2032 2020 2020 2020 2020 2020 2032     2           2
│ │ │ │ +0007aab0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007aad0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +0007aae0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  0007aaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ab00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007ab10: 7c78 2078 2020 2b20 3778 2078 2020 2d20  |x x  + 7x x  - 
│ │ │ │ -0007ab20: 3278 2078 202c 202d 2035 7820 7820 202b  2x x , - 5x x  +
│ │ │ │ -0007ab30: 2032 7820 202b 2078 2078 2020 2d20 7820   2x  + x x  - x 
│ │ │ │ -0007ab40: 202d 2078 2078 2020 2b20 3878 2078 2020   - x x  + 8x x  
│ │ │ │ -0007ab50: 2d20 3130 7820 7820 202b 2020 2020 7c0a  - 10x x  +    |.
│ │ │ │ -0007ab60: 7c20 3420 3820 2020 2020 3520 3820 2020  | 4 8     5 8   
│ │ │ │ -0007ab70: 2020 3620 3820 2020 2020 2030 2032 2020    6 8      0 2  
│ │ │ │ -0007ab80: 2020 2032 2020 2020 3220 3420 2020 2034     2    2 4    4
│ │ │ │ -0007ab90: 2020 2020 3220 3520 2020 2020 3420 3520      2 5     4 5 
│ │ │ │ -0007aba0: 2020 2020 2030 2036 2020 2020 2020 7c0a       0 6      |.
│ │ │ │ -0007abb0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +0007ab00: 2020 2020 2020 2020 7c0a 7c78 2078 2020          |.|x x  
│ │ │ │ +0007ab10: 2b20 3778 2078 2020 2d20 3278 2078 202c  + 7x x  - 2x x ,
│ │ │ │ +0007ab20: 202d 2035 7820 7820 202b 2032 7820 202b   - 5x x  + 2x  +
│ │ │ │ +0007ab30: 2078 2078 2020 2d20 7820 202d 2078 2078   x x  - x  - x x
│ │ │ │ +0007ab40: 2020 2b20 3878 2078 2020 2d20 3130 7820    + 8x x  - 10x 
│ │ │ │ +0007ab50: 7820 202b 2020 2020 7c0a 7c20 3420 3820  x  +    |.| 4 8 
│ │ │ │ +0007ab60: 2020 2020 3520 3820 2020 2020 3620 3820      5 8     6 8 
│ │ │ │ +0007ab70: 2020 2020 2030 2032 2020 2020 2032 2020       0 2     2  
│ │ │ │ +0007ab80: 2020 3220 3420 2020 2034 2020 2020 3220    2 4    4    2 
│ │ │ │ +0007ab90: 3520 2020 2020 3420 3520 2020 2020 2030  5     4 5      0
│ │ │ │ +0007aba0: 2036 2020 2020 2020 7c0a 7c2d 2d2d 2d2d   6      |.|-----
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│ │ │ │  0007abc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007abd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007abe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007abf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007ac00: 7c32 7820 7820 202b 2032 7820 7820 202d  |2x x  + 2x x  -
│ │ │ │ -0007ac10: 2032 7820 7820 202b 2031 3478 2078 2020   2x x  + 14x x  
│ │ │ │ -0007ac20: 2d20 3478 2078 2020 2b20 3578 2078 2020  - 4x x  + 5x x  
│ │ │ │ -0007ac30: 2d20 3378 2078 2020 2d20 3278 2078 2020  - 3x x  - 2x x  
│ │ │ │ -0007ac40: 2b20 3778 2078 2020 2d20 2020 2020 7c0a  + 7x x  -     |.
│ │ │ │ -0007ac50: 7c20 2035 2036 2020 2020 2032 2037 2020  |  5 6     2 7  
│ │ │ │ -0007ac60: 2020 2034 2037 2020 2020 2020 3520 3720     4 7      5 7 
│ │ │ │ -0007ac70: 2020 2020 3620 3720 2020 2020 3020 3820      6 7     0 8 
│ │ │ │ -0007ac80: 2020 2020 3220 3820 2020 2020 3420 3820      2 8     4 8 
│ │ │ │ -0007ac90: 2020 2020 3520 3820 2020 2020 2020 7c0a      5 8       |.
│ │ │ │ -0007aca0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +0007abf0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c32 7820 7820  --------|.|2x x 
│ │ │ │ +0007ac00: 202b 2032 7820 7820 202d 2032 7820 7820   + 2x x  - 2x x 
│ │ │ │ +0007ac10: 202b 2031 3478 2078 2020 2d20 3478 2078   + 14x x  - 4x x
│ │ │ │ +0007ac20: 2020 2b20 3578 2078 2020 2d20 3378 2078    + 5x x  - 3x x
│ │ │ │ +0007ac30: 2020 2d20 3278 2078 2020 2b20 3778 2078    - 2x x  + 7x x
│ │ │ │ +0007ac40: 2020 2d20 2020 2020 7c0a 7c20 2035 2036    -     |.|  5 6
│ │ │ │ +0007ac50: 2020 2020 2032 2037 2020 2020 2034 2037       2 7     4 7
│ │ │ │ +0007ac60: 2020 2020 2020 3520 3720 2020 2020 3620        5 7     6 
│ │ │ │ +0007ac70: 3720 2020 2020 3020 3820 2020 2020 3220  7     0 8     2 
│ │ │ │ +0007ac80: 3820 2020 2020 3420 3820 2020 2020 3520  8     4 8     5 
│ │ │ │ +0007ac90: 3820 2020 2020 2020 7c0a 7c2d 2d2d 2d2d  8       |.|-----
│ │ │ │ +0007aca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007acb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007acc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007acd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007ace0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007acf0: 7c32 7820 7820 202d 2033 7820 7820 2c20  |2x x  - 3x x , 
│ │ │ │ -0007ad00: 2d20 3578 2078 2020 2b20 7820 7820 202b  - 5x x  + x x  +
│ │ │ │ -0007ad10: 2078 2078 2020 2d20 3478 2078 2020 2d20   x x  - 4x x  - 
│ │ │ │ -0007ad20: 7820 7820 202b 2078 2078 2020 2b20 7820  x x  + x x  + x 
│ │ │ │ -0007ad30: 7820 2c20 7820 7820 202d 2020 2020 7c0a  x , x x  -    |.
│ │ │ │ -0007ad40: 7c20 2036 2038 2020 2020 2037 2038 2020  |  6 8     7 8  
│ │ │ │ -0007ad50: 2020 2020 3020 3120 2020 2031 2032 2020      0 1    1 2  
│ │ │ │ -0007ad60: 2020 3120 3420 2020 2020 3020 3620 2020    1 4     0 6   
│ │ │ │ -0007ad70: 2031 2036 2020 2020 3420 3620 2020 2030   1 6    4 6    0
│ │ │ │ -0007ad80: 2037 2020 2030 2032 2020 2020 2020 7c0a   7   0 2      |.
│ │ │ │ -0007ad90: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +0007ace0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c32 7820 7820  --------|.|2x x 
│ │ │ │ +0007acf0: 202d 2033 7820 7820 2c20 2d20 3578 2078   - 3x x , - 5x x
│ │ │ │ +0007ad00: 2020 2b20 7820 7820 202b 2078 2078 2020    + x x  + x x  
│ │ │ │ +0007ad10: 2d20 3478 2078 2020 2d20 7820 7820 202b  - 4x x  - x x  +
│ │ │ │ +0007ad20: 2078 2078 2020 2b20 7820 7820 2c20 7820   x x  + x x , x 
│ │ │ │ +0007ad30: 7820 202d 2020 2020 7c0a 7c20 2036 2038  x  -    |.|  6 8
│ │ │ │ +0007ad40: 2020 2020 2037 2038 2020 2020 2020 3020       7 8      0 
│ │ │ │ +0007ad50: 3120 2020 2031 2032 2020 2020 3120 3420  1    1 2    1 4 
│ │ │ │ +0007ad60: 2020 2020 3020 3620 2020 2031 2036 2020      0 6    1 6  
│ │ │ │ +0007ad70: 2020 3420 3620 2020 2030 2037 2020 2030    4 6    0 7   0
│ │ │ │ +0007ad80: 2032 2020 2020 2020 7c0a 7c2d 2d2d 2d2d   2      |.|-----
│ │ │ │ +0007ad90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007ada0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007adb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007adc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007add0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007ade0: 7c78 2078 2020 2b20 3578 2078 2020 2b20  |x x  + 5x x  + 
│ │ │ │ -0007adf0: 7820 7820 202d 2031 3478 2078 2020 2d20  x x  - 14x x  - 
│ │ │ │ -0007ae00: 7820 7820 202d 2038 7820 7820 202d 2038  x x  - 8x x  - 8
│ │ │ │ -0007ae10: 7820 7820 202b 2032 7820 7820 202b 2034  x x  + 2x x  + 4
│ │ │ │ -0007ae20: 7820 7820 202b 2032 7820 7820 202b 7c0a  x x  + 2x x  +|.
│ │ │ │ -0007ae30: 7c20 3120 3220 2020 2020 3020 3420 2020  | 1 2     0 4   
│ │ │ │ -0007ae40: 2031 2034 2020 2020 2020 3120 3520 2020   1 4      1 5   
│ │ │ │ -0007ae50: 2032 2035 2020 2020 2034 2035 2020 2020   2 5     4 5    
│ │ │ │ -0007ae60: 2030 2036 2020 2020 2031 2036 2020 2020   0 6     1 6    
│ │ │ │ -0007ae70: 2034 2036 2020 2020 2032 2037 2020 7c0a   4 6     2 7  |.
│ │ │ │ -0007ae80: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +0007add0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c78 2078 2020  --------|.|x x  
│ │ │ │ +0007ade0: 2b20 3578 2078 2020 2b20 7820 7820 202d  + 5x x  + x x  -
│ │ │ │ +0007adf0: 2031 3478 2078 2020 2d20 7820 7820 202d   14x x  - x x  -
│ │ │ │ +0007ae00: 2038 7820 7820 202d 2038 7820 7820 202b   8x x  - 8x x  +
│ │ │ │ +0007ae10: 2032 7820 7820 202b 2034 7820 7820 202b   2x x  + 4x x  +
│ │ │ │ +0007ae20: 2032 7820 7820 202b 7c0a 7c20 3120 3220   2x x  +|.| 1 2 
│ │ │ │ +0007ae30: 2020 2020 3020 3420 2020 2031 2034 2020      0 4    1 4  
│ │ │ │ +0007ae40: 2020 2020 3120 3520 2020 2032 2035 2020      1 5    2 5  
│ │ │ │ +0007ae50: 2020 2034 2035 2020 2020 2030 2036 2020     4 5     0 6  
│ │ │ │ +0007ae60: 2020 2031 2036 2020 2020 2034 2036 2020     1 6     4 6  
│ │ │ │ +0007ae70: 2020 2032 2037 2020 7c0a 7c2d 2d2d 2d2d     2 7  |.|-----
│ │ │ │ +0007ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007ae90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007aea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007aeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007aec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007aed0: 7c34 7820 7820 202b 2033 7820 7820 202d  |4x x  + 3x x  -
│ │ │ │ -0007aee0: 2037 7820 7820 202b 2032 7820 7820 202d   7x x  + 2x x  -
│ │ │ │ -0007aef0: 2033 7820 7820 7d29 2020 2020 2020 2020   3x x })        
│ │ │ │ +0007aec0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c34 7820 7820  --------|.|4x x 
│ │ │ │ +0007aed0: 202b 2033 7820 7820 202d 2037 7820 7820   + 3x x  - 7x x 
│ │ │ │ +0007aee0: 202b 2032 7820 7820 202d 2033 7820 7820   + 2x x  - 3x x 
│ │ │ │ +0007aef0: 7d29 2020 2020 2020 2020 2020 2020 2020  })              
│ │ │ │  0007af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007af10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007af20: 7c20 2030 2038 2020 2020 2031 2038 2020  |  0 8     1 8  
│ │ │ │ -0007af30: 2020 2035 2038 2020 2020 2036 2038 2020     5 8     6 8  
│ │ │ │ -0007af40: 2020 2037 2038 2020 2020 2020 2020 2020     7 8          
│ │ │ │ +0007af10: 2020 2020 2020 2020 7c0a 7c20 2030 2038          |.|  0 8
│ │ │ │ +0007af20: 2020 2020 2031 2038 2020 2020 2035 2038       1 8     5 8
│ │ │ │ +0007af30: 2020 2020 2036 2038 2020 2020 2037 2038       6 8     7 8
│ │ │ │ +0007af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007af60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007af70: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0007af60: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0007af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007af80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007af90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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)        
│ │ │ │ +0007afb0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ +0007afc0: 7469 6d65 206b 6572 6e65 6c28 7068 692c  time kernel(phi,
│ │ │ │ +0007afd0: 3129 2020 2020 2020 2020 2020 2020 2020  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 3734  | -- used 0.0174
│ │ │ │ -0007b020: 3434 3673 2028 6370 7529 3b20 302e 3031  446s (cpu); 0.01
│ │ │ │ -0007b030: 3734 3431 3173 2028 7468 7265 6164 293b  74411s (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  |               
│ │ │ │ +0007b000: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ +0007b010: 7365 6420 302e 3032 3135 3430 3473 2028  sed 0.0215404s (
│ │ │ │ +0007b020: 6370 7529 3b20 302e 3032 3135 3431 3873  cpu); 0.0215418s
│ │ │ │ +0007b030: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ +0007b040: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │ +0007b050: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007b060: 2020 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                |.
│ │ │ │ -0007b0b0: 7c6f 3220 3d20 6964 6561 6c20 2829 2020  |o2 = ideal ()  
│ │ │ │ +0007b0a0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ +0007b0b0: 6964 6561 6c20 2829 2020 2020 2020 2020  ideal ()        
│ │ │ │  0007b0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007b0f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007b100: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007b0f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007b100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007b140: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007b150: 7c6f 3220 3a20 4964 6561 6c20 6f66 2051  |o2 : Ideal of Q
│ │ │ │ -0007b160: 515b 7920 2e2e 7920 205d 2020 2020 2020  Q[y ..y  ]      
│ │ │ │ +0007b140: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ +0007b150: 4964 6561 6c20 6f66 2051 515b 7920 2e2e  Ideal of QQ[y ..
│ │ │ │ +0007b160: 7920 205d 2020 2020 2020 2020 2020 2020  y  ]            
│ │ │ │  0007b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007b190: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007b1a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007b1b0: 2020 2030 2020 2031 3120 2020 2020 2020     0   11       
│ │ │ │ +0007b190: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007b1a0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +0007b1b0: 2031 3120 2020 2020 2020 2020 2020 2020   11             
│ │ │ │  0007b1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007b1e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007b1f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0007b1e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0007b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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)        
│ │ │ │ +0007b230: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +0007b240: 7469 6d65 206b 6572 6e65 6c28 7068 692c  time kernel(phi,
│ │ │ │ +0007b250: 3229 2020 2020 2020 2020 2020 2020 2020  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 302e 3931 3136  | -- used 0.9116
│ │ │ │ -0007b2a0: 3835 7320 2863 7075 293b 2030 2e34 3439  85s (cpu); 0.449
│ │ │ │ -0007b2b0: 3333 3573 2028 7468 7265 6164 293b 2030  335s (thread); 0
│ │ │ │ -0007b2c0: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ -0007b2d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007b2e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007b280: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ +0007b290: 7365 6420 312e 3134 3533 3673 2028 6370  sed 1.14536s (cp
│ │ │ │ +0007b2a0: 7529 3b20 302e 3533 3832 3938 7320 2874  u); 0.538298s (t
│ │ │ │ +0007b2b0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +0007b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007b2d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007b2e0: 2020 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                |.
│ │ │ │ -0007b330: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007b340: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +0007b320: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007b340: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  0007b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007b370: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007b380: 7c6f 3320 3d20 6964 6561 6c20 2879 2079  |o3 = ideal (y y
│ │ │ │ -0007b390: 2020 2b20 7920 7920 202b 2079 2020 2b20    + y y  + y  + 
│ │ │ │ -0007b3a0: 3579 2079 2020 2b20 7920 7920 202b 2035  5y y  + y y  + 5
│ │ │ │ -0007b3b0: 7920 7920 202d 2079 2079 2020 2d20 3479  y y  - y y  - 4y
│ │ │ │ -0007b3c0: 2079 2020 2d20 3579 2079 2020 2d20 7c0a   y  - 5y y  - |.
│ │ │ │ -0007b3d0: 7c20 2020 2020 2020 2020 2020 2020 3220  |             2 
│ │ │ │ -0007b3e0: 3420 2020 2033 2034 2020 2020 3420 2020  4    3 4    4   
│ │ │ │ -0007b3f0: 2020 3220 3520 2020 2033 2035 2020 2020    2 5    3 5    
│ │ │ │ -0007b400: 2034 2035 2020 2020 3120 3620 2020 2020   4 5    1 6     
│ │ │ │ -0007b410: 3220 3620 2020 2020 3520 3620 2020 7c0a  2 6     5 6   |.
│ │ │ │ -0007b420: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +0007b370: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ +0007b380: 6964 6561 6c20 2879 2079 2020 2b20 7920  ideal (y y  + y 
│ │ │ │ +0007b390: 7920 202b 2079 2020 2b20 3579 2079 2020  y  + y  + 5y y  
│ │ │ │ +0007b3a0: 2b20 7920 7920 202b 2035 7920 7920 202d  + y y  + 5y y  -
│ │ │ │ +0007b3b0: 2079 2079 2020 2d20 3479 2079 2020 2d20   y y  - 4y y  - 
│ │ │ │ +0007b3c0: 3579 2079 2020 2d20 7c0a 7c20 2020 2020  5y y  - |.|     
│ │ │ │ +0007b3d0: 2020 2020 2020 2020 3220 3420 2020 2033          2 4    3
│ │ │ │ +0007b3e0: 2034 2020 2020 3420 2020 2020 3220 3520   4    4     2 5 
│ │ │ │ +0007b3f0: 2020 2033 2035 2020 2020 2034 2035 2020     3 5     4 5  
│ │ │ │ +0007b400: 2020 3120 3620 2020 2020 3220 3620 2020    1 6     2 6   
│ │ │ │ +0007b410: 2020 3520 3620 2020 7c0a 7c20 2020 2020    5 6   |.|     
│ │ │ │ +0007b420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007b460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007b470: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007b460: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007b4b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007b4c0: 7c20 2020 2020 3479 2079 2020 2d20 3279  |     4y y  - 2y
│ │ │ │ -0007b4d0: 2079 2020 2d20 7920 7920 202b 2034 7920   y  - y y  + 4y 
│ │ │ │ -0007b4e0: 7920 202d 2035 7920 7920 202d 2034 7920  y  - 5y y  - 4y 
│ │ │ │ -0007b4f0: 7920 202b 2033 7920 7920 202d 2034 7920  y  + 3y y  - 4y 
│ │ │ │ -0007b500: 7920 202d 2079 2079 2020 202d 2020 7c0a  y  - y y   -  |.
│ │ │ │ -0007b510: 7c20 2020 2020 2020 3220 3720 2020 2020  |       2 7     
│ │ │ │ -0007b520: 3420 3720 2020 2031 2038 2020 2020 2034  4 7    1 8     4
│ │ │ │ -0007b530: 2038 2020 2020 2035 2038 2020 2020 2035   8     5 8     5
│ │ │ │ -0007b540: 2039 2020 2020 2037 2039 2020 2020 2038   9     7 9     8
│ │ │ │ -0007b550: 2039 2020 2020 3320 3130 2020 2020 7c0a   9    3 10    |.
│ │ │ │ -0007b560: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +0007b4b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007b4c0: 3479 2079 2020 2d20 3279 2079 2020 2d20  4y y  - 2y y  - 
│ │ │ │ +0007b4d0: 7920 7920 202b 2034 7920 7920 202d 2035  y y  + 4y y  - 5
│ │ │ │ +0007b4e0: 7920 7920 202d 2034 7920 7920 202b 2033  y y  - 4y y  + 3
│ │ │ │ +0007b4f0: 7920 7920 202d 2034 7920 7920 202d 2079  y y  - 4y y  - y
│ │ │ │ +0007b500: 2079 2020 202d 2020 7c0a 7c20 2020 2020   y   -  |.|     
│ │ │ │ +0007b510: 2020 3220 3720 2020 2020 3420 3720 2020    2 7     4 7   
│ │ │ │ +0007b520: 2031 2038 2020 2020 2034 2038 2020 2020   1 8     4 8    
│ │ │ │ +0007b530: 2035 2038 2020 2020 2035 2039 2020 2020   5 8     5 9    
│ │ │ │ +0007b540: 2037 2039 2020 2020 2038 2039 2020 2020   7 9     8 9    
│ │ │ │ +0007b550: 3320 3130 2020 2020 7c0a 7c20 2020 2020  3 10    |.|     
│ │ │ │ +0007b560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007b5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007b5b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007b5a0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007b5f0: 2020 2020 2020 2020 2032 2020 2020 7c0a           2    |.
│ │ │ │ -0007b600: 7c20 2020 2020 3379 2079 2020 202d 2035  |     3y y   - 5
│ │ │ │ -0007b610: 7920 7920 2020 2d20 7920 7920 2020 2b20  y y   - y y   + 
│ │ │ │ -0007b620: 3479 2079 2020 202b 2035 7920 7920 202c  4y y   + 5y y  ,
│ │ │ │ -0007b630: 2033 7920 7920 202d 2079 2079 2020 2d20   3y y  - y y  - 
│ │ │ │ -0007b640: 3379 2079 2020 2d20 7920 202b 2020 7c0a  3y y  - y  +  |.
│ │ │ │ -0007b650: 7c20 2020 2020 2020 3620 3130 2020 2020  |       6 10    
│ │ │ │ -0007b660: 2038 2031 3020 2020 2034 2031 3120 2020   8 10    4 11   
│ │ │ │ -0007b670: 2020 3620 3131 2020 2020 2038 2031 3120    6 11     8 11 
│ │ │ │ -0007b680: 2020 2031 2033 2020 2020 3220 3320 2020     1 3    2 3   
│ │ │ │ -0007b690: 2020 3320 3420 2020 2034 2020 2020 7c0a    3 4    4    |.
│ │ │ │ -0007b6a0: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +0007b5f0: 2020 2032 2020 2020 7c0a 7c20 2020 2020     2    |.|     
│ │ │ │ +0007b600: 3379 2079 2020 202d 2035 7920 7920 2020  3y y   - 5y y   
│ │ │ │ +0007b610: 2d20 7920 7920 2020 2b20 3479 2079 2020  - y y   + 4y y  
│ │ │ │ +0007b620: 202b 2035 7920 7920 202c 2033 7920 7920   + 5y y  , 3y y 
│ │ │ │ +0007b630: 202d 2079 2079 2020 2d20 3379 2079 2020   - y y  - 3y y  
│ │ │ │ +0007b640: 2d20 7920 202b 2020 7c0a 7c20 2020 2020  - y  +  |.|     
│ │ │ │ +0007b650: 2020 3620 3130 2020 2020 2038 2031 3020    6 10     8 10 
│ │ │ │ +0007b660: 2020 2034 2031 3120 2020 2020 3620 3131     4 11     6 11
│ │ │ │ +0007b670: 2020 2020 2038 2031 3120 2020 2031 2033       8 11    1 3
│ │ │ │ +0007b680: 2020 2020 3220 3320 2020 2020 3320 3420      2 3     3 4 
│ │ │ │ +0007b690: 2020 2034 2020 2020 7c0a 7c20 2020 2020     4    |.|     
│ │ │ │ +0007b6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007b6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007b6f0: 7c20 2020 2020 3279 2079 2020 2d20 7920  |     2y y  - y 
│ │ │ │ -0007b700: 7920 202b 2079 2079 2020 2b20 3279 2079  y  + y y  + 2y y
│ │ │ │ -0007b710: 2020 2b20 3379 2079 2020 2d20 3779 2079    + 3y y  - 7y y
│ │ │ │ -0007b720: 2020 2d20 3479 2079 2020 2b20 3779 2079    - 4y y  + 7y y
│ │ │ │ -0007b730: 2020 2d20 3279 2079 2020 2b20 2020 7c0a    - 2y y  +   |.
│ │ │ │ -0007b740: 7c20 2020 2020 2020 3020 3520 2020 2033  |       0 5    3
│ │ │ │ -0007b750: 2035 2020 2020 3120 3620 2020 2020 3220   5    1 6     2 
│ │ │ │ -0007b760: 3620 2020 2020 3520 3620 2020 2020 3220  6     5 6     2 
│ │ │ │ -0007b770: 3720 2020 2020 3420 3720 2020 2020 3120  7     4 7     1 
│ │ │ │ -0007b780: 3820 2020 2020 3420 3820 2020 2020 7c0a  8     4 8     |.
│ │ │ │ -0007b790: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +0007b6e0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007b6f0: 3279 2079 2020 2d20 7920 7920 202b 2079  2y y  - y y  + y
│ │ │ │ +0007b700: 2079 2020 2b20 3279 2079 2020 2b20 3379   y  + 2y y  + 3y
│ │ │ │ +0007b710: 2079 2020 2d20 3779 2079 2020 2d20 3479   y  - 7y y  - 4y
│ │ │ │ +0007b720: 2079 2020 2b20 3779 2079 2020 2d20 3279   y  + 7y y  - 2y
│ │ │ │ +0007b730: 2079 2020 2b20 2020 7c0a 7c20 2020 2020   y  +   |.|     
│ │ │ │ +0007b740: 2020 3020 3520 2020 2033 2035 2020 2020    0 5    3 5    
│ │ │ │ +0007b750: 3120 3620 2020 2020 3220 3620 2020 2020  1 6     2 6     
│ │ │ │ +0007b760: 3520 3620 2020 2020 3220 3720 2020 2020  5 6     2 7     
│ │ │ │ +0007b770: 3420 3720 2020 2020 3120 3820 2020 2020  4 7     1 8     
│ │ │ │ +0007b780: 3420 3820 2020 2020 7c0a 7c20 2020 2020  4 8     |.|     
│ │ │ │ +0007b790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007b7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007b7e0: 7c20 2020 2020 7920 7920 202d 2079 2079  |     y y  - y y
│ │ │ │ -0007b7f0: 2020 2b20 3279 2079 2020 2b20 3279 2079    + 2y y  + 2y y
│ │ │ │ -0007b800: 2020 2b20 7920 7920 202d 2037 7920 7920    + y y  - 7y y 
│ │ │ │ -0007b810: 2020 2b20 3579 2079 2020 202d 2033 7920    + 5y y   - 3y 
│ │ │ │ -0007b820: 7920 2020 2d20 7920 7920 2020 2d20 7c0a  y   - y y   - |.
│ │ │ │ -0007b830: 7c20 2020 2020 2030 2039 2020 2020 3420  |      0 9    4 
│ │ │ │ -0007b840: 3920 2020 2020 3520 3920 2020 2020 3720  9     5 9     7 
│ │ │ │ -0007b850: 3920 2020 2038 2039 2020 2020 2030 2031  9    8 9     0 1
│ │ │ │ -0007b860: 3020 2020 2020 3320 3130 2020 2020 2036  0     3 10     6
│ │ │ │ -0007b870: 2031 3020 2020 2030 2031 3120 2020 7c0a   10    0 11   |.
│ │ │ │ -0007b880: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +0007b7d0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007b7e0: 7920 7920 202d 2079 2079 2020 2b20 3279  y y  - y y  + 2y
│ │ │ │ +0007b7f0: 2079 2020 2b20 3279 2079 2020 2b20 7920   y  + 2y y  + y 
│ │ │ │ +0007b800: 7920 202d 2037 7920 7920 2020 2b20 3579  y  - 7y y   + 5y
│ │ │ │ +0007b810: 2079 2020 202d 2033 7920 7920 2020 2d20   y   - 3y y   - 
│ │ │ │ +0007b820: 7920 7920 2020 2d20 7c0a 7c20 2020 2020  y y   - |.|     
│ │ │ │ +0007b830: 2030 2039 2020 2020 3420 3920 2020 2020   0 9    4 9     
│ │ │ │ +0007b840: 3520 3920 2020 2020 3720 3920 2020 2038  5 9     7 9    8
│ │ │ │ +0007b850: 2039 2020 2020 2030 2031 3020 2020 2020   9     0 10     
│ │ │ │ +0007b860: 3320 3130 2020 2020 2036 2031 3020 2020  3 10     6 10   
│ │ │ │ +0007b870: 2030 2031 3120 2020 7c0a 7c20 2020 2020   0 11   |.|     
│ │ │ │ +0007b880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007b8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007b8d0: 7c20 2020 2020 3279 2079 2020 202d 2032  |     2y y   - 2
│ │ │ │ -0007b8e0: 7920 7920 202c 2037 7920 7920 202b 2079  y y  , 7y y  + y
│ │ │ │ -0007b8f0: 2079 2020 2b20 3779 2079 2020 2d20 7920   y  + 7y y  - y 
│ │ │ │ -0007b900: 7920 202b 2038 7920 7920 202d 2079 2079  y  + 8y y  - y y
│ │ │ │ -0007b910: 2020 2d20 7920 7920 202b 2020 2020 7c0a    - y y  +    |.
│ │ │ │ -0007b920: 7c20 2020 2020 2020 3320 3131 2020 2020  |       3 11    
│ │ │ │ -0007b930: 2034 2031 3120 2020 2030 2031 2020 2020   4 11    0 1    
│ │ │ │ -0007b940: 3020 3420 2020 2020 3120 3420 2020 2033  0 4     1 4    3
│ │ │ │ -0007b950: 2034 2020 2020 2030 2035 2020 2020 3320   4     0 5    3 
│ │ │ │ -0007b960: 3520 2020 2031 2036 2020 2020 2020 7c0a  5    1 6      |.
│ │ │ │ -0007b970: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +0007b8c0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007b8d0: 3279 2079 2020 202d 2032 7920 7920 202c  2y y   - 2y y  ,
│ │ │ │ +0007b8e0: 2037 7920 7920 202b 2079 2079 2020 2b20   7y y  + y y  + 
│ │ │ │ +0007b8f0: 3779 2079 2020 2d20 7920 7920 202b 2038  7y y  - y y  + 8
│ │ │ │ +0007b900: 7920 7920 202d 2079 2079 2020 2d20 7920  y y  - y y  - y 
│ │ │ │ +0007b910: 7920 202b 2020 2020 7c0a 7c20 2020 2020  y  +    |.|     
│ │ │ │ +0007b920: 2020 3320 3131 2020 2020 2034 2031 3120    3 11     4 11 
│ │ │ │ +0007b930: 2020 2030 2031 2020 2020 3020 3420 2020     0 1    0 4   
│ │ │ │ +0007b940: 2020 3120 3420 2020 2033 2034 2020 2020    1 4    3 4    
│ │ │ │ +0007b950: 2030 2035 2020 2020 3320 3520 2020 2031   0 5    3 5    1
│ │ │ │ +0007b960: 2036 2020 2020 2020 7c0a 7c20 2020 2020   6      |.|     
│ │ │ │ +0007b970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  0007b990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007b9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007b9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007b9c0: 7c20 2020 2020 3779 2079 2020 2b20 3879  |     7y y  + 8y
│ │ │ │ -0007b9d0: 2079 2020 2b20 7920 7920 202b 2038 7920   y  + y y  + 8y 
│ │ │ │ -0007b9e0: 7920 202d 2079 2079 2020 2d20 3879 2079  y  - y y  - 8y y
│ │ │ │ -0007b9f0: 2020 2b20 3779 2079 2020 2d20 3879 2079    + 7y y  - 8y y
│ │ │ │ -0007ba00: 2020 2b20 3779 2079 2020 2b20 2020 7c0a    + 7y y  +   |.
│ │ │ │ -0007ba10: 7c20 2020 2020 2020 3220 3620 2020 2020  |       2 6     
│ │ │ │ -0007ba20: 3520 3620 2020 2032 2037 2020 2020 2034  5 6    2 7     4
│ │ │ │ -0007ba30: 2037 2020 2020 3120 3820 2020 2020 3420   7    1 8     4 
│ │ │ │ -0007ba40: 3820 2020 2020 3520 3920 2020 2020 3720  8     5 9     7 
│ │ │ │ -0007ba50: 3920 2020 2020 3820 3920 2020 2020 7c0a  9     8 9     |.
│ │ │ │ -0007ba60: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +0007b9b0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007b9c0: 3779 2079 2020 2b20 3879 2079 2020 2b20  7y y  + 8y y  + 
│ │ │ │ +0007b9d0: 7920 7920 202b 2038 7920 7920 202d 2079  y y  + 8y y  - y
│ │ │ │ +0007b9e0: 2079 2020 2d20 3879 2079 2020 2b20 3779   y  - 8y y  + 7y
│ │ │ │ +0007b9f0: 2079 2020 2d20 3879 2079 2020 2b20 3779   y  - 8y y  + 7y
│ │ │ │ +0007ba00: 2079 2020 2b20 2020 7c0a 7c20 2020 2020   y  +   |.|     
│ │ │ │ +0007ba10: 2020 3220 3620 2020 2020 3520 3620 2020    2 6     5 6   
│ │ │ │ +0007ba20: 2032 2037 2020 2020 2034 2037 2020 2020   2 7     4 7    
│ │ │ │ +0007ba30: 3120 3820 2020 2020 3420 3820 2020 2020  1 8     4 8     
│ │ │ │ +0007ba40: 3520 3920 2020 2020 3720 3920 2020 2020  5 9     7 9     
│ │ │ │ +0007ba50: 3820 3920 2020 2020 7c0a 7c20 2020 2020  8 9     |.|     
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│ │ │ │  0007ba70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007ba80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007ba90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007baa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007bab0: 7c20 2020 2020 7920 7920 2020 2d20 7920  |     y y   - y 
│ │ │ │ -0007bac0: 7920 2020 2b20 3879 2079 2020 202d 2037  y   + 8y y   - 7
│ │ │ │ -0007bad0: 7920 7920 2020 2d20 3779 2079 2020 202d  y y   - 7y y   -
│ │ │ │ -0007bae0: 2037 7920 7920 2029 2020 2020 2020 2020   7y y  )        
│ │ │ │ -0007baf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007bb00: 7c20 2020 2020 2030 2031 3020 2020 2033  |      0 10    3
│ │ │ │ -0007bb10: 2031 3020 2020 2020 3620 3130 2020 2020   10     6 10    
│ │ │ │ -0007bb20: 2030 2031 3120 2020 2020 3420 3131 2020   0 11     4 11  
│ │ │ │ -0007bb30: 2020 2036 2031 3120 2020 2020 2020 2020     6 11         
│ │ │ │ -0007bb40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007bb50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007baa0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007bab0: 7920 7920 2020 2d20 7920 7920 2020 2b20  y y   - y y   + 
│ │ │ │ +0007bac0: 3879 2079 2020 202d 2037 7920 7920 2020  8y y   - 7y y   
│ │ │ │ +0007bad0: 2d20 3779 2079 2020 202d 2037 7920 7920  - 7y y   - 7y y 
│ │ │ │ +0007bae0: 2029 2020 2020 2020 2020 2020 2020 2020   )              
│ │ │ │ +0007baf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007bb00: 2030 2031 3020 2020 2033 2031 3020 2020   0 10    3 10   
│ │ │ │ +0007bb10: 2020 3620 3130 2020 2020 2030 2031 3120    6 10     0 11 
│ │ │ │ +0007bb20: 2020 2020 3420 3131 2020 2020 2036 2031      4 11     6 1
│ │ │ │ +0007bb30: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +0007bb40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007bb90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007bba0: 7c6f 3320 3a20 4964 6561 6c20 6f66 2051  |o3 : Ideal of Q
│ │ │ │ -0007bbb0: 515b 7920 2e2e 7920 205d 2020 2020 2020  Q[y ..y  ]      
│ │ │ │ +0007bb90: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ +0007bba0: 4964 6561 6c20 6f66 2051 515b 7920 2e2e  Ideal of QQ[y ..
│ │ │ │ +0007bbb0: 7920 205d 2020 2020 2020 2020 2020 2020  y  ]            
│ │ │ │  0007bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007bbe0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007bbf0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007bc00: 2020 2030 2020 2031 3120 2020 2020 2020     0   11       
│ │ │ │ +0007bbe0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007bbf0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +0007bc00: 2031 3120 2020 2020 2020 2020 2020 2020   11             
│ │ │ │  0007bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007bc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007bc30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007bc40: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0007bc30: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0007bc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007bc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007bc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007bc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0007bc90: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -0007bca0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6b65  ==..  * *note ke
│ │ │ │ -0007bcb0: 726e 656c 2852 696e 674d 6170 293a 2028  rnel(RingMap): (
│ │ │ │ -0007bcc0: 4d61 6361 756c 6179 3244 6f63 296b 6572  Macaulay2Doc)ker
│ │ │ │ -0007bcd0: 6e65 6c5f 6c70 5269 6e67 4d61 705f 7270  nel_lpRingMap_rp
│ │ │ │ -0007bce0: 2c20 2d2d 206b 6572 6e65 6c20 6f66 2061  , -- kernel of a
│ │ │ │ -0007bcf0: 0a20 2020 2072 696e 676d 6170 0a0a 5761  .    ringmap..Wa
│ │ │ │ -0007bd00: 7973 2074 6f20 7573 6520 7468 6973 206d  ys to use this m
│ │ │ │ -0007bd10: 6574 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d  ethod:.=========
│ │ │ │ -0007bd20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -0007bd30: 0a20 202a 202a 6e6f 7465 206b 6572 6e65  .  * *note kerne
│ │ │ │ -0007bd40: 6c28 5269 6e67 4d61 702c 5a5a 293a 206b  l(RingMap,ZZ): k
│ │ │ │ -0007bd50: 6572 6e65 6c5f 6c70 5269 6e67 4d61 705f  ernel_lpRingMap_
│ │ │ │ -0007bd60: 636d 5a5a 5f72 702c 202d 2d20 686f 6d6f  cmZZ_rp, -- homo
│ │ │ │ -0007bd70: 6765 6e65 6f75 730a 2020 2020 636f 6d70  geneous.    comp
│ │ │ │ -0007bd80: 6f6e 656e 7473 206f 6620 7468 6520 6b65  onents of the ke
│ │ │ │ -0007bd90: 726e 656c 206f 6620 6120 686f 6d6f 6765  rnel of a homoge
│ │ │ │ -0007bda0: 6e65 6f75 7320 7269 6e67 206d 6170 0a2d  neous ring map.-
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│ │ │ │  0007bf40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0007bf50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0007bf60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -0007bf80: 4675 6e63 7469 6f6e 3a20 2a6e 6f74 6520  Function: *note 
│ │ │ │ -0007bf90: 6d61 703a 2028 4d61 6361 756c 6179 3244  map: (Macaulay2D
│ │ │ │ -0007bfa0: 6f63 296d 6170 2c0a 2020 2a20 5573 6167  oc)map,.  * Usag
│ │ │ │ -0007bfb0: 653a 200a 2020 2020 2020 2020 6d61 7020  e: .        map 
│ │ │ │ -0007bfc0: 5068 690a 2020 2a20 496e 7075 7473 3a0a  Phi.  * Inputs:.
│ │ │ │ -0007bfd0: 2020 2020 2020 2a20 5068 692c 2061 202a        * Phi, a *
│ │ │ │ -0007bfe0: 6e6f 7465 2072 6174 696f 6e61 6c20 6d61  note rational ma
│ │ │ │ -0007bff0: 703a 2052 6174 696f 6e61 6c4d 6170 2c0a  p: RationalMap,.
│ │ │ │ -0007c000: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e    * *note Option
│ │ │ │ -0007c010: 616c 2069 6e70 7574 733a 2028 4d61 6361  al inputs: (Maca
│ │ │ │ -0007c020: 756c 6179 3244 6f63 2975 7369 6e67 2066  ulay2Doc)using f
│ │ │ │ -0007c030: 756e 6374 696f 6e73 2077 6974 6820 6f70  unctions with op
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│ │ │ │ -0007c060: 6772 6565 3a20 284d 6163 6175 6c61 7932  gree: (Macaulay2
│ │ │ │ -0007c070: 446f 6329 6d61 705f 6c70 5f70 645f 7064  Doc)map_lp_pd_pd
│ │ │ │ -0007c080: 5f70 645f 636d 4465 6772 6565 3d3e 5f70  _pd_cmDegree=>_p
│ │ │ │ -0007c090: 645f 7064 5f70 645f 7270 2c20 3d3e 0a20  d_pd_pd_rp, =>. 
│ │ │ │ -0007c0a0: 2020 2020 2020 202e 2e2e 2c20 6465 6661         ..., defa
│ │ │ │ -0007c0b0: 756c 7420 7661 6c75 6520 6e75 6c6c 2c0a  ult value null,.
│ │ │ │ -0007c0c0: 2020 2020 2020 2a20 2a6e 6f74 6520 4465        * *note De
│ │ │ │ -0007c0d0: 6772 6565 4c69 6674 3a20 284d 6163 6175  greeLift: (Macau
│ │ │ │ -0007c0e0: 6c61 7932 446f 6329 6d61 705f 6c70 5269  lay2Doc)map_lpRi
│ │ │ │ -0007c0f0: 6e67 5f63 6d52 696e 675f 636d 4d61 7472  ng_cmRing_cmMatr
│ │ │ │ -0007c100: 6978 5f72 702c 203d 3e20 2e2e 2e2c 0a20  ix_rp, => ...,. 
│ │ │ │ -0007c110: 2020 2020 2020 2064 6566 6175 6c74 2076         default v
│ │ │ │ -0007c120: 616c 7565 206e 756c 6c2c 0a20 2020 2020  alue null,.     
│ │ │ │ -0007c130: 202a 202a 6e6f 7465 2044 6567 7265 654d   * *note DegreeM
│ │ │ │ -0007c140: 6170 3a20 284d 6163 6175 6c61 7932 446f  ap: (Macaulay2Do
│ │ │ │ -0007c150: 6329 6d61 705f 6c70 5269 6e67 5f63 6d52  c)map_lpRing_cmR
│ │ │ │ -0007c160: 696e 675f 636d 4d61 7472 6978 5f72 702c  ing_cmMatrix_rp,
│ │ │ │ -0007c170: 203d 3e20 2e2e 2e2c 0a20 2020 2020 2020   => ...,.       
│ │ │ │ -0007c180: 2064 6566 6175 6c74 2076 616c 7565 206e   default value n
│ │ │ │ -0007c190: 756c 6c2c 0a20 202a 204f 7574 7075 7473  ull,.  * Outputs
│ │ │ │ -0007c1a0: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74  :.      * a *not
│ │ │ │ -0007c1b0: 6520 7269 6e67 206d 6170 3a20 284d 6163  e ring map: (Mac
│ │ │ │ -0007c1c0: 6175 6c61 7932 446f 6329 5269 6e67 4d61  aulay2Doc)RingMa
│ │ │ │ -0007c1d0: 702c 2c20 7468 6520 7269 6e67 206d 6170  p,, the ring map
│ │ │ │ -0007c1e0: 2066 726f 6d20 7768 6963 6820 7468 650a   from which the.
│ │ │ │ -0007c1f0: 2020 2020 2020 2020 7261 7469 6f6e 616c          rational
│ │ │ │ -0007c200: 206d 6170 2050 6869 2077 6173 2064 6566   map Phi was def
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│ │ │ │ +0007bf70: 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 7469  ****..  * Functi
│ │ │ │ +0007bf80: 6f6e 3a20 2a6e 6f74 6520 6d61 703a 2028  on: *note map: (
│ │ │ │ +0007bf90: 4d61 6361 756c 6179 3244 6f63 296d 6170  Macaulay2Doc)map
│ │ │ │ +0007bfa0: 2c0a 2020 2a20 5573 6167 653a 200a 2020  ,.  * Usage: .  
│ │ │ │ +0007bfb0: 2020 2020 2020 6d61 7020 5068 690a 2020        map Phi.  
│ │ │ │ +0007bfc0: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ +0007bfd0: 2a20 5068 692c 2061 202a 6e6f 7465 2072  * Phi, a *note r
│ │ │ │ +0007bfe0: 6174 696f 6e61 6c20 6d61 703a 2052 6174  ational map: Rat
│ │ │ │ +0007bff0: 696f 6e61 6c4d 6170 2c0a 2020 2a20 2a6e  ionalMap,.  * *n
│ │ │ │ +0007c000: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70  ote Optional inp
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│ │ │ │ +0007c020: 6f63 2975 7369 6e67 2066 756e 6374 696f  oc)using functio
│ │ │ │ +0007c030: 6e73 2077 6974 6820 6f70 7469 6f6e 616c  ns with optional
│ │ │ │ +0007c040: 2069 6e70 7574 732c 3a0a 2020 2020 2020   inputs,:.      
│ │ │ │ +0007c050: 2a20 2a6e 6f74 6520 4465 6772 6565 3a20  * *note Degree: 
│ │ │ │ +0007c060: 284d 6163 6175 6c61 7932 446f 6329 6d61  (Macaulay2Doc)ma
│ │ │ │ +0007c070: 705f 6c70 5f70 645f 7064 5f70 645f 636d  p_lp_pd_pd_pd_cm
│ │ │ │ +0007c080: 4465 6772 6565 3d3e 5f70 645f 7064 5f70  Degree=>_pd_pd_p
│ │ │ │ +0007c090: 645f 7270 2c20 3d3e 0a20 2020 2020 2020  d_rp, =>.       
│ │ │ │ +0007c0a0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ +0007c0b0: 6c75 6520 6e75 6c6c 2c0a 2020 2020 2020  lue null,.      
│ │ │ │ +0007c0c0: 2a20 2a6e 6f74 6520 4465 6772 6565 4c69  * *note DegreeLi
│ │ │ │ +0007c0d0: 6674 3a20 284d 6163 6175 6c61 7932 446f  ft: (Macaulay2Do
│ │ │ │ +0007c0e0: 6329 6d61 705f 6c70 5269 6e67 5f63 6d52  c)map_lpRing_cmR
│ │ │ │ +0007c0f0: 696e 675f 636d 4d61 7472 6978 5f72 702c  ing_cmMatrix_rp,
│ │ │ │ +0007c100: 203d 3e20 2e2e 2e2c 0a20 2020 2020 2020   => ...,.       
│ │ │ │ +0007c110: 2064 6566 6175 6c74 2076 616c 7565 206e   default value n
│ │ │ │ +0007c120: 756c 6c2c 0a20 2020 2020 202a 202a 6e6f  ull,.      * *no
│ │ │ │ +0007c130: 7465 2044 6567 7265 654d 6170 3a20 284d  te DegreeMap: (M
│ │ │ │ +0007c140: 6163 6175 6c61 7932 446f 6329 6d61 705f  acaulay2Doc)map_
│ │ │ │ +0007c150: 6c70 5269 6e67 5f63 6d52 696e 675f 636d  lpRing_cmRing_cm
│ │ │ │ +0007c160: 4d61 7472 6978 5f72 702c 203d 3e20 2e2e  Matrix_rp, => ..
│ │ │ │ +0007c170: 2e2c 0a20 2020 2020 2020 2064 6566 6175  .,.        defau
│ │ │ │ +0007c180: 6c74 2076 616c 7565 206e 756c 6c2c 0a20  lt value null,. 
│ │ │ │ +0007c190: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +0007c1a0: 2020 2a20 6120 2a6e 6f74 6520 7269 6e67    * a *note ring
│ │ │ │ +0007c1b0: 206d 6170 3a20 284d 6163 6175 6c61 7932   map: (Macaulay2
│ │ │ │ +0007c1c0: 446f 6329 5269 6e67 4d61 702c 2c20 7468  Doc)RingMap,, th
│ │ │ │ +0007c1d0: 6520 7269 6e67 206d 6170 2066 726f 6d20  e ring map from 
│ │ │ │ +0007c1e0: 7768 6963 6820 7468 650a 2020 2020 2020  which the.      
│ │ │ │ +0007c1f0: 2020 7261 7469 6f6e 616c 206d 6170 2050    rational map P
│ │ │ │ +0007c200: 6869 2077 6173 2064 6566 696e 6564 0a0a  hi was defined..
│ │ │ │ +0007c210: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +0007c220: 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d  =======..+------
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│ │ │ │  0007c250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007c260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007c270: 2d2d 2d2b 0a7c 6931 203a 2051 515b 745f  ---+.|i1 : QQ[t_
│ │ │ │ -0007c280: 302e 2e74 5f33 5d20 2020 2020 2020 2020  0..t_3]         
│ │ │ │ +0007c260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0007c270: 6931 203a 2051 515b 745f 302e 2e74 5f33  i1 : QQ[t_0..t_3
│ │ │ │ +0007c280: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │  0007c290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c2b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0007c2b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0007c2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0007c2f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0007c300: 0a7c 6f31 203d 2051 515b 7420 2e2e 7420  .|o1 = QQ[t ..t 
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│ │ │ │ +0007c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c340: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007c350: 2030 2020 2033 2020 2020 2020 2020 2020   0   3          
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│ │ │ │ -0007c3e0: 6961 6c52 696e 6720 2020 2020 2020 2020  ialRing         
│ │ │ │ +0007c3c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0007c3d0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ +0007c3e0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │  0007c3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0007c430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007c440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007c450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0007c460: 6932 203a 2050 6869 203d 2072 6174 696f  i2 : Phi = ratio
│ │ │ │ -0007c470: 6e61 6c4d 6170 207b 745f 315e 322b 745f  nalMap {t_1^2+t_
│ │ │ │ -0007c480: 325e 322b 745f 335e 322c 745f 302a 745f  2^2+t_3^2,t_0*t_
│ │ │ │ -0007c490: 312c 745f 302a 745f 322c 745f 302a 745f  1,t_0*t_2,t_0*t_
│ │ │ │ -0007c4a0: 337d 207c 0a7c 2020 2020 2020 2020 2020  3} |.|          
│ │ │ │ +0007c450: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2050  -------+.|i2 : P
│ │ │ │ +0007c460: 6869 203d 2072 6174 696f 6e61 6c4d 6170  hi = rationalMap
│ │ │ │ +0007c470: 207b 745f 315e 322b 745f 325e 322b 745f   {t_1^2+t_2^2+t_
│ │ │ │ +0007c480: 335e 322c 745f 302a 745f 312c 745f 302a  3^2,t_0*t_1,t_0*
│ │ │ │ +0007c490: 745f 322c 745f 302a 745f 337d 207c 0a7c  t_2,t_0*t_3} |.|
│ │ │ │ +0007c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c4e0: 2020 2020 2020 2020 207c 0a7c 6f32 203d           |.|o2 =
│ │ │ │ -0007c4f0: 202d 2d20 7261 7469 6f6e 616c 206d 6170   -- rational map
│ │ │ │ -0007c500: 202d 2d20 2020 2020 2020 2020 2020 2020   --             
│ │ │ │ +0007c4e0: 2020 207c 0a7c 6f32 203d 202d 2d20 7261     |.|o2 = -- ra
│ │ │ │ +0007c4f0: 7469 6f6e 616c 206d 6170 202d 2d20 2020  tional map --   
│ │ │ │ +0007c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c520: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0007c530: 0a7c 2020 2020 2073 6f75 7263 653a 2050  .|     source: P
│ │ │ │ -0007c540: 726f 6a28 5151 5b74 202c 2074 202c 2074  roj(QQ[t , t , t
│ │ │ │ -0007c550: 202c 2074 205d 2920 2020 2020 2020 2020   , t ])         
│ │ │ │ -0007c560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c570: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007c580: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -0007c590: 2020 3120 2020 3220 2020 3320 2020 2020    1   2   3     
│ │ │ │ +0007c520: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0007c530: 2073 6f75 7263 653a 2050 726f 6a28 5151   source: Proj(QQ
│ │ │ │ +0007c540: 5b74 202c 2074 202c 2074 202c 2074 205d  [t , t , t , t ]
│ │ │ │ +0007c550: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0007c560: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0007c570: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0007c580: 2020 2020 2020 2020 3020 2020 3120 2020          0   1   
│ │ │ │ +0007c590: 3220 2020 3320 2020 2020 2020 2020 2020  2   3           
│ │ │ │  0007c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c5b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0007c5c0: 2020 2074 6172 6765 743a 2050 726f 6a28     target: Proj(
│ │ │ │ -0007c5d0: 5151 5b74 202c 2074 202c 2074 202c 2074  QQ[t , t , t , t
│ │ │ │ -0007c5e0: 205d 2920 2020 2020 2020 2020 2020 2020   ])             
│ │ │ │ -0007c5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c600: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0007c610: 2020 2020 2020 2020 2020 3020 2020 3120            0   1 
│ │ │ │ -0007c620: 2020 3220 2020 3320 2020 2020 2020 2020    2   3         
│ │ │ │ +0007c5b0: 2020 2020 207c 0a7c 2020 2020 2074 6172       |.|     tar
│ │ │ │ +0007c5c0: 6765 743a 2050 726f 6a28 5151 5b74 202c  get: Proj(QQ[t ,
│ │ │ │ +0007c5d0: 2074 202c 2074 202c 2074 205d 2920 2020   t , t , t ])   
│ │ │ │ +0007c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007c5f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007c600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007c610: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ +0007c620: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │  0007c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c640: 2020 2020 2020 207c 0a7c 2020 2020 2064         |.|     d
│ │ │ │ -0007c650: 6566 696e 696e 6720 666f 726d 733a 207b  efining forms: {
│ │ │ │ +0007c640: 207c 0a7c 2020 2020 2064 6566 696e 696e   |.|     definin
│ │ │ │ +0007c650: 6720 666f 726d 733a 207b 2020 2020 2020  g forms: {      
│ │ │ │  0007c660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c680: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0007c680: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0007c690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c6a0: 2020 2020 2020 2032 2020 2020 3220 2020         2    2   
│ │ │ │ -0007c6b0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0007c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c6d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0007c6e0: 2020 2020 2020 2020 2020 2020 7420 202b              t  +
│ │ │ │ -0007c6f0: 2074 2020 2b20 7420 2c20 2020 2020 2020   t  + t ,       
│ │ │ │ +0007c6a0: 2032 2020 2020 3220 2020 2032 2020 2020   2    2    2    
│ │ │ │ +0007c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007c6c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0007c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007c6e0: 2020 2020 2020 7420 202b 2074 2020 2b20        t  + t  + 
│ │ │ │ +0007c6f0: 7420 2c20 2020 2020 2020 2020 2020 2020  t ,             
│ │ │ │  0007c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c710: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0007c720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c730: 2020 2031 2020 2020 3220 2020 2033 2020     1    2    3  
│ │ │ │ +0007c710: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0007c720: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +0007c730: 2020 3220 2020 2033 2020 2020 2020 2020    2    3        
│ │ │ │  0007c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0007c760: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0007c750: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0007c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c7a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007c7b0: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ -0007c7c0: 7420 2c20 2020 2020 2020 2020 2020 2020  t ,             
│ │ │ │ +0007c790: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0007c7a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0007c7b0: 2020 2020 2020 2020 7420 7420 2c20 2020          t t ,   
│ │ │ │ +0007c7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c7e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0007c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c800: 2020 2020 2030 2031 2020 2020 2020 2020       0 1        
│ │ │ │ +0007c7e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0007c7f0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +0007c800: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │  0007c810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c830: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0007c820: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007c830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c870: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0007c880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c890: 7420 7420 2c20 2020 2020 2020 2020 2020  t t ,           
│ │ │ │ +0007c870: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0007c880: 2020 2020 2020 2020 2020 7420 7420 2c20            t t , 
│ │ │ │ +0007c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c8b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0007c8b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0007c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c8d0: 2020 2020 2020 2030 2032 2020 2020 2020         0 2      
│ │ │ │ +0007c8d0: 2030 2032 2020 2020 2020 2020 2020 2020   0 2            
│ │ │ │  0007c8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c900: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0007c8f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0007c900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c940: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0007c950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c960: 2020 7420 7420 2020 2020 2020 2020 2020    t t           
│ │ │ │ +0007c940: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0007c950: 2020 2020 2020 2020 2020 2020 7420 7420              t t 
│ │ │ │ +0007c960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007c970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0007c990: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0007c9a0: 2020 2020 2020 2020 2030 2033 2020 2020           0 3    
│ │ │ │ +0007c980: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0007c990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007c9a0: 2020 2030 2033 2020 2020 2020 2020 2020     0 3          
│ │ │ │  0007c9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007c9d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0007c9e0: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ +0007c9c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0007c9d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0007c9e0: 2020 2020 2020 207d 2020 2020 2020 2020         }        
│ │ │ │  0007c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ca00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ca10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0007ca10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0007ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ca60: 207c 0a7c 6f32 203a 2052 6174 696f 6e61   |.|o2 : Rationa
│ │ │ │ -0007ca70: 6c4d 6170 2028 7175 6164 7261 7469 6320  lMap (quadratic 
│ │ │ │ -0007ca80: 7261 7469 6f6e 616c 206d 6170 2066 726f  rational map fro
│ │ │ │ -0007ca90: 6d20 5050 5e33 2074 6f20 5050 5e33 2920  m PP^3 to PP^3) 
│ │ │ │ -0007caa0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0007ca50: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0007ca60: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ +0007ca70: 7175 6164 7261 7469 6320 7261 7469 6f6e  quadratic ration
│ │ │ │ +0007ca80: 616c 206d 6170 2066 726f 6d20 5050 5e33  al map from PP^3
│ │ │ │ +0007ca90: 2074 6f20 5050 5e33 2920 2020 2020 2020   to PP^3)       
│ │ │ │ +0007caa0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0007cab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007cac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007cad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007cae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0007caf0: 6933 203a 206d 6170 2050 6869 2020 2020  i3 : map Phi    
│ │ │ │ +0007cae0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 206d  -------+.|i3 : m
│ │ │ │ +0007caf0: 6170 2050 6869 2020 2020 2020 2020 2020  ap Phi          
│ │ │ │  0007cb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cb30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0007cb20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0007cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cb70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0007cb70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0007cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cba0: 3220 2020 2032 2020 2020 3220 2020 2020  2    2    2     
│ │ │ │ -0007cbb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0007cbc0: 0a7c 6f33 203d 206d 6170 2028 5151 5b74  .|o3 = map (QQ[t
│ │ │ │ -0007cbd0: 202e 2e74 205d 2c20 5151 5b74 202e 2e74   ..t ], QQ[t ..t
│ │ │ │ -0007cbe0: 205d 2c20 7b74 2020 2b20 7420 202b 2074   ], {t  + t  + t
│ │ │ │ -0007cbf0: 202c 2074 2074 202c 2074 2074 202c 2074   , t t , t t , t
│ │ │ │ -0007cc00: 2074 207d 297c 0a7c 2020 2020 2020 2020   t })|.|        
│ │ │ │ -0007cc10: 2020 2020 2020 3020 2020 3320 2020 2020        0   3     
│ │ │ │ -0007cc20: 2020 3020 2020 3320 2020 2020 3120 2020    0   3     1   
│ │ │ │ -0007cc30: 2032 2020 2020 3320 2020 3020 3120 2020   2    3   0 1   
│ │ │ │ -0007cc40: 3020 3220 2020 3020 3320 207c 0a7c 2020  0 2   0 3  |.|  
│ │ │ │ +0007cb90: 2020 2020 2020 2020 2020 3220 2020 2032            2    2
│ │ │ │ +0007cba0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0007cbb0: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ +0007cbc0: 206d 6170 2028 5151 5b74 202e 2e74 205d   map (QQ[t ..t ]
│ │ │ │ +0007cbd0: 2c20 5151 5b74 202e 2e74 205d 2c20 7b74  , QQ[t ..t ], {t
│ │ │ │ +0007cbe0: 2020 2b20 7420 202b 2074 202c 2074 2074    + t  + t , t t
│ │ │ │ +0007cbf0: 202c 2074 2074 202c 2074 2074 207d 297c   , t t , t t })|
│ │ │ │ +0007cc00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0007cc10: 3020 2020 3320 2020 2020 2020 3020 2020  0   3       0   
│ │ │ │ +0007cc20: 3320 2020 2020 3120 2020 2032 2020 2020  3     1    2    
│ │ │ │ +0007cc30: 3320 2020 3020 3120 2020 3020 3220 2020  3   0 1   0 2   
│ │ │ │ +0007cc40: 3020 3320 207c 0a7c 2020 2020 2020 2020  0 3  |.|        
│ │ │ │  0007cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cc90: 207c 0a7c 6f33 203a 2052 696e 674d 6170   |.|o3 : RingMap
│ │ │ │ -0007cca0: 2051 515b 7420 2e2e 7420 5d20 3c2d 2d20   QQ[t ..t ] <-- 
│ │ │ │ -0007ccb0: 5151 5b74 202e 2e74 205d 2020 2020 2020  QQ[t ..t ]      
│ │ │ │ +0007cc80: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +0007cc90: 203a 2052 696e 674d 6170 2051 515b 7420   : RingMap QQ[t 
│ │ │ │ +0007cca0: 2e2e 7420 5d20 3c2d 2d20 5151 5b74 202e  ..t ] <-- QQ[t .
│ │ │ │ +0007ccb0: 2e74 205d 2020 2020 2020 2020 2020 2020  .t ]            
│ │ │ │  0007ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ccd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0007cce0: 2020 2020 2020 2020 2020 2030 2020 2033             0   3
│ │ │ │ -0007ccf0: 2020 2020 2020 2020 2020 3020 2020 3320            0   3 
│ │ │ │ +0007ccd0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0007cce0: 2020 2020 2030 2020 2033 2020 2020 2020       0   3      
│ │ │ │ +0007ccf0: 2020 2020 3020 2020 3320 2020 2020 2020      0   3       
│ │ │ │  0007cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007cd10: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0007cd10: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0007cd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007cd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007cd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007cd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007cd60: 2d2d 2d2b 0a0a 5468 6520 636f 6d6d 616e  ---+..The comman
│ │ │ │ -0007cd70: 6420 6d61 7020 5068 6920 6973 2065 7175  d map Phi is equ
│ │ │ │ -0007cd80: 6976 616c 656e 7420 746f 206d 6170 2830  ivalent to map(0
│ │ │ │ -0007cd90: 2c50 6869 292e 204d 6f72 6520 6765 6e65  ,Phi). More gene
│ │ │ │ -0007cda0: 7261 6c6c 792c 2074 6865 2063 6f6d 6d61  rally, the comma
│ │ │ │ -0007cdb0: 6e64 0a6d 6170 2869 2c50 6869 2920 7265  nd.map(i,Phi) re
│ │ │ │ -0007cdc0: 7475 726e 7320 7468 6520 692d 7468 2072  turns the i-th r
│ │ │ │ -0007cdd0: 6570 7265 7365 6e74 6174 6976 6520 6f66  epresentative of
│ │ │ │ -0007cde0: 2074 6865 206d 6170 2050 6869 2e0a 0a53   the map Phi...S
│ │ │ │ -0007cdf0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ -0007ce00: 0a0a 2020 2a20 2a6e 6f74 6520 6d61 7472  ..  * *note matr
│ │ │ │ -0007ce10: 6978 2852 6174 696f 6e61 6c4d 6170 293a  ix(RationalMap):
│ │ │ │ -0007ce20: 206d 6174 7269 785f 6c70 5261 7469 6f6e   matrix_lpRation
│ │ │ │ -0007ce30: 616c 4d61 705f 7270 2c20 2d2d 2074 6865  alMap_rp, -- the
│ │ │ │ -0007ce40: 206d 6174 7269 780a 2020 2020 6173 736f   matrix.    asso
│ │ │ │ -0007ce50: 6369 6174 6564 2074 6f20 6120 7261 7469  ciated to a rati
│ │ │ │ -0007ce60: 6f6e 616c 206d 6170 0a0a 5761 7973 2074  onal map..Ways t
│ │ │ │ -0007ce70: 6f20 7573 6520 7468 6973 206d 6574 686f  o use this metho
│ │ │ │ -0007ce80: 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  d:.=============
│ │ │ │ -0007ce90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -0007cea0: 202a 6e6f 7465 206d 6170 2852 6174 696f   *note map(Ratio
│ │ │ │ -0007ceb0: 6e61 6c4d 6170 293a 206d 6170 5f6c 7052  nalMap): map_lpR
│ │ │ │ -0007cec0: 6174 696f 6e61 6c4d 6170 5f72 702c 202d  ationalMap_rp, -
│ │ │ │ -0007ced0: 2d20 6765 7420 7468 6520 7269 6e67 206d  - get the ring m
│ │ │ │ -0007cee0: 6170 2064 6566 696e 696e 670a 2020 2020  ap defining.    
│ │ │ │ -0007cef0: 6120 7261 7469 6f6e 616c 206d 6170 0a20  a rational map. 
│ │ │ │ -0007cf00: 202a 2022 6d61 7028 5a5a 2c52 6174 696f   * "map(ZZ,Ratio
│ │ │ │ -0007cf10: 6e61 6c4d 6170 2922 0a2d 2d2d 2d2d 2d2d  nalMap)".-------
│ │ │ │ +0007cd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +0007cd60: 5468 6520 636f 6d6d 616e 6420 6d61 7020  The command map 
│ │ │ │ +0007cd70: 5068 6920 6973 2065 7175 6976 616c 656e  Phi is equivalen
│ │ │ │ +0007cd80: 7420 746f 206d 6170 2830 2c50 6869 292e  t to map(0,Phi).
│ │ │ │ +0007cd90: 204d 6f72 6520 6765 6e65 7261 6c6c 792c   More generally,
│ │ │ │ +0007cda0: 2074 6865 2063 6f6d 6d61 6e64 0a6d 6170   the command.map
│ │ │ │ +0007cdb0: 2869 2c50 6869 2920 7265 7475 726e 7320  (i,Phi) returns 
│ │ │ │ +0007cdc0: 7468 6520 692d 7468 2072 6570 7265 7365  the i-th represe
│ │ │ │ +0007cdd0: 6e74 6174 6976 6520 6f66 2074 6865 206d  ntative of the m
│ │ │ │ +0007cde0: 6170 2050 6869 2e0a 0a53 6565 2061 6c73  ap Phi...See als
│ │ │ │ +0007cdf0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ +0007ce00: 2a6e 6f74 6520 6d61 7472 6978 2852 6174  *note matrix(Rat
│ │ │ │ +0007ce10: 696f 6e61 6c4d 6170 293a 206d 6174 7269  ionalMap): matri
│ │ │ │ +0007ce20: 785f 6c70 5261 7469 6f6e 616c 4d61 705f  x_lpRationalMap_
│ │ │ │ +0007ce30: 7270 2c20 2d2d 2074 6865 206d 6174 7269  rp, -- the matri
│ │ │ │ +0007ce40: 780a 2020 2020 6173 736f 6369 6174 6564  x.    associated
│ │ │ │ +0007ce50: 2074 6f20 6120 7261 7469 6f6e 616c 206d   to a rational m
│ │ │ │ +0007ce60: 6170 0a0a 5761 7973 2074 6f20 7573 6520  ap..Ways to use 
│ │ │ │ +0007ce70: 7468 6973 206d 6574 686f 643a 0a3d 3d3d  this method:.===
│ │ │ │ +0007ce80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0007ce90: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +0007cea0: 206d 6170 2852 6174 696f 6e61 6c4d 6170   map(RationalMap
│ │ │ │ +0007ceb0: 293a 206d 6170 5f6c 7052 6174 696f 6e61  ): map_lpRationa
│ │ │ │ +0007cec0: 6c4d 6170 5f72 702c 202d 2d20 6765 7420  lMap_rp, -- get 
│ │ │ │ +0007ced0: 7468 6520 7269 6e67 206d 6170 2064 6566  the ring map def
│ │ │ │ +0007cee0: 696e 696e 670a 2020 2020 6120 7261 7469  ining.    a rati
│ │ │ │ +0007cef0: 6f6e 616c 206d 6170 0a20 202a 2022 6d61  onal map.  * "ma
│ │ │ │ +0007cf00: 7028 5a5a 2c52 6174 696f 6e61 6c4d 6170  p(ZZ,RationalMap
│ │ │ │ +0007cf10: 2922 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  )".-------------
│ │ │ │  0007cf20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007cf30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007cf40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007cf50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007cf60: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ -0007cf70: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ -0007cf80: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ -0007cf90: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ -0007cfa0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ -0007cfb0: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163  .25.11+ds/M2/Mac
│ │ │ │ -0007cfc0: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ -0007cfd0: 4372 656d 6f6e 612f 0a64 6f63 756d 656e  Cremona/.documen
│ │ │ │ -0007cfe0: 7461 7469 6f6e 2e6d 323a 3731 383a 302e  tation.m2:718:0.
│ │ │ │ -0007cff0: 0a1f 0a46 696c 653a 2043 7265 6d6f 6e61  ...File: Cremona
│ │ │ │ -0007d000: 2e69 6e66 6f2c 204e 6f64 653a 206d 6174  .info, Node: mat
│ │ │ │ -0007d010: 7269 785f 6c70 5261 7469 6f6e 616c 4d61  rix_lpRationalMa
│ │ │ │ -0007d020: 705f 7270 2c20 4e65 7874 3a20 4e75 6d44  p_rp, Next: NumD
│ │ │ │ -0007d030: 6567 7265 6573 2c20 5072 6576 3a20 6d61  egrees, Prev: ma
│ │ │ │ -0007d040: 705f 6c70 5261 7469 6f6e 616c 4d61 705f  p_lpRationalMap_
│ │ │ │ -0007d050: 7270 2c20 5570 3a20 546f 700a 0a6d 6174  rp, Up: Top..mat
│ │ │ │ -0007d060: 7269 7828 5261 7469 6f6e 616c 4d61 7029  rix(RationalMap)
│ │ │ │ -0007d070: 202d 2d20 7468 6520 6d61 7472 6978 2061   -- the matrix a
│ │ │ │ -0007d080: 7373 6f63 6961 7465 6420 746f 2061 2072  ssociated to a r
│ │ │ │ -0007d090: 6174 696f 6e61 6c20 6d61 700a 2a2a 2a2a  ational map.****
│ │ │ │ +0007cf60: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ +0007cf70: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ +0007cf80: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ +0007cf90: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ +0007cfa0: 6163 6175 6c61 7932 2d31 2e32 352e 3131  acaulay2-1.25.11
│ │ │ │ +0007cfb0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ +0007cfc0: 2f70 6163 6b61 6765 732f 4372 656d 6f6e  /packages/Cremon
│ │ │ │ +0007cfd0: 612f 0a64 6f63 756d 656e 7461 7469 6f6e  a/.documentation
│ │ │ │ +0007cfe0: 2e6d 323a 3731 383a 302e 0a1f 0a46 696c  .m2:718:0....Fil
│ │ │ │ +0007cff0: 653a 2043 7265 6d6f 6e61 2e69 6e66 6f2c  e: Cremona.info,
│ │ │ │ +0007d000: 204e 6f64 653a 206d 6174 7269 785f 6c70   Node: matrix_lp
│ │ │ │ +0007d010: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ +0007d020: 4e65 7874 3a20 4e75 6d44 6567 7265 6573  Next: NumDegrees
│ │ │ │ +0007d030: 2c20 5072 6576 3a20 6d61 705f 6c70 5261  , Prev: map_lpRa
│ │ │ │ +0007d040: 7469 6f6e 616c 4d61 705f 7270 2c20 5570  tionalMap_rp, Up
│ │ │ │ +0007d050: 3a20 546f 700a 0a6d 6174 7269 7828 5261  : Top..matrix(Ra
│ │ │ │ +0007d060: 7469 6f6e 616c 4d61 7029 202d 2d20 7468  tionalMap) -- th
│ │ │ │ +0007d070: 6520 6d61 7472 6978 2061 7373 6f63 6961  e matrix associa
│ │ │ │ +0007d080: 7465 6420 746f 2061 2072 6174 696f 6e61  ted to a rationa
│ │ │ │ +0007d090: 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a  l map.**********
│ │ │ │  0007d0a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0007d0b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0007d0c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0007d0d0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -0007d0e0: 4675 6e63 7469 6f6e 3a20 2a6e 6f74 6520  Function: *note 
│ │ │ │ -0007d0f0: 6d61 7472 6978 3a20 284d 6163 6175 6c61  matrix: (Macaula
│ │ │ │ -0007d100: 7932 446f 6329 6d61 7472 6978 2c0a 2020  y2Doc)matrix,.  
│ │ │ │ -0007d110: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ -0007d120: 2020 6d61 7472 6978 2050 6869 0a20 202a    matrix Phi.  *
│ │ │ │ -0007d130: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -0007d140: 2050 6869 2c20 6120 2a6e 6f74 6520 7261   Phi, a *note ra
│ │ │ │ -0007d150: 7469 6f6e 616c 206d 6170 3a20 5261 7469  tional map: Rati
│ │ │ │ -0007d160: 6f6e 616c 4d61 702c 0a20 202a 202a 6e6f  onalMap,.  * *no
│ │ │ │ -0007d170: 7465 204f 7074 696f 6e61 6c20 696e 7075  te Optional inpu
│ │ │ │ -0007d180: 7473 3a20 284d 6163 6175 6c61 7932 446f  ts: (Macaulay2Do
│ │ │ │ -0007d190: 6329 7573 696e 6720 6675 6e63 7469 6f6e  c)using function
│ │ │ │ -0007d1a0: 7320 7769 7468 206f 7074 696f 6e61 6c20  s with optional 
│ │ │ │ -0007d1b0: 696e 7075 7473 2c3a 0a20 2020 2020 202a  inputs,:.      *
│ │ │ │ -0007d1c0: 202a 6e6f 7465 2044 6567 7265 653a 2028   *note Degree: (
│ │ │ │ -0007d1d0: 4d61 6361 756c 6179 3244 6f63 296d 6174  Macaulay2Doc)mat
│ │ │ │ -0007d1e0: 7269 785f 6c70 4c69 7374 5f72 702c 203d  rix_lpList_rp, =
│ │ │ │ -0007d1f0: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ -0007d200: 616c 7565 0a20 2020 2020 2020 206e 756c  alue.        nul
│ │ │ │ -0007d210: 6c2c 0a20 202a 204f 7574 7075 7473 3a0a  l,.  * Outputs:.
│ │ │ │ -0007d220: 2020 2020 2020 2a20 6120 2a6e 6f74 6520        * a *note 
│ │ │ │ -0007d230: 6d61 7472 6978 3a20 284d 6163 6175 6c61  matrix: (Macaula
│ │ │ │ -0007d240: 7932 446f 6329 4d61 7472 6978 2c2c 2074  y2Doc)Matrix,, t
│ │ │ │ -0007d250: 6865 206d 6174 7269 7820 6173 736f 6369  he matrix associ
│ │ │ │ -0007d260: 6174 6564 2074 6f20 7468 650a 2020 2020  ated to the.    
│ │ │ │ -0007d270: 2020 2020 7269 6e67 206d 6170 2064 6566      ring map def
│ │ │ │ -0007d280: 696e 696e 6720 7468 6520 7261 7469 6f6e  ining the ration
│ │ │ │ -0007d290: 616c 206d 6170 2050 6869 0a0a 4465 7363  al map Phi..Desc
│ │ │ │ -0007d2a0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ -0007d2b0: 3d3d 3d0a 0a54 6869 7320 6973 2065 7175  ===..This is equ
│ │ │ │ -0007d2c0: 6976 616c 656e 7420 746f 206d 6174 7269  ivalent to matri
│ │ │ │ -0007d2d0: 7820 6d61 7020 5068 692e 204d 6f72 656f  x map Phi. Moreo
│ │ │ │ -0007d2e0: 7665 722c 2074 6865 2063 6f6d 6d61 6e64  ver, the command
│ │ │ │ -0007d2f0: 206d 6174 7269 7820 5068 6920 6973 0a65   matrix Phi is.e
│ │ │ │ -0007d300: 7175 6976 616c 656e 7420 746f 206d 6174  quivalent to mat
│ │ │ │ -0007d310: 7269 7828 302c 5068 6929 2c20 616e 6420  rix(0,Phi), and 
│ │ │ │ -0007d320: 6d6f 7265 2067 656e 6572 616c 6c79 2074  more generally t
│ │ │ │ -0007d330: 6865 2063 6f6d 6d61 6e64 206d 6174 7269  he command matri
│ │ │ │ -0007d340: 7828 692c 5068 6929 0a72 6574 7572 6e73  x(i,Phi).returns
│ │ │ │ -0007d350: 2074 6865 206d 6174 7269 7820 6f66 2074   the matrix of t
│ │ │ │ -0007d360: 6865 2069 2d74 6820 7265 7072 6573 656e  he i-th represen
│ │ │ │ -0007d370: 7461 7469 7665 206f 6620 5068 692e 0a0a  tative of Phi...
│ │ │ │ -0007d380: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ -0007d390: 3d0a 0a20 202a 202a 6e6f 7465 206d 6170  =..  * *note map
│ │ │ │ -0007d3a0: 2852 6174 696f 6e61 6c4d 6170 293a 206d  (RationalMap): m
│ │ │ │ -0007d3b0: 6170 5f6c 7052 6174 696f 6e61 6c4d 6170  ap_lpRationalMap
│ │ │ │ -0007d3c0: 5f72 702c 202d 2d20 6765 7420 7468 6520  _rp, -- get the 
│ │ │ │ -0007d3d0: 7269 6e67 206d 6170 2064 6566 696e 696e  ring map definin
│ │ │ │ -0007d3e0: 670a 2020 2020 6120 7261 7469 6f6e 616c  g.    a rational
│ │ │ │ -0007d3f0: 206d 6170 0a20 202a 202a 6e6f 7465 206d   map.  * *note m
│ │ │ │ -0007d400: 6174 7269 7828 5269 6e67 4d61 7029 3a20  atrix(RingMap): 
│ │ │ │ -0007d410: 284d 6163 6175 6c61 7932 446f 6329 6d61  (Macaulay2Doc)ma
│ │ │ │ -0007d420: 7472 6978 5f6c 7052 696e 674d 6170 5f72  trix_lpRingMap_r
│ │ │ │ -0007d430: 702c 202d 2d20 7468 6520 6d61 7472 6978  p, -- the matrix
│ │ │ │ -0007d440: 0a20 2020 2061 7373 6f63 6961 7465 6420  .    associated 
│ │ │ │ -0007d450: 746f 2061 2072 696e 6720 6d61 700a 0a57  to a ring map..W
│ │ │ │ -0007d460: 6179 7320 746f 2075 7365 2074 6869 7320  ays to use this 
│ │ │ │ -0007d470: 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d  method:.========
│ │ │ │ -0007d480: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0007d490: 0a0a 2020 2a20 2a6e 6f74 6520 6d61 7472  ..  * *note matr
│ │ │ │ -0007d4a0: 6978 2852 6174 696f 6e61 6c4d 6170 293a  ix(RationalMap):
│ │ │ │ -0007d4b0: 206d 6174 7269 785f 6c70 5261 7469 6f6e   matrix_lpRation
│ │ │ │ -0007d4c0: 616c 4d61 705f 7270 2c20 2d2d 2074 6865  alMap_rp, -- the
│ │ │ │ -0007d4d0: 206d 6174 7269 780a 2020 2020 6173 736f   matrix.    asso
│ │ │ │ -0007d4e0: 6369 6174 6564 2074 6f20 6120 7261 7469  ciated to a rati
│ │ │ │ -0007d4f0: 6f6e 616c 206d 6170 0a20 202a 2022 6d61  onal map.  * "ma
│ │ │ │ -0007d500: 7472 6978 285a 5a2c 5261 7469 6f6e 616c  trix(ZZ,Rational
│ │ │ │ -0007d510: 4d61 7029 220a 2d2d 2d2d 2d2d 2d2d 2d2d  Map)".----------
│ │ │ │ +0007d0d0: 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 7469  ****..  * Functi
│ │ │ │ +0007d0e0: 6f6e 3a20 2a6e 6f74 6520 6d61 7472 6978  on: *note matrix
│ │ │ │ +0007d0f0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0007d100: 6d61 7472 6978 2c0a 2020 2a20 5573 6167  matrix,.  * Usag
│ │ │ │ +0007d110: 653a 200a 2020 2020 2020 2020 6d61 7472  e: .        matr
│ │ │ │ +0007d120: 6978 2050 6869 0a20 202a 2049 6e70 7574  ix Phi.  * Input
│ │ │ │ +0007d130: 733a 0a20 2020 2020 202a 2050 6869 2c20  s:.      * Phi, 
│ │ │ │ +0007d140: 6120 2a6e 6f74 6520 7261 7469 6f6e 616c  a *note rational
│ │ │ │ +0007d150: 206d 6170 3a20 5261 7469 6f6e 616c 4d61   map: RationalMa
│ │ │ │ +0007d160: 702c 0a20 202a 202a 6e6f 7465 204f 7074  p,.  * *note Opt
│ │ │ │ +0007d170: 696f 6e61 6c20 696e 7075 7473 3a20 284d  ional inputs: (M
│ │ │ │ +0007d180: 6163 6175 6c61 7932 446f 6329 7573 696e  acaulay2Doc)usin
│ │ │ │ +0007d190: 6720 6675 6e63 7469 6f6e 7320 7769 7468  g functions with
│ │ │ │ +0007d1a0: 206f 7074 696f 6e61 6c20 696e 7075 7473   optional inputs
│ │ │ │ +0007d1b0: 2c3a 0a20 2020 2020 202a 202a 6e6f 7465  ,:.      * *note
│ │ │ │ +0007d1c0: 2044 6567 7265 653a 2028 4d61 6361 756c   Degree: (Macaul
│ │ │ │ +0007d1d0: 6179 3244 6f63 296d 6174 7269 785f 6c70  ay2Doc)matrix_lp
│ │ │ │ +0007d1e0: 4c69 7374 5f72 702c 203d 3e20 2e2e 2e2c  List_rp, => ...,
│ │ │ │ +0007d1f0: 2064 6566 6175 6c74 2076 616c 7565 0a20   default value. 
│ │ │ │ +0007d200: 2020 2020 2020 206e 756c 6c2c 0a20 202a         null,.  *
│ │ │ │ +0007d210: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +0007d220: 2a20 6120 2a6e 6f74 6520 6d61 7472 6978  * a *note matrix
│ │ │ │ +0007d230: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0007d240: 4d61 7472 6978 2c2c 2074 6865 206d 6174  Matrix,, the mat
│ │ │ │ +0007d250: 7269 7820 6173 736f 6369 6174 6564 2074  rix associated t
│ │ │ │ +0007d260: 6f20 7468 650a 2020 2020 2020 2020 7269  o the.        ri
│ │ │ │ +0007d270: 6e67 206d 6170 2064 6566 696e 696e 6720  ng map defining 
│ │ │ │ +0007d280: 7468 6520 7261 7469 6f6e 616c 206d 6170  the rational map
│ │ │ │ +0007d290: 2050 6869 0a0a 4465 7363 7269 7074 696f   Phi..Descriptio
│ │ │ │ +0007d2a0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  n.===========..T
│ │ │ │ +0007d2b0: 6869 7320 6973 2065 7175 6976 616c 656e  his is equivalen
│ │ │ │ +0007d2c0: 7420 746f 206d 6174 7269 7820 6d61 7020  t to matrix map 
│ │ │ │ +0007d2d0: 5068 692e 204d 6f72 656f 7665 722c 2074  Phi. Moreover, t
│ │ │ │ +0007d2e0: 6865 2063 6f6d 6d61 6e64 206d 6174 7269  he command matri
│ │ │ │ +0007d2f0: 7820 5068 6920 6973 0a65 7175 6976 616c  x Phi is.equival
│ │ │ │ +0007d300: 656e 7420 746f 206d 6174 7269 7828 302c  ent to matrix(0,
│ │ │ │ +0007d310: 5068 6929 2c20 616e 6420 6d6f 7265 2067  Phi), and more g
│ │ │ │ +0007d320: 656e 6572 616c 6c79 2074 6865 2063 6f6d  enerally the com
│ │ │ │ +0007d330: 6d61 6e64 206d 6174 7269 7828 692c 5068  mand matrix(i,Ph
│ │ │ │ +0007d340: 6929 0a72 6574 7572 6e73 2074 6865 206d  i).returns the m
│ │ │ │ +0007d350: 6174 7269 7820 6f66 2074 6865 2069 2d74  atrix of the i-t
│ │ │ │ +0007d360: 6820 7265 7072 6573 656e 7461 7469 7665  h representative
│ │ │ │ +0007d370: 206f 6620 5068 692e 0a0a 5365 6520 616c   of Phi...See al
│ │ │ │ +0007d380: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ +0007d390: 202a 6e6f 7465 206d 6170 2852 6174 696f   *note map(Ratio
│ │ │ │ +0007d3a0: 6e61 6c4d 6170 293a 206d 6170 5f6c 7052  nalMap): map_lpR
│ │ │ │ +0007d3b0: 6174 696f 6e61 6c4d 6170 5f72 702c 202d  ationalMap_rp, -
│ │ │ │ +0007d3c0: 2d20 6765 7420 7468 6520 7269 6e67 206d  - get the ring m
│ │ │ │ +0007d3d0: 6170 2064 6566 696e 696e 670a 2020 2020  ap defining.    
│ │ │ │ +0007d3e0: 6120 7261 7469 6f6e 616c 206d 6170 0a20  a rational map. 
│ │ │ │ +0007d3f0: 202a 202a 6e6f 7465 206d 6174 7269 7828   * *note matrix(
│ │ │ │ +0007d400: 5269 6e67 4d61 7029 3a20 284d 6163 6175  RingMap): (Macau
│ │ │ │ +0007d410: 6c61 7932 446f 6329 6d61 7472 6978 5f6c  lay2Doc)matrix_l
│ │ │ │ +0007d420: 7052 696e 674d 6170 5f72 702c 202d 2d20  pRingMap_rp, -- 
│ │ │ │ +0007d430: 7468 6520 6d61 7472 6978 0a20 2020 2061  the matrix.    a
│ │ │ │ +0007d440: 7373 6f63 6961 7465 6420 746f 2061 2072  ssociated to a r
│ │ │ │ +0007d450: 696e 6720 6d61 700a 0a57 6179 7320 746f  ing map..Ways to
│ │ │ │ +0007d460: 2075 7365 2074 6869 7320 6d65 7468 6f64   use this method
│ │ │ │ +0007d470: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +0007d480: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +0007d490: 2a6e 6f74 6520 6d61 7472 6978 2852 6174  *note matrix(Rat
│ │ │ │ +0007d4a0: 696f 6e61 6c4d 6170 293a 206d 6174 7269  ionalMap): matri
│ │ │ │ +0007d4b0: 785f 6c70 5261 7469 6f6e 616c 4d61 705f  x_lpRationalMap_
│ │ │ │ +0007d4c0: 7270 2c20 2d2d 2074 6865 206d 6174 7269  rp, -- the matri
│ │ │ │ +0007d4d0: 780a 2020 2020 6173 736f 6369 6174 6564  x.    associated
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│ │ │ │ +0007d500: 5a2c 5261 7469 6f6e 616c 4d61 7029 220a  Z,RationalMap)".
│ │ │ │ +0007d510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007d520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007d530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007d540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007d550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007d560: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
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│ │ │ │ -0007d590: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ -0007d5a0: 682f 6d61 6361 756c 6179 322d 312e 3235  h/macaulay2-1.25
│ │ │ │ -0007d5b0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c  .11+ds/M2/Macaul
│ │ │ │ -0007d5c0: 6179 322f 7061 636b 6167 6573 2f43 7265  ay2/packages/Cre
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│ │ │ │ -0007d620: 6d65 7472 697a 652c 2050 7265 763a 206d  metrize, Prev: m
│ │ │ │ -0007d630: 6174 7269 785f 6c70 5261 7469 6f6e 616c  atrix_lpRational
│ │ │ │ -0007d640: 4d61 705f 7270 2c20 5570 3a20 546f 700a  Map_rp, Up: Top.
│ │ │ │ -0007d650: 0a4e 756d 4465 6772 6565 730a 2a2a 2a2a  .NumDegrees.****
│ │ │ │ -0007d660: 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 7074  ******..Descript
│ │ │ │ -0007d670: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ -0007d680: 0a54 6869 7320 6973 2061 6e20 6f70 7469  .This is an opti
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│ │ │ │ -0007d6a0: 7220 2a6e 6f74 6520 7072 6f6a 6563 7469  r *note projecti
│ │ │ │ -0007d6b0: 7665 4465 6772 6565 733a 2070 726f 6a65  veDegrees: proje
│ │ │ │ -0007d6c0: 6374 6976 6544 6567 7265 6573 2c0a 616e  ctiveDegrees,.an
│ │ │ │ -0007d6d0: 6420 6163 6365 7074 7320 6120 6e6f 6e2d  d accepts a non-
│ │ │ │ -0007d6e0: 6e65 6761 7469 7665 2069 6e74 6567 6572  negative integer
│ │ │ │ -0007d6f0: 2c20 3120 6c65 7373 2074 6861 6e20 7468  , 1 less than th
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│ │ │ │ -0007d710: 6563 7469 7665 0a64 6567 7265 6573 2074  ective.degrees t
│ │ │ │ -0007d720: 6f20 6265 2063 6f6d 7075 7465 642e 0a0a  o be computed...
│ │ │ │ -0007d730: 4675 6e63 7469 6f6e 7320 7769 7468 206f  Functions with o
│ │ │ │ -0007d740: 7074 696f 6e61 6c20 6172 6775 6d65 6e74  ptional argument
│ │ │ │ -0007d750: 206e 616d 6564 204e 756d 4465 6772 6565   named NumDegree
│ │ │ │ -0007d760: 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  s:.=============
│ │ │ │ +0007d550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ +0007d560: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
│ │ │ │ +0007d570: 6869 7320 646f 6375 6d65 6e74 2069 7320  his document is 
│ │ │ │ +0007d580: 696e 0a2f 6275 696c 642f 7265 7072 6f64  in./build/reprod
│ │ │ │ +0007d590: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ +0007d5a0: 756c 6179 322d 312e 3235 2e31 312b 6473  ulay2-1.25.11+ds
│ │ │ │ +0007d5b0: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ +0007d5c0: 636b 6167 6573 2f43 7265 6d6f 6e61 2f0a  ckages/Cremona/.
│ │ │ │ +0007d5d0: 646f 6375 6d65 6e74 6174 696f 6e2e 6d32  documentation.m2
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│ │ │ │ +0007d5f0: 4372 656d 6f6e 612e 696e 666f 2c20 4e6f  Cremona.info, No
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│ │ │ │ +0007d610: 4e65 7874 3a20 7061 7261 6d65 7472 697a  Next: parametriz
│ │ │ │ +0007d620: 652c 2050 7265 763a 206d 6174 7269 785f  e, Prev: matrix_
│ │ │ │ +0007d630: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ +0007d640: 2c20 5570 3a20 546f 700a 0a4e 756d 4465  , Up: Top..NumDe
│ │ │ │ +0007d650: 6772 6565 730a 2a2a 2a2a 2a2a 2a2a 2a2a  grees.**********
│ │ │ │ +0007d660: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +0007d670: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320  =========..This 
│ │ │ │ +0007d680: 6973 2061 6e20 6f70 7469 6f6e 616c 2061  is an optional a
│ │ │ │ +0007d690: 7267 756d 656e 7420 666f 7220 2a6e 6f74  rgument for *not
│ │ │ │ +0007d6a0: 6520 7072 6f6a 6563 7469 7665 4465 6772  e projectiveDegr
│ │ │ │ +0007d6b0: 6565 733a 2070 726f 6a65 6374 6976 6544  ees: projectiveD
│ │ │ │ +0007d6c0: 6567 7265 6573 2c0a 616e 6420 6163 6365  egrees,.and acce
│ │ │ │ +0007d6d0: 7074 7320 6120 6e6f 6e2d 6e65 6761 7469  pts a non-negati
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│ │ │ │ +0007d6f0: 7373 2074 6861 6e20 7468 6520 6e75 6d62  ss than the numb
│ │ │ │ +0007d700: 6572 206f 6620 7072 6f6a 6563 7469 7665  er of projective
│ │ │ │ +0007d710: 0a64 6567 7265 6573 2074 6f20 6265 2063  .degrees to be c
│ │ │ │ +0007d720: 6f6d 7075 7465 642e 0a0a 4675 6e63 7469  omputed...Functi
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│ │ │ │ +0007d740: 6c20 6172 6775 6d65 6e74 206e 616d 6564  l argument named
│ │ │ │ +0007d750: 204e 756d 4465 6772 6565 733a 0a3d 3d3d   NumDegrees:.===
│ │ │ │ +0007d760: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  0007d770: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0007d780: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0007d790: 3d3d 3d3d 3d0a 0a20 202a 2022 7072 6f6a  =====..  * "proj
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│ │ │ │ -0007d7c0: 2e29 220a 0a46 6f72 2074 6865 2070 726f  .)"..For the pro
│ │ │ │ -0007d7d0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ -0007d7e0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
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│ │ │ │ -0007d810: 6565 732c 2069 7320 6120 2a6e 6f74 6520  ees, is a *note 
│ │ │ │ -0007d820: 7379 6d62 6f6c 3a0a 284d 6163 6175 6c61  symbol:.(Macaula
│ │ │ │ -0007d830: 7932 446f 6329 5379 6d62 6f6c 2c2e 0a0a  y2Doc)Symbol,...
│ │ │ │ +0007d780: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0007d790: 0a20 202a 2022 7072 6f6a 6563 7469 7665  .  * "projective
│ │ │ │ +0007d7a0: 4465 6772 6565 7328 2e2e 2e2c 4e75 6d44  Degrees(...,NumD
│ │ │ │ +0007d7b0: 6567 7265 6573 3d3e 2e2e 2e29 220a 0a46  egrees=>...)"..F
│ │ │ │ +0007d7c0: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
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│ │ │ │ +0007d7f0: 202a 6e6f 7465 204e 756d 4465 6772 6565   *note NumDegree
│ │ │ │ +0007d800: 733a 204e 756d 4465 6772 6565 732c 2069  s: NumDegrees, i
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│ │ │ │ +0007d830: 5379 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d  Symbol,...------
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│ │ │ │  0007d850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007d860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007d870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007d880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
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│ │ │ │ -0007d940: 4e65 7874 3a20 7061 7261 6d65 7472 697a  Next: parametriz
│ │ │ │ -0007d950: 655f 6c70 4964 6561 6c5f 7270 2c20 5072  e_lpIdeal_rp, Pr
│ │ │ │ -0007d960: 6576 3a20 4e75 6d44 6567 7265 6573 2c20  ev: NumDegrees, 
│ │ │ │ -0007d970: 5570 3a20 546f 700a 0a70 6172 616d 6574  Up: Top..paramet
│ │ │ │ -0007d980: 7269 7a65 202d 2d20 7061 7261 6d65 7472  rize -- parametr
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│ │ │ │  0007d9c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0007d9d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0007d9e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0007d9f0: 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 7269  ********..Descri
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│ │ │ │ -0007da90: 6172 616d 6574 7269 7a65 2851 756f 7469  arametrize(Quoti
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│ │ │ │ -0007dac0: 2c2e 0a0a 5761 7973 2074 6f20 7573 6520  ,...Ways to use 
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│ │ │ │ -0007dae0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0007daf0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
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│ │ │ │ -0007db20: 5f6c 7049 6465 616c 5f72 702c 202d 2d20  _lpIdeal_rp, -- 
│ │ │ │ -0007db30: 7061 7261 6d65 7472 697a 6174 696f 6e20  parametrization 
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│ │ │ │ -0007db90: 6565 202a 6e6f 7465 2070 6172 616d 6574  ee *note paramet
│ │ │ │ -0007dba0: 7269 7a65 2849 6465 616c 293a 0a20 2020  rize(Ideal):.   
│ │ │ │ -0007dbb0: 2070 6172 616d 6574 7269 7a65 5f6c 7049   parametrize_lpI
│ │ │ │ -0007dbc0: 6465 616c 5f72 702c 202d 2d20 7061 7261  deal_rp, -- para
│ │ │ │ -0007dbd0: 6d65 7472 697a 6174 696f 6e20 6f66 206c  metrization of l
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│ │ │ │ -0007dc60: 702c 202d 2d20 7061 7261 6d65 7472 697a  p, -- parametriz
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│ │ │ │ -0007dc90: 2020 6879 7065 7271 7561 6472 6963 730a    hyperquadrics.
│ │ │ │ -0007dca0: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ -0007dcb0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ -0007dcc0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ -0007dcd0: 6374 202a 6e6f 7465 2070 6172 616d 6574  ct *note paramet
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│ │ │ │  0007dd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007dd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007dd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007dd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0007de30: 7049 6465 616c 5f72 702c 204e 6578 743a  pIdeal_rp, Next:
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│ │ │ │ -0007de50: 7261 6d65 7472 697a 652c 2055 703a 2054  rametrize, Up: T
│ │ │ │ -0007de60: 6f70 0a0a 7061 7261 6d65 7472 697a 6528  op..parametrize(
│ │ │ │ -0007de70: 4964 6561 6c29 202d 2d20 7061 7261 6d65  Ideal) -- parame
│ │ │ │ -0007de80: 7472 697a 6174 696f 6e20 6f66 206c 696e  trization of lin
│ │ │ │ -0007de90: 6561 7220 7661 7269 6574 6965 7320 616e  ear varieties an
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│ │ │ │ +0007dde0: 4372 656d 6f6e 612f 0a64 6f63 756d 656e  Cremona/.documen
│ │ │ │ +0007ddf0: 7461 7469 6f6e 2e6d 323a 3638 363a 302e  tation.m2:686:0.
│ │ │ │ +0007de00: 0a1f 0a46 696c 653a 2043 7265 6d6f 6e61  ...File: Cremona
│ │ │ │ +0007de10: 2e69 6e66 6f2c 204e 6f64 653a 2070 6172  .info, Node: par
│ │ │ │ +0007de20: 616d 6574 7269 7a65 5f6c 7049 6465 616c  ametrize_lpIdeal
│ │ │ │ +0007de30: 5f72 702c 204e 6578 743a 2070 6f69 6e74  _rp, Next: point
│ │ │ │ +0007de40: 2c20 5072 6576 3a20 7061 7261 6d65 7472  , Prev: parametr
│ │ │ │ +0007de50: 697a 652c 2055 703a 2054 6f70 0a0a 7061  ize, Up: Top..pa
│ │ │ │ +0007de60: 7261 6d65 7472 697a 6528 4964 6561 6c29  rametrize(Ideal)
│ │ │ │ +0007de70: 202d 2d20 7061 7261 6d65 7472 697a 6174   -- parametrizat
│ │ │ │ +0007de80: 696f 6e20 6f66 206c 696e 6561 7220 7661  ion of linear va
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│ │ │ │ +0007dea0: 7271 7561 6472 6963 730a 2a2a 2a2a 2a2a  rquadrics.******
│ │ │ │  0007deb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0007dec0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0007ded0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0007dee0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0007def0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ -0007df00: 2046 756e 6374 696f 6e3a 202a 6e6f 7465   Function: *note
│ │ │ │ -0007df10: 2070 6172 616d 6574 7269 7a65 3a20 7061   parametrize: pa
│ │ │ │ -0007df20: 7261 6d65 7472 697a 652c 0a20 202a 2055  rametrize,.  * U
│ │ │ │ -0007df30: 7361 6765 3a20 0a20 2020 2020 2020 2070  sage: .        p
│ │ │ │ -0007df40: 6172 616d 6574 7269 7a65 2049 0a20 202a  arametrize I.  *
│ │ │ │ -0007df50: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -0007df60: 2049 2c20 616e 202a 6e6f 7465 2069 6465   I, an *note ide
│ │ │ │ -0007df70: 616c 3a20 284d 6163 6175 6c61 7932 446f  al: (Macaulay2Do
│ │ │ │ -0007df80: 6329 4964 6561 6c2c 2c20 7468 6520 6964  c)Ideal,, the id
│ │ │ │ -0007df90: 6561 6c20 6f66 2061 206c 696e 6561 7220  eal of a linear 
│ │ │ │ -0007dfa0: 7661 7269 6574 790a 2020 2020 2020 2020  variety.        
│ │ │ │ -0007dfb0: 6f72 206f 6620 6120 6879 7065 7271 7561  or of a hyperqua
│ │ │ │ -0007dfc0: 6472 6963 0a20 202a 204f 7574 7075 7473  dric.  * Outputs
│ │ │ │ -0007dfd0: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74  :.      * a *not
│ │ │ │ -0007dfe0: 6520 7261 7469 6f6e 616c 206d 6170 3a20  e rational map: 
│ │ │ │ -0007dff0: 5261 7469 6f6e 616c 4d61 702c 2c20 6120  RationalMap,, a 
│ │ │ │ -0007e000: 6269 7261 7469 6f6e 616c 206d 6170 2070  birational map p
│ │ │ │ -0007e010: 6869 2073 7563 6820 7468 6174 2049 203d  hi such that I =
│ │ │ │ -0007e020: 3d0a 2020 2020 2020 2020 696d 6167 6520  =.        image 
│ │ │ │ -0007e030: 7068 690a 0a44 6573 6372 6970 7469 6f6e  phi..Description
│ │ │ │ -0007e040: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  .===========..Th
│ │ │ │ -0007e050: 6973 2066 756e 6374 696f 6e20 6861 7320  is function has 
│ │ │ │ -0007e060: 6265 656e 2069 6d70 726f 7665 6420 616e  been improved an
│ │ │ │ -0007e070: 6420 6578 7465 6e64 6564 2069 6e20 7468  d extended in th
│ │ │ │ -0007e080: 6520 7061 636b 6167 650a 4d75 6c74 6970  e package.Multip
│ │ │ │ -0007e090: 726f 6a65 6374 6976 6556 6172 6965 7469  rojectiveVarieti
│ │ │ │ -0007e0a0: 6573 2028 6d69 7373 696e 6720 646f 6375  es (missing docu
│ │ │ │ -0007e0b0: 6d65 6e74 6174 696f 6e29 2c20 7365 6520  mentation), see 
│ │ │ │ -0007e0c0: 2a6e 6f74 650a 7061 7261 6d65 7472 697a  *note.parametriz
│ │ │ │ -0007e0d0: 6528 4d75 6c74 6970 726f 6a65 6374 6976  e(Multiprojectiv
│ │ │ │ -0007e0e0: 6556 6172 6965 7479 293a 0a28 4d75 6c74  eVariety):.(Mult
│ │ │ │ -0007e0f0: 6970 726f 6a65 6374 6976 6556 6172 6965  iprojectiveVarie
│ │ │ │ -0007e100: 7469 6573 2970 6172 616d 6574 7269 7a65  ties)parametrize
│ │ │ │ -0007e110: 5f6c 704d 756c 7469 7072 6f6a 6563 7469  _lpMultiprojecti
│ │ │ │ -0007e120: 7665 5661 7269 6574 795f 7270 2c2e 0a0a  veVariety_rp,...
│ │ │ │ -0007e130: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0007def0: 2a2a 2a2a 2a0a 0a20 202a 2046 756e 6374  *****..  * Funct
│ │ │ │ +0007df00: 696f 6e3a 202a 6e6f 7465 2070 6172 616d  ion: *note param
│ │ │ │ +0007df10: 6574 7269 7a65 3a20 7061 7261 6d65 7472  etrize: parametr
│ │ │ │ +0007df20: 697a 652c 0a20 202a 2055 7361 6765 3a20  ize,.  * Usage: 
│ │ │ │ +0007df30: 0a20 2020 2020 2020 2070 6172 616d 6574  .        paramet
│ │ │ │ +0007df40: 7269 7a65 2049 0a20 202a 2049 6e70 7574  rize I.  * Input
│ │ │ │ +0007df50: 733a 0a20 2020 2020 202a 2049 2c20 616e  s:.      * I, an
│ │ │ │ +0007df60: 202a 6e6f 7465 2069 6465 616c 3a20 284d   *note ideal: (M
│ │ │ │ +0007df70: 6163 6175 6c61 7932 446f 6329 4964 6561  acaulay2Doc)Idea
│ │ │ │ +0007df80: 6c2c 2c20 7468 6520 6964 6561 6c20 6f66  l,, the ideal of
│ │ │ │ +0007df90: 2061 206c 696e 6561 7220 7661 7269 6574   a linear variet
│ │ │ │ +0007dfa0: 790a 2020 2020 2020 2020 6f72 206f 6620  y.        or of 
│ │ │ │ +0007dfb0: 6120 6879 7065 7271 7561 6472 6963 0a20  a hyperquadric. 
│ │ │ │ +0007dfc0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +0007dfd0: 2020 2a20 6120 2a6e 6f74 6520 7261 7469    * a *note rati
│ │ │ │ +0007dfe0: 6f6e 616c 206d 6170 3a20 5261 7469 6f6e  onal map: Ration
│ │ │ │ +0007dff0: 616c 4d61 702c 2c20 6120 6269 7261 7469  alMap,, a birati
│ │ │ │ +0007e000: 6f6e 616c 206d 6170 2070 6869 2073 7563  onal map phi suc
│ │ │ │ +0007e010: 6820 7468 6174 2049 203d 3d0a 2020 2020  h that I ==.    
│ │ │ │ +0007e020: 2020 2020 696d 6167 6520 7068 690a 0a44      image phi..D
│ │ │ │ +0007e030: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +0007e040: 3d3d 3d3d 3d3d 0a0a 5468 6973 2066 756e  ======..This fun
│ │ │ │ +0007e050: 6374 696f 6e20 6861 7320 6265 656e 2069  ction has been i
│ │ │ │ +0007e060: 6d70 726f 7665 6420 616e 6420 6578 7465  mproved and exte
│ │ │ │ +0007e070: 6e64 6564 2069 6e20 7468 6520 7061 636b  nded in the pack
│ │ │ │ +0007e080: 6167 650a 4d75 6c74 6970 726f 6a65 6374  age.Multiproject
│ │ │ │ +0007e090: 6976 6556 6172 6965 7469 6573 2028 6d69  iveVarieties (mi
│ │ │ │ +0007e0a0: 7373 696e 6720 646f 6375 6d65 6e74 6174  ssing documentat
│ │ │ │ +0007e0b0: 696f 6e29 2c20 7365 6520 2a6e 6f74 650a  ion), see *note.
│ │ │ │ +0007e0c0: 7061 7261 6d65 7472 697a 6528 4d75 6c74  parametrize(Mult
│ │ │ │ +0007e0d0: 6970 726f 6a65 6374 6976 6556 6172 6965  iprojectiveVarie
│ │ │ │ +0007e0e0: 7479 293a 0a28 4d75 6c74 6970 726f 6a65  ty):.(Multiproje
│ │ │ │ +0007e0f0: 6374 6976 6556 6172 6965 7469 6573 2970  ctiveVarieties)p
│ │ │ │ +0007e100: 6172 616d 6574 7269 7a65 5f6c 704d 756c  arametrize_lpMul
│ │ │ │ +0007e110: 7469 7072 6f6a 6563 7469 7665 5661 7269  tiprojectiveVari
│ │ │ │ +0007e120: 6574 795f 7270 2c2e 0a0a 2b2d 2d2d 2d2d  ety_rp,...+-----
│ │ │ │ +0007e130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0007e180: 7c69 3120 3a20 5039 203a 3d20 5a5a 2f31  |i1 : P9 := ZZ/1
│ │ │ │ -0007e190: 3030 3030 3031 395b 785f 302e 2e78 5f39  0000019[x_0..x_9
│ │ │ │ -0007e1a0: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ +0007e170: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +0007e180: 5039 203a 3d20 5a5a 2f31 3030 3030 3031  P9 := ZZ/1000001
│ │ │ │ +0007e190: 395b 785f 302e 2e78 5f39 5d20 2020 2020  9[x_0..x_9]     
│ │ │ │ +0007e1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e1c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007e1d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007e1c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007e1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e210: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007e220: 7c20 2020 2020 2020 205a 5a20 2020 2020  |        ZZ     
│ │ │ │ +0007e210: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007e220: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │  0007e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e260: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007e270: 7c6f 3120 3d20 2d2d 2d2d 2d2d 2d2d 5b78  |o1 = --------[x
│ │ │ │ -0007e280: 202e 2e78 205d 2020 2020 2020 2020 2020   ..x ]          
│ │ │ │ +0007e260: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ +0007e270: 2d2d 2d2d 2d2d 2d2d 5b78 202e 2e78 205d  --------[x ..x ]
│ │ │ │ +0007e280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e2b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007e2c0: 7c20 2020 2020 3130 3030 3030 3139 2020  |     10000019  
│ │ │ │ -0007e2d0: 3020 2020 3920 2020 2020 2020 2020 2020  0   9           
│ │ │ │ +0007e2b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007e2c0: 3130 3030 3030 3139 2020 3020 2020 3920  10000019  0   9 
│ │ │ │ +0007e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e300: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007e310: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007e300: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e350: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007e360: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c  |o1 : Polynomial
│ │ │ │ -0007e370: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +0007e350: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ +0007e360: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ +0007e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e3a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007e3b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0007e3a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0007e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0007e400: 7c69 3220 3a20 4c20 3d20 7472 696d 2069  |i2 : L = trim i
│ │ │ │ -0007e410: 6465 616c 2872 616e 646f 6d28 312c 5039  deal(random(1,P9
│ │ │ │ -0007e420: 292c 7261 6e64 6f6d 2831 2c50 3929 2c72  ),random(1,P9),r
│ │ │ │ -0007e430: 616e 646f 6d28 312c 5039 292c 7261 6e64  andom(1,P9),rand
│ │ │ │ -0007e440: 6f6d 2831 2c50 3929 2920 2020 2020 7c0a  om(1,P9))     |.
│ │ │ │ -0007e450: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007e3f0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ +0007e400: 4c20 3d20 7472 696d 2069 6465 616c 2872  L = trim ideal(r
│ │ │ │ +0007e410: 616e 646f 6d28 312c 5039 292c 7261 6e64  andom(1,P9),rand
│ │ │ │ +0007e420: 6f6d 2831 2c50 3929 2c72 616e 646f 6d28  om(1,P9),random(
│ │ │ │ +0007e430: 312c 5039 292c 7261 6e64 6f6d 2831 2c50  1,P9),random(1,P
│ │ │ │ +0007e440: 3929 2920 2020 2020 7c0a 7c20 2020 2020  9))     |.|     
│ │ │ │ +0007e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e490: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007e4a0: 7c6f 3220 3d20 6964 6561 6c20 2878 2020  |o2 = ideal (x  
│ │ │ │ -0007e4b0: 2d20 3131 3132 3031 3678 2020 2d20 3339  - 1112016x  - 39
│ │ │ │ -0007e4c0: 3031 3336 3178 2020 2d20 3331 3933 3836  01361x  - 319386
│ │ │ │ -0007e4d0: 3378 2020 2b20 3431 3433 3034 3078 2020  3x  + 4143040x  
│ │ │ │ -0007e4e0: 2d20 3139 3634 3431 3778 2020 2b20 7c0a  - 1964417x  + |.
│ │ │ │ -0007e4f0: 7c20 2020 2020 2020 2020 2020 2020 3320  |             3 
│ │ │ │ -0007e500: 2020 2020 2020 2020 2020 3420 2020 2020            4     
│ │ │ │ -0007e510: 2020 2020 2020 3520 2020 2020 2020 2020        5         
│ │ │ │ -0007e520: 2020 3620 2020 2020 2020 2020 2020 3720    6           7 
│ │ │ │ -0007e530: 2020 2020 2020 2020 2020 3820 2020 7c0a            8   |.
│ │ │ │ -0007e540: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +0007e490: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ +0007e4a0: 6964 6561 6c20 2878 2020 2d20 3131 3132  ideal (x  - 1112
│ │ │ │ +0007e4b0: 3031 3678 2020 2d20 3339 3031 3336 3178  016x  - 3901361x
│ │ │ │ +0007e4c0: 2020 2d20 3331 3933 3836 3378 2020 2b20    - 3193863x  + 
│ │ │ │ +0007e4d0: 3431 3433 3034 3078 2020 2d20 3139 3634  4143040x  - 1964
│ │ │ │ +0007e4e0: 3431 3778 2020 2b20 7c0a 7c20 2020 2020  417x  + |.|     
│ │ │ │ +0007e4f0: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ +0007e500: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ +0007e510: 3520 2020 2020 2020 2020 2020 3620 2020  5           6   
│ │ │ │ +0007e520: 2020 2020 2020 2020 3720 2020 2020 2020          7       
│ │ │ │ +0007e530: 2020 2020 3820 2020 7c0a 7c20 2020 2020      8   |.|     
│ │ │ │ +0007e540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007e580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007e590: 7c20 2020 2020 3130 3734 3935 3878 202c  |     1074958x ,
│ │ │ │ -0007e5a0: 2078 2020 2b20 3633 3232 3834 7820 202b   x  + 632284x  +
│ │ │ │ -0007e5b0: 2034 3932 3435 3878 2020 2b20 3338 3639   492458x  + 3869
│ │ │ │ -0007e5c0: 3235 3478 2020 2b20 3238 3430 3236 3678  254x  + 2840266x
│ │ │ │ -0007e5d0: 2020 2b20 3438 3833 3937 3478 2020 7c0a    + 4883974x  |.
│ │ │ │ -0007e5e0: 7c20 2020 2020 2020 2020 2020 2020 3920  |             9 
│ │ │ │ -0007e5f0: 2020 3220 2020 2020 2020 2020 2034 2020    2          4  
│ │ │ │ -0007e600: 2020 2020 2020 2020 3520 2020 2020 2020          5       
│ │ │ │ -0007e610: 2020 2020 3620 2020 2020 2020 2020 2020      6           
│ │ │ │ -0007e620: 3720 2020 2020 2020 2020 2020 3820 7c0a  7           8 |.
│ │ │ │ -0007e630: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +0007e580: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007e590: 3130 3734 3935 3878 202c 2078 2020 2b20  1074958x , x  + 
│ │ │ │ +0007e5a0: 3633 3232 3834 7820 202b 2034 3932 3435  632284x  + 49245
│ │ │ │ +0007e5b0: 3878 2020 2b20 3338 3639 3235 3478 2020  8x  + 3869254x  
│ │ │ │ +0007e5c0: 2b20 3238 3430 3236 3678 2020 2b20 3438  + 2840266x  + 48
│ │ │ │ +0007e5d0: 3833 3937 3478 2020 7c0a 7c20 2020 2020  83974x  |.|     
│ │ │ │ +0007e5e0: 2020 2020 2020 2020 3920 2020 3220 2020          9   2   
│ │ │ │ +0007e5f0: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ +0007e600: 2020 3520 2020 2020 2020 2020 2020 3620    5           6 
│ │ │ │ +0007e610: 2020 2020 2020 2020 2020 3720 2020 2020            7     
│ │ │ │ +0007e620: 2020 2020 2020 3820 7c0a 7c20 2020 2020        8 |.|     
│ │ │ │ +0007e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007e670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007e680: 7c20 2020 2020 2b20 3333 3430 3936 3178  |     + 3340961x
│ │ │ │ -0007e690: 202c 2078 2020 2b20 3437 3234 3730 3978   , x  + 4724709x
│ │ │ │ -0007e6a0: 2020 2d20 3335 3035 3338 3678 2020 2b20    - 3505386x  + 
│ │ │ │ -0007e6b0: 3234 3639 3230 3678 2020 2d20 3133 3831  2469206x  - 1381
│ │ │ │ -0007e6c0: 3531 3578 2020 2b20 2020 2020 2020 7c0a  515x  +       |.
│ │ │ │ -0007e6d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007e6e0: 3920 2020 3120 2020 2020 2020 2020 2020  9   1           
│ │ │ │ -0007e6f0: 3420 2020 2020 2020 2020 2020 3520 2020  4           5   
│ │ │ │ -0007e700: 2020 2020 2020 2020 3620 2020 2020 2020          6       
│ │ │ │ -0007e710: 2020 2020 3720 2020 2020 2020 2020 7c0a      7         |.
│ │ │ │ -0007e720: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +0007e670: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007e680: 2b20 3333 3430 3936 3178 202c 2078 2020  + 3340961x , x  
│ │ │ │ +0007e690: 2b20 3437 3234 3730 3978 2020 2d20 3335  + 4724709x  - 35
│ │ │ │ +0007e6a0: 3035 3338 3678 2020 2b20 3234 3639 3230  05386x  + 246920
│ │ │ │ +0007e6b0: 3678 2020 2d20 3133 3831 3531 3578 2020  6x  - 1381515x  
│ │ │ │ +0007e6c0: 2b20 2020 2020 2020 7c0a 7c20 2020 2020  +       |.|     
│ │ │ │ +0007e6d0: 2020 2020 2020 2020 2020 3920 2020 3120            9   1 
│ │ │ │ +0007e6e0: 2020 2020 2020 2020 2020 3420 2020 2020            4     
│ │ │ │ +0007e6f0: 2020 2020 2020 3520 2020 2020 2020 2020        5         
│ │ │ │ +0007e700: 2020 3620 2020 2020 2020 2020 2020 3720    6           7 
│ │ │ │ +0007e710: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007e720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007e770: 7c20 2020 2020 3233 3331 3238 3078 2020  |     2331280x  
│ │ │ │ -0007e780: 2d20 3439 3336 3232 3978 202c 2078 2020  - 4936229x , x  
│ │ │ │ -0007e790: 2d20 3230 3934 3435 3678 2020 2d20 3339  - 2094456x  - 39
│ │ │ │ -0007e7a0: 3336 3439 3878 2020 2d20 3436 3635 3430  36498x  - 466540
│ │ │ │ -0007e7b0: 3478 2020 2d20 3733 3639 3433 7820 7c0a  4x  - 736943x |.
│ │ │ │ -0007e7c0: 7c20 2020 2020 2020 2020 2020 2020 3820  |             8 
│ │ │ │ -0007e7d0: 2020 2020 2020 2020 2020 3920 2020 3020            9   0 
│ │ │ │ -0007e7e0: 2020 2020 2020 2020 2020 3420 2020 2020            4     
│ │ │ │ -0007e7f0: 2020 2020 2020 3520 2020 2020 2020 2020        5         
│ │ │ │ -0007e800: 2020 3620 2020 2020 2020 2020 2037 7c0a    6          7|.
│ │ │ │ -0007e810: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +0007e760: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007e770: 3233 3331 3238 3078 2020 2d20 3439 3336  2331280x  - 4936
│ │ │ │ +0007e780: 3232 3978 202c 2078 2020 2d20 3230 3934  229x , x  - 2094
│ │ │ │ +0007e790: 3435 3678 2020 2d20 3339 3336 3439 3878  456x  - 3936498x
│ │ │ │ +0007e7a0: 2020 2d20 3436 3635 3430 3478 2020 2d20    - 4665404x  - 
│ │ │ │ +0007e7b0: 3733 3639 3433 7820 7c0a 7c20 2020 2020  736943x |.|     
│ │ │ │ +0007e7c0: 2020 2020 2020 2020 3820 2020 2020 2020          8       
│ │ │ │ +0007e7d0: 2020 2020 3920 2020 3020 2020 2020 2020      9   0       
│ │ │ │ +0007e7e0: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ +0007e7f0: 3520 2020 2020 2020 2020 2020 3620 2020  5           6   
│ │ │ │ +0007e800: 2020 2020 2020 2037 7c0a 7c20 2020 2020         7|.|     
│ │ │ │ +0007e810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007e860: 7c20 2020 2020 2d20 3834 3936 3731 7820  |     - 849671x 
│ │ │ │ -0007e870: 202b 2033 3033 3431 3337 7820 2920 2020   + 3034137x )   
│ │ │ │ +0007e850: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007e860: 2d20 3834 3936 3731 7820 202b 2033 3033  - 849671x  + 303
│ │ │ │ +0007e870: 3431 3337 7820 2920 2020 2020 2020 2020  4137x )         
│ │ │ │  0007e880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e8a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007e8b0: 7c20 2020 2020 2020 2020 2020 2020 2038  |              8
│ │ │ │ -0007e8c0: 2020 2020 2020 2020 2020 2039 2020 2020             9    
│ │ │ │ +0007e8a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007e8b0: 2020 2020 2020 2020 2038 2020 2020 2020           8      
│ │ │ │ +0007e8c0: 2020 2020 2039 2020 2020 2020 2020 2020       9          
│ │ │ │  0007e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e8f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007e900: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007e8f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007e900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e940: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007e950: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007e960: 2020 5a5a 2020 2020 2020 2020 2020 2020    ZZ            
│ │ │ │ +0007e940: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007e950: 2020 2020 2020 2020 2020 2020 5a5a 2020              ZZ  
│ │ │ │ +0007e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e990: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007e9a0: 7c6f 3220 3a20 4964 6561 6c20 6f66 202d  |o2 : Ideal of -
│ │ │ │ -0007e9b0: 2d2d 2d2d 2d2d 2d5b 7820 2e2e 7820 5d20  -------[x ..x ] 
│ │ │ │ +0007e990: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ +0007e9a0: 4964 6561 6c20 6f66 202d 2d2d 2d2d 2d2d  Ideal of -------
│ │ │ │ +0007e9b0: 2d5b 7820 2e2e 7820 5d20 2020 2020 2020  -[x ..x ]       
│ │ │ │  0007e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007e9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007e9e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007e9f0: 7c20 2020 2020 2020 2020 2020 2020 2031  |              1
│ │ │ │ -0007ea00: 3030 3030 3031 3920 2030 2020 2039 2020  0000019  0   9  
│ │ │ │ +0007e9e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007e9f0: 2020 2020 2020 2020 2031 3030 3030 3031           1000001
│ │ │ │ +0007ea00: 3920 2030 2020 2039 2020 2020 2020 2020  9  0   9        
│ │ │ │  0007ea10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ea20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ea30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007ea40: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0007ea30: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0007ea40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007ea50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007ea60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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        
│ │ │ │ +0007ea80: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +0007ea90: 7469 6d65 2070 6172 616d 6574 7269 7a65  time parametrize
│ │ │ │ +0007eaa0: 204c 2020 2020 2020 2020 2020 2020 2020   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 3530  | -- used 0.0050
│ │ │ │ -0007eaf0: 3036 3138 7320 2863 7075 293b 2030 2e30  0618s (cpu); 0.0
│ │ │ │ -0007eb00: 3035 3030 3136 3873 2028 7468 7265 6164  0500168s (thread
│ │ │ │ -0007eb10: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ -0007eb20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007eb30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007ead0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ +0007eae0: 7365 6420 302e 3030 3633 3634 3434 7320  sed 0.00636444s 
│ │ │ │ +0007eaf0: 2863 7075 293b 2030 2e30 3036 3338 3333  (cpu); 0.0063833
│ │ │ │ +0007eb00: 3873 2028 7468 7265 6164 293b 2030 7320  8s (thread); 0s 
│ │ │ │ +0007eb10: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ +0007eb20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007eb30: 2020 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
│ │ │ │ -0007eb90: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │ +0007eb70: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ +0007eb80: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ +0007eb90: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │  0007eba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ebc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007ebd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007ebe0: 2020 2020 2020 5a5a 2020 2020 2020 2020        ZZ        
│ │ │ │ +0007ebc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007ebd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007ebe0: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │  0007ebf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ec10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007ec20: 7c20 2020 2020 736f 7572 6365 3a20 5072  |     source: Pr
│ │ │ │ -0007ec30: 6f6a 282d 2d2d 2d2d 2d2d 2d5b 7420 2c20  oj(--------[t , 
│ │ │ │ -0007ec40: 7420 2c20 7420 2c20 7420 2c20 7420 2c20  t , t , t , t , 
│ │ │ │ -0007ec50: 7420 5d29 2020 2020 2020 2020 2020 2020  t ])            
│ │ │ │ -0007ec60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007ec70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007ec80: 2020 2031 3030 3030 3031 3920 2030 2020     10000019  0  
│ │ │ │ -0007ec90: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -0007eca0: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ -0007ecb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007ecc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007ecd0: 2020 2020 2020 5a5a 2020 2020 2020 2020        ZZ        
│ │ │ │ +0007ec10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007ec20: 736f 7572 6365 3a20 5072 6f6a 282d 2d2d  source: Proj(---
│ │ │ │ +0007ec30: 2d2d 2d2d 2d5b 7420 2c20 7420 2c20 7420  -----[t , t , t 
│ │ │ │ +0007ec40: 2c20 7420 2c20 7420 2c20 7420 5d29 2020  , t , t , t ])  
│ │ │ │ +0007ec50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007ec60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007ec70: 2020 2020 2020 2020 2020 2020 2031 3030               100
│ │ │ │ +0007ec80: 3030 3031 3920 2030 2020 2031 2020 2032  00019  0   1   2
│ │ │ │ +0007ec90: 2020 2033 2020 2034 2020 2035 2020 2020     3   4   5    
│ │ │ │ +0007eca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007ecb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007ecc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007ecd0: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │  0007ece0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ed00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007ed10: 7c20 2020 2020 7461 7267 6574 3a20 5072  |     target: Pr
│ │ │ │ -0007ed20: 6f6a 282d 2d2d 2d2d 2d2d 2d5b 7820 2c20  oj(--------[x , 
│ │ │ │ -0007ed30: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -0007ed40: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -0007ed50: 7820 5d29 2020 2020 2020 2020 2020 7c0a  x ])          |.
│ │ │ │ -0007ed60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007ed70: 2020 2031 3030 3030 3031 3920 2030 2020     10000019  0  
│ │ │ │ -0007ed80: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -0007ed90: 2035 2020 2036 2020 2037 2020 2038 2020   5   6   7   8  
│ │ │ │ -0007eda0: 2039 2020 2020 2020 2020 2020 2020 7c0a   9            |.
│ │ │ │ -0007edb0: 7c20 2020 2020 6465 6669 6e69 6e67 2066  |     defining f
│ │ │ │ -0007edc0: 6f72 6d73 3a20 7b20 2020 2020 2020 2020  orms: {         
│ │ │ │ +0007ed00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007ed10: 7461 7267 6574 3a20 5072 6f6a 282d 2d2d  target: Proj(---
│ │ │ │ +0007ed20: 2d2d 2d2d 2d5b 7820 2c20 7820 2c20 7820  -----[x , x , x 
│ │ │ │ +0007ed30: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ +0007ed40: 2c20 7820 2c20 7820 2c20 7820 5d29 2020  , x , x , x ])  
│ │ │ │ +0007ed50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007ed60: 2020 2020 2020 2020 2020 2020 2031 3030               100
│ │ │ │ +0007ed70: 3030 3031 3920 2030 2020 2031 2020 2032  00019  0   1   2
│ │ │ │ +0007ed80: 2020 2033 2020 2034 2020 2035 2020 2036     3   4   5   6
│ │ │ │ +0007ed90: 2020 2037 2020 2038 2020 2039 2020 2020     7   8   9    
│ │ │ │ +0007eda0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007edb0: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20  defining forms: 
│ │ │ │ +0007edc0: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │  0007edd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ede0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007edf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007ee00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007ee10: 2020 2020 2020 2032 3039 3434 3536 7420         2094456t 
│ │ │ │ -0007ee20: 202b 2033 3933 3634 3938 7420 202b 2034   + 3936498t  + 4
│ │ │ │ -0007ee30: 3636 3534 3034 7420 202b 2037 3336 3934  665404t  + 73694
│ │ │ │ -0007ee40: 3374 2020 2b20 2020 2020 2020 2020 7c0a  3t  +         |.
│ │ │ │ -0007ee50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007ee60: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -0007ee70: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -0007ee80: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0007ee90: 2020 3320 2020 2020 2020 2020 2020 7c0a    3           |.
│ │ │ │ -0007eea0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007edf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007ee00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007ee10: 2032 3039 3434 3536 7420 202b 2033 3933   2094456t  + 393
│ │ │ │ +0007ee20: 3634 3938 7420 202b 2034 3636 3534 3034  6498t  + 4665404
│ │ │ │ +0007ee30: 7420 202b 2037 3336 3934 3374 2020 2b20  t  + 736943t  + 
│ │ │ │ +0007ee40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007ee50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007ee60: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ +0007ee70: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +0007ee80: 2032 2020 2020 2020 2020 2020 3320 2020   2          3   
│ │ │ │ +0007ee90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007eea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007eed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007eee0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007eef0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007ef00: 2020 2020 2020 202d 2034 3732 3437 3039         - 4724709
│ │ │ │ -0007ef10: 7420 202b 2033 3530 3533 3836 7420 202d  t  + 3505386t  -
│ │ │ │ -0007ef20: 2032 3436 3932 3036 7420 202b 2031 3338   2469206t  + 138
│ │ │ │ -0007ef30: 3135 3135 7420 2020 2020 2020 2020 7c0a  1515t         |.
│ │ │ │ -0007ef40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ef60: 2030 2020 2020 2020 2020 2020 2031 2020   0           1  
│ │ │ │ -0007ef70: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -0007ef80: 2020 2020 2033 2020 2020 2020 2020 7c0a       3        |.
│ │ │ │ -0007ef90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007eee0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007eef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007ef00: 202d 2034 3732 3437 3039 7420 202b 2033   - 4724709t  + 3
│ │ │ │ +0007ef10: 3530 3533 3836 7420 202d 2032 3436 3932  505386t  - 24692
│ │ │ │ +0007ef20: 3036 7420 202b 2031 3338 3135 3135 7420  06t  + 1381515t 
│ │ │ │ +0007ef30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007ef40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007ef50: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ +0007ef60: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ +0007ef70: 2020 2032 2020 2020 2020 2020 2020 2033     2           3
│ │ │ │ +0007ef80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007efd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007efe0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007eff0: 2020 2020 2020 202d 2036 3332 3238 3474         - 632284t
│ │ │ │ -0007f000: 2020 2d20 3439 3234 3538 7420 202d 2033    - 492458t  - 3
│ │ │ │ -0007f010: 3836 3932 3534 7420 202d 2032 3834 3032  869254t  - 28402
│ │ │ │ -0007f020: 3636 7420 202d 2020 2020 2020 2020 7c0a  66t  -        |.
│ │ │ │ -0007f030: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f050: 3020 2020 2020 2020 2020 2031 2020 2020  0          1    
│ │ │ │ -0007f060: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0007f070: 2020 2033 2020 2020 2020 2020 2020 7c0a     3          |.
│ │ │ │ -0007f080: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007efd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007eff0: 202d 2036 3332 3238 3474 2020 2d20 3439   - 632284t  - 49
│ │ │ │ +0007f000: 3234 3538 7420 202d 2033 3836 3932 3534  2458t  - 3869254
│ │ │ │ +0007f010: 7420 202d 2032 3834 3032 3636 7420 202d  t  - 2840266t  -
│ │ │ │ +0007f020: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007f040: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +0007f050: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +0007f060: 2032 2020 2020 2020 2020 2020 2033 2020   2           3  
│ │ │ │ +0007f070: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f0c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f0d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007f0e0: 2020 2020 2020 2031 3131 3230 3136 7420         1112016t 
│ │ │ │ -0007f0f0: 202b 2033 3930 3133 3631 7420 202b 2033   + 3901361t  + 3
│ │ │ │ -0007f100: 3139 3338 3633 7420 202d 2034 3134 3330  193863t  - 41430
│ │ │ │ -0007f110: 3430 7420 202b 2020 2020 2020 2020 7c0a  40t  +        |.
│ │ │ │ -0007f120: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007f130: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -0007f140: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -0007f150: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0007f160: 2020 2033 2020 2020 2020 2020 2020 7c0a     3          |.
│ │ │ │ -0007f170: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007f0c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007f0e0: 2031 3131 3230 3136 7420 202b 2033 3930   1112016t  + 390
│ │ │ │ +0007f0f0: 3133 3631 7420 202b 2033 3139 3338 3633  1361t  + 3193863
│ │ │ │ +0007f100: 7420 202d 2034 3134 3330 3430 7420 202b  t  - 4143040t  +
│ │ │ │ +0007f110: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007f130: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ +0007f140: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +0007f150: 2032 2020 2020 2020 2020 2020 2033 2020   2           3  
│ │ │ │ +0007f160: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f1b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f1c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007f1d0: 2020 2020 2020 2074 202c 2020 2020 2020         t ,      
│ │ │ │ +0007f1b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007f1d0: 2074 202c 2020 2020 2020 2020 2020 2020   t ,            
│ │ │ │  0007f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f200: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f210: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007f220: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ +0007f200: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007f220: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │  0007f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f250: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f260: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007f250: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f2a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f2b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007f2c0: 2020 2020 2020 2074 202c 2020 2020 2020         t ,      
│ │ │ │ +0007f2a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007f2c0: 2074 202c 2020 2020 2020 2020 2020 2020   t ,            
│ │ │ │  0007f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f2f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f300: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007f310: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +0007f2f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007f310: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │  0007f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f340: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f350: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007f340: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f390: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f3a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007f3b0: 2020 2020 2020 2074 202c 2020 2020 2020         t ,      
│ │ │ │ +0007f390: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007f3b0: 2074 202c 2020 2020 2020 2020 2020 2020   t ,            
│ │ │ │  0007f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f3e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f3f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007f400: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0007f3e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007f400: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0007f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f430: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f440: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007f430: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f480: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f490: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007f4a0: 2020 2020 2020 2074 202c 2020 2020 2020         t ,      
│ │ │ │ +0007f480: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007f4a0: 2074 202c 2020 2020 2020 2020 2020 2020   t ,            
│ │ │ │  0007f4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f4d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f4e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007f4f0: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ +0007f4d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007f4f0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │  0007f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f520: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f530: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007f520: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f570: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f580: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007f590: 2020 2020 2020 2074 202c 2020 2020 2020         t ,      
│ │ │ │ +0007f570: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007f590: 2074 202c 2020 2020 2020 2020 2020 2020   t ,            
│ │ │ │  0007f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f5c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f5d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007f5e0: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +0007f5c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007f5e0: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │  0007f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f610: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f620: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007f610: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f660: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f670: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007f680: 2020 2020 2020 2074 2020 2020 2020 2020         t        
│ │ │ │ +0007f660: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007f680: 2074 2020 2020 2020 2020 2020 2020 2020   t              
│ │ │ │  0007f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f6b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f6c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007f6d0: 2020 2020 2020 2020 3520 2020 2020 2020          5       
│ │ │ │ +0007f6b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007f6d0: 2020 3520 2020 2020 2020 2020 2020 2020    5             
│ │ │ │  0007f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f700: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f710: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0007f720: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │ +0007f700: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0007f720: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │  0007f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f750: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f760: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007f750: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f7a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f7b0: 7c6f 3320 3a20 5261 7469 6f6e 616c 4d61  |o3 : RationalMa
│ │ │ │ -0007f7c0: 7020 286c 696e 6561 7220 7261 7469 6f6e  p (linear ration
│ │ │ │ -0007f7d0: 616c 206d 6170 2066 726f 6d20 5050 5e35  al map from PP^5
│ │ │ │ -0007f7e0: 2074 6f20 5050 5e39 2920 2020 2020 2020   to PP^9)       
│ │ │ │ -0007f7f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f800: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +0007f7a0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ +0007f7b0: 5261 7469 6f6e 616c 4d61 7020 286c 696e  RationalMap (lin
│ │ │ │ +0007f7c0: 6561 7220 7261 7469 6f6e 616c 206d 6170  ear rational map
│ │ │ │ +0007f7d0: 2066 726f 6d20 5050 5e35 2074 6f20 5050   from PP^5 to PP
│ │ │ │ +0007f7e0: 5e39 2920 2020 2020 2020 2020 2020 2020  ^9)             
│ │ │ │ +0007f7f0: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d          |.|-----
│ │ │ │ +0007f800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007f810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007f820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007f830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007f840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007f850: 7c38 3439 3637 3174 2020 2d20 3330 3334  |849671t  - 3034
│ │ │ │ -0007f860: 3133 3774 202c 2020 2020 2020 2020 2020  137t ,          
│ │ │ │ +0007f840: 2d2d 2d2d 2d2d 2d2d 7c0a 7c38 3439 3637  --------|.|84967
│ │ │ │ +0007f850: 3174 2020 2d20 3330 3334 3133 3774 202c  1t  - 3034137t ,
│ │ │ │ +0007f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f890: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f8a0: 7c20 2020 2020 2020 3420 2020 2020 2020  |       4       
│ │ │ │ -0007f8b0: 2020 2020 3520 2020 2020 2020 2020 2020      5           
│ │ │ │ +0007f890: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f8a0: 2020 3420 2020 2020 2020 2020 2020 3520    4           5 
│ │ │ │ +0007f8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f8e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f8f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007f8e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f930: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f940: 7c20 2d20 3233 3331 3238 3074 2020 2b20  | - 2331280t  + 
│ │ │ │ -0007f950: 3439 3336 3232 3974 202c 2020 2020 2020  4936229t ,      
│ │ │ │ +0007f930: 2020 2020 2020 2020 7c0a 7c20 2d20 3233          |.| - 23
│ │ │ │ +0007f940: 3331 3238 3074 2020 2b20 3439 3336 3232  31280t  + 493622
│ │ │ │ +0007f950: 3974 202c 2020 2020 2020 2020 2020 2020  9t ,            
│ │ │ │  0007f960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f980: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f990: 7c20 2020 2020 2020 2020 2020 3420 2020  |           4   
│ │ │ │ -0007f9a0: 2020 2020 2020 2020 3520 2020 2020 2020          5       
│ │ │ │ +0007f980: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f990: 2020 2020 2020 3420 2020 2020 2020 2020        4         
│ │ │ │ +0007f9a0: 2020 3520 2020 2020 2020 2020 2020 2020    5             
│ │ │ │  0007f9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007f9d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007f9e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007f9d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007f9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007f9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007fa20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007fa30: 7c20 3438 3833 3937 3474 2020 2d20 3333  | 4883974t  - 33
│ │ │ │ -0007fa40: 3430 3936 3174 202c 2020 2020 2020 2020  40961t ,        
│ │ │ │ +0007fa20: 2020 2020 2020 2020 7c0a 7c20 3438 3833          |.| 4883
│ │ │ │ +0007fa30: 3937 3474 2020 2d20 3333 3430 3936 3174  974t  - 3340961t
│ │ │ │ +0007fa40: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  0007fa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007fa70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007fa80: 7c20 2020 2020 2020 2020 3420 2020 2020  |         4     
│ │ │ │ -0007fa90: 2020 2020 2020 3520 2020 2020 2020 2020        5         
│ │ │ │ +0007fa70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007fa80: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ +0007fa90: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │  0007faa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007fac0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007fad0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007fac0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007fad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007faf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007fb10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007fb20: 7c20 3139 3634 3431 3774 2020 2d20 3130  | 1964417t  - 10
│ │ │ │ -0007fb30: 3734 3935 3874 202c 2020 2020 2020 2020  74958t ,        
│ │ │ │ +0007fb10: 2020 2020 2020 2020 7c0a 7c20 3139 3634          |.| 1964
│ │ │ │ +0007fb20: 3431 3774 2020 2d20 3130 3734 3935 3874  417t  - 1074958t
│ │ │ │ +0007fb30: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  0007fb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007fb60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007fb70: 7c20 2020 2020 2020 2020 3420 2020 2020  |         4     
│ │ │ │ -0007fb80: 2020 2020 2020 3520 2020 2020 2020 2020        5         
│ │ │ │ +0007fb60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007fb70: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ +0007fb80: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │  0007fb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007fbb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007fbc0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0007fbb0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0007fbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007fbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007fbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007fbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007fc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0007fc10: 7c69 3420 3a20 5120 3d20 7472 696d 2069  |i4 : Q = trim i
│ │ │ │ -0007fc20: 6465 616c 2872 616e 646f 6d28 322c 5039  deal(random(2,P9
│ │ │ │ -0007fc30: 292c 7261 6e64 6f6d 2831 2c50 3929 2c72  ),random(1,P9),r
│ │ │ │ -0007fc40: 616e 646f 6d28 312c 5039 2929 2020 2020  andom(1,P9))    
│ │ │ │ -0007fc50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007fc60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007fc00: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ +0007fc10: 5120 3d20 7472 696d 2069 6465 616c 2872  Q = trim ideal(r
│ │ │ │ +0007fc20: 616e 646f 6d28 322c 5039 292c 7261 6e64  andom(2,P9),rand
│ │ │ │ +0007fc30: 6f6d 2831 2c50 3929 2c72 616e 646f 6d28  om(1,P9),random(
│ │ │ │ +0007fc40: 312c 5039 2929 2020 2020 2020 2020 2020  1,P9))          
│ │ │ │ +0007fc50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007fc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007fca0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007fcb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007fca0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007fcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007fcf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007fd00: 7c6f 3420 3d20 6964 6561 6c20 2878 2020  |o4 = ideal (x  
│ │ │ │ -0007fd10: 2d20 3337 3331 3238 3578 2020 2b20 3536  - 3731285x  + 56
│ │ │ │ -0007fd20: 3934 3835 7820 202b 2034 3235 3532 3031  9485x  + 4255201
│ │ │ │ -0007fd30: 7820 202d 2032 3039 3837 3132 7820 202d  x  - 2098712x  -
│ │ │ │ -0007fd40: 2034 3234 3839 3930 7820 202d 2020 7c0a   4248990x  -  |.
│ │ │ │ -0007fd50: 7c20 2020 2020 2020 2020 2020 2020 3120  |             1 
│ │ │ │ -0007fd60: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0007fd70: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ -0007fd80: 2034 2020 2020 2020 2020 2020 2035 2020   4           5  
│ │ │ │ -0007fd90: 2020 2020 2020 2020 2036 2020 2020 7c0a           6    |.
│ │ │ │ -0007fda0: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +0007fcf0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ +0007fd00: 6964 6561 6c20 2878 2020 2d20 3337 3331  ideal (x  - 3731
│ │ │ │ +0007fd10: 3238 3578 2020 2b20 3536 3934 3835 7820  285x  + 569485x 
│ │ │ │ +0007fd20: 202b 2034 3235 3532 3031 7820 202d 2032   + 4255201x  - 2
│ │ │ │ +0007fd30: 3039 3837 3132 7820 202d 2034 3234 3839  098712x  - 42489
│ │ │ │ +0007fd40: 3930 7820 202d 2020 7c0a 7c20 2020 2020  90x  -  |.|     
│ │ │ │ +0007fd50: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +0007fd60: 2020 2020 3220 2020 2020 2020 2020 2033      2          3
│ │ │ │ +0007fd70: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ +0007fd80: 2020 2020 2020 2035 2020 2020 2020 2020         5        
│ │ │ │ +0007fd90: 2020 2036 2020 2020 7c0a 7c20 2020 2020     6    |.|     
│ │ │ │ +0007fda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007fdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007fdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007fdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007fde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007fdf0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007fde0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007fdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fe00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fe10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007fe20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007fe30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007fe40: 7c20 2020 2020 3138 3031 3334 3278 2020  |     1801342x  
│ │ │ │ -0007fe50: 2b20 3430 3530 3232 3978 2020 2d20 3233  + 4050229x  - 23
│ │ │ │ -0007fe60: 3139 3236 3378 202c 2078 2020 2d20 3330  19263x , x  - 30
│ │ │ │ -0007fe70: 3934 3638 3978 2020 2d20 3434 3130 3138  94689x  - 441018
│ │ │ │ -0007fe80: 3678 2020 2b20 2020 2020 2020 2020 7c0a  6x  +         |.
│ │ │ │ -0007fe90: 7c20 2020 2020 2020 2020 2020 2020 3720  |             7 
│ │ │ │ -0007fea0: 2020 2020 2020 2020 2020 3820 2020 2020            8     
│ │ │ │ -0007feb0: 2020 2020 2020 3920 2020 3020 2020 2020        9   0     
│ │ │ │ -0007fec0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -0007fed0: 2020 3320 2020 2020 2020 2020 2020 7c0a    3           |.
│ │ │ │ -0007fee0: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +0007fe30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007fe40: 3138 3031 3334 3278 2020 2b20 3430 3530  1801342x  + 4050
│ │ │ │ +0007fe50: 3232 3978 2020 2d20 3233 3139 3236 3378  229x  - 2319263x
│ │ │ │ +0007fe60: 202c 2078 2020 2d20 3330 3934 3638 3978   , x  - 3094689x
│ │ │ │ +0007fe70: 2020 2d20 3434 3130 3138 3678 2020 2b20    - 4410186x  + 
│ │ │ │ +0007fe80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007fe90: 2020 2020 2020 2020 3720 2020 2020 2020          7       
│ │ │ │ +0007fea0: 2020 2020 3820 2020 2020 2020 2020 2020      8           
│ │ │ │ +0007feb0: 3920 2020 3020 2020 2020 2020 2020 2020  9   0           
│ │ │ │ +0007fec0: 3220 2020 2020 2020 2020 2020 3320 2020  2           3   
│ │ │ │ +0007fed0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0007fee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007fef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007ff00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007ff10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0007ff20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0007ff30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0007ff20: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0007ff30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ff40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ff50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ff60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0007ff70: 2020 2020 2020 2020 2020 2020 2032 7c0a               2|.
│ │ │ │ -0007ff80: 7c20 2020 2020 3331 3936 3134 3678 2020  |     3196146x  
│ │ │ │ -0007ff90: 2b20 3237 3133 3737 3178 2020 2b20 3232  + 2713771x  + 22
│ │ │ │ -0007ffa0: 3631 3431 3278 2020 2d20 3132 3637 3139  61412x  - 126719
│ │ │ │ -0007ffb0: 3678 2020 2d20 3432 3130 3430 3378 2020  6x  - 4210403x  
│ │ │ │ -0007ffc0: 2b20 3238 3539 3332 7820 2c20 7820 7c0a  + 285932x , x |.
│ │ │ │ -0007ffd0: 7c20 2020 2020 2020 2020 2020 2020 3420  |             4 
│ │ │ │ -0007ffe0: 2020 2020 2020 2020 2020 3520 2020 2020            5     
│ │ │ │ -0007fff0: 2020 2020 2020 3620 2020 2020 2020 2020        6         
│ │ │ │ -00080000: 2020 3720 2020 2020 2020 2020 2020 3820    7           8 
│ │ │ │ -00080010: 2020 2020 2020 2020 2039 2020 2032 7c0a           9   2|.
│ │ │ │ -00080020: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +0007ff70: 2020 2020 2020 2032 7c0a 7c20 2020 2020         2|.|     
│ │ │ │ +0007ff80: 3331 3936 3134 3678 2020 2b20 3237 3133  3196146x  + 2713
│ │ │ │ +0007ff90: 3737 3178 2020 2b20 3232 3631 3431 3278  771x  + 2261412x
│ │ │ │ +0007ffa0: 2020 2d20 3132 3637 3139 3678 2020 2d20    - 1267196x  - 
│ │ │ │ +0007ffb0: 3432 3130 3430 3378 2020 2b20 3238 3539  4210403x  + 2859
│ │ │ │ +0007ffc0: 3332 7820 2c20 7820 7c0a 7c20 2020 2020  32x , x |.|     
│ │ │ │ +0007ffd0: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +0007ffe0: 2020 2020 3520 2020 2020 2020 2020 2020      5           
│ │ │ │ +0007fff0: 3620 2020 2020 2020 2020 2020 3720 2020  6           7   
│ │ │ │ +00080000: 2020 2020 2020 2020 3820 2020 2020 2020          8       
│ │ │ │ +00080010: 2020 2039 2020 2032 7c0a 7c20 2020 2020     9   2|.|     
│ │ │ │ +00080020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00080030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00080040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00080050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00080060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00080070: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00080080: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00080060: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +00080070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00080080: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  00080090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000800a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000800b0: 2020 2020 2032 2020 2020 2020 2020 7c0a       2        |.
│ │ │ │ -000800c0: 7c20 2020 2020 2b20 3130 3435 3432 3178  |     + 1045421x
│ │ │ │ -000800d0: 2078 2020 2b20 3731 3835 3332 7820 202d   x  + 718532x  -
│ │ │ │ -000800e0: 2033 3730 3136 3238 7820 7820 202b 2033   3701628x x  + 3
│ │ │ │ -000800f0: 3930 3337 3938 7820 7820 202b 2032 3834  903798x x  + 284
│ │ │ │ -00080100: 3233 3937 7820 202d 2020 2020 2020 7c0a  2397x  -      |.
│ │ │ │ -00080110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00080120: 3220 3320 2020 2020 2020 2020 2033 2020  2 3          3  
│ │ │ │ -00080130: 2020 2020 2020 2020 2032 2034 2020 2020           2 4    
│ │ │ │ -00080140: 2020 2020 2020 2033 2034 2020 2020 2020         3 4      
│ │ │ │ -00080150: 2020 2020 2034 2020 2020 2020 2020 7c0a       4        |.
│ │ │ │ -00080160: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +000800a0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +000800b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000800c0: 2b20 3130 3435 3432 3178 2078 2020 2b20  + 1045421x x  + 
│ │ │ │ +000800d0: 3731 3835 3332 7820 202d 2033 3730 3136  718532x  - 37016
│ │ │ │ +000800e0: 3238 7820 7820 202b 2033 3930 3337 3938  28x x  + 3903798
│ │ │ │ +000800f0: 7820 7820 202b 2032 3834 3233 3937 7820  x x  + 2842397x 
│ │ │ │ +00080100: 202d 2020 2020 2020 7c0a 7c20 2020 2020   -      |.|     
│ │ │ │ +00080110: 2020 2020 2020 2020 2020 3220 3320 2020            2 3   
│ │ │ │ +00080120: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +00080130: 2020 2032 2034 2020 2020 2020 2020 2020     2 4          
│ │ │ │ +00080140: 2033 2034 2020 2020 2020 2020 2020 2034   3 4           4
│ │ │ │ +00080150: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │  000801d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000801e0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -000801f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ -00080240: 3738 3278 2078 2020 2b20 2020 2020 7c0a  782x x  +     |.
│ │ │ │ -00080250: 7c20 2020 2020 2020 2020 2020 2020 3220  |             2 
│ │ │ │ -00080260: 3520 2020 2020 2020 2020 2020 3320 3520  5           3 5 
│ │ │ │ -00080270: 2020 2020 2020 2020 2020 3420 3520 2020            4 5   
│ │ │ │ -00080280: 2020 2020 2020 2020 3520 2020 2020 2020          5       
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│ │ │ │ -00080380: 3637 7820 7820 202b 2020 2020 2020 7c0a  67x x  +      |.
│ │ │ │ -00080390: 7c20 2020 2020 2020 2020 2020 2020 3320  |             3 
│ │ │ │ -000803a0: 3620 2020 2020 2020 2020 2020 3420 3620  6           4 6 
│ │ │ │ -000803b0: 2020 2020 2020 2020 2035 2036 2020 2020           5 6    
│ │ │ │ -000803c0: 2020 2020 2020 2036 2020 2020 2020 2020         6        
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│ │ │ │ -000803e0: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
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│ │ │ │ -00080460: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00080490: 2020 2b20 3137 3632 3532 3878 2078 2020    + 1762528x x  
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│ │ │ │ -000804c0: 3839 3637 7820 202b 2020 2020 2020 7c0a  8967x  +      |.
│ │ │ │ -000804d0: 7c20 2020 2020 2020 2020 2020 2020 3320  |             3 
│ │ │ │ -000804e0: 3720 2020 2020 2020 2020 2020 3420 3720  7           4 7 
│ │ │ │ -000804f0: 2020 2020 2020 2020 2035 2037 2020 2020           5 7    
│ │ │ │ -00080500: 2020 2020 2020 2036 2037 2020 2020 2020         6 7      
│ │ │ │ -00080510: 2020 2020 2037 2020 2020 2020 2020 7c0a       7        |.
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│ │ │ │ +00080470: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │ +00080490: 3632 3532 3878 2078 2020 2b20 3133 3736  62528x x  + 1376
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│ │ │ │ +000804b0: 7820 7820 202d 2034 3031 3839 3637 7820  x x  - 4018967x 
│ │ │ │ +000804c0: 202b 2020 2020 2020 7c0a 7c20 2020 2020   +      |.|     
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│ │ │ │ +00080510: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │  000805a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000805b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ -000805d0: 2020 2d20 3235 3431 3330 3978 2078 2020    - 2541309x x  
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│ │ │ │ -00080600: 3433 3536 3078 2078 2020 2b20 2020 7c0a  43560x x  +   |.
│ │ │ │ -00080610: 7c20 2020 2020 2020 2020 2020 2020 3220  |             2 
│ │ │ │ -00080620: 3820 2020 2020 2020 2020 2020 3320 3820  8           3 8 
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│ │ │ │ +000805f0: 3478 2078 2020 2d20 3136 3433 3536 3078  4x x  - 1643560x
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│ │ │ │ -00080710: 2020 2d20 3232 3830 3531 3678 2020 2d20    - 2280516x  - 
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│ │ │ │ -00080740: 3636 3778 2078 2020 2b20 2020 2020 7c0a  667x x  +     |.
│ │ │ │ -00080750: 7c20 2020 2020 2020 2020 2020 2020 3720  |             7 
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│ │ │ │ -00080780: 2020 2020 2020 3320 3920 2020 2020 2020        3 9       
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│ │ │ │ +00080740: 2020 2b20 2020 2020 7c0a 7c20 2020 2020    +     |.|     
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│ │ │ │  00080820: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00080870: 3330 3232 3234 3278 2078 2020 2b20 3336  3022242x x  + 36
│ │ │ │ -00080880: 3138 3738 3978 2029 2020 2020 2020 7c0a  18789x )      |.
│ │ │ │ -00080890: 7c20 2020 2020 2020 2020 2020 2020 3520  |             5 
│ │ │ │ -000808a0: 3920 2020 2020 2020 2020 2020 3620 3920  9           6 9 
│ │ │ │ -000808b0: 2020 2020 2020 2020 2020 3720 3920 2020            7 9   
│ │ │ │ -000808c0: 2020 2020 2020 2020 3820 3920 2020 2020          8 9     
│ │ │ │ -000808d0: 2020 2020 2020 3920 2020 2020 2020 7c0a        9       |.
│ │ │ │ -000808e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ +00080840: 3132 3131 3630 3178 2078 2020 2d20 3231  1211601x x  - 21
│ │ │ │ +00080850: 3638 3539 3478 2078 2020 2d20 3138 3031  68594x x  - 1801
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│ │ │ │ +00080870: 3278 2078 2020 2b20 3336 3138 3738 3978  2x x  + 3618789x
│ │ │ │ +00080880: 2029 2020 2020 2020 7c0a 7c20 2020 2020   )      |.|     
│ │ │ │ +00080890: 2020 2020 2020 2020 3520 3920 2020 2020          5 9     
│ │ │ │ +000808a0: 2020 2020 2020 3620 3920 2020 2020 2020        6 9       
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│ │ │ │  000808f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00080910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080920: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00080930: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00080940: 2020 5a5a 2020 2020 2020 2020 2020 2020    ZZ            
│ │ │ │ +00080920: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00080930: 2020 2020 2020 2020 2020 2020 5a5a 2020              ZZ  
│ │ │ │ +00080940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080970: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00080980: 7c6f 3420 3a20 4964 6561 6c20 6f66 202d  |o4 : Ideal of -
│ │ │ │ -00080990: 2d2d 2d2d 2d2d 2d5b 7820 2e2e 7820 5d20  -------[x ..x ] 
│ │ │ │ +00080970: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ +00080980: 4964 6561 6c20 6f66 202d 2d2d 2d2d 2d2d  Ideal of -------
│ │ │ │ +00080990: 2d5b 7820 2e2e 7820 5d20 2020 2020 2020  -[x ..x ]       
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│ │ │ │  000809b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000809c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000809d0: 7c20 2020 2020 2020 2020 2020 2020 2031  |              1
│ │ │ │ -000809e0: 3030 3030 3031 3920 2030 2020 2039 2020  0000019  0   9  
│ │ │ │ +000809c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000809d0: 2020 2020 2020 2020 2031 3030 3030 3031           1000001
│ │ │ │ +000809e0: 3920 2030 2020 2039 2020 2020 2020 2020  9  0   9        
│ │ │ │  000809f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080a10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00080a20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00080a10: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
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│ │ │ │  00080a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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        
│ │ │ │ +00080a60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ +00080a70: 7469 6d65 2070 6172 616d 6574 7269 7a65  time parametrize
│ │ │ │ +00080a80: 2051 2020 2020 2020 2020 2020 2020 2020   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 3534 3636  | -- used 0.5466
│ │ │ │ -00080ad0: 7320 2863 7075 293b 2030 2e33 3934 3339  s (cpu); 0.39439
│ │ │ │ -00080ae0: 3273 2028 7468 7265 6164 293b 2030 7320  2s (thread); 0s 
│ │ │ │ -00080af0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ -00080b00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00080b10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00080ab0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ +00080ac0: 7365 6420 302e 3633 3233 3037 7320 2863  sed 0.632307s (c
│ │ │ │ +00080ad0: 7075 293b 2030 2e34 3636 3235 3473 2028  pu); 0.466254s (
│ │ │ │ +00080ae0: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ +00080af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00080b00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00080b10: 2020 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                |.
│ │ │ │ -00080b60: 7c6f 3520 3d20 2d2d 2072 6174 696f 6e61  |o5 = -- rationa
│ │ │ │ -00080b70: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │ +00080b50: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ +00080b60: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ +00080b70: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │  00080b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080ba0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00080bb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00080bc0: 2020 2020 2020 5a5a 2020 2020 2020 2020        ZZ        
│ │ │ │ +00080ba0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00080bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00080bc0: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │  00080bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080bf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00080c00: 7c20 2020 2020 736f 7572 6365 3a20 5072  |     source: Pr
│ │ │ │ -00080c10: 6f6a 282d 2d2d 2d2d 2d2d 2d5b 7420 2c20  oj(--------[t , 
│ │ │ │ -00080c20: 7420 2c20 7420 2c20 7420 2c20 7420 2c20  t , t , t , t , 
│ │ │ │ -00080c30: 7420 2c20 7420 5d29 2020 2020 2020 2020  t , t ])        
│ │ │ │ -00080c40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00080c50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00080c60: 2020 2031 3030 3030 3031 3920 2030 2020     10000019  0  
│ │ │ │ -00080c70: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -00080c80: 2035 2020 2036 2020 2020 2020 2020 2020   5   6          
│ │ │ │ -00080c90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00080ca0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00080cb0: 2020 2020 2020 5a5a 2020 2020 2020 2020        ZZ        
│ │ │ │ +00080bf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00080c00: 736f 7572 6365 3a20 5072 6f6a 282d 2d2d  source: Proj(---
│ │ │ │ +00080c10: 2d2d 2d2d 2d5b 7420 2c20 7420 2c20 7420  -----[t , t , t 
│ │ │ │ +00080c20: 2c20 7420 2c20 7420 2c20 7420 2c20 7420  , t , t , t , t 
│ │ │ │ +00080c30: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ +00080c40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00080c50: 2020 2020 2020 2020 2020 2020 2031 3030               100
│ │ │ │ +00080c60: 3030 3031 3920 2030 2020 2031 2020 2032  00019  0   1   2
│ │ │ │ +00080c70: 2020 2033 2020 2034 2020 2035 2020 2036     3   4   5   6
│ │ │ │ +00080c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00080c90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00080ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00080cb0: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │  00080cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080ce0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00080cf0: 7c20 2020 2020 7461 7267 6574 3a20 5072  |     target: Pr
│ │ │ │ -00080d00: 6f6a 282d 2d2d 2d2d 2d2d 2d5b 7820 2c20  oj(--------[x , 
│ │ │ │ -00080d10: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -00080d20: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -00080d30: 7820 5d29 2020 2020 2020 2020 2020 7c0a  x ])          |.
│ │ │ │ -00080d40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00080d50: 2020 2031 3030 3030 3031 3920 2030 2020     10000019  0  
│ │ │ │ -00080d60: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -00080d70: 2035 2020 2036 2020 2037 2020 2038 2020   5   6   7   8  
│ │ │ │ -00080d80: 2039 2020 2020 2020 2020 2020 2020 7c0a   9            |.
│ │ │ │ -00080d90: 7c20 2020 2020 6465 6669 6e69 6e67 2066  |     defining f
│ │ │ │ -00080da0: 6f72 6d73 3a20 7b20 2020 2020 2020 2020  orms: {         
│ │ │ │ +00080ce0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00080cf0: 7461 7267 6574 3a20 5072 6f6a 282d 2d2d  target: Proj(---
│ │ │ │ +00080d00: 2d2d 2d2d 2d5b 7820 2c20 7820 2c20 7820  -----[x , x , x 
│ │ │ │ +00080d10: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ +00080d20: 2c20 7820 2c20 7820 2c20 7820 5d29 2020  , x , x , x ])  
│ │ │ │ +00080d30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00080d40: 2020 2020 2020 2020 2020 2020 2031 3030               100
│ │ │ │ +00080d50: 3030 3031 3920 2030 2020 2031 2020 2032  00019  0   1   2
│ │ │ │ +00080d60: 2020 2033 2020 2034 2020 2035 2020 2036     3   4   5   6
│ │ │ │ +00080d70: 2020 2037 2020 2038 2020 2039 2020 2020     7   8   9    
│ │ │ │ +00080d80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00080d90: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20  defining forms: 
│ │ │ │ +00080da0: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │  00080db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080dd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00080de0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00080df0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00080dd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00080de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00080df0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  00080e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080e10: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -00080e20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00080e30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00080e40: 2020 2020 2020 2038 3137 3939 3874 2020         817998t  
│ │ │ │ -00080e50: 2b20 3133 3239 3231 3674 2074 2020 2d20  + 1329216t t  - 
│ │ │ │ -00080e60: 3131 3134 3138 3274 2020 2b20 3133 3434  1114182t  + 1344
│ │ │ │ -00080e70: 3832 3174 2074 2020 2d20 3137 3830 7c0a  821t t  - 1780|.
│ │ │ │ -00080e80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00080e90: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00080ea0: 2020 2020 2020 2020 2020 3020 3120 2020            0 1   
│ │ │ │ -00080eb0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -00080ec0: 2020 2020 3020 3220 2020 2020 2020 7c0a      0 2       |.
│ │ │ │ -00080ed0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00080e10: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00080e20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00080e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00080e40: 2038 3137 3939 3874 2020 2b20 3133 3239   817998t  + 1329
│ │ │ │ +00080e50: 3231 3674 2074 2020 2d20 3131 3134 3138  216t t  - 111418
│ │ │ │ +00080e60: 3274 2020 2b20 3133 3434 3832 3174 2074  2t  + 1344821t t
│ │ │ │ +00080e70: 2020 2d20 3137 3830 7c0a 7c20 2020 2020    - 1780|.|     
│ │ │ │ +00080e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00080e90: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ +00080ea0: 2020 2020 3020 3120 2020 2020 2020 2020      0 1         
│ │ │ │ +00080eb0: 2020 3120 2020 2020 2020 2020 2020 3020    1           0 
│ │ │ │ +00080ec0: 3220 2020 2020 2020 7c0a 7c20 2020 2020  2       |.|     
│ │ │ │ +00080ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00080f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080f10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00080f20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00080f30: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00080f10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00080f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00080f30: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  00080f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00080f50: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -00080f60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00080f70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00080f80: 2020 2020 2020 2032 3639 3932 3932 7420         2699292t 
│ │ │ │ -00080f90: 202d 2039 3133 3238 3674 2074 2020 2b20   - 913286t t  + 
│ │ │ │ -00080fa0: 3133 3038 3536 3774 2020 2b20 3337 3432  1308567t  + 3742
│ │ │ │ -00080fb0: 3136 3574 2074 2020 2b20 3232 3334 7c0a  165t t  + 2234|.
│ │ │ │ -00080fc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00080fd0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00080fe0: 2020 2020 2020 2020 2020 3020 3120 2020            0 1   
│ │ │ │ -00080ff0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -00081000: 2020 2020 3020 3220 2020 2020 2020 7c0a      0 2       |.
│ │ │ │ -00081010: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00080f50: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00080f60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00080f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00080f80: 2032 3639 3932 3932 7420 202d 2039 3133   2699292t  - 913
│ │ │ │ +00080f90: 3238 3674 2074 2020 2b20 3133 3038 3536  286t t  + 130856
│ │ │ │ +00080fa0: 3774 2020 2b20 3337 3432 3136 3574 2074  7t  + 3742165t t
│ │ │ │ +00080fb0: 2020 2b20 3232 3334 7c0a 7c20 2020 2020    + 2234|.|     
│ │ │ │ +00080fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00080fd0: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ +00080fe0: 2020 2020 3020 3120 2020 2020 2020 2020      0 1         
│ │ │ │ +00080ff0: 2020 3120 2020 2020 2020 2020 2020 3020    1           0 
│ │ │ │ +00081000: 3220 2020 2020 2020 7c0a 7c20 2020 2020  2       |.|     
│ │ │ │ +00081010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081050: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00081060: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081070: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00081050: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00081060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081070: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  00081080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081090: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -000810a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000810b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000810c0: 2020 2020 2020 2032 3430 3937 3235 7420         2409725t 
│ │ │ │ -000810d0: 202d 2035 3238 3334 3174 2074 2020 2b20   - 528341t t  + 
│ │ │ │ -000810e0: 3136 3735 3430 3274 2020 2d20 3730 3736  1675402t  - 7076
│ │ │ │ -000810f0: 3136 7420 7420 202d 2031 3238 3138 7c0a  16t t  - 12818|.
│ │ │ │ -00081100: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081110: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00081120: 2020 2020 2020 2020 2020 3020 3120 2020            0 1   
│ │ │ │ -00081130: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -00081140: 2020 2030 2032 2020 2020 2020 2020 7c0a     0 2        |.
│ │ │ │ -00081150: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00081090: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +000810a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000810b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000810c0: 2032 3430 3937 3235 7420 202d 2035 3238   2409725t  - 528
│ │ │ │ +000810d0: 3334 3174 2074 2020 2b20 3136 3735 3430  341t t  + 167540
│ │ │ │ +000810e0: 3274 2020 2d20 3730 3736 3136 7420 7420  2t  - 707616t t 
│ │ │ │ +000810f0: 202d 2031 3238 3138 7c0a 7c20 2020 2020   - 12818|.|     
│ │ │ │ +00081100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081110: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ +00081120: 2020 2020 3020 3120 2020 2020 2020 2020      0 1         
│ │ │ │ +00081130: 2020 3120 2020 2020 2020 2020 2030 2032    1          0 2
│ │ │ │ +00081140: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00081150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081190: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000811a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000811b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000811c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -000811d0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -000811e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000811f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081200: 2020 2020 2020 202d 2033 3235 3733 3737         - 3257377
│ │ │ │ -00081210: 7420 202d 2031 3839 3532 3532 7420 7420  t  - 1895252t t 
│ │ │ │ -00081220: 202b 2032 3539 3237 3330 7420 202b 2032   + 2592730t  + 2
│ │ │ │ -00081230: 3630 3138 3332 7420 7420 202d 2032 7c0a  601832t t  - 2|.
│ │ │ │ -00081240: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081260: 2030 2020 2020 2020 2020 2020 2030 2031   0           0 1
│ │ │ │ -00081270: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -00081280: 2020 2020 2020 2030 2032 2020 2020 7c0a         0 2    |.
│ │ │ │ -00081290: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00081190: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000811a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000811b0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +000811c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000811d0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +000811e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000811f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081200: 202d 2033 3235 3733 3737 7420 202d 2031   - 3257377t  - 1
│ │ │ │ +00081210: 3839 3532 3532 7420 7420 202b 2032 3539  895252t t  + 259
│ │ │ │ +00081220: 3237 3330 7420 202b 2032 3630 3138 3332  2730t  + 2601832
│ │ │ │ +00081230: 7420 7420 202d 2032 7c0a 7c20 2020 2020  t t  - 2|.|     
│ │ │ │ +00081240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081250: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ +00081260: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ +00081270: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +00081280: 2030 2032 2020 2020 7c0a 7c20 2020 2020   0 2    |.|     
│ │ │ │ +00081290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000812a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000812b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000812c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000812d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000812e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000812f0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +000812d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000812e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000812f0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  00081300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081310: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -00081320: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00081330: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081340: 2020 2020 2020 2033 3636 3434 3674 2020         366446t  
│ │ │ │ -00081350: 2b20 3432 3937 3231 3374 2074 2020 2b20  + 4297213t t  + 
│ │ │ │ -00081360: 3437 3038 3634 3374 2020 2b20 3237 3436  4708643t  + 2746
│ │ │ │ -00081370: 3439 3374 2074 2020 2d20 3330 3134 7c0a  493t t  - 3014|.
│ │ │ │ -00081380: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081390: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -000813a0: 2020 2020 2020 2020 2020 3020 3120 2020            0 1   
│ │ │ │ -000813b0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -000813c0: 2020 2020 3020 3220 2020 2020 2020 7c0a      0 2       |.
│ │ │ │ -000813d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00081310: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00081320: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00081330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081340: 2033 3636 3434 3674 2020 2b20 3432 3937   366446t  + 4297
│ │ │ │ +00081350: 3231 3374 2074 2020 2b20 3437 3038 3634  213t t  + 470864
│ │ │ │ +00081360: 3374 2020 2b20 3237 3436 3439 3374 2074  3t  + 2746493t t
│ │ │ │ +00081370: 2020 2d20 3330 3134 7c0a 7c20 2020 2020    - 3014|.|     
│ │ │ │ +00081380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081390: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ +000813a0: 2020 2020 3020 3120 2020 2020 2020 2020      0 1         
│ │ │ │ +000813b0: 2020 3120 2020 2020 2020 2020 2020 3020    1           0 
│ │ │ │ +000813c0: 3220 2020 2020 2020 7c0a 7c20 2020 2020  2       |.|     
│ │ │ │ +000813d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000813e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000813f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081410: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00081420: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081430: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00081410: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00081420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081430: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  00081440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081450: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -00081460: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00081470: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081480: 2020 2020 2020 2031 3633 3237 3374 2020         163273t  
│ │ │ │ -00081490: 2d20 3131 3830 3037 3574 2074 2020 2d20  - 1180075t t  - 
│ │ │ │ -000814a0: 3231 3530 3533 3274 2020 2b20 3239 3838  2150532t  + 2988
│ │ │ │ -000814b0: 3539 3874 2074 2020 2b20 3433 3835 7c0a  598t t  + 4385|.
│ │ │ │ -000814c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000814d0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -000814e0: 2020 2020 2020 2020 2020 3020 3120 2020            0 1   
│ │ │ │ -000814f0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -00081500: 2020 2020 3020 3220 2020 2020 2020 7c0a      0 2       |.
│ │ │ │ -00081510: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00081450: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00081460: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00081470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081480: 2031 3633 3237 3374 2020 2d20 3131 3830   163273t  - 1180
│ │ │ │ +00081490: 3037 3574 2074 2020 2d20 3231 3530 3533  075t t  - 215053
│ │ │ │ +000814a0: 3274 2020 2b20 3239 3838 3539 3874 2074  2t  + 2988598t t
│ │ │ │ +000814b0: 2020 2b20 3433 3835 7c0a 7c20 2020 2020    + 4385|.|     
│ │ │ │ +000814c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000814d0: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ +000814e0: 2020 2020 3020 3120 2020 2020 2020 2020      0 1         
│ │ │ │ +000814f0: 2020 3120 2020 2020 2020 2020 2020 3020    1           0 
│ │ │ │ +00081500: 3220 2020 2020 2020 7c0a 7c20 2020 2020  2       |.|     
│ │ │ │ +00081510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081550: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00081560: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081570: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00081550: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00081560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081570: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  00081580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081590: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -000815a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000815b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000815c0: 2020 2020 2020 2032 3132 3933 3932 7420         2129392t 
│ │ │ │ -000815d0: 202d 2032 3736 3236 3430 7420 7420 202b   - 2762640t t  +
│ │ │ │ -000815e0: 2032 3036 3839 3335 7420 202d 2036 3635   2068935t  - 665
│ │ │ │ -000815f0: 3333 3574 2074 2020 2b20 3136 3534 7c0a  335t t  + 1654|.
│ │ │ │ -00081600: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081610: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00081620: 2020 2020 2020 2020 2020 2030 2031 2020             0 1  
│ │ │ │ -00081630: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ -00081640: 2020 2020 3020 3220 2020 2020 2020 7c0a      0 2       |.
│ │ │ │ -00081650: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00081590: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +000815a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000815b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000815c0: 2032 3132 3933 3932 7420 202d 2032 3736   2129392t  - 276
│ │ │ │ +000815d0: 3236 3430 7420 7420 202b 2032 3036 3839  2640t t  + 20689
│ │ │ │ +000815e0: 3335 7420 202d 2036 3635 3333 3574 2074  35t  - 665335t t
│ │ │ │ +000815f0: 2020 2b20 3136 3534 7c0a 7c20 2020 2020    + 1654|.|     
│ │ │ │ +00081600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081610: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ +00081620: 2020 2020 2030 2031 2020 2020 2020 2020       0 1        
│ │ │ │ +00081630: 2020 2031 2020 2020 2020 2020 2020 3020     1          0 
│ │ │ │ +00081640: 3220 2020 2020 2020 7c0a 7c20 2020 2020  2       |.|     
│ │ │ │ +00081650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081690: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000816a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000816b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000816c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -000816d0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -000816e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000816f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081700: 2020 2020 2020 202d 2032 3134 3035 3436         - 2140546
│ │ │ │ -00081710: 7420 202d 2034 3335 3532 3131 7420 7420  t  - 4355211t t 
│ │ │ │ -00081720: 202d 2033 3436 3539 3674 2020 2d20 3538   - 346596t  - 58
│ │ │ │ -00081730: 3836 3532 7420 7420 202b 2034 3438 7c0a  8652t t  + 448|.
│ │ │ │ -00081740: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081760: 2030 2020 2020 2020 2020 2020 2030 2031   0           0 1
│ │ │ │ -00081770: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ -00081780: 2020 2020 2030 2032 2020 2020 2020 7c0a       0 2      |.
│ │ │ │ -00081790: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00081690: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000816a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000816b0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +000816c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000816d0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +000816e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000816f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081700: 202d 2032 3134 3035 3436 7420 202d 2034   - 2140546t  - 4
│ │ │ │ +00081710: 3335 3532 3131 7420 7420 202d 2033 3436  355211t t  - 346
│ │ │ │ +00081720: 3539 3674 2020 2d20 3538 3836 3532 7420  596t  - 588652t 
│ │ │ │ +00081730: 7420 202b 2034 3438 7c0a 7c20 2020 2020  t  + 448|.|     
│ │ │ │ +00081740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081750: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ +00081760: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ +00081770: 2020 2020 3120 2020 2020 2020 2020 2030      1          0
│ │ │ │ +00081780: 2032 2020 2020 2020 7c0a 7c20 2020 2020   2      |.|     
│ │ │ │ +00081790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000817a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000817b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000817c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000817d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000817e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000817f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081800: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00081810: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00081820: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00081830: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081840: 2020 2020 2020 202d 2032 3735 3939 3233         - 2759923
│ │ │ │ -00081850: 7420 202b 2033 3731 3937 3637 7420 7420  t  + 3719767t t 
│ │ │ │ -00081860: 202d 2031 3936 3930 3135 7420 202d 2039   - 1969015t  - 9
│ │ │ │ -00081870: 3639 3236 3274 2074 2020 2d20 3831 7c0a  69262t t  - 81|.
│ │ │ │ -00081880: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000818a0: 2030 2020 2020 2020 2020 2020 2030 2031   0           0 1
│ │ │ │ -000818b0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -000818c0: 2020 2020 2020 3020 3220 2020 2020 7c0a        0 2     |.
│ │ │ │ -000818d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000817d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000817e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000817f0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +00081800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081810: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00081820: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00081830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081840: 202d 2032 3735 3939 3233 7420 202b 2033   - 2759923t  + 3
│ │ │ │ +00081850: 3731 3937 3637 7420 7420 202d 2031 3936  719767t t  - 196
│ │ │ │ +00081860: 3930 3135 7420 202d 2039 3639 3236 3274  9015t  - 969262t
│ │ │ │ +00081870: 2074 2020 2d20 3831 7c0a 7c20 2020 2020   t  - 81|.|     
│ │ │ │ +00081880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081890: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ +000818a0: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ +000818b0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +000818c0: 3020 3220 2020 2020 7c0a 7c20 2020 2020  0 2     |.|     
│ │ │ │ +000818d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000818e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000818f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081910: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00081920: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081930: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -00081940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081950: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00081960: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00081970: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081980: 2020 2020 2020 2074 2020 2b20 3235 3537         t  + 2557
│ │ │ │ -00081990: 3230 3174 2074 2020 2b20 3839 3735 3832  201t t  + 897582
│ │ │ │ -000819a0: 7420 202d 2031 3935 3536 3637 7420 7420  t  - 1955667t t 
│ │ │ │ -000819b0: 202d 2034 3136 3238 3332 7420 7420 7c0a   - 4162832t t |.
│ │ │ │ -000819c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000819d0: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ -000819e0: 2020 2020 3020 3120 2020 2020 2020 2020      0 1         
│ │ │ │ -000819f0: 2031 2020 2020 2020 2020 2020 2030 2032   1           0 2
│ │ │ │ -00081a00: 2020 2020 2020 2020 2020 2031 2032 7c0a             1 2|.
│ │ │ │ -00081a10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081a20: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │ +00081910: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00081920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081930: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00081940: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +00081950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081960: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00081970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081980: 2074 2020 2b20 3235 3537 3230 3174 2074   t  + 2557201t t
│ │ │ │ +00081990: 2020 2b20 3839 3735 3832 7420 202d 2031    + 897582t  - 1
│ │ │ │ +000819a0: 3935 3536 3637 7420 7420 202d 2034 3136  955667t t  - 416
│ │ │ │ +000819b0: 3238 3332 7420 7420 7c0a 7c20 2020 2020  2832t t |.|     
│ │ │ │ +000819c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000819d0: 2020 3020 2020 2020 2020 2020 2020 3020    0           0 
│ │ │ │ +000819e0: 3120 2020 2020 2020 2020 2031 2020 2020  1          1    
│ │ │ │ +000819f0: 2020 2020 2020 2030 2032 2020 2020 2020         0 2      
│ │ │ │ +00081a00: 2020 2020 2031 2032 7c0a 7c20 2020 2020       1 2|.|     
│ │ │ │ +00081a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081a20: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │  00081a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081a50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00081a60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00081a50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00081a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081aa0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00081ab0: 7c6f 3520 3a20 5261 7469 6f6e 616c 4d61  |o5 : RationalMa
│ │ │ │ -00081ac0: 7020 2871 7561 6472 6174 6963 2072 6174  p (quadratic rat
│ │ │ │ -00081ad0: 696f 6e61 6c20 6d61 7020 6672 6f6d 2050  ional map from P
│ │ │ │ -00081ae0: 505e 3620 746f 2050 505e 3929 2020 2020  P^6 to PP^9)    
│ │ │ │ -00081af0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00081b00: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00081aa0: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20          |.|o5 : 
│ │ │ │ +00081ab0: 5261 7469 6f6e 616c 4d61 7020 2871 7561  RationalMap (qua
│ │ │ │ +00081ac0: 6472 6174 6963 2072 6174 696f 6e61 6c20  dratic rational 
│ │ │ │ +00081ad0: 6d61 7020 6672 6f6d 2050 505e 3620 746f  map from PP^6 to
│ │ │ │ +00081ae0: 2050 505e 3929 2020 2020 2020 2020 2020   PP^9)          
│ │ │ │ +00081af0: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d          |.|-----
│ │ │ │ +00081b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00081b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00081b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00081b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00081b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00081b50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081b60: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00081b40: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +00081b50: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00081b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081b90: 2020 2020 2020 2032 2020 2020 2020 7c0a         2      |.
│ │ │ │ -00081ba0: 7c34 3934 7420 7420 202b 2032 3336 3039  |494t t  + 23609
│ │ │ │ -00081bb0: 3631 7420 202d 2032 3330 3031 3874 2074  61t  - 230018t t
│ │ │ │ -00081bc0: 2020 2b20 3431 3238 3931 3674 2074 2020    + 4128916t t  
│ │ │ │ -00081bd0: 2d20 3432 3130 3032 7420 7420 202b 2034  - 421002t t  + 4
│ │ │ │ -00081be0: 3832 3335 3130 7420 202b 2031 3930 7c0a  823510t  + 190|.
│ │ │ │ -00081bf0: 7c20 2020 2031 2032 2020 2020 2020 2020  |    1 2        
│ │ │ │ -00081c00: 2020 2032 2020 2020 2020 2020 2020 3020     2          0 
│ │ │ │ -00081c10: 3320 2020 2020 2020 2020 2020 3120 3320  3           1 3 
│ │ │ │ -00081c20: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
│ │ │ │ -00081c30: 2020 2020 2020 2033 2020 2020 2020 7c0a         3      |.
│ │ │ │ -00081c40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00081b90: 2032 2020 2020 2020 7c0a 7c34 3934 7420   2      |.|494t 
│ │ │ │ +00081ba0: 7420 202b 2032 3336 3039 3631 7420 202d  t  + 2360961t  -
│ │ │ │ +00081bb0: 2032 3330 3031 3874 2074 2020 2b20 3431   230018t t  + 41
│ │ │ │ +00081bc0: 3238 3931 3674 2074 2020 2d20 3432 3130  28916t t  - 4210
│ │ │ │ +00081bd0: 3032 7420 7420 202b 2034 3832 3335 3130  02t t  + 4823510
│ │ │ │ +00081be0: 7420 202b 2031 3930 7c0a 7c20 2020 2031  t  + 190|.|    1
│ │ │ │ +00081bf0: 2032 2020 2020 2020 2020 2020 2032 2020   2           2  
│ │ │ │ +00081c00: 2020 2020 2020 2020 3020 3320 2020 2020          0 3     
│ │ │ │ +00081c10: 2020 2020 2020 3120 3320 2020 2020 2020        1 3       
│ │ │ │ +00081c20: 2020 2032 2033 2020 2020 2020 2020 2020     2 3          
│ │ │ │ +00081c30: 2033 2020 2020 2020 7c0a 7c20 2020 2020   3      |.|     
│ │ │ │ +00081c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081c80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00081c90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081ca0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00081c80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00081c90: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00081ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081cd0: 2020 2020 2020 3220 2020 2020 2020 7c0a        2       |.
│ │ │ │ -00081ce0: 7c37 3438 7420 7420 202d 2031 3039 3038  |748t t  - 10908
│ │ │ │ -00081cf0: 3437 7420 202b 2032 3436 3430 7420 7420  47t  + 24640t t 
│ │ │ │ -00081d00: 202b 2033 3335 3732 3338 7420 7420 202b   + 3357238t t  +
│ │ │ │ -00081d10: 2033 3430 3134 3531 7420 7420 202d 2036   3401451t t  - 6
│ │ │ │ -00081d20: 3930 3639 3174 2020 2d20 3130 3932 7c0a  90691t  - 1092|.
│ │ │ │ -00081d30: 7c20 2020 2031 2032 2020 2020 2020 2020  |    1 2        
│ │ │ │ -00081d40: 2020 2032 2020 2020 2020 2020 2030 2033     2         0 3
│ │ │ │ -00081d50: 2020 2020 2020 2020 2020 2031 2033 2020             1 3  
│ │ │ │ -00081d60: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
│ │ │ │ -00081d70: 2020 2020 2020 3320 2020 2020 2020 7c0a        3       |.
│ │ │ │ -00081d80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00081cd0: 3220 2020 2020 2020 7c0a 7c37 3438 7420  2       |.|748t 
│ │ │ │ +00081ce0: 7420 202d 2031 3039 3038 3437 7420 202b  t  - 1090847t  +
│ │ │ │ +00081cf0: 2032 3436 3430 7420 7420 202b 2033 3335   24640t t  + 335
│ │ │ │ +00081d00: 3732 3338 7420 7420 202b 2033 3430 3134  7238t t  + 34014
│ │ │ │ +00081d10: 3531 7420 7420 202d 2036 3930 3639 3174  51t t  - 690691t
│ │ │ │ +00081d20: 2020 2d20 3130 3932 7c0a 7c20 2020 2031    - 1092|.|    1
│ │ │ │ +00081d30: 2032 2020 2020 2020 2020 2020 2032 2020   2           2  
│ │ │ │ +00081d40: 2020 2020 2020 2030 2033 2020 2020 2020         0 3      
│ │ │ │ +00081d50: 2020 2020 2031 2033 2020 2020 2020 2020       1 3        
│ │ │ │ +00081d60: 2020 2032 2033 2020 2020 2020 2020 2020     2 3          
│ │ │ │ +00081d70: 3320 2020 2020 2020 7c0a 7c20 2020 2020  3       |.|     
│ │ │ │ +00081d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081dc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00081dd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081de0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00081dc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00081dd0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00081de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081e10: 2020 2020 2020 2020 3220 2020 2020 7c0a          2     |.
│ │ │ │ -00081e20: 7c30 3674 2074 2020 2b20 3135 3337 3330  |06t t  + 153730
│ │ │ │ -00081e30: 3074 2020 2d20 3335 3335 3030 3274 2074  0t  - 3535002t t
│ │ │ │ -00081e40: 2020 2d20 3239 3133 3137 3574 2074 2020    - 2913175t t  
│ │ │ │ -00081e50: 2b20 3138 3639 3539 3674 2074 2020 2d20  + 1869596t t  - 
│ │ │ │ -00081e60: 3237 3531 3234 3674 2020 2d20 3635 7c0a  2751246t  - 65|.
│ │ │ │ -00081e70: 7c20 2020 3120 3220 2020 2020 2020 2020  |   1 2         
│ │ │ │ -00081e80: 2020 3220 2020 2020 2020 2020 2020 3020    2           0 
│ │ │ │ -00081e90: 3320 2020 2020 2020 2020 2020 3120 3320  3           1 3 
│ │ │ │ -00081ea0: 2020 2020 2020 2020 2020 3220 3320 2020            2 3   
│ │ │ │ -00081eb0: 2020 2020 2020 2020 3320 2020 2020 7c0a          3     |.
│ │ │ │ -00081ec0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00081e10: 2020 3220 2020 2020 7c0a 7c30 3674 2074    2     |.|06t t
│ │ │ │ +00081e20: 2020 2b20 3135 3337 3330 3074 2020 2d20    + 1537300t  - 
│ │ │ │ +00081e30: 3335 3335 3030 3274 2074 2020 2d20 3239  3535002t t  - 29
│ │ │ │ +00081e40: 3133 3137 3574 2074 2020 2b20 3138 3639  13175t t  + 1869
│ │ │ │ +00081e50: 3539 3674 2074 2020 2d20 3237 3531 3234  596t t  - 275124
│ │ │ │ +00081e60: 3674 2020 2d20 3635 7c0a 7c20 2020 3120  6t  - 65|.|   1 
│ │ │ │ +00081e70: 3220 2020 2020 2020 2020 2020 3220 2020  2           2   
│ │ │ │ +00081e80: 2020 2020 2020 2020 3020 3320 2020 2020          0 3     
│ │ │ │ +00081e90: 2020 2020 2020 3120 3320 2020 2020 2020        1 3       
│ │ │ │ +00081ea0: 2020 2020 3220 3320 2020 2020 2020 2020      2 3         
│ │ │ │ +00081eb0: 2020 3320 2020 2020 7c0a 7c20 2020 2020    3     |.|     
│ │ │ │ +00081ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081f00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00081f10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00081f20: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00081f00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00081f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00081f20: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  00081f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00081f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00081f50: 2020 2020 2020 2020 2020 2020 3220 7c0a              2 |.
│ │ │ │ -00081f60: 7c35 3532 3132 3774 2074 2020 2d20 3333  |552127t t  - 33
│ │ │ │ -00081f70: 3334 3835 3074 2020 2d20 3235 3436 3233  34850t  - 254623
│ │ │ │ -00081f80: 3174 2074 2020 2d20 3436 3636 3033 3874  1t t  - 4666038t
│ │ │ │ -00081f90: 2074 2020 2d20 3332 3734 3739 3374 2074   t  - 3274793t t
│ │ │ │ -00081fa0: 2020 2d20 3235 3731 3338 3774 2020 7c0a    - 2571387t  |.
│ │ │ │ -00081fb0: 7c20 2020 2020 2020 3120 3220 2020 2020  |       1 2     
│ │ │ │ -00081fc0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -00081fd0: 2020 3020 3320 2020 2020 2020 2020 2020    0 3           
│ │ │ │ -00081fe0: 3120 3320 2020 2020 2020 2020 2020 3220  1 3           2 
│ │ │ │ -00081ff0: 3320 2020 2020 2020 2020 2020 3320 7c0a  3           3 |.
│ │ │ │ -00082000: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00081f50: 2020 2020 2020 3220 7c0a 7c35 3532 3132        2 |.|55212
│ │ │ │ +00081f60: 3774 2074 2020 2d20 3333 3334 3835 3074  7t t  - 3334850t
│ │ │ │ +00081f70: 2020 2d20 3235 3436 3233 3174 2074 2020    - 2546231t t  
│ │ │ │ +00081f80: 2d20 3436 3636 3033 3874 2074 2020 2d20  - 4666038t t  - 
│ │ │ │ +00081f90: 3332 3734 3739 3374 2074 2020 2d20 3235  3274793t t  - 25
│ │ │ │ +00081fa0: 3731 3338 3774 2020 7c0a 7c20 2020 2020  71387t  |.|     
│ │ │ │ +00081fb0: 2020 3120 3220 2020 2020 2020 2020 2020    1 2           
│ │ │ │ +00081fc0: 3220 2020 2020 2020 2020 2020 3020 3320  2           0 3 
│ │ │ │ +00081fd0: 2020 2020 2020 2020 2020 3120 3320 2020            1 3   
│ │ │ │ +00081fe0: 2020 2020 2020 2020 3220 3320 2020 2020          2 3     
│ │ │ │ +00081ff0: 2020 2020 2020 3320 7c0a 7c20 2020 2020        3 |.|     
│ │ │ │ +00082000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082040: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082050: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00082060: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00082040: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082050: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00082060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082090: 2020 2020 2020 2032 2020 2020 2020 7c0a         2      |.
│ │ │ │ -000820a0: 7c36 3930 7420 7420 202d 2039 3637 3833  |690t t  - 96783
│ │ │ │ -000820b0: 3574 2020 2b20 3435 3436 3730 3874 2074  5t  + 4546708t t
│ │ │ │ -000820c0: 2020 2d20 3233 3537 3230 7420 7420 202b    - 235720t t  +
│ │ │ │ -000820d0: 2033 3934 3037 3033 7420 7420 202b 2034   3940703t t  + 4
│ │ │ │ -000820e0: 3731 3431 3139 7420 202d 2032 3236 7c0a  714119t  - 226|.
│ │ │ │ -000820f0: 7c20 2020 2031 2032 2020 2020 2020 2020  |    1 2        
│ │ │ │ -00082100: 2020 3220 2020 2020 2020 2020 2020 3020    2           0 
│ │ │ │ -00082110: 3320 2020 2020 2020 2020 2031 2033 2020  3          1 3  
│ │ │ │ -00082120: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
│ │ │ │ -00082130: 2020 2020 2020 2033 2020 2020 2020 7c0a         3      |.
│ │ │ │ -00082140: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082090: 2032 2020 2020 2020 7c0a 7c36 3930 7420   2      |.|690t 
│ │ │ │ +000820a0: 7420 202d 2039 3637 3833 3574 2020 2b20  t  - 967835t  + 
│ │ │ │ +000820b0: 3435 3436 3730 3874 2074 2020 2d20 3233  4546708t t  - 23
│ │ │ │ +000820c0: 3537 3230 7420 7420 202b 2033 3934 3037  5720t t  + 39407
│ │ │ │ +000820d0: 3033 7420 7420 202b 2034 3731 3431 3139  03t t  + 4714119
│ │ │ │ +000820e0: 7420 202d 2032 3236 7c0a 7c20 2020 2031  t  - 226|.|    1
│ │ │ │ +000820f0: 2032 2020 2020 2020 2020 2020 3220 2020   2          2   
│ │ │ │ +00082100: 2020 2020 2020 2020 3020 3320 2020 2020          0 3     
│ │ │ │ +00082110: 2020 2020 2031 2033 2020 2020 2020 2020       1 3        
│ │ │ │ +00082120: 2020 2032 2033 2020 2020 2020 2020 2020     2 3          
│ │ │ │ +00082130: 2033 2020 2020 2020 7c0a 7c20 2020 2020   3      |.|     
│ │ │ │ +00082140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082180: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082190: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000821a0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00082180: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082190: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +000821a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000821b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000821c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000821d0: 2020 2020 2020 2032 2020 2020 2020 7c0a         2      |.
│ │ │ │ -000821e0: 7c36 3439 7420 7420 202d 2034 3533 3633  |649t t  - 45363
│ │ │ │ -000821f0: 3135 7420 202b 2032 3337 3936 3030 7420  15t  + 2379600t 
│ │ │ │ -00082200: 7420 202d 2033 3132 3031 3374 2074 2020  t  - 312013t t  
│ │ │ │ -00082210: 2d20 3632 3939 3736 7420 7420 202b 2031  - 629976t t  + 1
│ │ │ │ -00082220: 3737 3038 3937 7420 202d 2034 3533 7c0a  770897t  - 453|.
│ │ │ │ -00082230: 7c20 2020 2031 2032 2020 2020 2020 2020  |    1 2        
│ │ │ │ -00082240: 2020 2032 2020 2020 2020 2020 2020 2030     2           0
│ │ │ │ -00082250: 2033 2020 2020 2020 2020 2020 3120 3320   3          1 3 
│ │ │ │ -00082260: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
│ │ │ │ -00082270: 2020 2020 2020 2033 2020 2020 2020 7c0a         3      |.
│ │ │ │ -00082280: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000821d0: 2032 2020 2020 2020 7c0a 7c36 3439 7420   2      |.|649t 
│ │ │ │ +000821e0: 7420 202d 2034 3533 3633 3135 7420 202b  t  - 4536315t  +
│ │ │ │ +000821f0: 2032 3337 3936 3030 7420 7420 202d 2033   2379600t t  - 3
│ │ │ │ +00082200: 3132 3031 3374 2074 2020 2d20 3632 3939  12013t t  - 6299
│ │ │ │ +00082210: 3736 7420 7420 202b 2031 3737 3038 3937  76t t  + 1770897
│ │ │ │ +00082220: 7420 202d 2034 3533 7c0a 7c20 2020 2031  t  - 453|.|    1
│ │ │ │ +00082230: 2032 2020 2020 2020 2020 2020 2032 2020   2           2  
│ │ │ │ +00082240: 2020 2020 2020 2020 2030 2033 2020 2020           0 3    
│ │ │ │ +00082250: 2020 2020 2020 3120 3320 2020 2020 2020        1 3       
│ │ │ │ +00082260: 2020 2032 2033 2020 2020 2020 2020 2020     2 3          
│ │ │ │ +00082270: 2033 2020 2020 2020 7c0a 7c20 2020 2020   3      |.|     
│ │ │ │ +00082280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000822a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000822b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000822c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000822d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000822e0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +000822c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000822d0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +000822e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000822f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082310: 2020 2020 2020 2020 2032 2020 2020 7c0a           2    |.
│ │ │ │ -00082320: 7c38 3133 7420 7420 202b 2033 3039 3738  |813t t  + 30978
│ │ │ │ -00082330: 3730 7420 202b 2034 3333 3037 3732 7420  70t  + 4330772t 
│ │ │ │ -00082340: 7420 202b 2034 3130 3634 3830 7420 7420  t  + 4106480t t 
│ │ │ │ -00082350: 202d 2032 3035 3838 3338 7420 7420 202b   - 2058838t t  +
│ │ │ │ -00082360: 2034 3932 3238 3035 7420 202b 2033 7c0a   4922805t  + 3|.
│ │ │ │ -00082370: 7c20 2020 2031 2032 2020 2020 2020 2020  |    1 2        
│ │ │ │ -00082380: 2020 2032 2020 2020 2020 2020 2020 2030     2           0
│ │ │ │ -00082390: 2033 2020 2020 2020 2020 2020 2031 2033   3           1 3
│ │ │ │ -000823a0: 2020 2020 2020 2020 2020 2032 2033 2020             2 3  
│ │ │ │ -000823b0: 2020 2020 2020 2020 2033 2020 2020 7c0a           3    |.
│ │ │ │ -000823c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082310: 2020 2032 2020 2020 7c0a 7c38 3133 7420     2    |.|813t 
│ │ │ │ +00082320: 7420 202b 2033 3039 3738 3730 7420 202b  t  + 3097870t  +
│ │ │ │ +00082330: 2034 3333 3037 3732 7420 7420 202b 2034   4330772t t  + 4
│ │ │ │ +00082340: 3130 3634 3830 7420 7420 202d 2032 3035  106480t t  - 205
│ │ │ │ +00082350: 3838 3338 7420 7420 202b 2034 3932 3238  8838t t  + 49228
│ │ │ │ +00082360: 3035 7420 202b 2033 7c0a 7c20 2020 2031  05t  + 3|.|    1
│ │ │ │ +00082370: 2032 2020 2020 2020 2020 2020 2032 2020   2           2  
│ │ │ │ +00082380: 2020 2020 2020 2020 2030 2033 2020 2020           0 3    
│ │ │ │ +00082390: 2020 2020 2020 2031 2033 2020 2020 2020         1 3      
│ │ │ │ +000823a0: 2020 2020 2032 2033 2020 2020 2020 2020       2 3        
│ │ │ │ +000823b0: 2020 2033 2020 2020 7c0a 7c20 2020 2020     3    |.|     
│ │ │ │ +000823c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000823d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000823e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000823f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082400: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082410: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00082420: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00082400: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082410: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00082420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082450: 2020 2020 2020 2020 2020 3220 2020 7c0a            2   |.
│ │ │ │ -00082460: 7c31 3530 3174 2074 2020 2b20 3230 3931  |1501t t  + 2091
│ │ │ │ -00082470: 3133 3074 2020 2d20 3139 3735 3639 3474  130t  - 1975694t
│ │ │ │ -00082480: 2074 2020 2b20 3234 3132 3030 3274 2074   t  + 2412002t t
│ │ │ │ -00082490: 2020 2b20 3437 3539 3939 3974 2074 2020    + 4759999t t  
│ │ │ │ -000824a0: 2b20 3239 3332 3631 3074 2020 2b20 7c0a  + 2932610t  + |.
│ │ │ │ -000824b0: 7c20 2020 2020 3120 3220 2020 2020 2020  |     1 2       
│ │ │ │ -000824c0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -000824d0: 3020 3320 2020 2020 2020 2020 2020 3120  0 3           1 
│ │ │ │ -000824e0: 3320 2020 2020 2020 2020 2020 3220 3320  3           2 3 
│ │ │ │ -000824f0: 2020 2020 2020 2020 2020 3320 2020 7c0a            3   |.
│ │ │ │ -00082500: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082450: 2020 2020 3220 2020 7c0a 7c31 3530 3174      2   |.|1501t
│ │ │ │ +00082460: 2074 2020 2b20 3230 3931 3133 3074 2020   t  + 2091130t  
│ │ │ │ +00082470: 2d20 3139 3735 3639 3474 2074 2020 2b20  - 1975694t t  + 
│ │ │ │ +00082480: 3234 3132 3030 3274 2074 2020 2b20 3437  2412002t t  + 47
│ │ │ │ +00082490: 3539 3939 3974 2074 2020 2b20 3239 3332  59999t t  + 2932
│ │ │ │ +000824a0: 3631 3074 2020 2b20 7c0a 7c20 2020 2020  610t  + |.|     
│ │ │ │ +000824b0: 3120 3220 2020 2020 2020 2020 2020 3220  1 2           2 
│ │ │ │ +000824c0: 2020 2020 2020 2020 2020 3020 3320 2020            0 3   
│ │ │ │ +000824d0: 2020 2020 2020 2020 3120 3320 2020 2020          1 3     
│ │ │ │ +000824e0: 2020 2020 2020 3220 3320 2020 2020 2020        2 3       
│ │ │ │ +000824f0: 2020 2020 3320 2020 7c0a 7c20 2020 2020      3   |.|     
│ │ │ │ +00082500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082540: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082550: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00082560: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00082540: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082550: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00082560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082590: 2020 2020 2020 2020 2020 3220 2020 7c0a            2   |.
│ │ │ │ -000825a0: 7c39 3239 3174 2074 2020 2b20 3336 3238  |9291t t  + 3628
│ │ │ │ -000825b0: 3132 3474 2020 2d20 3437 3831 3139 3574  124t  - 4781195t
│ │ │ │ -000825c0: 2074 2020 2d20 3431 3639 3535 3774 2074   t  - 4169557t t
│ │ │ │ -000825d0: 2020 2d20 3237 3037 3938 3474 2074 2020    - 2707984t t  
│ │ │ │ -000825e0: 2d20 3236 3733 3339 3574 2020 2d20 7c0a  - 2673395t  - |.
│ │ │ │ -000825f0: 7c20 2020 2020 3120 3220 2020 2020 2020  |     1 2       
│ │ │ │ -00082600: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -00082610: 3020 3320 2020 2020 2020 2020 2020 3120  0 3           1 
│ │ │ │ -00082620: 3320 2020 2020 2020 2020 2020 3220 3320  3           2 3 
│ │ │ │ -00082630: 2020 2020 2020 2020 2020 3320 2020 7c0a            3   |.
│ │ │ │ -00082640: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082590: 2020 2020 3220 2020 7c0a 7c39 3239 3174      2   |.|9291t
│ │ │ │ +000825a0: 2074 2020 2b20 3336 3238 3132 3474 2020   t  + 3628124t  
│ │ │ │ +000825b0: 2d20 3437 3831 3139 3574 2074 2020 2d20  - 4781195t t  - 
│ │ │ │ +000825c0: 3431 3639 3535 3774 2074 2020 2d20 3237  4169557t t  - 27
│ │ │ │ +000825d0: 3037 3938 3474 2074 2020 2d20 3236 3733  07984t t  - 2673
│ │ │ │ +000825e0: 3339 3574 2020 2d20 7c0a 7c20 2020 2020  395t  - |.|     
│ │ │ │ +000825f0: 3120 3220 2020 2020 2020 2020 2020 3220  1 2           2 
│ │ │ │ +00082600: 2020 2020 2020 2020 2020 3020 3320 2020            0 3   
│ │ │ │ +00082610: 2020 2020 2020 2020 3120 3320 2020 2020          1 3     
│ │ │ │ +00082620: 2020 2020 2020 3220 3320 2020 2020 2020        2 3       
│ │ │ │ +00082630: 2020 2020 3320 2020 7c0a 7c20 2020 2020      3   |.|     
│ │ │ │ +00082640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082680: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082690: 7c20 2020 2020 2020 2020 2020 3220 2020  |           2   
│ │ │ │ +00082680: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082690: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  000826a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000826b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000826c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000826d0: 3220 2020 2020 2020 2020 2020 2020 7c0a  2             |.
│ │ │ │ -000826e0: 7c20 2b20 3239 3137 3937 3474 2020 2d20  | + 2917974t  - 
│ │ │ │ -000826f0: 3234 3134 3034 3574 2074 2020 2d20 3331  2414045t t  - 31
│ │ │ │ -00082700: 3333 3435 3674 2074 2020 2b20 3133 3032  33456t t  + 1302
│ │ │ │ -00082710: 3639 7420 7420 202d 2039 3231 3032 3574  69t t  - 921025t
│ │ │ │ -00082720: 2020 2b20 3439 3234 3736 3774 2074 7c0a    + 4924767t t|.
│ │ │ │ -00082730: 7c20 2020 2020 2020 2020 2020 3220 2020  |           2   
│ │ │ │ -00082740: 2020 2020 2020 2020 3020 3320 2020 2020          0 3     
│ │ │ │ -00082750: 2020 2020 2020 3120 3320 2020 2020 2020        1 3       
│ │ │ │ -00082760: 2020 2032 2033 2020 2020 2020 2020 2020     2 3          
│ │ │ │ -00082770: 3320 2020 2020 2020 2020 2020 3020 7c0a  3           0 |.
│ │ │ │ -00082780: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +000826c0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +000826d0: 2020 2020 2020 2020 7c0a 7c20 2b20 3239          |.| + 29
│ │ │ │ +000826e0: 3137 3937 3474 2020 2d20 3234 3134 3034  17974t  - 241404
│ │ │ │ +000826f0: 3574 2074 2020 2d20 3331 3333 3435 3674  5t t  - 3133456t
│ │ │ │ +00082700: 2074 2020 2b20 3133 3032 3639 7420 7420   t  + 130269t t 
│ │ │ │ +00082710: 202d 2039 3231 3032 3574 2020 2b20 3439   - 921025t  + 49
│ │ │ │ +00082720: 3234 3736 3774 2074 7c0a 7c20 2020 2020  24767t t|.|     
│ │ │ │ +00082730: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00082740: 2020 3020 3320 2020 2020 2020 2020 2020    0 3           
│ │ │ │ +00082750: 3120 3320 2020 2020 2020 2020 2032 2033  1 3          2 3
│ │ │ │ +00082760: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ +00082770: 2020 2020 2020 3020 7c0a 7c2d 2d2d 2d2d        0 |.|-----
│ │ │ │ +00082780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00082790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000827a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000827b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000827c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -000827d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000827c0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +000827d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000827e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000827f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082810: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082820: 7c32 3433 3074 2074 2020 2b20 3334 3336  |2430t t  + 3436
│ │ │ │ -00082830: 3438 3374 2074 2020 2b20 3135 3335 3532  483t t  + 153552
│ │ │ │ -00082840: 3874 2074 2020 2d20 3130 3833 3538 3774  8t t  - 1083587t
│ │ │ │ -00082850: 2074 2020 2020 2020 2020 2020 2020 2020   t              
│ │ │ │ -00082860: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082870: 7c20 2020 2020 3020 3420 2020 2020 2020  |     0 4       
│ │ │ │ -00082880: 2020 2020 3120 3420 2020 2020 2020 2020      1 4         
│ │ │ │ -00082890: 2020 3220 3420 2020 2020 2020 2020 2020    2 4           
│ │ │ │ -000828a0: 3320 3420 2020 2020 2020 2020 2020 2020  3 4             
│ │ │ │ -000828b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000828c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082810: 2020 2020 2020 2020 7c0a 7c32 3433 3074          |.|2430t
│ │ │ │ +00082820: 2074 2020 2b20 3334 3336 3438 3374 2074   t  + 3436483t t
│ │ │ │ +00082830: 2020 2b20 3135 3335 3532 3874 2074 2020    + 1535528t t  
│ │ │ │ +00082840: 2d20 3130 3833 3538 3774 2074 2020 2020  - 1083587t t    
│ │ │ │ +00082850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00082860: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082870: 3020 3420 2020 2020 2020 2020 2020 3120  0 4           1 
│ │ │ │ +00082880: 3420 2020 2020 2020 2020 2020 3220 3420  4           2 4 
│ │ │ │ +00082890: 2020 2020 2020 2020 2020 3320 3420 2020            3 4   
│ │ │ │ +000828a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000828b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000828c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000828d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000828e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000828f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082900: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082910: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082900: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082920: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00082940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082950: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082960: 7c34 3230 7420 7420 202d 2032 3932 3033  |420t t  - 29203
│ │ │ │ -00082970: 3633 7420 7420 202b 2032 3730 3639 3434  63t t  + 2706944
│ │ │ │ -00082980: 7420 7420 202b 2034 3632 3639 3439 7420  t t  + 4626949t 
│ │ │ │ -00082990: 7420 202d 2020 2020 2020 2020 2020 2020  t  -            
│ │ │ │ -000829a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000829b0: 7c20 2020 2030 2034 2020 2020 2020 2020  |    0 4        
│ │ │ │ -000829c0: 2020 2031 2034 2020 2020 2020 2020 2020     1 4          
│ │ │ │ -000829d0: 2032 2034 2020 2020 2020 2020 2020 2033   2 4           3
│ │ │ │ -000829e0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -000829f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082a00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082950: 2020 2020 2020 2020 7c0a 7c34 3230 7420          |.|420t 
│ │ │ │ +00082960: 7420 202d 2032 3932 3033 3633 7420 7420  t  - 2920363t t 
│ │ │ │ +00082970: 202b 2032 3730 3639 3434 7420 7420 202b   + 2706944t t  +
│ │ │ │ +00082980: 2034 3632 3639 3439 7420 7420 202d 2020   4626949t t  -  
│ │ │ │ +00082990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000829a0: 2020 2020 2020 2020 7c0a 7c20 2020 2030          |.|    0
│ │ │ │ +000829b0: 2034 2020 2020 2020 2020 2020 2031 2034   4           1 4
│ │ │ │ +000829c0: 2020 2020 2020 2020 2020 2032 2034 2020             2 4  
│ │ │ │ +000829d0: 2020 2020 2020 2020 2033 2034 2020 2020           3 4    
│ │ │ │ +000829e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000829f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082a40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082a50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082a40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082a90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082aa0: 7c34 3437 3474 2074 2020 2d20 3339 3030  |4474t t  - 3900
│ │ │ │ -00082ab0: 3936 3974 2074 2020 2d20 3231 3737 3239  969t t  - 217729
│ │ │ │ -00082ac0: 3074 2074 2020 2b20 3439 3131 3536 3674  0t t  + 4911566t
│ │ │ │ -00082ad0: 2074 2020 2020 2020 2020 2020 2020 2020   t              
│ │ │ │ -00082ae0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082af0: 7c20 2020 2020 3020 3420 2020 2020 2020  |     0 4       
│ │ │ │ -00082b00: 2020 2020 3120 3420 2020 2020 2020 2020      1 4         
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│ │ │ │ -00082b20: 3320 3420 2020 2020 2020 2020 2020 2020  3 4             
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│ │ │ │ -00082b40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082a90: 2020 2020 2020 2020 7c0a 7c34 3437 3474          |.|4474t
│ │ │ │ +00082aa0: 2074 2020 2d20 3339 3030 3936 3974 2074   t  - 3900969t t
│ │ │ │ +00082ab0: 2020 2d20 3231 3737 3239 3074 2074 2020    - 2177290t t  
│ │ │ │ +00082ac0: 2b20 3439 3131 3536 3674 2074 2020 2020  + 4911566t t    
│ │ │ │ +00082ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00082ae0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082af0: 3020 3420 2020 2020 2020 2020 2020 3120  0 4           1 
│ │ │ │ +00082b00: 3420 2020 2020 2020 2020 2020 3220 3420  4           2 4 
│ │ │ │ +00082b10: 2020 2020 2020 2020 2020 3320 3420 2020            3 4   
│ │ │ │ +00082b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00082b30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082b80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082b90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082b80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082bd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082be0: 7c2d 2032 3237 3637 3538 7420 7420 202d  |- 2276758t t  -
│ │ │ │ -00082bf0: 2034 3539 3833 3230 7420 7420 202d 2037   4598320t t  - 7
│ │ │ │ -00082c00: 3031 3532 7420 7420 202b 2032 3135 3032  0152t t  + 21502
│ │ │ │ -00082c10: 3933 7420 2020 2020 2020 2020 2020 2020  93t             
│ │ │ │ -00082c20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082c30: 7c20 2020 2020 2020 2020 2030 2034 2020  |          0 4  
│ │ │ │ -00082c40: 2020 2020 2020 2020 2031 2034 2020 2020           1 4    
│ │ │ │ -00082c50: 2020 2020 2032 2034 2020 2020 2020 2020       2 4        
│ │ │ │ -00082c60: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ -00082c70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082c80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082bd0: 2020 2020 2020 2020 7c0a 7c2d 2032 3237          |.|- 227
│ │ │ │ +00082be0: 3637 3538 7420 7420 202d 2034 3539 3833  6758t t  - 45983
│ │ │ │ +00082bf0: 3230 7420 7420 202d 2037 3031 3532 7420  20t t  - 70152t 
│ │ │ │ +00082c00: 7420 202b 2032 3135 3032 3933 7420 2020  t  + 2150293t   
│ │ │ │ +00082c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00082c20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082c30: 2020 2020 2030 2034 2020 2020 2020 2020       0 4        
│ │ │ │ +00082c40: 2020 2031 2034 2020 2020 2020 2020 2032     1 4         2
│ │ │ │ +00082c50: 2034 2020 2020 2020 2020 2020 2033 2020   4           3  
│ │ │ │ +00082c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00082c70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082cc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082cd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082cc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082d10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082d20: 7c30 3737 3274 2074 2020 2d20 3333 3738  |0772t t  - 3378
│ │ │ │ -00082d30: 3736 3474 2074 2020 2d20 3432 3437 3631  764t t  - 424761
│ │ │ │ -00082d40: 3774 2074 2020 2b20 3137 3331 3336 7420  7t t  + 173136t 
│ │ │ │ -00082d50: 7420 202b 2020 2020 2020 2020 2020 2020  t  +            
│ │ │ │ -00082d60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082d70: 7c20 2020 2020 3020 3420 2020 2020 2020  |     0 4       
│ │ │ │ -00082d80: 2020 2020 3120 3420 2020 2020 2020 2020      1 4         
│ │ │ │ -00082d90: 2020 3220 3420 2020 2020 2020 2020 2033    2 4          3
│ │ │ │ -00082da0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -00082db0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082dc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082d10: 2020 2020 2020 2020 7c0a 7c30 3737 3274          |.|0772t
│ │ │ │ +00082d20: 2074 2020 2d20 3333 3738 3736 3474 2074   t  - 3378764t t
│ │ │ │ +00082d30: 2020 2d20 3432 3437 3631 3774 2074 2020    - 4247617t t  
│ │ │ │ +00082d40: 2b20 3137 3331 3336 7420 7420 202b 2020  + 173136t t  +  
│ │ │ │ +00082d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00082d60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082d70: 3020 3420 2020 2020 2020 2020 2020 3120  0 4           1 
│ │ │ │ +00082d80: 3420 2020 2020 2020 2020 2020 3220 3420  4           2 4 
│ │ │ │ +00082d90: 2020 2020 2020 2020 2033 2034 2020 2020           3 4    
│ │ │ │ +00082da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00082db0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082e00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082e10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082e00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082e50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082e60: 7c36 3174 2074 2020 2b20 3234 3836 3834  |61t t  + 248684
│ │ │ │ -00082e70: 3674 2074 2020 2d20 3233 3434 3439 3774  6t t  - 2344497t
│ │ │ │ -00082e80: 2074 2020 2d20 3431 3631 3235 3374 2074   t  - 4161253t t
│ │ │ │ -00082e90: 2020 2b20 2020 2020 2020 2020 2020 2020    +             
│ │ │ │ -00082ea0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082eb0: 7c20 2020 3020 3420 2020 2020 2020 2020  |   0 4         
│ │ │ │ -00082ec0: 2020 3120 3420 2020 2020 2020 2020 2020    1 4           
│ │ │ │ -00082ed0: 3220 3420 2020 2020 2020 2020 2020 3320  2 4           3 
│ │ │ │ -00082ee0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00082ef0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082f00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082e50: 2020 2020 2020 2020 7c0a 7c36 3174 2074          |.|61t t
│ │ │ │ +00082e60: 2020 2b20 3234 3836 3834 3674 2074 2020    + 2486846t t  
│ │ │ │ +00082e70: 2d20 3233 3434 3439 3774 2074 2020 2d20  - 2344497t t  - 
│ │ │ │ +00082e80: 3431 3631 3235 3374 2074 2020 2b20 2020  4161253t t  +   
│ │ │ │ +00082e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00082ea0: 2020 2020 2020 2020 7c0a 7c20 2020 3020          |.|   0 
│ │ │ │ +00082eb0: 3420 2020 2020 2020 2020 2020 3120 3420  4           1 4 
│ │ │ │ +00082ec0: 2020 2020 2020 2020 2020 3220 3420 2020            2 4   
│ │ │ │ +00082ed0: 2020 2020 2020 2020 3320 3420 2020 2020          3 4     
│ │ │ │ +00082ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00082ef0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082f40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082f50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082f40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00082f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00082f90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082fa0: 7c35 3332 3034 3574 2074 2020 2b20 3131  |532045t t  + 11
│ │ │ │ -00082fb0: 3134 3838 3674 2074 2020 2b20 3135 3033  14886t t  + 1503
│ │ │ │ -00082fc0: 3838 3574 2074 2020 2d20 3431 3531 3435  885t t  - 415145
│ │ │ │ -00082fd0: 3174 2074 2020 2020 2020 2020 2020 2020  1t t            
│ │ │ │ -00082fe0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00082ff0: 7c20 2020 2020 2020 3020 3420 2020 2020  |       0 4     
│ │ │ │ -00083000: 2020 2020 2020 3120 3420 2020 2020 2020        1 4       
│ │ │ │ -00083010: 2020 2020 3220 3420 2020 2020 2020 2020      2 4         
│ │ │ │ -00083020: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -00083030: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083040: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00082f90: 2020 2020 2020 2020 7c0a 7c35 3332 3034          |.|53204
│ │ │ │ +00082fa0: 3574 2074 2020 2b20 3131 3134 3838 3674  5t t  + 1114886t
│ │ │ │ +00082fb0: 2074 2020 2b20 3135 3033 3838 3574 2074   t  + 1503885t t
│ │ │ │ +00082fc0: 2020 2d20 3431 3531 3435 3174 2074 2020    - 4151451t t  
│ │ │ │ +00082fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00082fe0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00082ff0: 2020 3020 3420 2020 2020 2020 2020 2020    0 4           
│ │ │ │ +00083000: 3120 3420 2020 2020 2020 2020 2020 3220  1 4           2 
│ │ │ │ +00083010: 3420 2020 2020 2020 2020 2020 3320 2020  4           3   
│ │ │ │ +00083020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00083030: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083080: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083090: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00083080: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000830a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000830c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000830d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000830e0: 7c34 3231 3839 3133 7420 7420 202d 2032  |4218913t t  - 2
│ │ │ │ -000830f0: 3239 3439 3674 2074 2020 2d20 3135 3136  29496t t  - 1516
│ │ │ │ -00083100: 3831 3974 2074 2020 2d20 3239 3836 3730  819t t  - 298670
│ │ │ │ -00083110: 3174 2074 2020 2020 2020 2020 2020 2020  1t t            
│ │ │ │ -00083120: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083130: 7c20 2020 2020 2020 2030 2034 2020 2020  |        0 4    
│ │ │ │ -00083140: 2020 2020 2020 3120 3420 2020 2020 2020        1 4       
│ │ │ │ -00083150: 2020 2020 3220 3420 2020 2020 2020 2020      2 4         
│ │ │ │ -00083160: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -00083170: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083180: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000830d0: 2020 2020 2020 2020 7c0a 7c34 3231 3839          |.|42189
│ │ │ │ +000830e0: 3133 7420 7420 202d 2032 3239 3439 3674  13t t  - 229496t
│ │ │ │ +000830f0: 2074 2020 2d20 3135 3136 3831 3974 2074   t  - 1516819t t
│ │ │ │ +00083100: 2020 2d20 3239 3836 3730 3174 2074 2020    - 2986701t t  
│ │ │ │ +00083110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00083120: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083130: 2020 2030 2034 2020 2020 2020 2020 2020     0 4          
│ │ │ │ +00083140: 3120 3420 2020 2020 2020 2020 2020 3220  1 4           2 
│ │ │ │ +00083150: 3420 2020 2020 2020 2020 2020 3320 2020  4           3   
│ │ │ │ +00083160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00083170: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000831a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000831b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000831c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000831d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000831c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000831d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000831e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000831f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083210: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083220: 7c32 3737 3930 3638 7420 7420 202d 2033  |2779068t t  - 3
│ │ │ │ -00083230: 3832 3934 3036 7420 7420 202b 2032 3739  829406t t  + 279
│ │ │ │ -00083240: 3139 3837 7420 7420 202b 2031 3232 3936  1987t t  + 12296
│ │ │ │ -00083250: 3032 7420 2020 2020 2020 2020 2020 2020  02t             
│ │ │ │ -00083260: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083270: 7c20 2020 2020 2020 2030 2034 2020 2020  |        0 4    
│ │ │ │ -00083280: 2020 2020 2020 2031 2034 2020 2020 2020         1 4      
│ │ │ │ -00083290: 2020 2020 2032 2034 2020 2020 2020 2020       2 4        
│ │ │ │ -000832a0: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ -000832b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000832c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00083210: 2020 2020 2020 2020 7c0a 7c32 3737 3930          |.|27790
│ │ │ │ +00083220: 3638 7420 7420 202d 2033 3832 3934 3036  68t t  - 3829406
│ │ │ │ +00083230: 7420 7420 202b 2032 3739 3139 3837 7420  t t  + 2791987t 
│ │ │ │ +00083240: 7420 202b 2031 3232 3936 3032 7420 2020  t  + 1229602t   
│ │ │ │ +00083250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00083260: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083270: 2020 2030 2034 2020 2020 2020 2020 2020     0 4          
│ │ │ │ +00083280: 2031 2034 2020 2020 2020 2020 2020 2032   1 4           2
│ │ │ │ +00083290: 2034 2020 2020 2020 2020 2020 2033 2020   4           3  
│ │ │ │ +000832a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000832b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000832c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000832d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000832e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000832f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083300: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083310: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00083300: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083350: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083360: 7c20 202b 2032 3638 3834 3474 2074 2020  |  + 268844t t  
│ │ │ │ -00083370: 2d20 3131 3739 3838 3174 2074 2020 2d20  - 1179881t t  - 
│ │ │ │ -00083380: 3139 3834 3235 3274 2074 2020 2b20 3835  1984252t t  + 85
│ │ │ │ -00083390: 3235 3974 2020 2020 2020 2020 2020 2020  259t            
│ │ │ │ -000833a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000833b0: 7c34 2020 2020 2020 2020 2020 3120 3420  |4          1 4 
│ │ │ │ -000833c0: 2020 2020 2020 2020 2020 3220 3420 2020            2 4   
│ │ │ │ -000833d0: 2020 2020 2020 2020 3320 3420 2020 2020          3 4     
│ │ │ │ +00083350: 2020 2020 2020 2020 7c0a 7c20 202b 2032          |.|  + 2
│ │ │ │ +00083360: 3638 3834 3474 2074 2020 2d20 3131 3739  68844t t  - 1179
│ │ │ │ +00083370: 3838 3174 2074 2020 2d20 3139 3834 3235  881t t  - 198425
│ │ │ │ +00083380: 3274 2074 2020 2b20 3835 3235 3974 2020  2t t  + 85259t  
│ │ │ │ +00083390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000833a0: 2020 2020 2020 2020 7c0a 7c34 2020 2020          |.|4    
│ │ │ │ +000833b0: 2020 2020 2020 3120 3420 2020 2020 2020        1 4       
│ │ │ │ +000833c0: 2020 2020 3220 3420 2020 2020 2020 2020      2 4         
│ │ │ │ +000833d0: 2020 3320 3420 2020 2020 2020 2020 2020    3 4           
│ │ │ │  000833e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000833f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083400: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +000833f0: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d          |.|-----
│ │ │ │ +00083400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00083410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00083420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00083430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00083440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00083450: 7c20 2020 2020 2020 2020 2032 2020 2020  |          2    
│ │ │ │ +00083440: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +00083450: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00083460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083490: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000834a0: 7c2b 2031 3535 3839 3736 7420 202d 2033  |+ 1558976t  - 3
│ │ │ │ -000834b0: 3130 3638 3930 7420 7420 202d 2031 3832  106890t t  - 182
│ │ │ │ -000834c0: 3734 3535 7420 7420 202b 2034 3935 3339  7455t t  + 49539
│ │ │ │ -000834d0: 3131 7420 7420 202b 2033 3636 3830 3933  11t t  + 3668093
│ │ │ │ -000834e0: 7420 7420 202d 2033 3838 3132 3737 7c0a  t t  - 3881277|.
│ │ │ │ -000834f0: 7c20 2020 2020 2020 2020 2034 2020 2020  |          4    
│ │ │ │ -00083500: 2020 2020 2020 2030 2035 2020 2020 2020         0 5      
│ │ │ │ -00083510: 2020 2020 2031 2035 2020 2020 2020 2020       1 5        
│ │ │ │ -00083520: 2020 2032 2035 2020 2020 2020 2020 2020     2 5          
│ │ │ │ -00083530: 2033 2035 2020 2020 2020 2020 2020 7c0a   3 5          |.
│ │ │ │ -00083540: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00083490: 2020 2020 2020 2020 7c0a 7c2b 2031 3535          |.|+ 155
│ │ │ │ +000834a0: 3839 3736 7420 202d 2033 3130 3638 3930  8976t  - 3106890
│ │ │ │ +000834b0: 7420 7420 202d 2031 3832 3734 3535 7420  t t  - 1827455t 
│ │ │ │ +000834c0: 7420 202b 2034 3935 3339 3131 7420 7420  t  + 4953911t t 
│ │ │ │ +000834d0: 202b 2033 3636 3830 3933 7420 7420 202d   + 3668093t t  -
│ │ │ │ +000834e0: 2033 3838 3132 3737 7c0a 7c20 2020 2020   3881277|.|     
│ │ │ │ +000834f0: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │ +00083500: 2030 2035 2020 2020 2020 2020 2020 2031   0 5           1
│ │ │ │ +00083510: 2035 2020 2020 2020 2020 2020 2032 2035   5           2 5
│ │ │ │ +00083520: 2020 2020 2020 2020 2020 2033 2035 2020             3 5  
│ │ │ │ +00083530: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083580: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083590: 7c20 2020 2020 2020 2020 3220 2020 2020  |         2     
│ │ │ │ +00083580: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083590: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  000835a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000835b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000835c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000835d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000835e0: 7c20 3135 3030 3633 3874 2020 2d20 3137  | 1500638t  - 17
│ │ │ │ -000835f0: 3633 3639 7420 7420 202d 2032 3139 3233  6369t t  - 21923
│ │ │ │ -00083600: 3239 7420 7420 202d 2032 3332 3534 3839  29t t  - 2325489
│ │ │ │ -00083610: 7420 7420 202b 2031 3138 3239 3532 7420  t t  + 1182952t 
│ │ │ │ -00083620: 7420 202b 2037 3136 3235 3974 2074 7c0a  t  + 716259t t|.
│ │ │ │ -00083630: 7c20 2020 2020 2020 2020 3420 2020 2020  |         4     
│ │ │ │ -00083640: 2020 2020 2030 2035 2020 2020 2020 2020       0 5        
│ │ │ │ -00083650: 2020 2031 2035 2020 2020 2020 2020 2020     1 5          
│ │ │ │ -00083660: 2032 2035 2020 2020 2020 2020 2020 2033   2 5           3
│ │ │ │ -00083670: 2035 2020 2020 2020 2020 2020 3420 7c0a   5          4 |.
│ │ │ │ -00083680: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000835d0: 2020 2020 2020 2020 7c0a 7c20 3135 3030          |.| 1500
│ │ │ │ +000835e0: 3633 3874 2020 2d20 3137 3633 3639 7420  638t  - 176369t 
│ │ │ │ +000835f0: 7420 202d 2032 3139 3233 3239 7420 7420  t  - 2192329t t 
│ │ │ │ +00083600: 202d 2032 3332 3534 3839 7420 7420 202b   - 2325489t t  +
│ │ │ │ +00083610: 2031 3138 3239 3532 7420 7420 202b 2037   1182952t t  + 7
│ │ │ │ +00083620: 3136 3235 3974 2074 7c0a 7c20 2020 2020  16259t t|.|     
│ │ │ │ +00083630: 2020 2020 3420 2020 2020 2020 2020 2030      4          0
│ │ │ │ +00083640: 2035 2020 2020 2020 2020 2020 2031 2035   5           1 5
│ │ │ │ +00083650: 2020 2020 2020 2020 2020 2032 2035 2020             2 5  
│ │ │ │ +00083660: 2020 2020 2020 2020 2033 2035 2020 2020           3 5    
│ │ │ │ +00083670: 2020 2020 2020 3420 7c0a 7c20 2020 2020        4 |.|     
│ │ │ │ +00083680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000836a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000836b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000836c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000836d0: 7c20 2020 2020 2020 2020 2032 2020 2020  |          2    
│ │ │ │ +000836c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000836d0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  000836e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000836f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083710: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083720: 7c2d 2032 3034 3237 3535 7420 202d 2032  |- 2042755t  - 2
│ │ │ │ -00083730: 3231 3830 3938 7420 7420 202d 2033 3739  218098t t  - 379
│ │ │ │ -00083740: 3538 3138 7420 7420 202b 2032 3830 3933  5818t t  + 28093
│ │ │ │ -00083750: 3136 7420 7420 202b 2031 3033 3139 3731  16t t  + 1031971
│ │ │ │ -00083760: 7420 7420 202d 2035 3639 3838 3774 7c0a  t t  - 569887t|.
│ │ │ │ -00083770: 7c20 2020 2020 2020 2020 2034 2020 2020  |          4    
│ │ │ │ -00083780: 2020 2020 2020 2030 2035 2020 2020 2020         0 5      
│ │ │ │ -00083790: 2020 2020 2031 2035 2020 2020 2020 2020       1 5        
│ │ │ │ -000837a0: 2020 2032 2035 2020 2020 2020 2020 2020     2 5          
│ │ │ │ -000837b0: 2033 2035 2020 2020 2020 2020 2020 7c0a   3 5          |.
│ │ │ │ -000837c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00083710: 2020 2020 2020 2020 7c0a 7c2d 2032 3034          |.|- 204
│ │ │ │ +00083720: 3237 3535 7420 202d 2032 3231 3830 3938  2755t  - 2218098
│ │ │ │ +00083730: 7420 7420 202d 2033 3739 3538 3138 7420  t t  - 3795818t 
│ │ │ │ +00083740: 7420 202b 2032 3830 3933 3136 7420 7420  t  + 2809316t t 
│ │ │ │ +00083750: 202b 2031 3033 3139 3731 7420 7420 202d   + 1031971t t  -
│ │ │ │ +00083760: 2035 3639 3838 3774 7c0a 7c20 2020 2020   569887t|.|     
│ │ │ │ +00083770: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │ +00083780: 2030 2035 2020 2020 2020 2020 2020 2031   0 5           1
│ │ │ │ +00083790: 2035 2020 2020 2020 2020 2020 2032 2035   5           2 5
│ │ │ │ +000837a0: 2020 2020 2020 2020 2020 2033 2035 2020             3 5  
│ │ │ │ +000837b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000837c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000837d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000837e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000837f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083800: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083810: 7c20 2020 2020 2020 2020 2020 2032 2020  |            2  
│ │ │ │ +00083800: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083810: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  00083820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083850: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083860: 7c74 2020 2d20 3137 3739 3735 7420 202d  |t  - 177975t  -
│ │ │ │ -00083870: 2038 3133 3035 7420 7420 202d 2034 3330   81305t t  - 430
│ │ │ │ -00083880: 3031 3036 7420 7420 202b 2037 3639 3939  0106t t  + 76999
│ │ │ │ -00083890: 7420 7420 202b 2031 3830 3431 3235 7420  t t  + 1804125t 
│ │ │ │ -000838a0: 7420 202b 2032 3138 3335 3635 7420 7c0a  t  + 2183565t |.
│ │ │ │ -000838b0: 7c20 3420 2020 2020 2020 2020 2034 2020  | 4          4  
│ │ │ │ -000838c0: 2020 2020 2020 2030 2035 2020 2020 2020         0 5      
│ │ │ │ -000838d0: 2020 2020 2031 2035 2020 2020 2020 2020       1 5        
│ │ │ │ -000838e0: 2032 2035 2020 2020 2020 2020 2020 2033   2 5           3
│ │ │ │ -000838f0: 2035 2020 2020 2020 2020 2020 2034 7c0a   5           4|.
│ │ │ │ -00083900: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00083850: 2020 2020 2020 2020 7c0a 7c74 2020 2d20          |.|t  - 
│ │ │ │ +00083860: 3137 3739 3735 7420 202d 2038 3133 3035  177975t  - 81305
│ │ │ │ +00083870: 7420 7420 202d 2034 3330 3031 3036 7420  t t  - 4300106t 
│ │ │ │ +00083880: 7420 202b 2037 3639 3939 7420 7420 202b  t  + 76999t t  +
│ │ │ │ +00083890: 2031 3830 3431 3235 7420 7420 202b 2032   1804125t t  + 2
│ │ │ │ +000838a0: 3138 3335 3635 7420 7c0a 7c20 3420 2020  183565t |.| 4   
│ │ │ │ +000838b0: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ +000838c0: 2030 2035 2020 2020 2020 2020 2020 2031   0 5           1
│ │ │ │ +000838d0: 2035 2020 2020 2020 2020 2032 2035 2020   5         2 5  
│ │ │ │ +000838e0: 2020 2020 2020 2020 2033 2035 2020 2020           3 5    
│ │ │ │ +000838f0: 2020 2020 2020 2034 7c0a 7c20 2020 2020         4|.|     
│ │ │ │ +00083900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083940: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083950: 7c20 2020 2020 2020 2020 3220 2020 2020  |         2     
│ │ │ │ +00083940: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083950: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  00083960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083990: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000839a0: 7c20 3237 3630 3135 3874 2020 2d20 3235  | 2760158t  - 25
│ │ │ │ -000839b0: 3436 3335 3074 2074 2020 2b20 3232 3637  46350t t  + 2267
│ │ │ │ -000839c0: 3334 3274 2074 2020 2b20 3337 3637 3035  342t t  + 376705
│ │ │ │ -000839d0: 3174 2074 2020 2d20 3137 3637 3933 3574  1t t  - 1767935t
│ │ │ │ -000839e0: 2074 2020 2d20 3430 3930 3033 3774 7c0a   t  - 4090037t|.
│ │ │ │ -000839f0: 7c20 2020 2020 2020 2020 3420 2020 2020  |         4     
│ │ │ │ -00083a00: 2020 2020 2020 3020 3520 2020 2020 2020        0 5       
│ │ │ │ -00083a10: 2020 2020 3120 3520 2020 2020 2020 2020      1 5         
│ │ │ │ -00083a20: 2020 3220 3520 2020 2020 2020 2020 2020    2 5           
│ │ │ │ -00083a30: 3320 3520 2020 2020 2020 2020 2020 7c0a  3 5           |.
│ │ │ │ -00083a40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00083990: 2020 2020 2020 2020 7c0a 7c20 3237 3630          |.| 2760
│ │ │ │ +000839a0: 3135 3874 2020 2d20 3235 3436 3335 3074  158t  - 2546350t
│ │ │ │ +000839b0: 2074 2020 2b20 3232 3637 3334 3274 2074   t  + 2267342t t
│ │ │ │ +000839c0: 2020 2b20 3337 3637 3035 3174 2074 2020    + 3767051t t  
│ │ │ │ +000839d0: 2d20 3137 3637 3933 3574 2074 2020 2d20  - 1767935t t  - 
│ │ │ │ +000839e0: 3430 3930 3033 3774 7c0a 7c20 2020 2020  4090037t|.|     
│ │ │ │ +000839f0: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ +00083a00: 3020 3520 2020 2020 2020 2020 2020 3120  0 5           1 
│ │ │ │ +00083a10: 3520 2020 2020 2020 2020 2020 3220 3520  5           2 5 
│ │ │ │ +00083a20: 2020 2020 2020 2020 2020 3320 3520 2020            3 5   
│ │ │ │ +00083a30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083a80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083a90: 7c20 2020 2020 2020 3220 2020 2020 2020  |       2       
│ │ │ │ +00083a80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083a90: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  00083aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083ad0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083ae0: 7c34 3636 3235 3974 2020 2b20 3537 3039  |466259t  + 5709
│ │ │ │ -00083af0: 3139 7420 7420 202b 2034 3432 3636 3836  19t t  + 4426686
│ │ │ │ -00083b00: 7420 7420 202d 2032 3438 3639 3574 2074  t t  - 248695t t
│ │ │ │ -00083b10: 2020 2d20 3435 3134 3333 7420 7420 202b    - 451433t t  +
│ │ │ │ -00083b20: 2034 3032 3332 3938 7420 7420 202b 7c0a   4023298t t  +|.
│ │ │ │ -00083b30: 7c20 2020 2020 2020 3420 2020 2020 2020  |       4       
│ │ │ │ -00083b40: 2020 2030 2035 2020 2020 2020 2020 2020     0 5          
│ │ │ │ -00083b50: 2031 2035 2020 2020 2020 2020 2020 3220   1 5          2 
│ │ │ │ -00083b60: 3520 2020 2020 2020 2020 2033 2035 2020  5          3 5  
│ │ │ │ -00083b70: 2020 2020 2020 2020 2034 2035 2020 7c0a           4 5  |.
│ │ │ │ -00083b80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00083ad0: 2020 2020 2020 2020 7c0a 7c34 3636 3235          |.|46625
│ │ │ │ +00083ae0: 3974 2020 2b20 3537 3039 3139 7420 7420  9t  + 570919t t 
│ │ │ │ +00083af0: 202b 2034 3432 3636 3836 7420 7420 202d   + 4426686t t  -
│ │ │ │ +00083b00: 2032 3438 3639 3574 2074 2020 2d20 3435   248695t t  - 45
│ │ │ │ +00083b10: 3134 3333 7420 7420 202b 2034 3032 3332  1433t t  + 40232
│ │ │ │ +00083b20: 3938 7420 7420 202b 7c0a 7c20 2020 2020  98t t  +|.|     
│ │ │ │ +00083b30: 2020 3420 2020 2020 2020 2020 2030 2035    4          0 5
│ │ │ │ +00083b40: 2020 2020 2020 2020 2020 2031 2035 2020             1 5  
│ │ │ │ +00083b50: 2020 2020 2020 2020 3220 3520 2020 2020          2 5     
│ │ │ │ +00083b60: 2020 2020 2033 2035 2020 2020 2020 2020       3 5        
│ │ │ │ +00083b70: 2020 2034 2035 2020 7c0a 7c20 2020 2020     4 5  |.|     
│ │ │ │ +00083b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083bc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083bd0: 7c20 2020 2020 2020 2020 2020 2032 2020  |            2  
│ │ │ │ +00083bc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083bd0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  00083be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083c10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083c20: 7c20 202d 2032 3937 3335 3030 7420 202d  |  - 2973500t  -
│ │ │ │ -00083c30: 2034 3231 3831 3336 7420 7420 202d 2033   4218136t t  - 3
│ │ │ │ -00083c40: 3830 3434 3233 7420 7420 202b 2032 3732  804423t t  + 272
│ │ │ │ -00083c50: 3632 3635 7420 7420 202b 2034 3532 3339  6265t t  + 45239
│ │ │ │ -00083c60: 3138 7420 7420 202d 2032 3130 3332 7c0a  18t t  - 21032|.
│ │ │ │ -00083c70: 7c34 2020 2020 2020 2020 2020 2034 2020  |4           4  
│ │ │ │ -00083c80: 2020 2020 2020 2020 2030 2035 2020 2020           0 5    
│ │ │ │ -00083c90: 2020 2020 2020 2031 2035 2020 2020 2020         1 5      
│ │ │ │ -00083ca0: 2020 2020 2032 2035 2020 2020 2020 2020       2 5        
│ │ │ │ -00083cb0: 2020 2033 2035 2020 2020 2020 2020 7c0a     3 5        |.
│ │ │ │ -00083cc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00083c10: 2020 2020 2020 2020 7c0a 7c20 202d 2032          |.|  - 2
│ │ │ │ +00083c20: 3937 3335 3030 7420 202d 2034 3231 3831  973500t  - 42181
│ │ │ │ +00083c30: 3336 7420 7420 202d 2033 3830 3434 3233  36t t  - 3804423
│ │ │ │ +00083c40: 7420 7420 202b 2032 3732 3632 3635 7420  t t  + 2726265t 
│ │ │ │ +00083c50: 7420 202b 2034 3532 3339 3138 7420 7420  t  + 4523918t t 
│ │ │ │ +00083c60: 202d 2032 3130 3332 7c0a 7c34 2020 2020   - 21032|.|4    
│ │ │ │ +00083c70: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ +00083c80: 2020 2030 2035 2020 2020 2020 2020 2020     0 5          
│ │ │ │ +00083c90: 2031 2035 2020 2020 2020 2020 2020 2032   1 5           2
│ │ │ │ +00083ca0: 2035 2020 2020 2020 2020 2020 2033 2035   5           3 5
│ │ │ │ +00083cb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083d00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083d10: 7c20 2020 2020 2020 2020 2020 2032 2020  |            2  
│ │ │ │ +00083d00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083d10: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  00083d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083d50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083d60: 7c20 202d 2033 3335 3235 3934 7420 202d  |  - 3352594t  -
│ │ │ │ -00083d70: 2034 3437 3138 3939 7420 7420 202d 2032   4471899t t  - 2
│ │ │ │ -00083d80: 3632 3935 3330 7420 7420 202b 2036 3736  629530t t  + 676
│ │ │ │ -00083d90: 3732 3974 2074 2020 2d20 3233 3338 3236  729t t  - 233826
│ │ │ │ -00083da0: 3574 2074 2020 2d20 3430 3738 3330 7c0a  5t t  - 407830|.
│ │ │ │ -00083db0: 7c34 2020 2020 2020 2020 2020 2034 2020  |4           4  
│ │ │ │ -00083dc0: 2020 2020 2020 2020 2030 2035 2020 2020           0 5    
│ │ │ │ -00083dd0: 2020 2020 2020 2031 2035 2020 2020 2020         1 5      
│ │ │ │ -00083de0: 2020 2020 3220 3520 2020 2020 2020 2020      2 5         
│ │ │ │ -00083df0: 2020 3320 3520 2020 2020 2020 2020 7c0a    3 5         |.
│ │ │ │ -00083e00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00083d50: 2020 2020 2020 2020 7c0a 7c20 202d 2033          |.|  - 3
│ │ │ │ +00083d60: 3335 3235 3934 7420 202d 2034 3437 3138  352594t  - 44718
│ │ │ │ +00083d70: 3939 7420 7420 202d 2032 3632 3935 3330  99t t  - 2629530
│ │ │ │ +00083d80: 7420 7420 202b 2036 3736 3732 3974 2074  t t  + 676729t t
│ │ │ │ +00083d90: 2020 2d20 3233 3338 3236 3574 2074 2020    - 2338265t t  
│ │ │ │ +00083da0: 2d20 3430 3738 3330 7c0a 7c34 2020 2020  - 407830|.|4    
│ │ │ │ +00083db0: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │ +00083dc0: 2020 2030 2035 2020 2020 2020 2020 2020     0 5          
│ │ │ │ +00083dd0: 2031 2035 2020 2020 2020 2020 2020 3220   1 5          2 
│ │ │ │ +00083de0: 3520 2020 2020 2020 2020 2020 3320 3520  5           3 5 
│ │ │ │ +00083df0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083e40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083e50: 7c20 2020 2020 2020 2020 2020 2020 3220  |             2 
│ │ │ │ +00083e40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083e50: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  00083e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083e90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083ea0: 7c74 2020 2b20 3439 3032 3837 3074 2020  |t  + 4902870t  
│ │ │ │ -00083eb0: 2b20 3830 3039 3532 7420 7420 202d 2033  + 800952t t  - 3
│ │ │ │ -00083ec0: 3832 3136 3231 7420 7420 202d 2038 3638  821621t t  - 868
│ │ │ │ -00083ed0: 3536 3774 2074 2020 2b20 3938 3439 3331  567t t  + 984931
│ │ │ │ -00083ee0: 7420 7420 202b 2039 3734 3534 3174 7c0a  t t  + 974541t|.
│ │ │ │ -00083ef0: 7c20 3420 2020 2020 2020 2020 2020 3420  | 4           4 
│ │ │ │ -00083f00: 2020 2020 2020 2020 2030 2035 2020 2020           0 5    
│ │ │ │ -00083f10: 2020 2020 2020 2031 2035 2020 2020 2020         1 5      
│ │ │ │ -00083f20: 2020 2020 3220 3520 2020 2020 2020 2020      2 5         
│ │ │ │ -00083f30: 2033 2035 2020 2020 2020 2020 2020 7c0a   3 5          |.
│ │ │ │ -00083f40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00083e90: 2020 2020 2020 2020 7c0a 7c74 2020 2b20          |.|t  + 
│ │ │ │ +00083ea0: 3439 3032 3837 3074 2020 2b20 3830 3039  4902870t  + 8009
│ │ │ │ +00083eb0: 3532 7420 7420 202d 2033 3832 3136 3231  52t t  - 3821621
│ │ │ │ +00083ec0: 7420 7420 202d 2038 3638 3536 3774 2074  t t  - 868567t t
│ │ │ │ +00083ed0: 2020 2b20 3938 3439 3331 7420 7420 202b    + 984931t t  +
│ │ │ │ +00083ee0: 2039 3734 3534 3174 7c0a 7c20 3420 2020   974541t|.| 4   
│ │ │ │ +00083ef0: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +00083f00: 2020 2030 2035 2020 2020 2020 2020 2020     0 5          
│ │ │ │ +00083f10: 2031 2035 2020 2020 2020 2020 2020 3220   1 5          2 
│ │ │ │ +00083f20: 3520 2020 2020 2020 2020 2033 2035 2020  5          3 5  
│ │ │ │ +00083f30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00083f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083f80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083f90: 7c32 2020 2020 2020 2020 2020 2020 2020  |2              
│ │ │ │ +00083f80: 2020 2020 2020 2020 7c0a 7c32 2020 2020          |.|2    
│ │ │ │ +00083f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00083fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00083fd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00083fe0: 7c20 202d 2039 3839 3536 3874 2074 2020  |  - 989568t t  
│ │ │ │ -00083ff0: 2d20 3233 3033 3931 3274 2074 2020 2d20  - 2303912t t  - 
│ │ │ │ -00084000: 3430 3734 3539 3674 2074 2020 2d20 3137  4074596t t  - 17
│ │ │ │ -00084010: 3431 3632 3974 2074 2020 2d20 3132 3531  41629t t  - 1251
│ │ │ │ -00084020: 3638 3574 2074 2020 2d20 3231 3935 7c0a  685t t  - 2195|.
│ │ │ │ -00084030: 7c34 2020 2020 2020 2020 2020 3020 3520  |4          0 5 
│ │ │ │ -00084040: 2020 2020 2020 2020 2020 3120 3520 2020            1 5   
│ │ │ │ -00084050: 2020 2020 2020 2020 3220 3520 2020 2020          2 5     
│ │ │ │ -00084060: 2020 2020 2020 3320 3520 2020 2020 2020        3 5       
│ │ │ │ -00084070: 2020 2020 3420 3520 2020 2020 2020 7c0a      4 5       |.
│ │ │ │ -00084080: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00083fd0: 2020 2020 2020 2020 7c0a 7c20 202d 2039          |.|  - 9
│ │ │ │ +00083fe0: 3839 3536 3874 2074 2020 2d20 3233 3033  89568t t  - 2303
│ │ │ │ +00083ff0: 3931 3274 2074 2020 2d20 3430 3734 3539  912t t  - 407459
│ │ │ │ +00084000: 3674 2074 2020 2d20 3137 3431 3632 3974  6t t  - 1741629t
│ │ │ │ +00084010: 2074 2020 2d20 3132 3531 3638 3574 2074   t  - 1251685t t
│ │ │ │ +00084020: 2020 2d20 3231 3935 7c0a 7c34 2020 2020    - 2195|.|4    
│ │ │ │ +00084030: 2020 2020 2020 3020 3520 2020 2020 2020        0 5       
│ │ │ │ +00084040: 2020 2020 3120 3520 2020 2020 2020 2020      1 5         
│ │ │ │ +00084050: 2020 3220 3520 2020 2020 2020 2020 2020    2 5           
│ │ │ │ +00084060: 3320 3520 2020 2020 2020 2020 2020 3420  3 5           4 
│ │ │ │ +00084070: 3520 2020 2020 2020 7c0a 7c2d 2d2d 2d2d  5       |.|-----
│ │ │ │ +00084080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00084090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000840a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000840b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000840c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -000840d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000840e0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000840c0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +000840d0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +000840e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000840f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084110: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084120: 7c74 2074 2020 2b20 3234 3433 3330 3774  |t t  + 2443307t
│ │ │ │ -00084130: 2020 2b20 3435 3134 3838 3974 2074 2020    + 4514889t t  
│ │ │ │ -00084140: 2d20 3230 3734 3437 3874 2074 2020 2b20  - 2074478t t  + 
│ │ │ │ -00084150: 3433 3238 3839 3374 2074 2020 2d20 3631  4328893t t  - 61
│ │ │ │ -00084160: 3034 3438 7420 7420 202d 2032 3736 7c0a  0448t t  - 276|.
│ │ │ │ -00084170: 7c20 3420 3520 2020 2020 2020 2020 2020  | 4 5           
│ │ │ │ -00084180: 3520 2020 2020 2020 2020 2020 3020 3620  5           0 6 
│ │ │ │ -00084190: 2020 2020 2020 2020 2020 3120 3620 2020            1 6   
│ │ │ │ -000841a0: 2020 2020 2020 2020 3220 3620 2020 2020          2 6     
│ │ │ │ -000841b0: 2020 2020 2033 2036 2020 2020 2020 7c0a       3 6      |.
│ │ │ │ -000841c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00084110: 2020 2020 2020 2020 7c0a 7c74 2074 2020          |.|t t  
│ │ │ │ +00084120: 2b20 3234 3433 3330 3774 2020 2b20 3435  + 2443307t  + 45
│ │ │ │ +00084130: 3134 3838 3974 2074 2020 2d20 3230 3734  14889t t  - 2074
│ │ │ │ +00084140: 3437 3874 2074 2020 2b20 3433 3238 3839  478t t  + 432889
│ │ │ │ +00084150: 3374 2074 2020 2d20 3631 3034 3438 7420  3t t  - 610448t 
│ │ │ │ +00084160: 7420 202d 2032 3736 7c0a 7c20 3420 3520  t  - 276|.| 4 5 
│ │ │ │ +00084170: 2020 2020 2020 2020 2020 3520 2020 2020            5     
│ │ │ │ +00084180: 2020 2020 2020 3020 3620 2020 2020 2020        0 6       
│ │ │ │ +00084190: 2020 2020 3120 3620 2020 2020 2020 2020      1 6         
│ │ │ │ +000841a0: 2020 3220 3620 2020 2020 2020 2020 2033    2 6          3
│ │ │ │ +000841b0: 2036 2020 2020 2020 7c0a 7c20 2020 2020   6      |.|     
│ │ │ │ +000841c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000841d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000841e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000841f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084200: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084210: 7c20 2020 2020 2020 2020 2020 2032 2020  |            2  
│ │ │ │ +00084200: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00084210: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  00084220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084250: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084260: 7c20 202b 2032 3032 3431 3435 7420 202b  |  + 2024145t  +
│ │ │ │ -00084270: 2038 3130 3233 3274 2074 2020 2d20 3436   810232t t  - 46
│ │ │ │ -00084280: 3334 3430 3374 2074 2020 2d20 3130 3739  34403t t  - 1079
│ │ │ │ -00084290: 3634 3774 2074 2020 2d20 3437 3239 3335  647t t  - 472935
│ │ │ │ -000842a0: 3674 2074 2020 2d20 3236 3230 3937 7c0a  6t t  - 262097|.
│ │ │ │ -000842b0: 7c35 2020 2020 2020 2020 2020 2035 2020  |5           5  
│ │ │ │ -000842c0: 2020 2020 2020 2020 3020 3620 2020 2020          0 6     
│ │ │ │ -000842d0: 2020 2020 2020 3120 3620 2020 2020 2020        1 6       
│ │ │ │ -000842e0: 2020 2020 3220 3620 2020 2020 2020 2020      2 6         
│ │ │ │ -000842f0: 2020 3320 3620 2020 2020 2020 2020 7c0a    3 6         |.
│ │ │ │ -00084300: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00084250: 2020 2020 2020 2020 7c0a 7c20 202b 2032          |.|  + 2
│ │ │ │ +00084260: 3032 3431 3435 7420 202b 2038 3130 3233  024145t  + 81023
│ │ │ │ +00084270: 3274 2074 2020 2d20 3436 3334 3430 3374  2t t  - 4634403t
│ │ │ │ +00084280: 2074 2020 2d20 3130 3739 3634 3774 2074   t  - 1079647t t
│ │ │ │ +00084290: 2020 2d20 3437 3239 3335 3674 2074 2020    - 4729356t t  
│ │ │ │ +000842a0: 2d20 3236 3230 3937 7c0a 7c35 2020 2020  - 262097|.|5    
│ │ │ │ +000842b0: 2020 2020 2020 2035 2020 2020 2020 2020         5        
│ │ │ │ +000842c0: 2020 3020 3620 2020 2020 2020 2020 2020    0 6           
│ │ │ │ +000842d0: 3120 3620 2020 2020 2020 2020 2020 3220  1 6           2 
│ │ │ │ +000842e0: 3620 2020 2020 2020 2020 2020 3320 3620  6           3 6 
│ │ │ │ +000842f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00084300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084340: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084350: 7c20 2020 2020 2020 2020 2020 2020 3220  |             2 
│ │ │ │ +00084340: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00084350: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  00084360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084390: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000843a0: 7c20 7420 202b 2036 3230 3132 3174 2020  | t  + 620121t  
│ │ │ │ -000843b0: 2d20 3133 3031 3236 7420 7420 202d 2033  - 130126t t  - 3
│ │ │ │ -000843c0: 3330 3134 3337 7420 7420 202d 2032 3739  301437t t  - 279
│ │ │ │ -000843d0: 3637 3937 7420 7420 202d 2032 3231 3730  6797t t  - 22170
│ │ │ │ -000843e0: 3834 7420 7420 202b 2033 3635 3133 7c0a  84t t  + 36513|.
│ │ │ │ -000843f0: 7c34 2035 2020 2020 2020 2020 2020 3520  |4 5          5 
│ │ │ │ -00084400: 2020 2020 2020 2020 2030 2036 2020 2020           0 6    
│ │ │ │ -00084410: 2020 2020 2020 2031 2036 2020 2020 2020         1 6      
│ │ │ │ -00084420: 2020 2020 2032 2036 2020 2020 2020 2020       2 6        
│ │ │ │ -00084430: 2020 2033 2036 2020 2020 2020 2020 7c0a     3 6        |.
│ │ │ │ -00084440: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00084390: 2020 2020 2020 2020 7c0a 7c20 7420 202b          |.| t  +
│ │ │ │ +000843a0: 2036 3230 3132 3174 2020 2d20 3133 3031   620121t  - 1301
│ │ │ │ +000843b0: 3236 7420 7420 202d 2033 3330 3134 3337  26t t  - 3301437
│ │ │ │ +000843c0: 7420 7420 202d 2032 3739 3637 3937 7420  t t  - 2796797t 
│ │ │ │ +000843d0: 7420 202d 2032 3231 3730 3834 7420 7420  t  - 2217084t t 
│ │ │ │ +000843e0: 202b 2033 3635 3133 7c0a 7c34 2035 2020   + 36513|.|4 5  
│ │ │ │ +000843f0: 2020 2020 2020 2020 3520 2020 2020 2020          5       
│ │ │ │ +00084400: 2020 2030 2036 2020 2020 2020 2020 2020     0 6          
│ │ │ │ +00084410: 2031 2036 2020 2020 2020 2020 2020 2032   1 6           2
│ │ │ │ +00084420: 2036 2020 2020 2020 2020 2020 2033 2036   6           3 6
│ │ │ │ +00084430: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00084440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084480: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084490: 7c20 2020 2020 2020 2020 2020 2020 3220  |             2 
│ │ │ │ +00084480: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00084490: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  000844a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000844b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000844c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000844d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000844e0: 7c74 2020 2b20 3237 3632 3936 3274 2020  |t  + 2762962t  
│ │ │ │ -000844f0: 2d20 3130 3733 3334 3574 2074 2020 2b20  - 1073345t t  + 
│ │ │ │ -00084500: 3232 3733 3030 3074 2074 2020 2d20 3136  2273000t t  - 16
│ │ │ │ -00084510: 3434 3139 3774 2074 2020 2d20 3838 3034  44197t t  - 8804
│ │ │ │ -00084520: 3038 7420 7420 202d 2032 3839 3035 7c0a  08t t  - 28905|.
│ │ │ │ -00084530: 7c20 3520 2020 2020 2020 2020 2020 3520  | 5           5 
│ │ │ │ -00084540: 2020 2020 2020 2020 2020 3020 3620 2020            0 6   
│ │ │ │ -00084550: 2020 2020 2020 2020 3120 3620 2020 2020          1 6     
│ │ │ │ -00084560: 2020 2020 2020 3220 3620 2020 2020 2020        2 6       
│ │ │ │ -00084570: 2020 2033 2036 2020 2020 2020 2020 7c0a     3 6        |.
│ │ │ │ -00084580: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000844d0: 2020 2020 2020 2020 7c0a 7c74 2020 2b20          |.|t  + 
│ │ │ │ +000844e0: 3237 3632 3936 3274 2020 2d20 3130 3733  2762962t  - 1073
│ │ │ │ +000844f0: 3334 3574 2074 2020 2b20 3232 3733 3030  345t t  + 227300
│ │ │ │ +00084500: 3074 2074 2020 2d20 3136 3434 3139 3774  0t t  - 1644197t
│ │ │ │ +00084510: 2074 2020 2d20 3838 3034 3038 7420 7420   t  - 880408t t 
│ │ │ │ +00084520: 202d 2032 3839 3035 7c0a 7c20 3520 2020   - 28905|.| 5   
│ │ │ │ +00084530: 2020 2020 2020 2020 3520 2020 2020 2020          5       
│ │ │ │ +00084540: 2020 2020 3020 3620 2020 2020 2020 2020      0 6         
│ │ │ │ +00084550: 2020 3120 3620 2020 2020 2020 2020 2020    1 6           
│ │ │ │ +00084560: 3220 3620 2020 2020 2020 2020 2033 2036  2 6          3 6
│ │ │ │ +00084570: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00084580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000845a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000845b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000845c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000845d0: 7c20 2020 2020 2020 2020 2020 2020 2032  |              2
│ │ │ │ +000845c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000845d0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  000845e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000845f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084610: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084620: 7c20 7420 202b 2032 3030 3637 3036 7420  | t  + 2006706t 
│ │ │ │ -00084630: 202d 2039 3732 3335 3174 2074 2020 2d20   - 972351t t  - 
│ │ │ │ -00084640: 3338 3833 3030 3974 2074 2020 2b20 3438  3883009t t  + 48
│ │ │ │ -00084650: 3331 3133 3174 2074 2020 2b20 3434 3338  31131t t  + 4438
│ │ │ │ -00084660: 3332 3474 2074 2020 2d20 3735 3637 7c0a  324t t  - 7567|.
│ │ │ │ -00084670: 7c34 2035 2020 2020 2020 2020 2020 2035  |4 5           5
│ │ │ │ -00084680: 2020 2020 2020 2020 2020 3020 3620 2020            0 6   
│ │ │ │ -00084690: 2020 2020 2020 2020 3120 3620 2020 2020          1 6     
│ │ │ │ -000846a0: 2020 2020 2020 3220 3620 2020 2020 2020        2 6       
│ │ │ │ -000846b0: 2020 2020 3320 3620 2020 2020 2020 7c0a      3 6       |.
│ │ │ │ -000846c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00084610: 2020 2020 2020 2020 7c0a 7c20 7420 202b          |.| t  +
│ │ │ │ +00084620: 2032 3030 3637 3036 7420 202d 2039 3732   2006706t  - 972
│ │ │ │ +00084630: 3335 3174 2074 2020 2d20 3338 3833 3030  351t t  - 388300
│ │ │ │ +00084640: 3974 2074 2020 2b20 3438 3331 3133 3174  9t t  + 4831131t
│ │ │ │ +00084650: 2074 2020 2b20 3434 3338 3332 3474 2074   t  + 4438324t t
│ │ │ │ +00084660: 2020 2d20 3735 3637 7c0a 7c34 2035 2020    - 7567|.|4 5  
│ │ │ │ +00084670: 2020 2020 2020 2020 2035 2020 2020 2020           5      
│ │ │ │ +00084680: 2020 2020 3020 3620 2020 2020 2020 2020      0 6         
│ │ │ │ +00084690: 2020 3120 3620 2020 2020 2020 2020 2020    1 6           
│ │ │ │ +000846a0: 3220 3620 2020 2020 2020 2020 2020 3320  2 6           3 
│ │ │ │ +000846b0: 3620 2020 2020 2020 7c0a 7c20 2020 2020  6       |.|     
│ │ │ │ +000846c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000846d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000846e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000846f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084700: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084710: 7c20 2020 2020 2020 2020 3220 2020 2020  |         2     
│ │ │ │ +00084700: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00084710: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  00084720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084750: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084760: 7c20 3135 3434 3133 3374 2020 2b20 3132  | 1544133t  + 12
│ │ │ │ -00084770: 3334 3735 3974 2074 2020 2b20 3930 3239  34759t t  + 9029
│ │ │ │ -00084780: 3074 2074 2020 2d20 3235 3435 3538 3674  0t t  - 2545586t
│ │ │ │ -00084790: 2074 2020 2d20 3131 3530 3733 3974 2074   t  - 1150739t t
│ │ │ │ -000847a0: 2020 2d20 3137 3837 3737 3774 2074 7c0a    - 1787777t t|.
│ │ │ │ -000847b0: 7c20 2020 2020 2020 2020 3520 2020 2020  |         5     
│ │ │ │ -000847c0: 2020 2020 2020 3020 3620 2020 2020 2020        0 6       
│ │ │ │ -000847d0: 2020 3120 3620 2020 2020 2020 2020 2020    1 6           
│ │ │ │ -000847e0: 3220 3620 2020 2020 2020 2020 2020 3320  2 6           3 
│ │ │ │ -000847f0: 3620 2020 2020 2020 2020 2020 3420 7c0a  6           4 |.
│ │ │ │ -00084800: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00084750: 2020 2020 2020 2020 7c0a 7c20 3135 3434          |.| 1544
│ │ │ │ +00084760: 3133 3374 2020 2b20 3132 3334 3735 3974  133t  + 1234759t
│ │ │ │ +00084770: 2074 2020 2b20 3930 3239 3074 2074 2020   t  + 90290t t  
│ │ │ │ +00084780: 2d20 3235 3435 3538 3674 2074 2020 2d20  - 2545586t t  - 
│ │ │ │ +00084790: 3131 3530 3733 3974 2074 2020 2d20 3137  1150739t t  - 17
│ │ │ │ +000847a0: 3837 3737 3774 2074 7c0a 7c20 2020 2020  87777t t|.|     
│ │ │ │ +000847b0: 2020 2020 3520 2020 2020 2020 2020 2020      5           
│ │ │ │ +000847c0: 3020 3620 2020 2020 2020 2020 3120 3620  0 6         1 6 
│ │ │ │ +000847d0: 2020 2020 2020 2020 2020 3220 3620 2020            2 6   
│ │ │ │ +000847e0: 2020 2020 2020 2020 3320 3620 2020 2020          3 6     
│ │ │ │ +000847f0: 2020 2020 2020 3420 7c0a 7c20 2020 2020        4 |.|     
│ │ │ │ +00084800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084850: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00084860: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00084840: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00084850: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +00084860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084890: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000848a0: 7c31 7420 7420 202b 2033 3736 3638 3339  |1t t  + 3766839
│ │ │ │ -000848b0: 7420 202b 2031 3639 3432 3739 7420 7420  t  + 1694279t t 
│ │ │ │ -000848c0: 202b 2032 3439 3035 3338 7420 7420 202b   + 2490538t t  +
│ │ │ │ -000848d0: 2037 3538 3437 3674 2074 2020 2b20 3435   758476t t  + 45
│ │ │ │ -000848e0: 3735 3534 3774 2074 2020 2d20 3431 7c0a  75547t t  - 41|.
│ │ │ │ -000848f0: 7c20 2034 2035 2020 2020 2020 2020 2020  |  4 5          
│ │ │ │ -00084900: 2035 2020 2020 2020 2020 2020 2030 2036   5           0 6
│ │ │ │ -00084910: 2020 2020 2020 2020 2020 2031 2036 2020             1 6  
│ │ │ │ -00084920: 2020 2020 2020 2020 3220 3620 2020 2020          2 6     
│ │ │ │ -00084930: 2020 2020 2020 3320 3620 2020 2020 7c0a        3 6     |.
│ │ │ │ -00084940: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00084890: 2020 2020 2020 2020 7c0a 7c31 7420 7420          |.|1t t 
│ │ │ │ +000848a0: 202b 2033 3736 3638 3339 7420 202b 2031   + 3766839t  + 1
│ │ │ │ +000848b0: 3639 3432 3739 7420 7420 202b 2032 3439  694279t t  + 249
│ │ │ │ +000848c0: 3035 3338 7420 7420 202b 2037 3538 3437  0538t t  + 75847
│ │ │ │ +000848d0: 3674 2074 2020 2b20 3435 3735 3534 3774  6t t  + 4575547t
│ │ │ │ +000848e0: 2074 2020 2d20 3431 7c0a 7c20 2034 2035   t  - 41|.|  4 5
│ │ │ │ +000848f0: 2020 2020 2020 2020 2020 2035 2020 2020             5    
│ │ │ │ +00084900: 2020 2020 2020 2030 2036 2020 2020 2020         0 6      
│ │ │ │ +00084910: 2020 2020 2031 2036 2020 2020 2020 2020       1 6        
│ │ │ │ +00084920: 2020 3220 3620 2020 2020 2020 2020 2020    2 6           
│ │ │ │ +00084930: 3320 3620 2020 2020 7c0a 7c20 2020 2020  3 6     |.|     
│ │ │ │ +00084940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084980: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084990: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000849a0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00084980: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00084990: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +000849a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000849b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000849c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000849d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000849e0: 7c38 7420 7420 202b 2033 3634 3435 3035  |8t t  + 3644505
│ │ │ │ -000849f0: 7420 202d 2031 3733 3830 3137 7420 7420  t  - 1738017t t 
│ │ │ │ -00084a00: 202d 2033 3838 3530 3331 7420 7420 202d   - 3885031t t  -
│ │ │ │ -00084a10: 2034 3734 3039 3539 7420 7420 202d 2033   4740959t t  - 3
│ │ │ │ -00084a20: 3435 3931 3433 7420 7420 202d 2031 7c0a  459143t t  - 1|.
│ │ │ │ -00084a30: 7c20 2034 2035 2020 2020 2020 2020 2020  |  4 5          
│ │ │ │ -00084a40: 2035 2020 2020 2020 2020 2020 2030 2036   5           0 6
│ │ │ │ -00084a50: 2020 2020 2020 2020 2020 2031 2036 2020             1 6  
│ │ │ │ -00084a60: 2020 2020 2020 2020 2032 2036 2020 2020           2 6    
│ │ │ │ -00084a70: 2020 2020 2020 2033 2036 2020 2020 7c0a         3 6    |.
│ │ │ │ -00084a80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000849d0: 2020 2020 2020 2020 7c0a 7c38 7420 7420          |.|8t t 
│ │ │ │ +000849e0: 202b 2033 3634 3435 3035 7420 202d 2031   + 3644505t  - 1
│ │ │ │ +000849f0: 3733 3830 3137 7420 7420 202d 2033 3838  738017t t  - 388
│ │ │ │ +00084a00: 3530 3331 7420 7420 202d 2034 3734 3039  5031t t  - 47409
│ │ │ │ +00084a10: 3539 7420 7420 202d 2033 3435 3931 3433  59t t  - 3459143
│ │ │ │ +00084a20: 7420 7420 202d 2031 7c0a 7c20 2034 2035  t t  - 1|.|  4 5
│ │ │ │ +00084a30: 2020 2020 2020 2020 2020 2035 2020 2020             5    
│ │ │ │ +00084a40: 2020 2020 2020 2030 2036 2020 2020 2020         0 6      
│ │ │ │ +00084a50: 2020 2020 2031 2036 2020 2020 2020 2020       1 6        
│ │ │ │ +00084a60: 2020 2032 2036 2020 2020 2020 2020 2020     2 6          
│ │ │ │ +00084a70: 2033 2036 2020 2020 7c0a 7c20 2020 2020   3 6    |.|     
│ │ │ │ +00084a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084ac0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084ad0: 7c20 2020 2020 2020 2020 2020 2020 2032  |              2
│ │ │ │ +00084ac0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00084ad0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  00084ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084b10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084b20: 7c20 7420 202d 2034 3239 3334 3334 7420  | t  - 4293434t 
│ │ │ │ -00084b30: 202d 2031 3633 3336 3037 7420 7420 202b   - 1633607t t  +
│ │ │ │ -00084b40: 2032 3033 3036 3531 7420 7420 202d 2033   2030651t t  - 3
│ │ │ │ -00084b50: 3536 3434 3134 7420 7420 202d 2034 3835  564414t t  - 485
│ │ │ │ -00084b60: 3934 3931 7420 7420 202b 2033 3135 7c0a  9491t t  + 315|.
│ │ │ │ -00084b70: 7c34 2035 2020 2020 2020 2020 2020 2035  |4 5           5
│ │ │ │ -00084b80: 2020 2020 2020 2020 2020 2030 2036 2020             0 6  
│ │ │ │ -00084b90: 2020 2020 2020 2020 2031 2036 2020 2020           1 6    
│ │ │ │ -00084ba0: 2020 2020 2020 2032 2036 2020 2020 2020         2 6      
│ │ │ │ -00084bb0: 2020 2020 2033 2036 2020 2020 2020 7c0a       3 6      |.
│ │ │ │ -00084bc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00084b10: 2020 2020 2020 2020 7c0a 7c20 7420 202d          |.| t  -
│ │ │ │ +00084b20: 2034 3239 3334 3334 7420 202d 2031 3633   4293434t  - 163
│ │ │ │ +00084b30: 3336 3037 7420 7420 202b 2032 3033 3036  3607t t  + 20306
│ │ │ │ +00084b40: 3531 7420 7420 202d 2033 3536 3434 3134  51t t  - 3564414
│ │ │ │ +00084b50: 7420 7420 202d 2034 3835 3934 3931 7420  t t  - 4859491t 
│ │ │ │ +00084b60: 7420 202b 2033 3135 7c0a 7c34 2035 2020  t  + 315|.|4 5  
│ │ │ │ +00084b70: 2020 2020 2020 2020 2035 2020 2020 2020           5      
│ │ │ │ +00084b80: 2020 2020 2030 2036 2020 2020 2020 2020       0 6        
│ │ │ │ +00084b90: 2020 2031 2036 2020 2020 2020 2020 2020     1 6          
│ │ │ │ +00084ba0: 2032 2036 2020 2020 2020 2020 2020 2033   2 6           3
│ │ │ │ +00084bb0: 2036 2020 2020 2020 7c0a 7c20 2020 2020   6      |.|     
│ │ │ │ +00084bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084c00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084c10: 7c20 2020 2032 2020 2020 2020 2020 2020  |    2          
│ │ │ │ +00084c00: 2020 2020 2020 2020 7c0a 7c20 2020 2032          |.|    2
│ │ │ │ +00084c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084c50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084c60: 7c35 3230 7420 202b 2031 3039 3033 3932  |520t  + 1090392
│ │ │ │ -00084c70: 7420 7420 202d 2034 3232 3133 3530 7420  t t  - 4221350t 
│ │ │ │ -00084c80: 7420 202d 2033 3633 3338 3735 7420 7420  t  - 3633875t t 
│ │ │ │ -00084c90: 202b 2032 3836 3338 3574 2074 2020 2d20   + 286385t t  - 
│ │ │ │ -00084ca0: 3431 3238 3235 3874 2074 2020 2d20 7c0a  4128258t t  - |.
│ │ │ │ -00084cb0: 7c20 2020 2035 2020 2020 2020 2020 2020  |    5          
│ │ │ │ -00084cc0: 2030 2036 2020 2020 2020 2020 2020 2031   0 6           1
│ │ │ │ -00084cd0: 2036 2020 2020 2020 2020 2020 2032 2036   6           2 6
│ │ │ │ -00084ce0: 2020 2020 2020 2020 2020 3320 3620 2020            3 6   
│ │ │ │ -00084cf0: 2020 2020 2020 2020 3420 3620 2020 7c0a          4 6   |.
│ │ │ │ -00084d00: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00084c50: 2020 2020 2020 2020 7c0a 7c35 3230 7420          |.|520t 
│ │ │ │ +00084c60: 202b 2031 3039 3033 3932 7420 7420 202d   + 1090392t t  -
│ │ │ │ +00084c70: 2034 3232 3133 3530 7420 7420 202d 2033   4221350t t  - 3
│ │ │ │ +00084c80: 3633 3338 3735 7420 7420 202b 2032 3836  633875t t  + 286
│ │ │ │ +00084c90: 3338 3574 2074 2020 2d20 3431 3238 3235  385t t  - 412825
│ │ │ │ +00084ca0: 3874 2074 2020 2d20 7c0a 7c20 2020 2035  8t t  - |.|    5
│ │ │ │ +00084cb0: 2020 2020 2020 2020 2020 2030 2036 2020             0 6  
│ │ │ │ +00084cc0: 2020 2020 2020 2020 2031 2036 2020 2020           1 6    
│ │ │ │ +00084cd0: 2020 2020 2020 2032 2036 2020 2020 2020         2 6      
│ │ │ │ +00084ce0: 2020 2020 3320 3620 2020 2020 2020 2020      3 6         
│ │ │ │ +00084cf0: 2020 3420 3620 2020 7c0a 7c2d 2d2d 2d2d    4 6   |.|-----
│ │ │ │ +00084d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00084d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00084d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00084d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00084d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00084d50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00084d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084d70: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00084d40: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +00084d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00084d60: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00084d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084d90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084da0: 7c37 3534 3974 2074 2020 2d20 3338 3930  |7549t t  - 3890
│ │ │ │ -00084db0: 3934 3874 2074 2020 2b20 3133 3330 3935  948t t  + 133095
│ │ │ │ -00084dc0: 3074 202c 2020 2020 2020 2020 2020 2020  0t ,            
│ │ │ │ +00084d90: 2020 2020 2020 2020 7c0a 7c37 3534 3974          |.|7549t
│ │ │ │ +00084da0: 2074 2020 2d20 3338 3930 3934 3874 2074   t  - 3890948t t
│ │ │ │ +00084db0: 2020 2b20 3133 3330 3935 3074 202c 2020    + 1330950t ,  
│ │ │ │ +00084dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084de0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084df0: 7c20 2020 2020 3420 3620 2020 2020 2020  |     4 6       
│ │ │ │ -00084e00: 2020 2020 3520 3620 2020 2020 2020 2020      5 6         
│ │ │ │ -00084e10: 2020 3620 2020 2020 2020 2020 2020 2020    6             
│ │ │ │ +00084de0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00084df0: 3420 3620 2020 2020 2020 2020 2020 3520  4 6           5 
│ │ │ │ +00084e00: 3620 2020 2020 2020 2020 2020 3620 2020  6           6   
│ │ │ │ +00084e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084e30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084e40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00084e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00084e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084e80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084e90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00084ea0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00084e80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00084e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00084ea0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  00084eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084ed0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084ee0: 7c31 7420 7420 202b 2033 3532 3437 3430  |1t t  + 3524740
│ │ │ │ -00084ef0: 7420 7420 202d 2031 3739 3633 3231 7420  t t  - 1796321t 
│ │ │ │ -00084f00: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00084ed0: 2020 2020 2020 2020 7c0a 7c31 7420 7420          |.|1t t 
│ │ │ │ +00084ee0: 202b 2033 3532 3437 3430 7420 7420 202d   + 3524740t t  -
│ │ │ │ +00084ef0: 2031 3739 3633 3231 7420 2c20 2020 2020   1796321t ,     
│ │ │ │ +00084f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084f20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084f30: 7c20 2034 2036 2020 2020 2020 2020 2020  |  4 6          
│ │ │ │ -00084f40: 2035 2036 2020 2020 2020 2020 2020 2036   5 6           6
│ │ │ │ +00084f20: 2020 2020 2020 2020 7c0a 7c20 2034 2036          |.|  4 6
│ │ │ │ +00084f30: 2020 2020 2020 2020 2020 2035 2036 2020             5 6  
│ │ │ │ +00084f40: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │  00084f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084f70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084f80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00084f70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00084f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00084fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084fc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00084fd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00084fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00084ff0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00084fc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00084fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00084fe0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00084ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085010: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085020: 7c35 3874 2074 2020 2d20 3434 3930 3633  |58t t  - 449063
│ │ │ │ -00085030: 3874 2074 2020 2d20 3436 3236 3337 3874  8t t  - 4626378t
│ │ │ │ -00085040: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │ +00085010: 2020 2020 2020 2020 7c0a 7c35 3874 2074          |.|58t t
│ │ │ │ +00085020: 2020 2d20 3434 3930 3633 3874 2074 2020    - 4490638t t  
│ │ │ │ +00085030: 2d20 3436 3236 3337 3874 202c 2020 2020  - 4626378t ,    
│ │ │ │ +00085040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085060: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085070: 7c20 2020 3420 3620 2020 2020 2020 2020  |   4 6         
│ │ │ │ -00085080: 2020 3520 3620 2020 2020 2020 2020 2020    5 6           
│ │ │ │ -00085090: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ +00085060: 2020 2020 2020 2020 7c0a 7c20 2020 3420          |.|   4 
│ │ │ │ +00085070: 3620 2020 2020 2020 2020 2020 3520 3620  6           5 6 
│ │ │ │ +00085080: 2020 2020 2020 2020 2020 3620 2020 2020            6     
│ │ │ │ +00085090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000850a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000850b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000850c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000850b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000850c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000850d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000850e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000850f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085100: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00085120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085130: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00085100: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00085110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00085120: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00085130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085150: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085160: 7c34 3774 2074 2020 2d20 3136 3138 3730  |47t t  - 161870
│ │ │ │ -00085170: 3174 2074 2020 2b20 3238 3835 3734 3574  1t t  + 2885745t
│ │ │ │ -00085180: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │ +00085150: 2020 2020 2020 2020 7c0a 7c34 3774 2074          |.|47t t
│ │ │ │ +00085160: 2020 2d20 3136 3138 3730 3174 2074 2020    - 1618701t t  
│ │ │ │ +00085170: 2b20 3238 3835 3734 3574 202c 2020 2020  + 2885745t ,    
│ │ │ │ +00085180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000851a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000851b0: 7c20 2020 3420 3620 2020 2020 2020 2020  |   4 6         
│ │ │ │ -000851c0: 2020 3520 3620 2020 2020 2020 2020 2020    5 6           
│ │ │ │ -000851d0: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ +000851a0: 2020 2020 2020 2020 7c0a 7c20 2020 3420          |.|   4 
│ │ │ │ +000851b0: 3620 2020 2020 2020 2020 2020 3520 3620  6           5 6 
│ │ │ │ +000851c0: 2020 2020 2020 2020 2020 3620 2020 2020            6     
│ │ │ │ +000851d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000851e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000851f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085200: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000851f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00085200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085240: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085250: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00085260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085270: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00085240: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00085250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00085260: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00085270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085290: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000852a0: 7c38 3674 2074 2020 2b20 3433 3735 3835  |86t t  + 437585
│ │ │ │ -000852b0: 3974 2074 2020 2d20 3135 3538 3931 3374  9t t  - 1558913t
│ │ │ │ -000852c0: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │ +00085290: 2020 2020 2020 2020 7c0a 7c38 3674 2074          |.|86t t
│ │ │ │ +000852a0: 2020 2b20 3433 3735 3835 3974 2074 2020    + 4375859t t  
│ │ │ │ +000852b0: 2d20 3135 3538 3931 3374 202c 2020 2020  - 1558913t ,    
│ │ │ │ +000852c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000852d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000852e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000852f0: 7c20 2020 3420 3620 2020 2020 2020 2020  |   4 6         
│ │ │ │ -00085300: 2020 3520 3620 2020 2020 2020 2020 2020    5 6           
│ │ │ │ -00085310: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ +000852e0: 2020 2020 2020 2020 7c0a 7c20 2020 3420          |.|   4 
│ │ │ │ +000852f0: 3620 2020 2020 2020 2020 2020 3520 3620  6           5 6 
│ │ │ │ +00085300: 2020 2020 2020 2020 2020 3620 2020 2020            6     
│ │ │ │ +00085310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085330: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085340: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00085330: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00085340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085380: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085390: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000853a0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +00085380: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00085390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000853a0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  000853b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000853c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000853d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000853e0: 7c20 202b 2031 3830 3239 3131 7420 7420  |  + 1802911t t 
│ │ │ │ -000853f0: 202b 2032 3333 3239 3035 7420 2c20 2020   + 2332905t ,   
│ │ │ │ +000853d0: 2020 2020 2020 2020 7c0a 7c20 202b 2031          |.|  + 1
│ │ │ │ +000853e0: 3830 3239 3131 7420 7420 202b 2032 3333  802911t t  + 233
│ │ │ │ +000853f0: 3239 3035 7420 2c20 2020 2020 2020 2020  2905t ,         
│ │ │ │  00085400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085420: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085430: 7c36 2020 2020 2020 2020 2020 2035 2036  |6           5 6
│ │ │ │ -00085440: 2020 2020 2020 2020 2020 2036 2020 2020             6    
│ │ │ │ +00085420: 2020 2020 2020 2020 7c0a 7c36 2020 2020          |.|6    
│ │ │ │ +00085430: 2020 2020 2020 2035 2036 2020 2020 2020         5 6      
│ │ │ │ +00085440: 2020 2020 2036 2020 2020 2020 2020 2020       6          
│ │ │ │  00085450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085470: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085480: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00085470: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00085480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000854a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000854b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000854c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000854d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000854e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000854f0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +000854c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000854d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000854e0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +000854f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085510: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085520: 7c30 3630 3633 7420 7420 202b 2031 3836  |06063t t  + 186
│ │ │ │ -00085530: 3433 3230 7420 7420 202b 2032 3132 3830  4320t t  + 21280
│ │ │ │ -00085540: 3932 7420 2c20 2020 2020 2020 2020 2020  92t ,           
│ │ │ │ +00085510: 2020 2020 2020 2020 7c0a 7c30 3630 3633          |.|06063
│ │ │ │ +00085520: 7420 7420 202b 2031 3836 3433 3230 7420  t t  + 1864320t 
│ │ │ │ +00085530: 7420 202b 2032 3132 3830 3932 7420 2c20  t  + 2128092t , 
│ │ │ │ +00085540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085560: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085570: 7c20 2020 2020 2034 2036 2020 2020 2020  |      4 6      
│ │ │ │ -00085580: 2020 2020 2035 2036 2020 2020 2020 2020       5 6        
│ │ │ │ -00085590: 2020 2036 2020 2020 2020 2020 2020 2020     6            
│ │ │ │ +00085560: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00085570: 2034 2036 2020 2020 2020 2020 2020 2035   4 6           5
│ │ │ │ +00085580: 2036 2020 2020 2020 2020 2020 2036 2020   6           6  
│ │ │ │ +00085590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000855a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000855b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000855c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000855b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000855c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000855d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000855e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000855f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085600: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085610: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00085620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085630: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00085600: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00085610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00085620: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00085630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085650: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085660: 7c31 3432 3334 3874 2074 2020 2d20 3236  |142348t t  - 26
│ │ │ │ -00085670: 3738 3234 3774 2074 2020 2b20 3531 3735  78247t t  + 5175
│ │ │ │ -00085680: 3134 7420 2c20 2020 2020 2020 2020 2020  14t ,           
│ │ │ │ +00085650: 2020 2020 2020 2020 7c0a 7c31 3432 3334          |.|14234
│ │ │ │ +00085660: 3874 2074 2020 2d20 3236 3738 3234 3774  8t t  - 2678247t
│ │ │ │ +00085670: 2074 2020 2b20 3531 3735 3134 7420 2c20   t  + 517514t , 
│ │ │ │ +00085680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000856a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000856b0: 7c20 2020 2020 2020 3420 3620 2020 2020  |       4 6     
│ │ │ │ -000856c0: 2020 2020 2020 3520 3620 2020 2020 2020        5 6       
│ │ │ │ -000856d0: 2020 2036 2020 2020 2020 2020 2020 2020     6            
│ │ │ │ +000856a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000856b0: 2020 3420 3620 2020 2020 2020 2020 2020    4 6           
│ │ │ │ +000856c0: 3520 3620 2020 2020 2020 2020 2036 2020  5 6          6  
│ │ │ │ +000856d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000856e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000856f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085700: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000856f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00085700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085740: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085750: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00085760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085770: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00085740: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00085750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00085760: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00085770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085790: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000857a0: 7c33 3433 3874 2074 2020 2d20 3635 3632  |3438t t  - 6562
│ │ │ │ -000857b0: 3139 7420 7420 202b 2033 3939 3934 3874  19t t  + 399948t
│ │ │ │ -000857c0: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │ +00085790: 2020 2020 2020 2020 7c0a 7c33 3433 3874          |.|3438t
│ │ │ │ +000857a0: 2074 2020 2d20 3635 3632 3139 7420 7420   t  - 656219t t 
│ │ │ │ +000857b0: 202b 2033 3939 3934 3874 202c 2020 2020   + 399948t ,    
│ │ │ │ +000857c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000857d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000857e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000857f0: 7c20 2020 2020 3420 3620 2020 2020 2020  |     4 6       
│ │ │ │ -00085800: 2020 2035 2036 2020 2020 2020 2020 2020     5 6          
│ │ │ │ -00085810: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ +000857e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000857f0: 3420 3620 2020 2020 2020 2020 2035 2036  4 6          5 6
│ │ │ │ +00085800: 2020 2020 2020 2020 2020 3620 2020 2020            6     
│ │ │ │ +00085810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085830: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085840: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00085830: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00085840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085880: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085890: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000858a0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00085880: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00085890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000858a0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  000858b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000858c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000858d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000858e0: 7c32 3831 3237 3638 7420 7420 202d 2032  |2812768t t  - 2
│ │ │ │ -000858f0: 3635 3235 3432 7420 2020 2020 2020 2020  652542t         
│ │ │ │ +000858d0: 2020 2020 2020 2020 7c0a 7c32 3831 3237          |.|28127
│ │ │ │ +000858e0: 3638 7420 7420 202d 2032 3635 3235 3432  68t t  - 2652542
│ │ │ │ +000858f0: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │  00085900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085920: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085930: 7c20 2020 2020 2020 2035 2036 2020 2020  |        5 6    
│ │ │ │ -00085940: 2020 2020 2020 2036 2020 2020 2020 2020         6        
│ │ │ │ +00085920: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00085930: 2020 2035 2036 2020 2020 2020 2020 2020     5 6          
│ │ │ │ +00085940: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │  00085950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00085960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00085970: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00085980: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00085970: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00085980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00085990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000859a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000859b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000859c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000859d0: 0a57 6179 7320 746f 2075 7365 2074 6869  .Ways to use thi
│ │ │ │ -000859e0: 7320 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d  s method:.======
│ │ │ │ -000859f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00085a00: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7061  ==..  * *note pa
│ │ │ │ -00085a10: 7261 6d65 7472 697a 6528 4964 6561 6c29  rametrize(Ideal)
│ │ │ │ -00085a20: 3a20 7061 7261 6d65 7472 697a 655f 6c70  : parametrize_lp
│ │ │ │ -00085a30: 4964 6561 6c5f 7270 2c20 2d2d 2070 6172  Ideal_rp, -- par
│ │ │ │ -00085a40: 616d 6574 7269 7a61 7469 6f6e 206f 660a  ametrization of.
│ │ │ │ -00085a50: 2020 2020 6c69 6e65 6172 2076 6172 6965      linear varie
│ │ │ │ -00085a60: 7469 6573 2061 6e64 2068 7970 6572 7175  ties and hyperqu
│ │ │ │ -00085a70: 6164 7269 6373 0a20 202a 2022 7061 7261  adrics.  * "para
│ │ │ │ -00085a80: 6d65 7472 697a 6528 506f 6c79 6e6f 6d69  metrize(Polynomi
│ │ │ │ -00085a90: 616c 5269 6e67 2922 0a20 202a 2022 7061  alRing)".  * "pa
│ │ │ │ -00085aa0: 7261 6d65 7472 697a 6528 5175 6f74 6965  rametrize(Quotie
│ │ │ │ -00085ab0: 6e74 5269 6e67 2922 0a2d 2d2d 2d2d 2d2d  ntRing)".-------
│ │ │ │ +000859c0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320  --------+..Ways 
│ │ │ │ +000859d0: 746f 2075 7365 2074 6869 7320 6d65 7468  to use this meth
│ │ │ │ +000859e0: 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  od:.============
│ │ │ │ +000859f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +00085a00: 2a20 2a6e 6f74 6520 7061 7261 6d65 7472  * *note parametr
│ │ │ │ +00085a10: 697a 6528 4964 6561 6c29 3a20 7061 7261  ize(Ideal): para
│ │ │ │ +00085a20: 6d65 7472 697a 655f 6c70 4964 6561 6c5f  metrize_lpIdeal_
│ │ │ │ +00085a30: 7270 2c20 2d2d 2070 6172 616d 6574 7269  rp, -- parametri
│ │ │ │ +00085a40: 7a61 7469 6f6e 206f 660a 2020 2020 6c69  zation of.    li
│ │ │ │ +00085a50: 6e65 6172 2076 6172 6965 7469 6573 2061  near varieties a
│ │ │ │ +00085a60: 6e64 2068 7970 6572 7175 6164 7269 6373  nd hyperquadrics
│ │ │ │ +00085a70: 0a20 202a 2022 7061 7261 6d65 7472 697a  .  * "parametriz
│ │ │ │ +00085a80: 6528 506f 6c79 6e6f 6d69 616c 5269 6e67  e(PolynomialRing
│ │ │ │ +00085a90: 2922 0a20 202a 2022 7061 7261 6d65 7472  )".  * "parametr
│ │ │ │ +00085aa0: 697a 6528 5175 6f74 6965 6e74 5269 6e67  ize(QuotientRing
│ │ │ │ +00085ab0: 2922 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  )".-------------
│ │ │ │  00085ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00085ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00085af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00085c10: 2072 6174 696f 6e61 6c20 706f 696e 7420   rational point 
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│ │ │ │ +00085bc0: 6965 6e74 5269 6e67 5f72 702c 2050 7265  ientRing_rp, Pre
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│ │ │ │ +00085be0: 7049 6465 616c 5f72 702c 2055 703a 2054  pIdeal_rp, Up: T
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│ │ │ │ +00085c00: 6b20 6120 7261 6e64 6f6d 2072 6174 696f  k a random ratio
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│ │ │ │ +00085c30: 790a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  y.**************
│ │ │ │  00085c40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00085c50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00085c60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00085c70: 2a2a 2a2a 2a0a 0a44 6573 6372 6970 7469  *****..Descripti
│ │ │ │ -00085c80: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -00085c90: 5365 6520 706f 696e 7428 4d75 6c74 6970  See point(Multip
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│ │ │ │ -00085cc0: 756d 656e 7461 7469 6f6e 2920 616e 6420  umentation) and 
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│ │ │ │ -00085ce0: 6965 6e74 5269 6e67 293a 2070 6f69 6e74  ientRing): point
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│ │ │ │ -00085d20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -00085d30: 2022 706f 696e 7428 506f 6c79 6e6f 6d69   "point(Polynomi
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│ │ │ │ -00085d80: 696e 675f 7270 2c20 2d2d 2070 6963 6b20  ing_rp, -- pick 
│ │ │ │ -00085d90: 6120 7261 6e64 6f6d 2072 6174 696f 6e61  a random rationa
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│ │ │ │ +00085cd0: 706f 696e 7428 5175 6f74 6965 6e74 5269  point(QuotientRi
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│ │ │ │ +00085cf0: 7469 656e 7452 696e 675f 7270 2c2e 0a0a  tientRing_rp,...
│ │ │ │ +00085d00: 5761 7973 2074 6f20 7573 6520 706f 696e  Ways to use poin
│ │ │ │ +00085d10: 743a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  t:.=============
│ │ │ │ +00085d20: 3d3d 3d3d 3d0a 0a20 202a 2022 706f 696e  =====..  * "poin
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│ │ │ │ +00085d50: 706f 696e 7428 5175 6f74 6965 6e74 5269  point(QuotientRi
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│ │ │ │ +00085d80: 2c20 2d2d 2070 6963 6b20 6120 7261 6e64  , -- pick a rand
│ │ │ │ +00085d90: 6f6d 2072 6174 696f 6e61 6c20 706f 696e  om rational poin
│ │ │ │ +00085da0: 7420 6f6e 2061 2070 726f 6a65 6374 6976  t on a projectiv
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│ │ │ │ +00085dc0: 2a20 2a6e 6f74 6520 706f 696e 7428 5175  * *note point(Qu
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│ │ │ │ +00085de0: 6e74 5f6c 7051 756f 7469 656e 7452 696e  nt_lpQuotientRin
│ │ │ │ +00085df0: 675f 7270 2c20 2d2d 2070 6963 6b20 6120  g_rp, -- pick a 
│ │ │ │ +00085e00: 7261 6e64 6f6d 0a20 2020 2072 6174 696f  random.    ratio
│ │ │ │ +00085e10: 6e61 6c20 706f 696e 7420 6f6e 2061 2070  nal point on a p
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│ │ │ │ +00086040: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00086050: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00086060: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00086070: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00086080: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ -00086090: 0a20 202a 2046 756e 6374 696f 6e3a 202a  .  * Function: *
│ │ │ │ -000860a0: 6e6f 7465 2070 6f69 6e74 3a20 706f 696e  note point: poin
│ │ │ │ -000860b0: 742c 0a20 202a 2055 7361 6765 3a20 0a20  t,.  * Usage: . 
│ │ │ │ -000860c0: 2020 2020 2020 2070 6f69 6e74 2052 0a20         point R. 
│ │ │ │ -000860d0: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -000860e0: 202a 2052 2c20 6120 2a6e 6f74 6520 7175   * R, a *note qu
│ │ │ │ -000860f0: 6f74 6965 6e74 2072 696e 673a 2028 4d61  otient ring: (Ma
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│ │ │ │ -00086130: 2020 636f 6f72 6469 6e61 7465 2072 696e    coordinate rin
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│ │ │ │ -00086170: 206f 7665 7220 610a 2020 2020 2020 2020   over a.        
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│ │ │ │ -000861d0: 6e20 6964 6561 6c20 696e 2052 2064 6566  n ideal in R def
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│ │ │ │ -00086f40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
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│ │ │ │ +00086e90: 202a 6e6f 7465 2072 616e 646f 6d4b 5261   *note randomKRa
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│ │ │ │ +00086f30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00086f40: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2270 6f69  ======..  * "poi
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│ │ │ │ +00087160: 6174 696f 6e61 6c20 6d61 7020 6265 7477  ational map betw
│ │ │ │ +00087170: 6565 6e20 7072 6f6a 6563 7469 7665 2076  een projective v
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│ │ │ │  00087190: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000871a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000871b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000871c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000871d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000871e0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ -000871f0: 3a20 0a20 2020 2020 2020 2070 726f 6a65  : .        proje
│ │ │ │ -00087200: 6374 6976 6544 6567 7265 6573 2070 6869  ctiveDegrees phi
│ │ │ │ -00087210: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ -00087220: 2020 202a 2070 6869 2c20 6120 2a6e 6f74     * phi, a *not
│ │ │ │ -00087230: 6520 7269 6e67 206d 6170 3a20 284d 6163  e ring map: (Mac
│ │ │ │ -00087240: 6175 6c61 7932 446f 6329 5269 6e67 4d61  aulay2Doc)RingMa
│ │ │ │ -00087250: 702c 2c20 7768 6963 6820 7265 7072 6573  p,, which repres
│ │ │ │ -00087260: 656e 7473 2061 0a20 2020 2020 2020 2072  ents a.        r
│ │ │ │ -00087270: 6174 696f 6e61 6c20 6d61 7020 245c 5068  ational map $\Ph
│ │ │ │ -00087280: 6924 2062 6574 7765 656e 2070 726f 6a65  i$ between proje
│ │ │ │ -00087290: 6374 6976 6520 7661 7269 6574 6965 730a  ctive varieties.
│ │ │ │ -000872a0: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e    * *note Option
│ │ │ │ -000872b0: 616c 2069 6e70 7574 733a 2028 4d61 6361  al inputs: (Maca
│ │ │ │ -000872c0: 756c 6179 3244 6f63 2975 7369 6e67 2066  ulay2Doc)using f
│ │ │ │ -000872d0: 756e 6374 696f 6e73 2077 6974 6820 6f70  unctions with op
│ │ │ │ -000872e0: 7469 6f6e 616c 2069 6e70 7574 732c 3a0a  tional inputs,:.
│ │ │ │ -000872f0: 2020 2020 2020 2a20 2a6e 6f74 6520 426c        * *note Bl
│ │ │ │ -00087300: 6f77 5570 5374 7261 7465 6779 3a20 426c  owUpStrategy: Bl
│ │ │ │ -00087310: 6f77 5570 5374 7261 7465 6779 2c20 3d3e  owUpStrategy, =>
│ │ │ │ -00087320: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ -00087330: 6c75 650a 2020 2020 2020 2020 2245 6c69  lue.        "Eli
│ │ │ │ -00087340: 6d69 6e61 7465 222c 0a20 2020 2020 202a  minate",.      *
│ │ │ │ -00087350: 202a 6e6f 7465 2043 6572 7469 6679 3a20   *note Certify: 
│ │ │ │ -00087360: 4365 7274 6966 792c 203d 3e20 2e2e 2e2c  Certify, => ...,
│ │ │ │ -00087370: 2064 6566 6175 6c74 2076 616c 7565 2066   default value f
│ │ │ │ -00087380: 616c 7365 2c20 7768 6574 6865 7220 746f  alse, whether to
│ │ │ │ -00087390: 2065 6e73 7572 650a 2020 2020 2020 2020   ensure.        
│ │ │ │ -000873a0: 636f 7272 6563 746e 6573 7320 6f66 206f  correctness of o
│ │ │ │ -000873b0: 7574 7075 740a 2020 2020 2020 2a20 2a6e  utput.      * *n
│ │ │ │ -000873c0: 6f74 6520 4e75 6d44 6567 7265 6573 3a20  ote NumDegrees: 
│ │ │ │ -000873d0: 4e75 6d44 6567 7265 6573 2c20 3d3e 202e  NumDegrees, => .
│ │ │ │ -000873e0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ -000873f0: 6520 696e 6669 6e69 7479 2c20 0a20 2020  e infinity, .   
│ │ │ │ -00087400: 2020 202a 202a 6e6f 7465 2056 6572 626f     * *note Verbo
│ │ │ │ -00087410: 7365 3a20 696e 7665 7273 654d 6170 5f6c  se: inverseMap_l
│ │ │ │ -00087420: 705f 7064 5f70 645f 7064 5f63 6d56 6572  p_pd_pd_pd_cmVer
│ │ │ │ -00087430: 626f 7365 3d3e 5f70 645f 7064 5f70 645f  bose=>_pd_pd_pd_
│ │ │ │ -00087440: 7270 2c20 3d3e 202e 2e2e 2c0a 2020 2020  rp, => ...,.    
│ │ │ │ -00087450: 2020 2020 6465 6661 756c 7420 7661 6c75      default valu
│ │ │ │ -00087460: 6520 7472 7565 2c0a 2020 2a20 4f75 7470  e true,.  * Outp
│ │ │ │ -00087470: 7574 733a 0a20 2020 2020 202a 2061 202a  uts:.      * a *
│ │ │ │ -00087480: 6e6f 7465 206c 6973 743a 2028 4d61 6361  note list: (Maca
│ │ │ │ -00087490: 756c 6179 3244 6f63 294c 6973 742c 2c20  ulay2Doc)List,, 
│ │ │ │ -000874a0: 7468 6520 6c69 7374 206f 6620 7468 6520  the list of the 
│ │ │ │ -000874b0: 7072 6f6a 6563 7469 7665 2064 6567 7265  projective degre
│ │ │ │ -000874c0: 6573 0a20 2020 2020 2020 206f 6620 245c  es.        of $\
│ │ │ │ -000874d0: 5068 6924 0a0a 4465 7363 7269 7074 696f  Phi$..Descriptio
│ │ │ │ -000874e0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a4c  n.===========..L
│ │ │ │ -000874f0: 6574 2024 5c70 6869 3a4b 5b79 5f30 2c5c  et $\phi:K[y_0,\
│ │ │ │ -00087500: 6c64 6f74 732c 795f 6d5d 2f4a 205c 746f  ldots,y_m]/J \to
│ │ │ │ -00087510: 204b 5b78 5f30 2c5c 6c64 6f74 732c 785f   K[x_0,\ldots,x_
│ │ │ │ -00087520: 6e5d 2f49 2420 6265 2061 2072 696e 6720  n]/I$ be a ring 
│ │ │ │ -00087530: 6d61 700a 7265 7072 6573 656e 7469 6e67  map.representing
│ │ │ │ -00087540: 2061 2072 6174 696f 6e61 6c20 6d61 7020   a rational map 
│ │ │ │ -00087550: 245c 5068 693a 2056 2849 2920 5c73 7562  $\Phi: V(I) \sub
│ │ │ │ -00087560: 7365 7465 710a 5c6d 6174 6862 627b 507d  seteq.\mathbb{P}
│ │ │ │ -00087570: 5e6e 3d50 726f 6a28 4b5b 785f 302c 5c6c  ^n=Proj(K[x_0,\l
│ │ │ │ -00087580: 646f 7473 2c78 5f6e 5d29 205c 6461 7368  dots,x_n]) \dash
│ │ │ │ -00087590: 7269 6768 7461 7272 6f77 2056 284a 2920  rightarrow V(J) 
│ │ │ │ -000875a0: 5c73 7562 7365 7465 710a 5c6d 6174 6862  \subseteq.\mathb
│ │ │ │ -000875b0: 627b 507d 5e6d 3d50 726f 6a28 4b5b 795f  b{P}^m=Proj(K[y_
│ │ │ │ -000875c0: 302c 5c6c 646f 7473 2c79 5f6d 5d29 242e  0,\ldots,y_m])$.
│ │ │ │ -000875d0: 2054 6865 2024 6924 2d74 6820 7072 6f6a   The $i$-th proj
│ │ │ │ -000875e0: 6563 7469 7665 2064 6567 7265 6520 6f66  ective degree of
│ │ │ │ -000875f0: 2024 5c50 6869 240a 6973 2064 6566 696e   $\Phi$.is defin
│ │ │ │ -00087600: 6564 2069 6e20 7465 726d 7320 6f66 2064  ed in terms of d
│ │ │ │ -00087610: 696d 656e 7369 6f6e 2061 6e64 2064 6567  imension and deg
│ │ │ │ -00087620: 7265 6520 6f66 2074 6865 2063 6c6f 7375  ree of the closu
│ │ │ │ -00087630: 7265 206f 6620 245c 5068 695e 7b2d 317d  re of $\Phi^{-1}
│ │ │ │ -00087640: 284c 2924 2c0a 7768 6572 6520 244c 2420  (L)$,.where $L$ 
│ │ │ │ -00087650: 6973 2061 2067 656e 6572 616c 206c 696e  is a general lin
│ │ │ │ -00087660: 6561 7220 7375 6273 7061 6365 206f 6620  ear subspace of 
│ │ │ │ -00087670: 245c 6d61 7468 6262 7b50 7d5e 6d24 206f  $\mathbb{P}^m$ o
│ │ │ │ -00087680: 6620 6120 6365 7274 6169 6e0a 6469 6d65  f a certain.dime
│ │ │ │ -00087690: 6e73 696f 6e3b 2066 6f72 2074 6865 2070  nsion; for the p
│ │ │ │ -000876a0: 7265 6369 7365 2064 6566 696e 6974 696f  recise definitio
│ │ │ │ -000876b0: 6e2c 2073 6565 2048 6172 7269 7327 7320  n, see Harris's 
│ │ │ │ -000876c0: 626f 6f6b 2028 416c 6765 6272 6169 6320  book (Algebraic 
│ │ │ │ -000876d0: 6765 6f6d 6574 7279 3a20 410a 6669 7273  geometry: A.firs
│ │ │ │ -000876e0: 7420 636f 7572 7365 202d 2056 6f6c 2e20  t course - Vol. 
│ │ │ │ -000876f0: 3133 3320 6f66 2047 7261 642e 2054 6578  133 of Grad. Tex
│ │ │ │ -00087700: 7473 2069 6e20 4d61 7468 2e2c 2070 2e20  ts in Math., p. 
│ │ │ │ -00087710: 3234 3029 2e20 4966 2024 5c50 6869 2420  240). If $\Phi$ 
│ │ │ │ -00087720: 6973 2064 6566 696e 6564 0a62 7920 656c  is defined.by el
│ │ │ │ -00087730: 656d 656e 7473 2024 465f 3028 785f 302c  ements $F_0(x_0,
│ │ │ │ -00087740: 5c6c 646f 7473 2c78 5f6e 292c 5c6c 646f  \ldots,x_n),\ldo
│ │ │ │ -00087750: 7473 2c46 5f6d 2878 5f30 2c5c 6c64 6f74  ts,F_m(x_0,\ldot
│ │ │ │ -00087760: 732c 785f 6e29 2420 616e 6420 2449 5f4c  s,x_n)$ and $I_L
│ │ │ │ -00087770: 2420 6465 6e6f 7465 730a 7468 6520 6964  $ denotes.the id
│ │ │ │ -00087780: 6561 6c20 6f66 2074 6865 2073 7562 7370  eal of the subsp
│ │ │ │ -00087790: 6163 6520 244c 5c73 7562 7365 7465 7120  ace $L\subseteq 
│ │ │ │ -000877a0: 5c6d 6174 6862 627b 507d 5e6d 242c 2074  \mathbb{P}^m$, t
│ │ │ │ -000877b0: 6865 6e20 7468 6520 6964 6561 6c20 6f66  hen the ideal of
│ │ │ │ -000877c0: 2074 6865 0a63 6c6f 7375 7265 206f 6620   the.closure of 
│ │ │ │ -000877d0: 245c 5068 695e 7b2d 317d 284c 2920 2420  $\Phi^{-1}(L) $ 
│ │ │ │ -000877e0: 6973 206e 6f74 6869 6e67 2062 7574 2074  is nothing but t
│ │ │ │ -000877f0: 6865 2073 6174 7572 6174 696f 6e20 6f66  he saturation of
│ │ │ │ -00087800: 2074 6865 2069 6465 616c 0a24 285c 7068   the ideal.$(\ph
│ │ │ │ -00087810: 6928 495f 4c29 2924 2062 7920 2428 465f  i(I_L))$ by $(F_
│ │ │ │ -00087820: 302c 2e2e 2e2e 2c46 5f6d 2924 2069 6e20  0,....,F_m)$ in 
│ │ │ │ -00087830: 7468 6520 7269 6e67 2024 4b5b 785f 302c  the ring $K[x_0,
│ │ │ │ -00087840: 5c6c 646f 7473 2c78 5f6e 5d2f 4924 2e20  \ldots,x_n]/I$. 
│ │ │ │ -00087850: 536f 2c0a 7265 706c 6163 696e 6720 696e  So,.replacing in
│ │ │ │ -00087860: 2074 6865 2064 6566 696e 6974 696f 6e2c   the definition,
│ │ │ │ -00087870: 2067 656e 6572 616c 206c 696e 6561 7220   general linear 
│ │ │ │ -00087880: 7375 6273 7061 6365 2062 7920 7261 6e64  subspace by rand
│ │ │ │ -00087890: 6f6d 206c 696e 6561 7220 7375 6273 7061  om linear subspa
│ │ │ │ -000878a0: 6365 2c0a 7765 2067 6574 2061 2070 726f  ce,.we get a pro
│ │ │ │ -000878b0: 6261 6269 6c69 7374 6963 2061 6c67 6f72  babilistic algor
│ │ │ │ -000878c0: 6974 686d 2074 6f20 636f 6d70 7574 6520  ithm to compute 
│ │ │ │ -000878d0: 616c 6c20 7072 6f6a 6563 7469 7665 2064  all projective d
│ │ │ │ -000878e0: 6567 7265 6573 2e0a 4675 7274 6865 726d  egrees..Furtherm
│ │ │ │ -000878f0: 6f72 652c 2077 6520 6361 6e20 636f 6e73  ore, we can cons
│ │ │ │ -00087900: 6964 6572 6162 6c79 2073 7065 6564 2075  iderably speed u
│ │ │ │ -00087910: 7020 7468 6973 2061 6c67 6f72 6974 686d  p this algorithm
│ │ │ │ -00087920: 2062 7920 7461 6b69 6e67 2069 6e74 6f20   by taking into 
│ │ │ │ -00087930: 6163 636f 756e 740a 7477 6f20 7369 6d70  account.two simp
│ │ │ │ -00087940: 6c65 2072 656d 6172 6b73 3a20 3129 2074  le remarks: 1) t
│ │ │ │ -00087950: 6865 2073 6174 7572 6174 696f 6e20 2428  he saturation $(
│ │ │ │ -00087960: 5c70 6869 2849 5f4c 2929 3a7b 2846 5f30  \phi(I_L)):{(F_0
│ │ │ │ -00087970: 2c5c 6c64 6f74 732c 465f 6d29 7d5e 7b5c  ,\ldots,F_m)}^{\
│ │ │ │ -00087980: 696e 6674 797d 240a 6973 2074 6865 2073  infty}$.is the s
│ │ │ │ -00087990: 616d 6520 6173 2024 285c 7068 6928 495f  ame as $(\phi(I_
│ │ │ │ -000879a0: 4c29 293a 7b28 5c6c 616d 6264 615f 3020  L)):{(\lambda_0 
│ │ │ │ -000879b0: 465f 302b 5c63 646f 7473 2b5c 6c61 6d62  F_0+\cdots+\lamb
│ │ │ │ -000879c0: 6461 5f6d 2046 5f6d 297d 5e7b 5c69 6e66  da_m F_m)}^{\inf
│ │ │ │ -000879d0: 7479 7d24 2c0a 7768 6572 6520 245c 6c61  ty}$,.where $\la
│ │ │ │ -000879e0: 6d62 6461 5f30 2c5c 6c64 6f74 732c 5c6c  mbda_0,\ldots,\l
│ │ │ │ -000879f0: 616d 6264 615f 6d5c 696e 5c6d 6174 6862  ambda_m\in\mathb
│ │ │ │ -00087a00: 627b 4b7d 2420 6172 6520 6765 6e65 7261  b{K}$ are genera
│ │ │ │ -00087a10: 6c20 7363 616c 6172 733b 2032 2920 7468  l scalars; 2) th
│ │ │ │ -00087a20: 650a 2469 242d 7468 2070 726f 6a65 6374  e.$i$-th project
│ │ │ │ -00087a30: 6976 6520 6465 6772 6565 206f 6620 245c  ive degree of $\
│ │ │ │ -00087a40: 5068 6924 2063 6f69 6e63 6964 6573 2077  Phi$ coincides w
│ │ │ │ -00087a50: 6974 6820 7468 6520 2428 692d 3129 242d  ith the $(i-1)$-
│ │ │ │ -00087a60: 7468 2070 726f 6a65 6374 6976 650a 6465  th projective.de
│ │ │ │ -00087a70: 6772 6565 206f 6620 7468 6520 7265 7374  gree of the rest
│ │ │ │ -00087a80: 7269 6374 696f 6e20 6f66 2024 5c50 6869  riction of $\Phi
│ │ │ │ -00087a90: 2420 746f 2061 2067 656e 6572 616c 2068  $ to a general h
│ │ │ │ -00087aa0: 7970 6572 706c 616e 6520 7365 6374 696f  yperplane sectio
│ │ │ │ -00087ab0: 6e20 6f66 2024 5824 2028 7365 650a 6c6f  n of $X$ (see.lo
│ │ │ │ -00087ac0: 632e 2063 6974 2e29 2e20 5468 6973 2069  c. cit.). This i
│ │ │ │ -00087ad0: 7320 7768 6174 2074 6865 206d 6574 686f  s what the metho
│ │ │ │ -00087ae0: 6420 7573 6573 2069 6620 2a6e 6f74 6520  d uses if *note 
│ │ │ │ -00087af0: 4365 7274 6966 793a 2043 6572 7469 6679  Certify: Certify
│ │ │ │ -00087b00: 2c20 6973 2073 6574 2074 6f0a 6661 6c73  , is set to.fals
│ │ │ │ -00087b10: 652e 2049 6620 696e 7374 6561 6420 2a6e  e. If instead *n
│ │ │ │ -00087b20: 6f74 6520 4365 7274 6966 793a 2043 6572  ote Certify: Cer
│ │ │ │ -00087b30: 7469 6679 2c20 6973 2073 6574 2074 6f20  tify, is set to 
│ │ │ │ -00087b40: 7472 7565 2c20 7468 656e 2074 6865 206d  true, then the m
│ │ │ │ -00087b50: 6574 686f 640a 7369 6d70 6c79 2063 6f6d  ethod.simply com
│ │ │ │ -00087b60: 7075 7465 7320 7468 6520 2a6e 6f74 6520  putes the *note 
│ │ │ │ -00087b70: 6d75 6c74 6964 6567 7265 653a 2028 4d61  multidegree: (Ma
│ │ │ │ -00087b80: 6361 756c 6179 3244 6f63 296d 756c 7469  caulay2Doc)multi
│ │ │ │ -00087b90: 6465 6772 6565 2c20 6f66 2074 6865 202a  degree, of the *
│ │ │ │ -00087ba0: 6e6f 7465 0a67 7261 7068 3a20 6772 6170  note.graph: grap
│ │ │ │ -00087bb0: 682c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  h,...+----------
│ │ │ │ +000871d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +000871e0: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +000871f0: 2020 2020 2070 726f 6a65 6374 6976 6544       projectiveD
│ │ │ │ +00087200: 6567 7265 6573 2070 6869 0a20 202a 2049  egrees phi.  * I
│ │ │ │ +00087210: 6e70 7574 733a 0a20 2020 2020 202a 2070  nputs:.      * p
│ │ │ │ +00087220: 6869 2c20 6120 2a6e 6f74 6520 7269 6e67  hi, a *note ring
│ │ │ │ +00087230: 206d 6170 3a20 284d 6163 6175 6c61 7932   map: (Macaulay2
│ │ │ │ +00087240: 446f 6329 5269 6e67 4d61 702c 2c20 7768  Doc)RingMap,, wh
│ │ │ │ +00087250: 6963 6820 7265 7072 6573 656e 7473 2061  ich represents a
│ │ │ │ +00087260: 0a20 2020 2020 2020 2072 6174 696f 6e61  .        rationa
│ │ │ │ +00087270: 6c20 6d61 7020 245c 5068 6924 2062 6574  l map $\Phi$ bet
│ │ │ │ +00087280: 7765 656e 2070 726f 6a65 6374 6976 6520  ween projective 
│ │ │ │ +00087290: 7661 7269 6574 6965 730a 2020 2a20 2a6e  varieties.  * *n
│ │ │ │ +000872a0: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70  ote Optional inp
│ │ │ │ +000872b0: 7574 733a 2028 4d61 6361 756c 6179 3244  uts: (Macaulay2D
│ │ │ │ +000872c0: 6f63 2975 7369 6e67 2066 756e 6374 696f  oc)using functio
│ │ │ │ +000872d0: 6e73 2077 6974 6820 6f70 7469 6f6e 616c  ns with optional
│ │ │ │ +000872e0: 2069 6e70 7574 732c 3a0a 2020 2020 2020   inputs,:.      
│ │ │ │ +000872f0: 2a20 2a6e 6f74 6520 426c 6f77 5570 5374  * *note BlowUpSt
│ │ │ │ +00087300: 7261 7465 6779 3a20 426c 6f77 5570 5374  rategy: BlowUpSt
│ │ │ │ +00087310: 7261 7465 6779 2c20 3d3e 202e 2e2e 2c20  rategy, => ..., 
│ │ │ │ +00087320: 6465 6661 756c 7420 7661 6c75 650a 2020  default value.  
│ │ │ │ +00087330: 2020 2020 2020 2245 6c69 6d69 6e61 7465        "Eliminate
│ │ │ │ +00087340: 222c 0a20 2020 2020 202a 202a 6e6f 7465  ",.      * *note
│ │ │ │ +00087350: 2043 6572 7469 6679 3a20 4365 7274 6966   Certify: Certif
│ │ │ │ +00087360: 792c 203d 3e20 2e2e 2e2c 2064 6566 6175  y, => ..., defau
│ │ │ │ +00087370: 6c74 2076 616c 7565 2066 616c 7365 2c20  lt value false, 
│ │ │ │ +00087380: 7768 6574 6865 7220 746f 2065 6e73 7572  whether to ensur
│ │ │ │ +00087390: 650a 2020 2020 2020 2020 636f 7272 6563  e.        correc
│ │ │ │ +000873a0: 746e 6573 7320 6f66 206f 7574 7075 740a  tness of output.
│ │ │ │ +000873b0: 2020 2020 2020 2a20 2a6e 6f74 6520 4e75        * *note Nu
│ │ │ │ +000873c0: 6d44 6567 7265 6573 3a20 4e75 6d44 6567  mDegrees: NumDeg
│ │ │ │ +000873d0: 7265 6573 2c20 3d3e 202e 2e2e 2c20 6465  rees, => ..., de
│ │ │ │ +000873e0: 6661 756c 7420 7661 6c75 6520 696e 6669  fault value infi
│ │ │ │ +000873f0: 6e69 7479 2c20 0a20 2020 2020 202a 202a  nity, .      * *
│ │ │ │ +00087400: 6e6f 7465 2056 6572 626f 7365 3a20 696e  note Verbose: in
│ │ │ │ +00087410: 7665 7273 654d 6170 5f6c 705f 7064 5f70  verseMap_lp_pd_p
│ │ │ │ +00087420: 645f 7064 5f63 6d56 6572 626f 7365 3d3e  d_pd_cmVerbose=>
│ │ │ │ +00087430: 5f70 645f 7064 5f70 645f 7270 2c20 3d3e  _pd_pd_pd_rp, =>
│ │ │ │ +00087440: 202e 2e2e 2c0a 2020 2020 2020 2020 6465   ...,.        de
│ │ │ │ +00087450: 6661 756c 7420 7661 6c75 6520 7472 7565  fault value true
│ │ │ │ +00087460: 2c0a 2020 2a20 4f75 7470 7574 733a 0a20  ,.  * Outputs:. 
│ │ │ │ +00087470: 2020 2020 202a 2061 202a 6e6f 7465 206c       * a *note l
│ │ │ │ +00087480: 6973 743a 2028 4d61 6361 756c 6179 3244  ist: (Macaulay2D
│ │ │ │ +00087490: 6f63 294c 6973 742c 2c20 7468 6520 6c69  oc)List,, the li
│ │ │ │ +000874a0: 7374 206f 6620 7468 6520 7072 6f6a 6563  st of the projec
│ │ │ │ +000874b0: 7469 7665 2064 6567 7265 6573 0a20 2020  tive degrees.   
│ │ │ │ +000874c0: 2020 2020 206f 6620 245c 5068 6924 0a0a       of $\Phi$..
│ │ │ │ +000874d0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +000874e0: 3d3d 3d3d 3d3d 3d0a 0a4c 6574 2024 5c70  =======..Let $\p
│ │ │ │ +000874f0: 6869 3a4b 5b79 5f30 2c5c 6c64 6f74 732c  hi:K[y_0,\ldots,
│ │ │ │ +00087500: 795f 6d5d 2f4a 205c 746f 204b 5b78 5f30  y_m]/J \to K[x_0
│ │ │ │ +00087510: 2c5c 6c64 6f74 732c 785f 6e5d 2f49 2420  ,\ldots,x_n]/I$ 
│ │ │ │ +00087520: 6265 2061 2072 696e 6720 6d61 700a 7265  be a ring map.re
│ │ │ │ +00087530: 7072 6573 656e 7469 6e67 2061 2072 6174  presenting a rat
│ │ │ │ +00087540: 696f 6e61 6c20 6d61 7020 245c 5068 693a  ional map $\Phi:
│ │ │ │ +00087550: 2056 2849 2920 5c73 7562 7365 7465 710a   V(I) \subseteq.
│ │ │ │ +00087560: 5c6d 6174 6862 627b 507d 5e6e 3d50 726f  \mathbb{P}^n=Pro
│ │ │ │ +00087570: 6a28 4b5b 785f 302c 5c6c 646f 7473 2c78  j(K[x_0,\ldots,x
│ │ │ │ +00087580: 5f6e 5d29 205c 6461 7368 7269 6768 7461  _n]) \dashrighta
│ │ │ │ +00087590: 7272 6f77 2056 284a 2920 5c73 7562 7365  rrow V(J) \subse
│ │ │ │ +000875a0: 7465 710a 5c6d 6174 6862 627b 507d 5e6d  teq.\mathbb{P}^m
│ │ │ │ +000875b0: 3d50 726f 6a28 4b5b 795f 302c 5c6c 646f  =Proj(K[y_0,\ldo
│ │ │ │ +000875c0: 7473 2c79 5f6d 5d29 242e 2054 6865 2024  ts,y_m])$. The $
│ │ │ │ +000875d0: 6924 2d74 6820 7072 6f6a 6563 7469 7665  i$-th projective
│ │ │ │ +000875e0: 2064 6567 7265 6520 6f66 2024 5c50 6869   degree of $\Phi
│ │ │ │ +000875f0: 240a 6973 2064 6566 696e 6564 2069 6e20  $.is defined in 
│ │ │ │ +00087600: 7465 726d 7320 6f66 2064 696d 656e 7369  terms of dimensi
│ │ │ │ +00087610: 6f6e 2061 6e64 2064 6567 7265 6520 6f66  on and degree of
│ │ │ │ +00087620: 2074 6865 2063 6c6f 7375 7265 206f 6620   the closure of 
│ │ │ │ +00087630: 245c 5068 695e 7b2d 317d 284c 2924 2c0a  $\Phi^{-1}(L)$,.
│ │ │ │ +00087640: 7768 6572 6520 244c 2420 6973 2061 2067  where $L$ is a g
│ │ │ │ +00087650: 656e 6572 616c 206c 696e 6561 7220 7375  eneral linear su
│ │ │ │ +00087660: 6273 7061 6365 206f 6620 245c 6d61 7468  bspace of $\math
│ │ │ │ +00087670: 6262 7b50 7d5e 6d24 206f 6620 6120 6365  bb{P}^m$ of a ce
│ │ │ │ +00087680: 7274 6169 6e0a 6469 6d65 6e73 696f 6e3b  rtain.dimension;
│ │ │ │ +00087690: 2066 6f72 2074 6865 2070 7265 6369 7365   for the precise
│ │ │ │ +000876a0: 2064 6566 696e 6974 696f 6e2c 2073 6565   definition, see
│ │ │ │ +000876b0: 2048 6172 7269 7327 7320 626f 6f6b 2028   Harris's book (
│ │ │ │ +000876c0: 416c 6765 6272 6169 6320 6765 6f6d 6574  Algebraic geomet
│ │ │ │ +000876d0: 7279 3a20 410a 6669 7273 7420 636f 7572  ry: A.first cour
│ │ │ │ +000876e0: 7365 202d 2056 6f6c 2e20 3133 3320 6f66  se - Vol. 133 of
│ │ │ │ +000876f0: 2047 7261 642e 2054 6578 7473 2069 6e20   Grad. Texts in 
│ │ │ │ +00087700: 4d61 7468 2e2c 2070 2e20 3234 3029 2e20  Math., p. 240). 
│ │ │ │ +00087710: 4966 2024 5c50 6869 2420 6973 2064 6566  If $\Phi$ is def
│ │ │ │ +00087720: 696e 6564 0a62 7920 656c 656d 656e 7473  ined.by elements
│ │ │ │ +00087730: 2024 465f 3028 785f 302c 5c6c 646f 7473   $F_0(x_0,\ldots
│ │ │ │ +00087740: 2c78 5f6e 292c 5c6c 646f 7473 2c46 5f6d  ,x_n),\ldots,F_m
│ │ │ │ +00087750: 2878 5f30 2c5c 6c64 6f74 732c 785f 6e29  (x_0,\ldots,x_n)
│ │ │ │ +00087760: 2420 616e 6420 2449 5f4c 2420 6465 6e6f  $ and $I_L$ deno
│ │ │ │ +00087770: 7465 730a 7468 6520 6964 6561 6c20 6f66  tes.the ideal of
│ │ │ │ +00087780: 2074 6865 2073 7562 7370 6163 6520 244c   the subspace $L
│ │ │ │ +00087790: 5c73 7562 7365 7465 7120 5c6d 6174 6862  \subseteq \mathb
│ │ │ │ +000877a0: 627b 507d 5e6d 242c 2074 6865 6e20 7468  b{P}^m$, then th
│ │ │ │ +000877b0: 6520 6964 6561 6c20 6f66 2074 6865 0a63  e ideal of the.c
│ │ │ │ +000877c0: 6c6f 7375 7265 206f 6620 245c 5068 695e  losure of $\Phi^
│ │ │ │ +000877d0: 7b2d 317d 284c 2920 2420 6973 206e 6f74  {-1}(L) $ is not
│ │ │ │ +000877e0: 6869 6e67 2062 7574 2074 6865 2073 6174  hing but the sat
│ │ │ │ +000877f0: 7572 6174 696f 6e20 6f66 2074 6865 2069  uration of the i
│ │ │ │ +00087800: 6465 616c 0a24 285c 7068 6928 495f 4c29  deal.$(\phi(I_L)
│ │ │ │ +00087810: 2924 2062 7920 2428 465f 302c 2e2e 2e2e  )$ by $(F_0,....
│ │ │ │ +00087820: 2c46 5f6d 2924 2069 6e20 7468 6520 7269  ,F_m)$ in the ri
│ │ │ │ +00087830: 6e67 2024 4b5b 785f 302c 5c6c 646f 7473  ng $K[x_0,\ldots
│ │ │ │ +00087840: 2c78 5f6e 5d2f 4924 2e20 536f 2c0a 7265  ,x_n]/I$. So,.re
│ │ │ │ +00087850: 706c 6163 696e 6720 696e 2074 6865 2064  placing in the d
│ │ │ │ +00087860: 6566 696e 6974 696f 6e2c 2067 656e 6572  efinition, gener
│ │ │ │ +00087870: 616c 206c 696e 6561 7220 7375 6273 7061  al linear subspa
│ │ │ │ +00087880: 6365 2062 7920 7261 6e64 6f6d 206c 696e  ce by random lin
│ │ │ │ +00087890: 6561 7220 7375 6273 7061 6365 2c0a 7765  ear subspace,.we
│ │ │ │ +000878a0: 2067 6574 2061 2070 726f 6261 6269 6c69   get a probabili
│ │ │ │ +000878b0: 7374 6963 2061 6c67 6f72 6974 686d 2074  stic algorithm t
│ │ │ │ +000878c0: 6f20 636f 6d70 7574 6520 616c 6c20 7072  o compute all pr
│ │ │ │ +000878d0: 6f6a 6563 7469 7665 2064 6567 7265 6573  ojective degrees
│ │ │ │ +000878e0: 2e0a 4675 7274 6865 726d 6f72 652c 2077  ..Furthermore, w
│ │ │ │ +000878f0: 6520 6361 6e20 636f 6e73 6964 6572 6162  e can considerab
│ │ │ │ +00087900: 6c79 2073 7065 6564 2075 7020 7468 6973  ly speed up this
│ │ │ │ +00087910: 2061 6c67 6f72 6974 686d 2062 7920 7461   algorithm by ta
│ │ │ │ +00087920: 6b69 6e67 2069 6e74 6f20 6163 636f 756e  king into accoun
│ │ │ │ +00087930: 740a 7477 6f20 7369 6d70 6c65 2072 656d  t.two simple rem
│ │ │ │ +00087940: 6172 6b73 3a20 3129 2074 6865 2073 6174  arks: 1) the sat
│ │ │ │ +00087950: 7572 6174 696f 6e20 2428 5c70 6869 2849  uration $(\phi(I
│ │ │ │ +00087960: 5f4c 2929 3a7b 2846 5f30 2c5c 6c64 6f74  _L)):{(F_0,\ldot
│ │ │ │ +00087970: 732c 465f 6d29 7d5e 7b5c 696e 6674 797d  s,F_m)}^{\infty}
│ │ │ │ +00087980: 240a 6973 2074 6865 2073 616d 6520 6173  $.is the same as
│ │ │ │ +00087990: 2024 285c 7068 6928 495f 4c29 293a 7b28   $(\phi(I_L)):{(
│ │ │ │ +000879a0: 5c6c 616d 6264 615f 3020 465f 302b 5c63  \lambda_0 F_0+\c
│ │ │ │ +000879b0: 646f 7473 2b5c 6c61 6d62 6461 5f6d 2046  dots+\lambda_m F
│ │ │ │ +000879c0: 5f6d 297d 5e7b 5c69 6e66 7479 7d24 2c0a  _m)}^{\infty}$,.
│ │ │ │ +000879d0: 7768 6572 6520 245c 6c61 6d62 6461 5f30  where $\lambda_0
│ │ │ │ +000879e0: 2c5c 6c64 6f74 732c 5c6c 616d 6264 615f  ,\ldots,\lambda_
│ │ │ │ +000879f0: 6d5c 696e 5c6d 6174 6862 627b 4b7d 2420  m\in\mathbb{K}$ 
│ │ │ │ +00087a00: 6172 6520 6765 6e65 7261 6c20 7363 616c  are general scal
│ │ │ │ +00087a10: 6172 733b 2032 2920 7468 650a 2469 242d  ars; 2) the.$i$-
│ │ │ │ +00087a20: 7468 2070 726f 6a65 6374 6976 6520 6465  th projective de
│ │ │ │ +00087a30: 6772 6565 206f 6620 245c 5068 6924 2063  gree of $\Phi$ c
│ │ │ │ +00087a40: 6f69 6e63 6964 6573 2077 6974 6820 7468  oincides with th
│ │ │ │ +00087a50: 6520 2428 692d 3129 242d 7468 2070 726f  e $(i-1)$-th pro
│ │ │ │ +00087a60: 6a65 6374 6976 650a 6465 6772 6565 206f  jective.degree o
│ │ │ │ +00087a70: 6620 7468 6520 7265 7374 7269 6374 696f  f the restrictio
│ │ │ │ +00087a80: 6e20 6f66 2024 5c50 6869 2420 746f 2061  n of $\Phi$ to a
│ │ │ │ +00087a90: 2067 656e 6572 616c 2068 7970 6572 706c   general hyperpl
│ │ │ │ +00087aa0: 616e 6520 7365 6374 696f 6e20 6f66 2024  ane section of $
│ │ │ │ +00087ab0: 5824 2028 7365 650a 6c6f 632e 2063 6974  X$ (see.loc. cit
│ │ │ │ +00087ac0: 2e29 2e20 5468 6973 2069 7320 7768 6174  .). This is what
│ │ │ │ +00087ad0: 2074 6865 206d 6574 686f 6420 7573 6573   the method uses
│ │ │ │ +00087ae0: 2069 6620 2a6e 6f74 6520 4365 7274 6966   if *note Certif
│ │ │ │ +00087af0: 793a 2043 6572 7469 6679 2c20 6973 2073  y: Certify, is s
│ │ │ │ +00087b00: 6574 2074 6f0a 6661 6c73 652e 2049 6620  et to.false. If 
│ │ │ │ +00087b10: 696e 7374 6561 6420 2a6e 6f74 6520 4365  instead *note Ce
│ │ │ │ +00087b20: 7274 6966 793a 2043 6572 7469 6679 2c20  rtify: Certify, 
│ │ │ │ +00087b30: 6973 2073 6574 2074 6f20 7472 7565 2c20  is set to true, 
│ │ │ │ +00087b40: 7468 656e 2074 6865 206d 6574 686f 640a  then the method.
│ │ │ │ +00087b50: 7369 6d70 6c79 2063 6f6d 7075 7465 7320  simply computes 
│ │ │ │ +00087b60: 7468 6520 2a6e 6f74 6520 6d75 6c74 6964  the *note multid
│ │ │ │ +00087b70: 6567 7265 653a 2028 4d61 6361 756c 6179  egree: (Macaulay
│ │ │ │ +00087b80: 3244 6f63 296d 756c 7469 6465 6772 6565  2Doc)multidegree
│ │ │ │ +00087b90: 2c20 6f66 2074 6865 202a 6e6f 7465 0a67  , of the *note.g
│ │ │ │ +00087ba0: 7261 7068 3a20 6772 6170 682c 2e0a 0a2b  raph: graph,...+
│ │ │ │ +00087bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00087bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00087bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00087be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00087bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00087c00: 2d2d 2d2b 0a7c 6931 203a 202d 2d20 6d61  ---+.|i1 : -- ma
│ │ │ │ -00087c10: 7020 6672 6f6d 2050 5e34 2074 6f20 4728  p from P^4 to G(
│ │ │ │ -00087c20: 312c 3329 2067 6976 656e 2062 7920 7468  1,3) given by th
│ │ │ │ -00087c30: 6520 7175 6164 7269 6373 2074 6872 6f75  e quadrics throu
│ │ │ │ -00087c40: 6768 2061 2072 6174 696f 6e61 6c20 2020  gh a rational   
│ │ │ │ -00087c50: 2020 207c 0a7c 2020 2020 2047 4628 3333     |.|     GF(33
│ │ │ │ -00087c60: 315e 3229 5b74 5f30 2e2e 745f 345d 3b20  1^2)[t_0..t_4]; 
│ │ │ │ -00087c70: 7068 693d 746f 4d61 7020 6d69 6e6f 7273  phi=toMap minors
│ │ │ │ -00087c80: 2832 2c6d 6174 7269 787b 7b74 5f30 2e2e  (2,matrix{{t_0..
│ │ │ │ -00087c90: 745f 337d 2c7b 745f 312e 2e74 5f34 2020  t_3},{t_1..t_4  
│ │ │ │ -00087ca0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00087bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00087c00: 6931 203a 202d 2d20 6d61 7020 6672 6f6d  i1 : -- map from
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│ │ │ │ +00087c20: 6976 656e 2062 7920 7468 6520 7175 6164  iven by the quad
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│ │ │ │ -00087da0: 2020 2020 2020 2020 2020 2030 2020 2034             0   4
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│ │ │ │ +00087d40: 6f32 203d 206d 6170 2028 4746 2031 3039  o2 = map (GF 109
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│ │ │ │ -00087e50: 2e74 205d 203c 2d2d 2047 4620 3130 3935  .t ] <-- GF 1095
│ │ │ │ -00087e60: 3631 5b78 202e 2e78 205d 2020 2020 2020  61[x ..x ]      
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│ │ │ │ -00087e90: 2020 2020 2020 2020 2020 2020 2020 3020                0 
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│ │ │ │ +00087e30: 6f32 203a 2052 696e 674d 6170 2047 4620  o2 : RingMap GF 
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│ │ │ │ +00087e60: 2e78 205d 2020 2020 2020 2020 2020 2020  .x ]            
│ │ │ │ +00087e70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00087e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00087e90: 2020 2020 2020 2020 3020 2020 3420 2020          0   4   
│ │ │ │ +00087ea0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00087eb0: 2020 3520 2020 2020 2020 2020 2020 2020    5             
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│ │ │ │ +00087ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00087ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00087ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00087f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00087f70: 7d7d 2920 2020 2020 2020 2020 2020 2020  }})             
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│ │ │ │ -00088050: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00088070: 2b20 7420 7420 2c20 2d20 7420 7420 202b  + t t , - t t  +
│ │ │ │ -00088080: 2074 2074 202c 202d 2074 2074 2020 2b20   t t , - t t  + 
│ │ │ │ -00088090: 7420 7420 2c20 2d20 7420 202b 2074 2074  t t , - t  + t t
│ │ │ │ -000880a0: 202c 2061 7d29 2020 2020 2020 2020 2020   , a})          
│ │ │ │ -000880b0: 2020 207c 0a7c 3020 3320 2020 2020 3220     |.|0 3     2 
│ │ │ │ -000880c0: 2020 2031 2033 2020 2020 2031 2033 2020     1 3     1 3  
│ │ │ │ -000880d0: 2020 3020 3420 2020 2020 3220 3320 2020    0 4     2 3   
│ │ │ │ -000880e0: 2031 2034 2020 2020 2033 2020 2020 3220   1 4     3    2 
│ │ │ │ -000880f0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00088100: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00088040: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00088050: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00088060: 2074 202c 202d 2074 2020 2b20 7420 7420   t , - t  + t t 
│ │ │ │ +00088070: 2c20 2d20 7420 7420 202b 2074 2074 202c  , - t t  + t t ,
│ │ │ │ +00088080: 202d 2074 2074 2020 2b20 7420 7420 2c20   - t t  + t t , 
│ │ │ │ +00088090: 2d20 7420 202b 2074 2074 202c 2061 7d29  - t  + t t , a})
│ │ │ │ +000880a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000880b0: 3020 3320 2020 2020 3220 2020 2031 2033  0 3     2    1 3
│ │ │ │ +000880c0: 2020 2020 2031 2033 2020 2020 3020 3420       1 3    0 4 
│ │ │ │ +000880d0: 2020 2020 3220 3320 2020 2031 2034 2020      2 3    1 4  
│ │ │ │ +000880e0: 2020 2033 2020 2020 3220 3420 2020 2020     3    2 4     
│ │ │ │ +000880f0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00088100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00088110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00088120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00088130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00088140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00088150: 2d2d 2d2b 0a7c 6933 203a 2074 696d 6520  ---+.|i3 : time 
│ │ │ │ -00088160: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │ -00088170: 7328 7068 692c 4365 7274 6966 793d 3e74  s(phi,Certify=>t
│ │ │ │ -00088180: 7275 6529 2020 2020 2020 2020 2020 2020  rue)            
│ │ │ │ -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!
│ │ │ │ +00088140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00088150: 6933 203a 2074 696d 6520 7072 6f6a 6563  i3 : time projec
│ │ │ │ +00088160: 7469 7665 4465 6772 6565 7328 7068 692c  tiveDegrees(phi,
│ │ │ │ +00088170: 4365 7274 6966 793d 3e74 7275 6529 2020  Certify=>true)  
│ │ │ │ +00088180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00088190: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000881a0: 4365 7274 6966 793a 206f 7574 7075 7420  Certify: output 
│ │ │ │ +000881b0: 6365 7274 6966 6965 6421 2020 2020 2020  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: 2e30 3135 3133 3736 7320 2863 7075 293b  .0151376s (cpu);
│ │ │ │ -00088210: 2030 2e30 3134 3830 3639 7320 2874 6872   0.0148069s (thr
│ │ │ │ -00088220: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ -00088230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088240: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000881e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000881f0: 202d 2d20 7573 6564 2030 2e30 3431 3235   -- used 0.04125
│ │ │ │ +00088200: 3734 7320 2863 7075 293b 2030 2e30 3231  74s (cpu); 0.021
│ │ │ │ +00088210: 3136 3134 7320 2874 6872 6561 6429 3b20  1614s (thread); 
│ │ │ │ +00088220: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ +00088230: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00088240: 2020 2020 2020 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                  
│ │ │ │ -00088290: 2020 207c 0a7c 6f33 203d 207b 312c 2032     |.|o3 = {1, 2
│ │ │ │ -000882a0: 2c20 342c 2034 2c20 327d 2020 2020 2020  , 4, 4, 2}      
│ │ │ │ +00088280: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00088290: 6f33 203d 207b 312c 2032 2c20 342c 2034  o3 = {1, 2, 4, 4
│ │ │ │ +000882a0: 2c20 327d 2020 2020 2020 2020 2020 2020  , 2}            
│ │ │ │  000882b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000882c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000882d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000882e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000882d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000882e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000882f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088330: 2020 207c 0a7c 6f33 203a 204c 6973 7420     |.|o3 : List 
│ │ │ │ +00088320: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00088330: 6f33 203a 204c 6973 7420 2020 2020 2020  o3 : List       
│ │ │ │  00088340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088380: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00088370: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00088380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00088390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000883a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000883b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000883c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000883d0: 2d2d 2d2b 0a7c 6934 203a 2070 7369 3d69  ---+.|i4 : psi=i
│ │ │ │ -000883e0: 6e76 6572 7365 4d61 7028 746f 4d61 7028  nverseMap(toMap(
│ │ │ │ -000883f0: 7068 692c 446f 6d69 6e61 6e74 3d3e 696e  phi,Dominant=>in
│ │ │ │ -00088400: 6669 6e69 7479 2929 2020 2020 2020 2020  finity))        
│ │ │ │ -00088410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088420: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000883c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000883d0: 6934 203a 2070 7369 3d69 6e76 6572 7365  i4 : psi=inverse
│ │ │ │ +000883e0: 4d61 7028 746f 4d61 7028 7068 692c 446f  Map(toMap(phi,Do
│ │ │ │ +000883f0: 6d69 6e61 6e74 3d3e 696e 6669 6e69 7479  minant=>infinity
│ │ │ │ +00088400: 2929 2020 2020 2020 2020 2020 2020 2020  ))              
│ │ │ │ +00088410: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00088420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088470: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00088480: 2047 4620 3130 3935 3631 5b78 202e 2e78   GF 109561[x ..x
│ │ │ │ -00088490: 205d 2020 2020 2020 2020 2020 2020 2020   ]              
│ │ │ │ +00088460: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00088470: 2020 2020 2020 2020 2020 2047 4620 3130             GF 10
│ │ │ │ +00088480: 3935 3631 5b78 202e 2e78 205d 2020 2020  9561[x ..x ]    
│ │ │ │ +00088490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000884a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000884b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000884c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000884d0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ -000884e0: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │ -000884f0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -00088500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088510: 2020 207c 0a7c 6f34 203d 206d 6170 2028     |.|o4 = map (
│ │ │ │ -00088520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00088530: 2d2d 2c20 4746 2031 3039 3536 315b 7420  --, GF 109561[t 
│ │ │ │ -00088540: 2e2e 7420 5d2c 207b 7820 202d 2078 2078  ..t ], {x  - x x
│ │ │ │ -00088550: 2020 2d20 7820 7820 2c20 7820 7820 202d    - x x , x x  -
│ │ │ │ -00088560: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00088570: 7820 7820 202d 2078 2078 2020 2b20 7820  x x  - x x  + x 
│ │ │ │ -00088580: 7820 2020 2020 2020 2020 2020 2020 2030  x              0
│ │ │ │ -00088590: 2020 2034 2020 2020 2031 2020 2020 3020     4     1    0 
│ │ │ │ -000885a0: 3220 2020 2030 2033 2020 2031 2032 2020  2    0 3   1 2  
│ │ │ │ -000885b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000885c0: 2032 2033 2020 2020 3120 3420 2020 2030   2 3    1 4    0
│ │ │ │ -000885d0: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ +000884b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000884c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000884d0: 2020 2020 2020 3020 2020 3520 2020 2020        0   5     
│ │ │ │ +000884e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000884f0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00088500: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00088510: 6f34 203d 206d 6170 2028 2d2d 2d2d 2d2d  o4 = map (------
│ │ │ │ +00088520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2c20 4746  ------------, GF
│ │ │ │ +00088530: 2031 3039 3536 315b 7420 2e2e 7420 5d2c   109561[t ..t ],
│ │ │ │ +00088540: 207b 7820 202d 2078 2078 2020 2d20 7820   {x  - x x  - x 
│ │ │ │ +00088550: 7820 2c20 7820 7820 202d 2020 207c 0a7c  x , x x  -   |.|
│ │ │ │ +00088560: 2020 2020 2020 2020 2020 7820 7820 202d            x x  -
│ │ │ │ +00088570: 2078 2078 2020 2b20 7820 7820 2020 2020   x x  + x x     
│ │ │ │ +00088580: 2020 2020 2020 2020 2030 2020 2034 2020           0   4  
│ │ │ │ +00088590: 2020 2031 2020 2020 3020 3220 2020 2030     1    0 2    0
│ │ │ │ +000885a0: 2033 2020 2031 2032 2020 2020 207c 0a7c   3   1 2     |.|
│ │ │ │ +000885b0: 2020 2020 2020 2020 2020 2032 2033 2020             2 3  
│ │ │ │ +000885c0: 2020 3120 3420 2020 2030 2035 2020 2020    1 4    0 5    
│ │ │ │ +000885d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000885e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000885f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088600: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000885f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00088600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088650: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00088660: 2020 2020 4746 2031 3039 3536 315b 7820      GF 109561[x 
│ │ │ │ -00088670: 2e2e 7820 5d20 2020 2020 2020 2020 2020  ..x ]           
│ │ │ │ +00088640: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00088650: 2020 2020 2020 2020 2020 2020 2020 4746                GF
│ │ │ │ +00088660: 2031 3039 3536 315b 7820 2e2e 7820 5d20   109561[x ..x ] 
│ │ │ │ +00088670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000886a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000886b0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -000886c0: 2020 2035 2020 2020 2020 2020 2020 2020     5            
│ │ │ │ +00088690: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000886a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000886b0: 2020 2020 2020 2020 2030 2020 2035 2020           0   5  
│ │ │ │ +000886c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000886d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000886e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000886f0: 2020 207c 0a7c 6f34 203a 2052 696e 674d     |.|o4 : RingM
│ │ │ │ -00088700: 6170 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ap -------------
│ │ │ │ -00088710: 2d2d 2d2d 2d20 3c2d 2d20 4746 2031 3039  ----- <-- GF 109
│ │ │ │ -00088720: 3536 315b 7420 2e2e 7420 5d20 2020 2020  561[t ..t ]     
│ │ │ │ -00088730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088740: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00088750: 2020 2078 2078 2020 2d20 7820 7820 202b     x x  - x x  +
│ │ │ │ -00088760: 2078 2078 2020 2020 2020 2020 2020 2020   x x            
│ │ │ │ -00088770: 2020 2020 2030 2020 2034 2020 2020 2020       0   4      
│ │ │ │ -00088780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088790: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000887a0: 2020 2020 3220 3320 2020 2031 2034 2020      2 3    1 4  
│ │ │ │ -000887b0: 2020 3020 3520 2020 2020 2020 2020 2020    0 5           
│ │ │ │ +000886e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000886f0: 6f34 203a 2052 696e 674d 6170 202d 2d2d  o4 : RingMap ---
│ │ │ │ +00088700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d20  --------------- 
│ │ │ │ +00088710: 3c2d 2d20 4746 2031 3039 3536 315b 7420  <-- GF 109561[t 
│ │ │ │ +00088720: 2e2e 7420 5d20 2020 2020 2020 2020 2020  ..t ]           
│ │ │ │ +00088730: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00088740: 2020 2020 2020 2020 2020 2020 2078 2078               x x
│ │ │ │ +00088750: 2020 2d20 7820 7820 202b 2078 2078 2020    - x x  + x x  
│ │ │ │ +00088760: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00088770: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +00088780: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00088790: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +000887a0: 3320 2020 2031 2034 2020 2020 3020 3520  3    1 4    0 5 
│ │ │ │ +000887b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000887c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000887d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000887e0: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │ +000887d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000887e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000887f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00088800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00088810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00088820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00088830: 2d2d 2d7c 0a7c 2020 2020 2020 2032 2020  ---|.|       2  
│ │ │ │ -00088840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088850: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00088820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00088830: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00088840: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00088850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088880: 2020 207c 0a7c 7820 7820 2c20 7820 202d     |.|x x , x  -
│ │ │ │ -00088890: 2078 2078 202c 2078 2078 2020 2d20 7820   x x , x x  - x 
│ │ │ │ -000888a0: 7820 2c20 7820 202d 2078 2078 2020 2d20  x , x  - x x  - 
│ │ │ │ -000888b0: 7820 7820 2c20 617d 2920 2020 2020 2020  x x , a})       
│ │ │ │ -000888c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000888d0: 2020 207c 0a7c 2030 2034 2020 2032 2020     |.| 0 4   2  
│ │ │ │ -000888e0: 2020 3020 3520 2020 3220 3420 2020 2031    0 5   2 4    1
│ │ │ │ -000888f0: 2035 2020 2034 2020 2020 3220 3520 2020   5   4    2 5   
│ │ │ │ -00088900: 2033 2035 2020 2020 2020 2020 2020 2020   3 5            
│ │ │ │ -00088910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00088920: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00088870: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00088880: 7820 7820 2c20 7820 202d 2078 2078 202c  x x , x  - x x ,
│ │ │ │ +00088890: 2078 2078 2020 2d20 7820 7820 2c20 7820   x x  - x x , x 
│ │ │ │ +000888a0: 202d 2078 2078 2020 2d20 7820 7820 2c20   - x x  - x x , 
│ │ │ │ +000888b0: 617d 2920 2020 2020 2020 2020 2020 2020  a})             
│ │ │ │ +000888c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000888d0: 2030 2034 2020 2032 2020 2020 3020 3520   0 4   2    0 5 
│ │ │ │ +000888e0: 2020 3220 3420 2020 2031 2035 2020 2034    2 4    1 5   4
│ │ │ │ +000888f0: 2020 2020 3220 3520 2020 2033 2035 2020      2 5    3 5  
│ │ │ │ +00088900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00088910: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00088920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00088930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00088940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00088950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00088960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00088970: 2d2d 2d2b 0a7c 6935 203a 2074 696d 6520  ---+.|i5 : time 
│ │ │ │ -00088980: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │ -00088990: 7328 7073 692c 4365 7274 6966 793d 3e74  s(psi,Certify=>t
│ │ │ │ -000889a0: 7275 6529 2020 2020 2020 2020 2020 2020  rue)            
│ │ │ │ -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!
│ │ │ │ +00088960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
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│ │ │ │ +000889b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │  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                  
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│ │ │ │ +00088a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000892b0: 2036 2020 2032 2033 2020 2020 3020 3520   6   2 3    0 5 
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│ │ │ │ -00089790: 2020 2020 2020 2020 2035 2020 2020 2020           5      
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│ │ │ │ +00089750: 202d 2031 3034 3033 3878 2078 2020 2d20   - 104038x x  - 
│ │ │ │ +00089760: 3130 3730 3478 2078 202c 202d 2035 3033  10704x x , - 503
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│ │ │ │ -00089a10: 2020 2020 2020 2020 2030 2036 2020 2020           0 6    
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│ │ │ │ +00089a10: 2020 2030 2036 2020 2020 2020 2020 2031     0 6         1
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│ │ │ │ -00089b30: 3635 3438 7820 7820 202b 2020 2020 2020  6548x x  +      
│ │ │ │ -00089b40: 2020 207c 0a7c 2020 2020 2020 3020 3220     |.|      0 2 
│ │ │ │ -00089b50: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
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│ │ │ │ -00089b70: 3420 2020 2020 2020 2020 2034 2020 2020  4          4    
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│ │ │ │ +00089b40: 2020 2020 2020 3020 3220 2020 2020 2020        0 2       
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│ │ │ │ +00089b60: 2020 2020 2020 2020 3120 3420 2020 2020          1 4     
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│ │ │ │ +00089b80: 2035 2020 2020 2020 2020 2020 207c 0a7c   5           |.|
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│ │ │ │  00089bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00089c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00089c40: 2d20 3132 3837 3237 7820 7820 202b 2031  - 128727x x  + 1
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│ │ │ │ -00089c60: 3978 2020 2d20 3135 3938 3478 2078 2020  9x  - 15984x x  
│ │ │ │ -00089c70: 2b20 3338 3537 3878 2078 2020 2d20 2020  + 38578x x  -   
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│ │ │ │ -00089c90: 2020 2020 2020 2020 2032 2035 2020 2020           2 5    
│ │ │ │ -00089ca0: 2020 2020 2020 3420 3520 2020 2020 2020        4 5       
│ │ │ │ -00089cb0: 2020 3520 2020 2020 2020 2020 3020 3620    5         0 6 
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│ │ │ │ -00089d50: 3878 2078 207d 2920 2020 2020 2020 2020  8x x })         
│ │ │ │ -00089d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00089d70: 2020 207c 0a7c 2020 2020 2020 2032 2036     |.|       2 6
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│ │ │ │ -00089d90: 2020 2020 2020 3420 3620 2020 2020 2020        4 6       
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│ │ │ │ -00089db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00089d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
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│ │ │ │ +00089d60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ +00089d80: 2020 2020 3320 3620 2020 2020 2020 2020      3 6         
│ │ │ │ +00089d90: 3420 3620 2020 2020 2020 2020 3520 3620  4 6         5 6 
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│ │ │ │ +00089dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00089dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00089de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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 
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│ │ │ │ -00089e30: 7320 7068 6920 2020 2020 2020 2020 2020  s phi           
│ │ │ │ +00089e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00089e10: 6937 203a 2074 696d 6520 7072 6f6a 6563  i7 : time projec
│ │ │ │ +00089e20: 7469 7665 4465 6772 6565 7320 7068 6920  tiveDegrees phi 
│ │ │ │ +00089e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2035     |.| -- used 5
│ │ │ │ -00089e70: 2e38 3539 652d 3035 7320 2863 7075 293b  .859e-05s (cpu);
│ │ │ │ -00089e80: 2035 2e33 3631 652d 3035 7320 2874 6872   5.361e-05s (thr
│ │ │ │ -00089e90: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ -00089ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00089eb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00089e50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00089e60: 202d 2d20 7573 6564 2037 2e33 3532 3765   -- used 7.3527e
│ │ │ │ +00089e70: 2d30 3573 2028 6370 7529 3b20 362e 3230  -05s (cpu); 6.20
│ │ │ │ +00089e80: 3336 652d 3035 7320 2874 6872 6561 6429  36e-05s (thread)
│ │ │ │ +00089e90: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +00089ea0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00089eb0: 2020 2020 2020 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                  
│ │ │ │ -00089f00: 2020 207c 0a7c 6f37 203d 207b 312c 2032     |.|o7 = {1, 2
│ │ │ │ -00089f10: 2c20 342c 2038 2c20 382c 2034 2c20 317d  , 4, 8, 8, 4, 1}
│ │ │ │ +00089ef0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00089f00: 6f37 203d 207b 312c 2032 2c20 342c 2038  o7 = {1, 2, 4, 8
│ │ │ │ +00089f10: 2c20 382c 2034 2c20 317d 2020 2020 2020  , 8, 4, 1}      
│ │ │ │  00089f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00089f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00089f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00089f50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00089f40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00089f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00089f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00089f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00089f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00089f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00089fa0: 2020 207c 0a7c 6f37 203a 204c 6973 7420     |.|o7 : List 
│ │ │ │ +00089f90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00089fa0: 6f37 203a 204c 6973 7420 2020 2020 2020  o7 : List       
│ │ │ │  00089fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00089fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00089fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00089fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00089ff0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00089fe0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00089ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008a000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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: 2e36 3130 3965 2d30 3573 2028 6370 7529  .6109e-05s (cpu)
│ │ │ │ -0008a0b0: 3b20 322e 3539 3339 652d 3035 7320 2874  ; 2.5939e-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     |.|          
│ │ │ │ +0008a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0008a040: 6938 203a 2074 696d 6520 7072 6f6a 6563  i8 : time projec
│ │ │ │ +0008a050: 7469 7665 4465 6772 6565 7328 7068 692c  tiveDegrees(phi,
│ │ │ │ +0008a060: 4e75 6d44 6567 7265 6573 3d3e 3129 2020  NumDegrees=>1)  
│ │ │ │ +0008a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008a080: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0008a090: 202d 2d20 7573 6564 2033 2e39 3930 3465   -- used 3.9904e
│ │ │ │ +0008a0a0: 2d30 3573 2028 6370 7529 3b20 332e 3734  -05s (cpu); 3.74
│ │ │ │ +0008a0b0: 3537 652d 3035 7320 2874 6872 6561 6429  57e-05s (thread)
│ │ │ │ +0008a0c0: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +0008a0d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0008a0e0: 2020 2020 2020 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
│ │ │ │ -0008a140: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +0008a120: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0008a130: 6f38 203d 207b 342c 2031 7d20 2020 2020  o8 = {4, 1}     
│ │ │ │ +0008a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008a180: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0008a170: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0008a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008a1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008a1d0: 2020 207c 0a7c 6f38 203a 204c 6973 7420     |.|o8 : List 
│ │ │ │ +0008a1c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0008a1d0: 6f38 203a 204c 6973 7420 2020 2020 2020  o8 : List       
│ │ │ │  0008a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008a210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008a220: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0008a210: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0008a220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008a230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008a240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008a250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008a260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0008a2f0: 6976 6544 6567 7265 6573 2870 6869 2c4e  iveDegrees(phi,N
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│ │ │ │ -0008a330: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ -0008a340: 0a0a 2020 2a20 2a6e 6f74 6520 6465 6772  ..  * *note degr
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│ │ │ │ -0008a360: 3a20 6465 6772 6565 735f 6c70 5261 7469  : degrees_lpRati
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│ │ │ │ -0008a390: 730a 2020 2020 6f66 2061 2072 6174 696f  s.    of a ratio
│ │ │ │ -0008a3a0: 6e61 6c20 6d61 700a 2020 2a20 2a6e 6f74  nal map.  * *not
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│ │ │ │ -0008a3d0: 6520 6f66 2061 2072 6174 696f 6e61 6c20  e of a rational 
│ │ │ │ -0008a3e0: 6d61 7020 6265 7477 6565 6e20 7072 6f6a  map between proj
│ │ │ │ -0008a3f0: 6563 7469 7665 0a20 2020 2076 6172 6965  ective.    varie
│ │ │ │ -0008a400: 7469 6573 0a20 202a 202a 6e6f 7465 2053  ties.  * *note S
│ │ │ │ -0008a410: 6567 7265 436c 6173 733a 2053 6567 7265  egreClass: Segre
│ │ │ │ -0008a420: 436c 6173 732c 202d 2d20 5365 6772 6520  Class, -- Segre 
│ │ │ │ -0008a430: 636c 6173 7320 6f66 2061 2063 6c6f 7365  class of a close
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│ │ │ │ -0008a460: 7661 7269 6574 790a 0a57 6179 7320 746f  variety..Ways to
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│ │ │ │ +0008a270: 416e 6f74 6865 7220 7761 7920 746f 2075  Another way to u
│ │ │ │ +0008a280: 7365 2074 6869 7320 6d65 7468 6f64 2069  se this method i
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│ │ │ │ +0008a2a0: 696e 7465 6765 7220 6920 6173 2073 6563  integer i as sec
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│ │ │ │ +0008a2e0: 7374 2070 726f 6a65 6374 6976 6544 6567  st projectiveDeg
│ │ │ │ +0008a2f0: 7265 6573 2870 6869 2c4e 756d 4465 6772  rees(phi,NumDegr
│ │ │ │ +0008a300: 6565 733d 3e69 2920 616e 640a 6765 6e65  ees=>i) and.gene
│ │ │ │ +0008a310: 7261 6c6c 7920 6974 2069 7320 6e6f 7420  rally it is not 
│ │ │ │ +0008a320: 6661 7374 6572 2e0a 0a53 6565 2061 6c73  faster...See als
│ │ │ │ +0008a330: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ +0008a340: 2a6e 6f74 6520 6465 6772 6565 7328 5261  *note degrees(Ra
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│ │ │ │ +0008a360: 6565 735f 6c70 5261 7469 6f6e 616c 4d61  ees_lpRationalMa
│ │ │ │ +0008a370: 705f 7270 2c20 2d2d 2070 726f 6a65 6374  p_rp, -- project
│ │ │ │ +0008a380: 6976 6520 6465 6772 6565 730a 2020 2020  ive degrees.    
│ │ │ │ +0008a390: 6f66 2061 2072 6174 696f 6e61 6c20 6d61  of a rational ma
│ │ │ │ +0008a3a0: 700a 2020 2a20 2a6e 6f74 6520 6465 6772  p.  * *note degr
│ │ │ │ +0008a3b0: 6565 4d61 703a 2064 6567 7265 654d 6170  eeMap: degreeMap
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│ │ │ │  0008a490: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │  0008a600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008a610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008a620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008a630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0008a720: 6154 7261 6e73 666f 726d 6174 696f 6e2c  aTransformation,
│ │ │ │ +0008a730: 2050 7265 763a 2070 726f 6a65 6374 6976   Prev: projectiv
│ │ │ │ +0008a740: 6544 6567 7265 6573 2c20 5570 3a20 546f  eDegrees, Up: To
│ │ │ │ +0008a750: 700a 0a70 726f 6a65 6374 6976 6544 6567  p..projectiveDeg
│ │ │ │ +0008a760: 7265 6573 2852 6174 696f 6e61 6c4d 6170  rees(RationalMap
│ │ │ │ +0008a770: 2920 2d2d 2070 726f 6a65 6374 6976 6520  ) -- projective 
│ │ │ │ +0008a780: 6465 6772 6565 7320 6f66 2061 2072 6174  degrees of a rat
│ │ │ │ +0008a790: 696f 6e61 6c20 6d61 700a 2a2a 2a2a 2a2a  ional map.******
│ │ │ │  0008a7a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0008a7b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0008a7c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0008a7d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0008a7e0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675 6e63  ******..  * Func
│ │ │ │ -0008a7f0: 7469 6f6e 3a20 2a6e 6f74 6520 7072 6f6a  tion: *note proj
│ │ │ │ -0008a800: 6563 7469 7665 4465 6772 6565 733a 2070  ectiveDegrees: p
│ │ │ │ -0008a810: 726f 6a65 6374 6976 6544 6567 7265 6573  rojectiveDegrees
│ │ │ │ -0008a820: 2c0a 2020 2a20 5573 6167 653a 200a 2020  ,.  * Usage: .  
│ │ │ │ -0008a830: 2020 2020 2020 7072 6f6a 6563 7469 7665        projective
│ │ │ │ -0008a840: 4465 6772 6565 7320 5068 690a 2020 2a20  Degrees Phi.  * 
│ │ │ │ -0008a850: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -0008a860: 5068 692c 2061 202a 6e6f 7465 2072 6174  Phi, a *note rat
│ │ │ │ -0008a870: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ -0008a880: 6e61 6c4d 6170 2c0a 2020 2a20 2a6e 6f74  nalMap,.  * *not
│ │ │ │ -0008a890: 6520 4f70 7469 6f6e 616c 2069 6e70 7574  e Optional input
│ │ │ │ -0008a8a0: 733a 2028 4d61 6361 756c 6179 3244 6f63  s: (Macaulay2Doc
│ │ │ │ -0008a8b0: 2975 7369 6e67 2066 756e 6374 696f 6e73  )using functions
│ │ │ │ -0008a8c0: 2077 6974 6820 6f70 7469 6f6e 616c 2069   with optional i
│ │ │ │ -0008a8d0: 6e70 7574 732c 3a0a 2020 2020 2020 2a20  nputs,:.      * 
│ │ │ │ -0008a8e0: 2a6e 6f74 6520 426c 6f77 5570 5374 7261  *note BlowUpStra
│ │ │ │ -0008a8f0: 7465 6779 3a20 426c 6f77 5570 5374 7261  tegy: BlowUpStra
│ │ │ │ -0008a900: 7465 6779 2c20 3d3e 202e 2e2e 2c20 6465  tegy, => ..., de
│ │ │ │ -0008a910: 6661 756c 7420 7661 6c75 650a 2020 2020  fault value.    
│ │ │ │ -0008a920: 2020 2020 2245 6c69 6d69 6e61 7465 222c      "Eliminate",
│ │ │ │ -0008a930: 0a20 2020 2020 202a 202a 6e6f 7465 2043  .      * *note C
│ │ │ │ -0008a940: 6572 7469 6679 3a20 4365 7274 6966 792c  ertify: Certify,
│ │ │ │ -0008a950: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ -0008a960: 2076 616c 7565 2066 616c 7365 2c20 7768   value false, wh
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│ │ │ │ -0008a980: 2020 2020 2020 2020 636f 7272 6563 746e          correctn
│ │ │ │ -0008a990: 6573 7320 6f66 206f 7574 7075 740a 2020  ess of output.  
│ │ │ │ -0008a9a0: 2020 2020 2a20 2a6e 6f74 6520 4e75 6d44      * *note NumD
│ │ │ │ -0008a9b0: 6567 7265 6573 3a20 4e75 6d44 6567 7265  egrees: NumDegre
│ │ │ │ -0008a9c0: 6573 2c20 3d3e 202e 2e2e 2c20 6465 6661  es, => ..., defa
│ │ │ │ -0008a9d0: 756c 7420 7661 6c75 6520 696e 6669 6e69  ult value infini
│ │ │ │ -0008a9e0: 7479 2c20 0a20 2020 2020 202a 202a 6e6f  ty, .      * *no
│ │ │ │ -0008a9f0: 7465 2056 6572 626f 7365 3a20 696e 7665  te Verbose: inve
│ │ │ │ -0008aa00: 7273 654d 6170 5f6c 705f 7064 5f70 645f  rseMap_lp_pd_pd_
│ │ │ │ -0008aa10: 7064 5f63 6d56 6572 626f 7365 3d3e 5f70  pd_cmVerbose=>_p
│ │ │ │ -0008aa20: 645f 7064 5f70 645f 7270 2c20 3d3e 202e  d_pd_pd_rp, => .
│ │ │ │ -0008aa30: 2e2e 2c0a 2020 2020 2020 2020 6465 6661  ..,.        defa
│ │ │ │ -0008aa40: 756c 7420 7661 6c75 6520 7472 7565 2c0a  ult value true,.
│ │ │ │ -0008aa50: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ -0008aa60: 2020 202a 2061 202a 6e6f 7465 206c 6973     * a *note lis
│ │ │ │ -0008aa70: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ -0008aa80: 294c 6973 742c 2c20 7468 6520 6c69 7374  )List,, the list
│ │ │ │ -0008aa90: 206f 6620 7072 6f6a 6563 7469 7665 2064   of projective d
│ │ │ │ -0008aaa0: 6567 7265 6573 206f 660a 2020 2020 2020  egrees of.      
│ │ │ │ -0008aab0: 2020 5068 690a 0a44 6573 6372 6970 7469    Phi..Descripti
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│ │ │ │ -0008aad0: 5468 6973 2063 6f6d 7075 7461 7469 6f6e  This computation
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│ │ │ │ -0008ab30: 7265 6573 2852 696e 674d 6170 293a 2070  rees(RingMap): p
│ │ │ │ -0008ab40: 726f 6a65 6374 6976 6544 6567 7265 6573  rojectiveDegrees
│ │ │ │ -0008ab50: 2c20 666f 7220 6d6f 7265 2064 6574 6169  , for more detai
│ │ │ │ -0008ab60: 6c73 2061 6e64 0a65 7861 6d70 6c65 732e  ls and.examples.
│ │ │ │ -0008ab70: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -0008ab80: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2070  ===..  * *note p
│ │ │ │ -0008ab90: 726f 6a65 6374 6976 6544 6567 7265 6573  rojectiveDegrees
│ │ │ │ -0008aba0: 2852 696e 674d 6170 293a 2070 726f 6a65  (RingMap): proje
│ │ │ │ -0008abb0: 6374 6976 6544 6567 7265 6573 2c20 2d2d  ctiveDegrees, --
│ │ │ │ -0008abc0: 2070 726f 6a65 6374 6976 6520 6465 6772   projective degr
│ │ │ │ -0008abd0: 6565 730a 2020 2020 6f66 2061 2072 6174  ees.    of a rat
│ │ │ │ -0008abe0: 696f 6e61 6c20 6d61 7020 6265 7477 6565  ional map betwee
│ │ │ │ -0008abf0: 6e20 7072 6f6a 6563 7469 7665 2076 6172  n projective var
│ │ │ │ -0008ac00: 6965 7469 6573 0a20 202a 202a 6e6f 7465  ieties.  * *note
│ │ │ │ -0008ac10: 2064 6567 7265 6573 2852 6174 696f 6e61   degrees(Rationa
│ │ │ │ -0008ac20: 6c4d 6170 293a 2064 6567 7265 6573 5f6c  lMap): degrees_l
│ │ │ │ -0008ac30: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ -0008ac40: 202d 2d20 7072 6f6a 6563 7469 7665 2064   -- projective d
│ │ │ │ -0008ac50: 6567 7265 6573 0a20 2020 206f 6620 6120  egrees.    of a 
│ │ │ │ -0008ac60: 7261 7469 6f6e 616c 206d 6170 0a20 202a  rational map.  *
│ │ │ │ -0008ac70: 202a 6e6f 7465 2064 6567 7265 6528 5261   *note degree(Ra
│ │ │ │ -0008ac80: 7469 6f6e 616c 4d61 7029 3a20 6465 6772  tionalMap): degr
│ │ │ │ -0008ac90: 6565 5f6c 7052 6174 696f 6e61 6c4d 6170  ee_lpRationalMap
│ │ │ │ -0008aca0: 5f72 702c 202d 2d20 6465 6772 6565 206f  _rp, -- degree o
│ │ │ │ -0008acb0: 6620 6120 7261 7469 6f6e 616c 0a20 2020  f a rational.   
│ │ │ │ -0008acc0: 206d 6170 0a0a 5761 7973 2074 6f20 7573   map..Ways to us
│ │ │ │ -0008acd0: 6520 7468 6973 206d 6574 686f 643a 0a3d  e this method:.=
│ │ │ │ +0008a7e0: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20  ..  * Function: 
│ │ │ │ +0008a7f0: 2a6e 6f74 6520 7072 6f6a 6563 7469 7665  *note projective
│ │ │ │ +0008a800: 4465 6772 6565 733a 2070 726f 6a65 6374  Degrees: project
│ │ │ │ +0008a810: 6976 6544 6567 7265 6573 2c0a 2020 2a20  iveDegrees,.  * 
│ │ │ │ +0008a820: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +0008a830: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │ +0008a840: 7320 5068 690a 2020 2a20 496e 7075 7473  s Phi.  * Inputs
│ │ │ │ +0008a850: 3a0a 2020 2020 2020 2a20 5068 692c 2061  :.      * Phi, a
│ │ │ │ +0008a860: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ +0008a870: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ +0008a880: 2c0a 2020 2a20 2a6e 6f74 6520 4f70 7469  ,.  * *note Opti
│ │ │ │ +0008a890: 6f6e 616c 2069 6e70 7574 733a 2028 4d61  onal inputs: (Ma
│ │ │ │ +0008a8a0: 6361 756c 6179 3244 6f63 2975 7369 6e67  caulay2Doc)using
│ │ │ │ +0008a8b0: 2066 756e 6374 696f 6e73 2077 6974 6820   functions with 
│ │ │ │ +0008a8c0: 6f70 7469 6f6e 616c 2069 6e70 7574 732c  optional inputs,
│ │ │ │ +0008a8d0: 3a0a 2020 2020 2020 2a20 2a6e 6f74 6520  :.      * *note 
│ │ │ │ +0008a8e0: 426c 6f77 5570 5374 7261 7465 6779 3a20  BlowUpStrategy: 
│ │ │ │ +0008a8f0: 426c 6f77 5570 5374 7261 7465 6779 2c20  BlowUpStrategy, 
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│ │ │ │ +0008af00: 4372 656d 6f6e 6120 7472 616e 7366 6f72  Cremona transfor
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│ │ │ │  0008af30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │  0008af50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -0008af70: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -0008af80: 7175 6164 726f 5175 6164 7269 6343 7265  quadroQuadricCre
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│ │ │ │ +0008b050: 7465 2069 6e74 6567 6572 3a20 284d 6163  te integer: (Mac
│ │ │ │ +0008b060: 6175 6c61 7932 446f 6329 5a5a 2c2c 2074  aulay2Doc)ZZ,, t
│ │ │ │ +0008b070: 6865 2069 2d74 6820 6361 7365 2069 6e20  he i-th case in 
│ │ │ │ +0008b080: 7468 650a 2020 2020 2020 2020 636c 6173  the.        clas
│ │ │ │ +0008b090: 7369 6669 6361 7469 6f6e 2066 6f72 2050  sification for P
│ │ │ │ +0008b0a0: 5e6e 2028 666f 7220 696e 7374 616e 6365  ^n (for instance
│ │ │ │ +0008b0b0: 2c20 6966 206e 3d35 2074 6865 6e20 313c  , if n=5 then 1<
│ │ │ │ +0008b0c0: 3d69 3c3d 3339 290a 2020 2020 2020 2a20  =i<=39).      * 
│ │ │ │ +0008b0d0: 4b2c 2061 202a 6e6f 7465 2072 696e 673a  K, a *note ring:
│ │ │ │ +0008b0e0: 2028 4d61 6361 756c 6179 3244 6f63 2952   (Macaulay2Doc)R
│ │ │ │ +0008b0f0: 696e 672c 2c20 7468 6520 6772 6f75 6e64  ing,, the ground
│ │ │ │ +0008b100: 2066 6965 6c64 2028 6f70 7469 6f6e 616c   field (optional
│ │ │ │ +0008b110: 2c20 7468 650a 2020 2020 2020 2020 6465  , the.        de
│ │ │ │ +0008b120: 6661 756c 7420 7661 6c75 6520 6973 202a  fault value is *
│ │ │ │ +0008b130: 6e6f 7465 2051 513a 2028 4d61 6361 756c  note QQ: (Macaul
│ │ │ │ +0008b140: 6179 3244 6f63 2951 512c 290a 2020 2a20  ay2Doc)QQ,).  * 
│ │ │ │ +0008b150: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +0008b160: 2061 202a 6e6f 7465 2072 6174 696f 6e61   a *note rationa
│ │ │ │ +0008b170: 6c20 6d61 703a 2052 6174 696f 6e61 6c4d  l map: RationalM
│ │ │ │ +0008b180: 6170 2c2c 2061 6e20 6578 616d 706c 6520  ap,, an example 
│ │ │ │ +0008b190: 6f66 2071 7561 6472 6f2d 7175 6164 7269  of quadro-quadri
│ │ │ │ +0008b1a0: 630a 2020 2020 2020 2020 4372 656d 6f6e  c.        Cremon
│ │ │ │ +0008b1b0: 6120 7472 616e 7366 6f72 6d61 7469 6f6e  a transformation
│ │ │ │ +0008b1c0: 206f 7665 7220 4b2c 2061 6363 6f72 6469   over K, accordi
│ │ │ │ +0008b1d0: 6e67 2074 6f20 7468 6520 636c 6173 7369  ng to the classi
│ │ │ │ +0008b1e0: 6669 6361 7469 6f6e 7320 6769 7665 6e0a  fications given.
│ │ │ │ +0008b1f0: 2020 2020 2020 2020 696e 2074 6865 2070          in the p
│ │ │ │ +0008b200: 6170 6572 2051 7561 6472 6f2d 7175 6164  aper Quadro-quad
│ │ │ │ +0008b210: 7269 6320 4372 656d 6f6e 6120 7472 616e  ric Cremona tran
│ │ │ │ +0008b220: 7366 6f72 6d61 7469 6f6e 7320 696e 206c  sformations in l
│ │ │ │ +0008b230: 6f77 2064 696d 656e 7369 6f6e 730a 2020  ow dimensions.  
│ │ │ │ +0008b240: 2020 2020 2020 7669 6120 7468 6520 4a43        via the JC
│ │ │ │ +0008b250: 2d63 6f72 7265 7370 6f6e 6465 6e63 6520  -correspondence 
│ │ │ │ +0008b260: 2873 6565 0a20 2020 2020 2020 2068 7474  (see.        htt
│ │ │ │ +0008b270: 7073 3a2f 2f61 6966 2e63 656e 7472 652d  ps://aif.centre-
│ │ │ │ +0008b280: 6d65 7273 656e 6e65 2e6f 7267 2f69 7465  mersenne.org/ite
│ │ │ │ +0008b290: 6d2f 4149 465f 3230 3134 5f5f 3634 5f31  m/AIF_2014__64_1
│ │ │ │ +0008b2a0: 5f37 315f 302f 2029 2c20 6279 2050 6972  _71_0/ ), by Pir
│ │ │ │ +0008b2b0: 696f 0a20 2020 2020 2020 2061 6e64 2052  io.        and R
│ │ │ │ +0008b2c0: 7573 736f 2e0a 0a44 6573 6372 6970 7469  usso...Descripti
│ │ │ │ +0008b2d0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ +0008b2e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0008b2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008b300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008b310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008b320: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7175  ------+.|i1 : qu
│ │ │ │ -0008b330: 6164 726f 5175 6164 7269 6343 7265 6d6f  adroQuadricCremo
│ │ │ │ -0008b340: 6e61 5472 616e 7366 6f72 6d61 7469 6f6e  naTransformation
│ │ │ │ -0008b350: 2835 2c32 3329 2020 2020 2020 2020 2020  (5,23)          
│ │ │ │ -0008b360: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0008b320: 2b0a 7c69 3120 3a20 7175 6164 726f 5175  +.|i1 : quadroQu
│ │ │ │ +0008b330: 6164 7269 6343 7265 6d6f 6e61 5472 616e  adricCremonaTran
│ │ │ │ +0008b340: 7366 6f72 6d61 7469 6f6e 2835 2c32 3329  sformation(5,23)
│ │ │ │ +0008b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b360: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0008b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b3a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ -0008b3b0: 3d20 2d2d 2072 6174 696f 6e61 6c20 6d61  = -- rational ma
│ │ │ │ -0008b3c0: 7020 2d2d 2020 2020 2020 2020 2020 2020  p --            
│ │ │ │ +0008b3a0: 2020 2020 7c0a 7c6f 3120 3d20 2d2d 2072      |.|o1 = -- r
│ │ │ │ +0008b3b0: 6174 696f 6e61 6c20 6d61 7020 2d2d 2020  ational map --  
│ │ │ │ +0008b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b3e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0008b3f0: 2020 2020 736f 7572 6365 3a20 5072 6f6a      source: Proj
│ │ │ │ -0008b400: 2851 515b 782c 2079 2c20 7a2c 2074 2c20  (QQ[x, y, z, t, 
│ │ │ │ -0008b410: 752c 2076 5d29 2020 2020 2020 2020 2020  u, v])          
│ │ │ │ -0008b420: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008b430: 7c20 2020 2020 7461 7267 6574 3a20 5072  |     target: Pr
│ │ │ │ -0008b440: 6f6a 2851 515b 782c 2079 2c20 7a2c 2074  oj(QQ[x, y, z, t
│ │ │ │ -0008b450: 2c20 752c 2076 5d29 2020 2020 2020 2020  , u, v])        
│ │ │ │ -0008b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b470: 7c0a 7c20 2020 2020 6465 6669 6e69 6e67  |.|     defining
│ │ │ │ -0008b480: 2066 6f72 6d73 3a20 7b20 2020 2020 2020   forms: {       
│ │ │ │ +0008b3e0: 2020 2020 2020 7c0a 7c20 2020 2020 736f        |.|     so
│ │ │ │ +0008b3f0: 7572 6365 3a20 5072 6f6a 2851 515b 782c  urce: Proj(QQ[x,
│ │ │ │ +0008b400: 2079 2c20 7a2c 2074 2c20 752c 2076 5d29   y, z, t, u, v])
│ │ │ │ +0008b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b420: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0008b430: 7461 7267 6574 3a20 5072 6f6a 2851 515b  target: Proj(QQ[
│ │ │ │ +0008b440: 782c 2079 2c20 7a2c 2074 2c20 752c 2076  x, y, z, t, u, v
│ │ │ │ +0008b450: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ +0008b460: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0008b470: 2020 6465 6669 6e69 6e67 2066 6f72 6d73    defining forms
│ │ │ │ +0008b480: 3a20 7b20 2020 2020 2020 2020 2020 2020  : {             
│ │ │ │  0008b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b4b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0008b4c0: 2020 2020 2020 2020 2020 2078 2a79 2c20             x*y, 
│ │ │ │ +0008b4a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0008b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b4c0: 2020 2020 2078 2a79 2c20 2020 2020 2020       x*y,       
│ │ │ │  0008b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b4f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0008b4e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008b4f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0008b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b530: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0008b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b550: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0008b530: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0008b540: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +0008b550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b570: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0008b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b590: 2078 202c 2020 2020 2020 2020 2020 2020   x ,            
│ │ │ │ +0008b570: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0008b580: 2020 2020 2020 2020 2020 2078 202c 2020             x ,  
│ │ │ │ +0008b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b5b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0008b5b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0008b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b5f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0008b5f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0008b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b610: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0008b620: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -0008b630: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008b640: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0008b650: 2020 2020 2020 202d 2079 2a7a 202b 2074         - y*z + t
│ │ │ │ -0008b660: 2020 2b20 7520 2c20 2020 2020 2020 2020    + u ,         
│ │ │ │ -0008b670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b680: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0008b610: 2020 2020 2020 2020 3220 2020 2032 2020          2    2  
│ │ │ │ +0008b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b630: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0008b640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b650: 202d 2079 2a7a 202b 2074 2020 2b20 7520   - y*z + t  + u 
│ │ │ │ +0008b660: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +0008b670: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0008b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b6c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0008b6d0: 2020 2020 2020 2020 2020 202d 782a 742c             -x*t,
│ │ │ │ +0008b6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0008b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008b6d0: 2020 2020 202d 782a 742c 2020 2020 2020       -x*t,      
│ │ │ │  0008b6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b700: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0008b6f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008b700: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0008b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b740: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0008b750: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ -0008b760: 782a 752c 2020 2020 2020 2020 2020 2020  x*u,            
│ │ │ │ +0008b740: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0008b750: 2020 2020 2020 2020 202d 782a 752c 2020           -x*u,  
│ │ │ │ +0008b760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b780: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0008b780: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0008b790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b7c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0008b7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b7e0: 2020 202d 782a 7620 2020 2020 2020 2020     -x*v         
│ │ │ │ +0008b7c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0008b7d0: 2020 2020 2020 2020 2020 2020 202d 782a               -x*
│ │ │ │ +0008b7e0: 7620 2020 2020 2020 2020 2020 2020 2020  v               
│ │ │ │  0008b7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b800: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0008b810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b820: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │ +0008b800: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0008b810: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
│ │ │ │ +0008b820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008b850: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008b840: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0008b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b890: 7c0a 7c6f 3120 3a20 5261 7469 6f6e 616c  |.|o1 : Rational
│ │ │ │ -0008b8a0: 4d61 7020 2843 7265 6d6f 6e61 2074 7261  Map (Cremona tra
│ │ │ │ -0008b8b0: 6e73 666f 726d 6174 696f 6e20 6f66 2050  nsformation of P
│ │ │ │ -0008b8c0: 505e 3520 6f66 2074 7970 6520 2832 2c32  P^5 of type (2,2
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│ │ │ │ +0008b880: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +0008b890: 3a20 5261 7469 6f6e 616c 4d61 7020 2843  : RationalMap (C
│ │ │ │ +0008b8a0: 7265 6d6f 6e61 2074 7261 6e73 666f 726d  remona transform
│ │ │ │ +0008b8b0: 6174 696f 6e20 6f66 2050 505e 3520 6f66  ation of PP^5 of
│ │ │ │ +0008b8c0: 2074 7970 6520 2832 2c32 2929 7c0a 2b2d   type (2,2))|.+-
│ │ │ │ +0008b8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008b8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008b900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008b910: 2d2d 2d2d 2b0a 7c69 3220 3a20 6465 7363  ----+.|i2 : desc
│ │ │ │ -0008b920: 7269 6265 206f 6f20 2020 2020 2020 2020  ribe oo         
│ │ │ │ +0008b900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0008b910: 7c69 3220 3a20 6465 7363 7269 6265 206f  |i2 : describe o
│ │ │ │ +0008b920: 6f20 2020 2020 2020 2020 2020 2020 2020  o               
│ │ │ │  0008b930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b950: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0008b950: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0008b960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008b980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008b990: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ -0008b9a0: 7261 7469 6f6e 616c 206d 6170 2064 6566  rational map def
│ │ │ │ -0008b9b0: 696e 6564 2062 7920 666f 726d 7320 6f66  ined by forms of
│ │ │ │ -0008b9c0: 2064 6567 7265 6520 3220 2020 2020 2020   degree 2       
│ │ │ │ -0008b9d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0008b9e0: 2020 736f 7572 6365 2076 6172 6965 7479    source variety
│ │ │ │ -0008b9f0: 3a20 5050 5e35 2020 2020 2020 2020 2020  : PP^5          
│ │ │ │ +0008b990: 2020 7c0a 7c6f 3220 3d20 7261 7469 6f6e    |.|o2 = ration
│ │ │ │ +0008b9a0: 616c 206d 6170 2064 6566 696e 6564 2062  al map defined b
│ │ │ │ +0008b9b0: 7920 666f 726d 7320 6f66 2064 6567 7265  y forms of degre
│ │ │ │ +0008b9c0: 6520 3220 2020 2020 2020 2020 2020 2020  e 2             
│ │ │ │ +0008b9d0: 2020 2020 7c0a 7c20 2020 2020 736f 7572      |.|     sour
│ │ │ │ +0008b9e0: 6365 2076 6172 6965 7479 3a20 5050 5e35  ce variety: PP^5
│ │ │ │ +0008b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008ba10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0008ba20: 2020 2020 7461 7267 6574 2076 6172 6965      target varie
│ │ │ │ -0008ba30: 7479 3a20 5050 5e35 2020 2020 2020 2020  ty: PP^5        
│ │ │ │ +0008ba10: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ +0008ba20: 7267 6574 2076 6172 6965 7479 3a20 5050  rget variety: PP
│ │ │ │ +0008ba30: 5e35 2020 2020 2020 2020 2020 2020 2020  ^5              
│ │ │ │  0008ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008ba50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008ba60: 7c20 2020 2020 646f 6d69 6e61 6e63 653a  |     dominance:
│ │ │ │ -0008ba70: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │ +0008ba50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0008ba60: 646f 6d69 6e61 6e63 653a 2074 7275 6520  dominance: true 
│ │ │ │ +0008ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008baa0: 7c0a 7c20 2020 2020 6269 7261 7469 6f6e  |.|     biration
│ │ │ │ -0008bab0: 616c 6974 793a 2074 7275 6520 2020 2020  ality: true     
│ │ │ │ +0008ba90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0008baa0: 2020 6269 7261 7469 6f6e 616c 6974 793a    birationality:
│ │ │ │ +0008bab0: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │  0008bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bae0: 2020 7c0a 7c20 2020 2020 7072 6f6a 6563    |.|     projec
│ │ │ │ -0008baf0: 7469 7665 2064 6567 7265 6573 3a20 7b31  tive degrees: {1
│ │ │ │ -0008bb00: 2c20 322c 2032 2c20 322c 2032 2c20 317d  , 2, 2, 2, 2, 1}
│ │ │ │ -0008bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bb20: 2020 2020 7c0a 7c20 2020 2020 6e75 6d62      |.|     numb
│ │ │ │ -0008bb30: 6572 206f 6620 6d69 6e69 6d61 6c20 7265  er of minimal re
│ │ │ │ -0008bb40: 7072 6573 656e 7461 7469 7665 733a 2031  presentatives: 1
│ │ │ │ +0008bad0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0008bae0: 2020 2020 7072 6f6a 6563 7469 7665 2064      projective d
│ │ │ │ +0008baf0: 6567 7265 6573 3a20 7b31 2c20 322c 2032  egrees: {1, 2, 2
│ │ │ │ +0008bb00: 2c20 322c 2032 2c20 317d 2020 2020 2020  , 2, 2, 1}      
│ │ │ │ +0008bb10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008bb20: 7c20 2020 2020 6e75 6d62 6572 206f 6620  |     number of 
│ │ │ │ +0008bb30: 6d69 6e69 6d61 6c20 7265 7072 6573 656e  minimal represen
│ │ │ │ +0008bb40: 7461 7469 7665 733a 2031 2020 2020 2020  tatives: 1      
│ │ │ │  0008bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bb60: 2020 2020 2020 7c0a 7c20 2020 2020 6469        |.|     di
│ │ │ │ -0008bb70: 6d65 6e73 696f 6e20 6261 7365 206c 6f63  mension base loc
│ │ │ │ -0008bb80: 7573 3a20 3320 2020 2020 2020 2020 2020  us: 3           
│ │ │ │ +0008bb60: 7c0a 7c20 2020 2020 6469 6d65 6e73 696f  |.|     dimensio
│ │ │ │ +0008bb70: 6e20 6261 7365 206c 6f63 7573 3a20 3320  n base locus: 3 
│ │ │ │ +0008bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008bb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bba0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0008bbb0: 6465 6772 6565 2062 6173 6520 6c6f 6375  degree base locu
│ │ │ │ -0008bbc0: 733a 2032 2020 2020 2020 2020 2020 2020  s: 2            
│ │ │ │ +0008bba0: 2020 7c0a 7c20 2020 2020 6465 6772 6565    |.|     degree
│ │ │ │ +0008bbb0: 2062 6173 6520 6c6f 6375 733a 2032 2020   base locus: 2  
│ │ │ │ +0008bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bbe0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0008bbf0: 2020 636f 6566 6669 6369 656e 7420 7269    coefficient ri
│ │ │ │ -0008bc00: 6e67 3a20 5151 2020 2020 2020 2020 2020  ng: QQ          
│ │ │ │ +0008bbe0: 2020 2020 7c0a 7c20 2020 2020 636f 6566      |.|     coef
│ │ │ │ +0008bbf0: 6669 6369 656e 7420 7269 6e67 3a20 5151  ficient ring: QQ
│ │ │ │ +0008bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bc20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0008bc20: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  0008bc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008bc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008bc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008bc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0008bc70: 0a49 6e20 6164 6469 7469 6f6e 2c20 7468  .In addition, th
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│ │ │ │ -0008bc90: 6929 3d28 352c 3129 2c28 382c 3129 2c28  i)=(5,1),(8,1),(
│ │ │ │ -0008bca0: 3134 2c31 292c 2832 362c 3129 2063 6f72  14,1),(26,1) cor
│ │ │ │ -0008bcb0: 7265 7370 6f6e 6420 746f 2074 6865 0a66  respond to the.f
│ │ │ │ -0008bcc0: 6f75 7220 6578 616d 706c 6573 206f 6620  our examples of 
│ │ │ │ -0008bcd0: 7370 6563 6961 6c20 7175 6164 726f 2d71  special quadro-q
│ │ │ │ -0008bce0: 7561 6472 6963 2043 7265 6d6f 6e61 2074  uadric Cremona t
│ │ │ │ -0008bcf0: 7261 6e73 666f 726d 6174 696f 6e73 3a0a  ransformations:.
│ │ │ │ -0008bd00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0008bc60: 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6e20 6164  --------+..In ad
│ │ │ │ +0008bc70: 6469 7469 6f6e 2c20 7468 6520 666f 7572  dition, the four
│ │ │ │ +0008bc80: 2070 6169 7273 2028 6e2c 6929 3d28 352c   pairs (n,i)=(5,
│ │ │ │ +0008bc90: 3129 2c28 382c 3129 2c28 3134 2c31 292c  1),(8,1),(14,1),
│ │ │ │ +0008bca0: 2832 362c 3129 2063 6f72 7265 7370 6f6e  (26,1) correspon
│ │ │ │ +0008bcb0: 6420 746f 2074 6865 0a66 6f75 7220 6578  d to the.four ex
│ │ │ │ +0008bcc0: 616d 706c 6573 206f 6620 7370 6563 6961  amples of specia
│ │ │ │ +0008bcd0: 6c20 7175 6164 726f 2d71 7561 6472 6963  l quadro-quadric
│ │ │ │ +0008bce0: 2043 7265 6d6f 6e61 2074 7261 6e73 666f   Cremona transfo
│ │ │ │ +0008bcf0: 726d 6174 696f 6e73 3a0a 0a2b 2d2d 2d2d  rmations:..+----
│ │ │ │ +0008bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008bd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008bd30: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ -0008bd40: 6465 7363 7269 6265 2071 7561 6472 6f51  describe quadroQ
│ │ │ │ -0008bd50: 7561 6472 6963 4372 656d 6f6e 6154 7261  uadricCremonaTra
│ │ │ │ -0008bd60: 6e73 666f 726d 6174 696f 6e28 352c 3129  nsformation(5,1)
│ │ │ │ -0008bd70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0008bd30: 2d2d 2b0a 7c69 3320 3a20 6465 7363 7269  --+.|i3 : descri
│ │ │ │ +0008bd40: 6265 2071 7561 6472 6f51 7561 6472 6963  be quadroQuadric
│ │ │ │ +0008bd50: 4372 656d 6f6e 6154 7261 6e73 666f 726d  CremonaTransform
│ │ │ │ +0008bd60: 6174 696f 6e28 352c 3129 207c 0a7c 2020  ation(5,1) |.|  
│ │ │ │ +0008bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008bd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bda0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ -0008bdb0: 3d20 7261 7469 6f6e 616c 206d 6170 2064  = rational map d
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│ │ │ │ -0008bdd0: 6f66 2064 6567 7265 6520 3220 2020 2020  of degree 2     
│ │ │ │ -0008bde0: 2020 207c 0a7c 2020 2020 2073 6f75 7263     |.|     sourc
│ │ │ │ -0008bdf0: 6520 7661 7269 6574 793a 2050 505e 3520  e variety: PP^5 
│ │ │ │ +0008bda0: 2020 2020 7c0a 7c6f 3320 3d20 7261 7469      |.|o3 = rati
│ │ │ │ +0008bdb0: 6f6e 616c 206d 6170 2064 6566 696e 6564  onal map defined
│ │ │ │ +0008bdc0: 2062 7920 666f 726d 7320 6f66 2064 6567   by forms of deg
│ │ │ │ +0008bdd0: 7265 6520 3220 2020 2020 2020 207c 0a7c  ree 2        |.|
│ │ │ │ +0008bde0: 2020 2020 2073 6f75 7263 6520 7661 7269       source vari
│ │ │ │ +0008bdf0: 6574 793a 2050 505e 3520 2020 2020 2020  ety: PP^5       
│ │ │ │  0008be00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008be10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0008be20: 2020 2020 7461 7267 6574 2076 6172 6965      target varie
│ │ │ │ -0008be30: 7479 3a20 5050 5e35 2020 2020 2020 2020  ty: PP^5        
│ │ │ │ -0008be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008be50: 2020 2020 207c 0a7c 2020 2020 2064 6f6d       |.|     dom
│ │ │ │ -0008be60: 696e 616e 6365 3a20 7472 7565 2020 2020  inance: true    
│ │ │ │ +0008be10: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ +0008be20: 7267 6574 2076 6172 6965 7479 3a20 5050  rget variety: PP
│ │ │ │ +0008be30: 5e35 2020 2020 2020 2020 2020 2020 2020  ^5              
│ │ │ │ +0008be40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0008be50: 0a7c 2020 2020 2064 6f6d 696e 616e 6365  .|     dominance
│ │ │ │ +0008be60: 3a20 7472 7565 2020 2020 2020 2020 2020  : true          
│ │ │ │  0008be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008be80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008be90: 7c20 2020 2020 6269 7261 7469 6f6e 616c  |     birational
│ │ │ │ -0008bea0: 6974 793a 2074 7275 6520 2020 2020 2020  ity: true       
│ │ │ │ +0008be80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0008be90: 6269 7261 7469 6f6e 616c 6974 793a 2074  birationality: t
│ │ │ │ +0008bea0: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │  0008beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bec0: 2020 2020 2020 207c 0a7c 2020 2020 2070         |.|     p
│ │ │ │ -0008bed0: 726f 6a65 6374 6976 6520 6465 6772 6565  rojective degree
│ │ │ │ -0008bee0: 733a 207b 312c 2032 2c20 342c 2034 2c20  s: {1, 2, 4, 4, 
│ │ │ │ -0008bef0: 322c 2031 7d20 2020 2020 2020 2020 2020  2, 1}           
│ │ │ │ -0008bf00: 7c0a 7c20 2020 2020 6e75 6d62 6572 206f  |.|     number o
│ │ │ │ -0008bf10: 6620 6d69 6e69 6d61 6c20 7265 7072 6573  f minimal repres
│ │ │ │ -0008bf20: 656e 7461 7469 7665 733a 2031 2020 2020  entatives: 1    
│ │ │ │ -0008bf30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0008bf40: 2064 696d 656e 7369 6f6e 2062 6173 6520   dimension base 
│ │ │ │ -0008bf50: 6c6f 6375 733a 2032 2020 2020 2020 2020  locus: 2        
│ │ │ │ -0008bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bf70: 2020 7c0a 7c20 2020 2020 6465 6772 6565    |.|     degree
│ │ │ │ -0008bf80: 2062 6173 6520 6c6f 6375 733a 2034 2020   base locus: 4  
│ │ │ │ +0008bec0: 207c 0a7c 2020 2020 2070 726f 6a65 6374   |.|     project
│ │ │ │ +0008bed0: 6976 6520 6465 6772 6565 733a 207b 312c  ive degrees: {1,
│ │ │ │ +0008bee0: 2032 2c20 342c 2034 2c20 322c 2031 7d20   2, 4, 4, 2, 1} 
│ │ │ │ +0008bef0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0008bf00: 2020 6e75 6d62 6572 206f 6620 6d69 6e69    number of mini
│ │ │ │ +0008bf10: 6d61 6c20 7265 7072 6573 656e 7461 7469  mal representati
│ │ │ │ +0008bf20: 7665 733a 2031 2020 2020 2020 2020 2020  ves: 1          
│ │ │ │ +0008bf30: 2020 207c 0a7c 2020 2020 2064 696d 656e     |.|     dimen
│ │ │ │ +0008bf40: 7369 6f6e 2062 6173 6520 6c6f 6375 733a  sion base locus:
│ │ │ │ +0008bf50: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0008bf60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0008bf70: 2020 2020 6465 6772 6565 2062 6173 6520      degree base 
│ │ │ │ +0008bf80: 6c6f 6375 733a 2034 2020 2020 2020 2020  locus: 4        
│ │ │ │  0008bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bfa0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0008bfb0: 2020 2063 6f65 6666 6963 6965 6e74 2072     coefficient r
│ │ │ │ -0008bfc0: 696e 673a 2051 5120 2020 2020 2020 2020  ing: QQ         
│ │ │ │ -0008bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008bfe0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0008bfa0: 2020 2020 207c 0a7c 2020 2020 2063 6f65       |.|     coe
│ │ │ │ +0008bfb0: 6666 6963 6965 6e74 2072 696e 673a 2051  fficient ring: Q
│ │ │ │ +0008bfc0: 5120 2020 2020 2020 2020 2020 2020 2020  Q               
│ │ │ │ +0008bfd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008bfe0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0008bff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008c000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008c010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0008c020: 6934 203a 2064 6573 6372 6962 6520 7175  i4 : describe qu
│ │ │ │ -0008c030: 6164 726f 5175 6164 7269 6343 7265 6d6f  adroQuadricCremo
│ │ │ │ -0008c040: 6e61 5472 616e 7366 6f72 6d61 7469 6f6e  naTransformation
│ │ │ │ -0008c050: 2838 2c31 2920 7c0a 7c20 2020 2020 2020  (8,1) |.|       
│ │ │ │ +0008c010: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2064  -------+.|i4 : d
│ │ │ │ +0008c020: 6573 6372 6962 6520 7175 6164 726f 5175  escribe quadroQu
│ │ │ │ +0008c030: 6164 7269 6343 7265 6d6f 6e61 5472 616e  adricCremonaTran
│ │ │ │ +0008c040: 7366 6f72 6d61 7469 6f6e 2838 2c31 2920  sformation(8,1) 
│ │ │ │ +0008c050: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0008c060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0008c090: 0a7c 6f34 203d 2072 6174 696f 6e61 6c20  .|o4 = rational 
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│ │ │ │ -0008c0b0: 6f72 6d73 206f 6620 6465 6772 6565 2032  orms of degree 2
│ │ │ │ -0008c0c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0008c0d0: 736f 7572 6365 2076 6172 6965 7479 3a20  source variety: 
│ │ │ │ -0008c0e0: 5050 5e38 2020 2020 2020 2020 2020 2020  PP^8            
│ │ │ │ -0008c0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c100: 207c 0a7c 2020 2020 2074 6172 6765 7420   |.|     target 
│ │ │ │ -0008c110: 7661 7269 6574 793a 2050 505e 3820 2020  variety: PP^8   
│ │ │ │ +0008c080: 2020 2020 2020 2020 207c 0a7c 6f34 203d           |.|o4 =
│ │ │ │ +0008c090: 2072 6174 696f 6e61 6c20 6d61 7020 6465   rational map de
│ │ │ │ +0008c0a0: 6669 6e65 6420 6279 2066 6f72 6d73 206f  fined by forms o
│ │ │ │ +0008c0b0: 6620 6465 6772 6565 2032 2020 2020 2020  f degree 2      
│ │ │ │ +0008c0c0: 2020 7c0a 7c20 2020 2020 736f 7572 6365    |.|     source
│ │ │ │ +0008c0d0: 2076 6172 6965 7479 3a20 5050 5e38 2020   variety: PP^8  
│ │ │ │ +0008c0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c0f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0008c100: 2020 2074 6172 6765 7420 7661 7269 6574     target variet
│ │ │ │ +0008c110: 793a 2050 505e 3820 2020 2020 2020 2020  y: PP^8         
│ │ │ │  0008c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c130: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0008c140: 2020 646f 6d69 6e61 6e63 653a 2074 7275    dominance: tru
│ │ │ │ -0008c150: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ -0008c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c170: 2020 207c 0a7c 2020 2020 2062 6972 6174     |.|     birat
│ │ │ │ -0008c180: 696f 6e61 6c69 7479 3a20 7472 7565 2020  ionality: true  
│ │ │ │ +0008c130: 2020 2020 7c0a 7c20 2020 2020 646f 6d69      |.|     domi
│ │ │ │ +0008c140: 6e61 6e63 653a 2074 7275 6520 2020 2020  nance: true     
│ │ │ │ +0008c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c160: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0008c170: 2020 2020 2062 6972 6174 696f 6e61 6c69       birationali
│ │ │ │ +0008c180: 7479 3a20 7472 7565 2020 2020 2020 2020  ty: true        
│ │ │ │  0008c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c1a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0008c1b0: 2020 2020 6e75 6d62 6572 206f 6620 6d69      number of mi
│ │ │ │ -0008c1c0: 6e69 6d61 6c20 7265 7072 6573 656e 7461  nimal representa
│ │ │ │ -0008c1d0: 7469 7665 733a 2031 2020 2020 2020 2020  tives: 1        
│ │ │ │ -0008c1e0: 2020 2020 207c 0a7c 2020 2020 2064 696d       |.|     dim
│ │ │ │ -0008c1f0: 656e 7369 6f6e 2062 6173 6520 6c6f 6375  ension base locu
│ │ │ │ -0008c200: 733a 2034 2020 2020 2020 2020 2020 2020  s: 4            
│ │ │ │ -0008c210: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0008c220: 7c20 2020 2020 6465 6772 6565 2062 6173  |     degree bas
│ │ │ │ -0008c230: 6520 6c6f 6375 733a 2036 2020 2020 2020  e locus: 6      
│ │ │ │ +0008c1a0: 2020 2020 2020 7c0a 7c20 2020 2020 6e75        |.|     nu
│ │ │ │ +0008c1b0: 6d62 6572 206f 6620 6d69 6e69 6d61 6c20  mber of minimal 
│ │ │ │ +0008c1c0: 7265 7072 6573 656e 7461 7469 7665 733a  representatives:
│ │ │ │ +0008c1d0: 2031 2020 2020 2020 2020 2020 2020 207c   1             |
│ │ │ │ +0008c1e0: 0a7c 2020 2020 2064 696d 656e 7369 6f6e  .|     dimension
│ │ │ │ +0008c1f0: 2062 6173 6520 6c6f 6375 733a 2034 2020   base locus: 4  
│ │ │ │ +0008c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c210: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0008c220: 6465 6772 6565 2062 6173 6520 6c6f 6375  degree base locu
│ │ │ │ +0008c230: 733a 2036 2020 2020 2020 2020 2020 2020  s: 6            
│ │ │ │  0008c240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c250: 2020 2020 2020 207c 0a7c 2020 2020 2063         |.|     c
│ │ │ │ -0008c260: 6f65 6666 6963 6965 6e74 2072 696e 673a  oefficient ring:
│ │ │ │ -0008c270: 2051 5120 2020 2020 2020 2020 2020 2020   QQ             
│ │ │ │ -0008c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c290: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0008c250: 207c 0a7c 2020 2020 2063 6f65 6666 6963   |.|     coeffic
│ │ │ │ +0008c260: 6965 6e74 2072 696e 673a 2051 5120 2020  ient ring: QQ   
│ │ │ │ +0008c270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008c280: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0008c290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008c2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008c2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008c2c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ -0008c2d0: 2064 6573 6372 6962 6520 7175 6164 726f   describe quadro
│ │ │ │ -0008c2e0: 5175 6164 7269 6343 7265 6d6f 6e61 5472  QuadricCremonaTr
│ │ │ │ -0008c2f0: 616e 7366 6f72 6d61 7469 6f6e 2831 342c  ansformation(14,
│ │ │ │ -0008c300: 3129 7c0a 7c20 2020 2020 2020 2020 2020  1)|.|           
│ │ │ │ +0008c2c0: 2d2d 2d2b 0a7c 6935 203a 2064 6573 6372  ---+.|i5 : descr
│ │ │ │ +0008c2d0: 6962 6520 7175 6164 726f 5175 6164 7269  ibe quadroQuadri
│ │ │ │ +0008c2e0: 6343 7265 6d6f 6e61 5472 616e 7366 6f72  cCremonaTransfor
│ │ │ │ +0008c2f0: 6d61 7469 6f6e 2831 342c 3129 7c0a 7c20  mation(14,1)|.| 
│ │ │ │ +0008c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008c330: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ -0008c340: 203d 2072 6174 696f 6e61 6c20 6d61 7020   = rational map 
│ │ │ │ -0008c350: 6465 6669 6e65 6420 6279 2066 6f72 6d73  defined by forms
│ │ │ │ -0008c360: 206f 6620 6465 6772 6565 2032 2020 2020   of degree 2    
│ │ │ │ -0008c370: 2020 2020 7c0a 7c20 2020 2020 736f 7572      |.|     sour
│ │ │ │ -0008c380: 6365 2076 6172 6965 7479 3a20 5050 5e31  ce variety: PP^1
│ │ │ │ -0008c390: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -0008c3a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0008c3b0: 2020 2020 2074 6172 6765 7420 7661 7269       target vari
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│ │ │ │ -0008c3e0: 2020 2020 2020 7c0a 7c20 2020 2020 646f        |.|     do
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│ │ │ │ +0008c4f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -0008c6d0: 2020 2062 6972 6174 696f 6e61 6c69 7479     birationality
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│ │ │ │ +0008ca30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  0008ca40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0008ca50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +0008d1a0: 6d65 7468 6f64 7320 7265 7475 726e 696e  methods returnin
│ │ │ │ +0008d1b0: 6720 6120 7261 7469 6f6e 616c 206d 6170  g a rational map
│ │ │ │ +0008d1c0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │  0008d1d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  0008d1e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0008d1f0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │ -0008d200: 7465 2071 7561 6472 6f51 7561 6472 6963  te quadroQuadric
│ │ │ │ -0008d210: 4372 656d 6f6e 6154 7261 6e73 666f 726d  CremonaTransform
│ │ │ │ -0008d220: 6174 696f 6e3a 0a20 2020 2071 7561 6472  ation:.    quadr
│ │ │ │ -0008d230: 6f51 7561 6472 6963 4372 656d 6f6e 6154  oQuadricCremonaT
│ │ │ │ -0008d240: 7261 6e73 666f 726d 6174 696f 6e2c 202d  ransformation, -
│ │ │ │ -0008d250: 2d20 7175 6164 726f 2d71 7561 6472 6963  - quadro-quadric
│ │ │ │ -0008d260: 2043 7265 6d6f 6e61 0a20 2020 2074 7261   Cremona.    tra
│ │ │ │ -0008d270: 6e73 666f 726d 6174 696f 6e73 0a20 202a  nsformations.  *
│ │ │ │ -0008d280: 202a 6e6f 7465 2073 6567 7265 3a20 7365   *note segre: se
│ │ │ │ -0008d290: 6772 652c 202d 2d20 5365 6772 6520 656d  gre, -- Segre em
│ │ │ │ -0008d2a0: 6265 6464 696e 670a 2020 2a20 2a6e 6f74  bedding.  * *not
│ │ │ │ -0008d2b0: 6520 7370 6563 6961 6c43 7265 6d6f 6e61  e specialCremona
│ │ │ │ -0008d2c0: 5472 616e 7366 6f72 6d61 7469 6f6e 3a20  Transformation: 
│ │ │ │ -0008d2d0: 7370 6563 6961 6c43 7265 6d6f 6e61 5472  specialCremonaTr
│ │ │ │ -0008d2e0: 616e 7366 6f72 6d61 7469 6f6e 2c20 2d2d  ansformation, --
│ │ │ │ -0008d2f0: 0a20 2020 2073 7065 6369 616c 2043 7265  .    special Cre
│ │ │ │ -0008d300: 6d6f 6e61 2074 7261 6e73 666f 726d 6174  mona transformat
│ │ │ │ -0008d310: 696f 6e73 2077 686f 7365 2062 6173 6520  ions whose base 
│ │ │ │ -0008d320: 6c6f 6375 7320 6861 7320 6469 6d65 6e73  locus has dimens
│ │ │ │ -0008d330: 696f 6e20 6174 206d 6f73 740a 2020 2020  ion at most.    
│ │ │ │ -0008d340: 7468 7265 650a 2020 2a20 2a6e 6f74 6520  three.  * *note 
│ │ │ │ -0008d350: 7370 6563 6961 6c43 7562 6963 5472 616e  specialCubicTran
│ │ │ │ -0008d360: 7366 6f72 6d61 7469 6f6e 3a20 7370 6563  sformation: spec
│ │ │ │ -0008d370: 6961 6c43 7562 6963 5472 616e 7366 6f72  ialCubicTransfor
│ │ │ │ -0008d380: 6d61 7469 6f6e 2c20 2d2d 2073 7065 6369  mation, -- speci
│ │ │ │ -0008d390: 616c 0a20 2020 2063 7562 6963 2074 7261  al.    cubic tra
│ │ │ │ -0008d3a0: 6e73 666f 726d 6174 696f 6e73 2077 686f  nsformations who
│ │ │ │ -0008d3b0: 7365 2062 6173 6520 6c6f 6375 7320 6861  se base locus ha
│ │ │ │ -0008d3c0: 7320 6469 6d65 6e73 696f 6e20 6174 206d  s dimension at m
│ │ │ │ -0008d3d0: 6f73 7420 7468 7265 650a 2020 2a20 2a6e  ost three.  * *n
│ │ │ │ -0008d3e0: 6f74 6520 7370 6563 6961 6c51 7561 6472  ote specialQuadr
│ │ │ │ -0008d3f0: 6174 6963 5472 616e 7366 6f72 6d61 7469  aticTransformati
│ │ │ │ -0008d400: 6f6e 3a20 7370 6563 6961 6c51 7561 6472  on: specialQuadr
│ │ │ │ -0008d410: 6174 6963 5472 616e 7366 6f72 6d61 7469  aticTransformati
│ │ │ │ -0008d420: 6f6e 2c20 2d2d 0a20 2020 2073 7065 6369  on, --.    speci
│ │ │ │ -0008d430: 616c 2071 7561 6472 6174 6963 2074 7261  al quadratic tra
│ │ │ │ -0008d440: 6e73 666f 726d 6174 696f 6e73 2077 686f  nsformations who
│ │ │ │ -0008d450: 7365 2062 6173 6520 6c6f 6375 7320 6861  se base locus ha
│ │ │ │ -0008d460: 7320 6469 6d65 6e73 696f 6e20 7468 7265  s dimension thre
│ │ │ │ -0008d470: 650a 0a4d 6574 686f 6473 2074 6861 7420  e..Methods that 
│ │ │ │ -0008d480: 7573 6520 6120 7261 7469 6f6e 616c 206d  use a rational m
│ │ │ │ -0008d490: 6170 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ap:.============
│ │ │ │ -0008d4a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0008d4b0: 3d3d 3d3d 0a0a 2020 2a20 2261 6273 7472  ====..  * "abstr
│ │ │ │ -0008d4c0: 6163 7452 6174 696f 6e61 6c4d 6170 2852  actRationalMap(R
│ │ │ │ -0008d4d0: 6174 696f 6e61 6c4d 6170 2922 202d 2d20  ationalMap)" -- 
│ │ │ │ -0008d4e0: 7365 6520 2a6e 6f74 6520 6162 7374 7261  see *note abstra
│ │ │ │ -0008d4f0: 6374 5261 7469 6f6e 616c 4d61 703a 0a20  ctRationalMap:. 
│ │ │ │ -0008d500: 2020 2061 6273 7472 6163 7452 6174 696f     abstractRatio
│ │ │ │ -0008d510: 6e61 6c4d 6170 2c20 2d2d 206d 616b 6520  nalMap, -- make 
│ │ │ │ -0008d520: 616e 2061 6273 7472 6163 7420 7261 7469  an abstract rati
│ │ │ │ -0008d530: 6f6e 616c 206d 6170 0a20 202a 2022 6170  onal map.  * "ap
│ │ │ │ -0008d540: 7072 6f78 696d 6174 6549 6e76 6572 7365  proximateInverse
│ │ │ │ -0008d550: 4d61 7028 5261 7469 6f6e 616c 4d61 7029  Map(RationalMap)
│ │ │ │ -0008d560: 2220 2d2d 2073 6565 202a 6e6f 7465 2061  " -- see *note a
│ │ │ │ -0008d570: 7070 726f 7869 6d61 7465 496e 7665 7273  pproximateInvers
│ │ │ │ -0008d580: 654d 6170 3a0a 2020 2020 6170 7072 6f78  eMap:.    approx
│ │ │ │ -0008d590: 696d 6174 6549 6e76 6572 7365 4d61 702c  imateInverseMap,
│ │ │ │ -0008d5a0: 202d 2d20 7261 6e64 6f6d 206d 6170 2072   -- random map r
│ │ │ │ -0008d5b0: 656c 6174 6564 2074 6f20 7468 6520 696e  elated to the in
│ │ │ │ -0008d5c0: 7665 7273 6520 6f66 2061 2062 6972 6174  verse of a birat
│ │ │ │ -0008d5d0: 696f 6e61 6c0a 2020 2020 6d61 700a 2020  ional.    map.  
│ │ │ │ -0008d5e0: 2a20 2261 7070 726f 7869 6d61 7465 496e  * "approximateIn
│ │ │ │ -0008d5f0: 7665 7273 654d 6170 2852 6174 696f 6e61  verseMap(Rationa
│ │ │ │ -0008d600: 6c4d 6170 2c5a 5a29 2220 2d2d 2073 6565  lMap,ZZ)" -- see
│ │ │ │ -0008d610: 202a 6e6f 7465 2061 7070 726f 7869 6d61   *note approxima
│ │ │ │ -0008d620: 7465 496e 7665 7273 654d 6170 3a0a 2020  teInverseMap:.  
│ │ │ │ -0008d630: 2020 6170 7072 6f78 696d 6174 6549 6e76    approximateInv
│ │ │ │ -0008d640: 6572 7365 4d61 702c 202d 2d20 7261 6e64  erseMap, -- rand
│ │ │ │ -0008d650: 6f6d 206d 6170 2072 656c 6174 6564 2074  om map related t
│ │ │ │ -0008d660: 6f20 7468 6520 696e 7665 7273 6520 6f66  o the inverse of
│ │ │ │ -0008d670: 2061 2062 6972 6174 696f 6e61 6c0a 2020   a birational.  
│ │ │ │ -0008d680: 2020 6d61 700a 2020 2a20 2a6e 6f74 6520    map.  * *note 
│ │ │ │ -0008d690: 636f 6566 6669 6369 656e 7452 696e 6728  coefficientRing(
│ │ │ │ -0008d6a0: 5261 7469 6f6e 616c 4d61 7029 3a20 636f  RationalMap): co
│ │ │ │ -0008d6b0: 6566 6669 6369 656e 7452 696e 675f 6c70  efficientRing_lp
│ │ │ │ -0008d6c0: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ -0008d6d0: 2d2d 0a20 2020 2063 6f65 6666 6963 6965  --.    coefficie
│ │ │ │ -0008d6e0: 6e74 2072 696e 6720 6f66 2061 2072 6174  nt ring of a rat
│ │ │ │ -0008d6f0: 696f 6e61 6c20 6d61 700a 2020 2a20 2a6e  ional map.  * *n
│ │ │ │ -0008d700: 6f74 6520 636f 6566 6669 6369 656e 7473  ote coefficients
│ │ │ │ -0008d710: 2852 6174 696f 6e61 6c4d 6170 293a 2063  (RationalMap): c
│ │ │ │ -0008d720: 6f65 6666 6963 6965 6e74 735f 6c70 5261  oefficients_lpRa
│ │ │ │ -0008d730: 7469 6f6e 616c 4d61 705f 7270 2c20 2d2d  tionalMap_rp, --
│ │ │ │ -0008d740: 0a20 2020 2063 6f65 6666 6963 6965 6e74  .    coefficient
│ │ │ │ -0008d750: 206d 6174 7269 7820 6f66 2061 2072 6174   matrix of a rat
│ │ │ │ -0008d760: 696f 6e61 6c20 6d61 700a 2020 2a20 2a6e  ional map.  * *n
│ │ │ │ -0008d770: 6f74 6520 6465 6772 6565 2852 6174 696f  ote degree(Ratio
│ │ │ │ -0008d780: 6e61 6c4d 6170 293a 2064 6567 7265 655f  nalMap): degree_
│ │ │ │ -0008d790: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ -0008d7a0: 2c20 2d2d 2064 6567 7265 6520 6f66 2061  , -- degree of a
│ │ │ │ -0008d7b0: 2072 6174 696f 6e61 6c0a 2020 2020 6d61   rational.    ma
│ │ │ │ -0008d7c0: 700a 2020 2a20 2a6e 6f74 6520 6465 6772  p.  * *note degr
│ │ │ │ -0008d7d0: 6565 4d61 7028 5261 7469 6f6e 616c 4d61  eeMap(RationalMa
│ │ │ │ -0008d7e0: 7029 3a20 6465 6772 6565 4d61 705f 6c70  p): degreeMap_lp
│ │ │ │ -0008d7f0: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ -0008d800: 2d2d 2064 6567 7265 6520 6f66 2061 0a20  -- degree of a. 
│ │ │ │ -0008d810: 2020 2072 6174 696f 6e61 6c20 6d61 700a     rational map.
│ │ │ │ -0008d820: 2020 2a20 2a6e 6f74 6520 6465 6772 6565    * *note degree
│ │ │ │ -0008d830: 7328 5261 7469 6f6e 616c 4d61 7029 3a20  s(RationalMap): 
│ │ │ │ -0008d840: 6465 6772 6565 735f 6c70 5261 7469 6f6e  degrees_lpRation
│ │ │ │ -0008d850: 616c 4d61 705f 7270 2c20 2d2d 2070 726f  alMap_rp, -- pro
│ │ │ │ -0008d860: 6a65 6374 6976 6520 6465 6772 6565 730a  jective degrees.
│ │ │ │ -0008d870: 2020 2020 6f66 2061 2072 6174 696f 6e61      of a rationa
│ │ │ │ -0008d880: 6c20 6d61 700a 2020 2a20 226d 756c 7469  l map.  * "multi
│ │ │ │ -0008d890: 6465 6772 6565 2852 6174 696f 6e61 6c4d  degree(RationalM
│ │ │ │ -0008d8a0: 6170 2922 202d 2d20 7365 6520 2a6e 6f74  ap)" -- see *not
│ │ │ │ -0008d8b0: 6520 6465 6772 6565 7328 5261 7469 6f6e  e degrees(Ration
│ │ │ │ -0008d8c0: 616c 4d61 7029 3a0a 2020 2020 6465 6772  alMap):.    degr
│ │ │ │ -0008d8d0: 6565 735f 6c70 5261 7469 6f6e 616c 4d61  ees_lpRationalMa
│ │ │ │ -0008d8e0: 705f 7270 2c20 2d2d 2070 726f 6a65 6374  p_rp, -- project
│ │ │ │ -0008d8f0: 6976 6520 6465 6772 6565 7320 6f66 2061  ive degrees of a
│ │ │ │ -0008d900: 2072 6174 696f 6e61 6c20 6d61 700a 2020   rational map.  
│ │ │ │ -0008d910: 2a20 2a6e 6f74 6520 6465 7363 7269 6265  * *note describe
│ │ │ │ -0008d920: 2852 6174 696f 6e61 6c4d 6170 293a 2064  (RationalMap): d
│ │ │ │ -0008d930: 6573 6372 6962 655f 6c70 5261 7469 6f6e  escribe_lpRation
│ │ │ │ -0008d940: 616c 4d61 705f 7270 2c20 2d2d 2064 6573  alMap_rp, -- des
│ │ │ │ -0008d950: 6372 6962 6520 610a 2020 2020 7261 7469  cribe a.    rati
│ │ │ │ -0008d960: 6f6e 616c 206d 6170 0a20 202a 202a 6e6f  onal map.  * *no
│ │ │ │ -0008d970: 7465 2065 6e74 7269 6573 2852 6174 696f  te entries(Ratio
│ │ │ │ -0008d980: 6e61 6c4d 6170 293a 2065 6e74 7269 6573  nalMap): entries
│ │ │ │ -0008d990: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ -0008d9a0: 702c 202d 2d20 7468 6520 656e 7472 6965  p, -- the entrie
│ │ │ │ -0008d9b0: 7320 6f66 2074 6865 0a20 2020 206d 6174  s of the.    mat
│ │ │ │ -0008d9c0: 7269 7820 6173 736f 6369 6174 6564 2074  rix associated t
│ │ │ │ -0008d9d0: 6f20 6120 7261 7469 6f6e 616c 206d 6170  o a rational map
│ │ │ │ -0008d9e0: 0a20 202a 2022 6578 6365 7074 696f 6e61  .  * "exceptiona
│ │ │ │ -0008d9f0: 6c4c 6f63 7573 2852 6174 696f 6e61 6c4d  lLocus(RationalM
│ │ │ │ -0008da00: 6170 2922 202d 2d20 7365 6520 2a6e 6f74  ap)" -- see *not
│ │ │ │ -0008da10: 6520 6578 6365 7074 696f 6e61 6c4c 6f63  e exceptionalLoc
│ │ │ │ -0008da20: 7573 3a0a 2020 2020 6578 6365 7074 696f  us:.    exceptio
│ │ │ │ -0008da30: 6e61 6c4c 6f63 7573 2c20 2d2d 2065 7863  nalLocus, -- exc
│ │ │ │ -0008da40: 6570 7469 6f6e 616c 206c 6f63 7573 206f  eptional locus o
│ │ │ │ -0008da50: 6620 6120 6269 7261 7469 6f6e 616c 206d  f a birational m
│ │ │ │ -0008da60: 6170 0a20 202a 202a 6e6f 7465 2066 6c61  ap.  * *note fla
│ │ │ │ -0008da70: 7474 656e 2852 6174 696f 6e61 6c4d 6170  tten(RationalMap
│ │ │ │ -0008da80: 293a 2066 6c61 7474 656e 5f6c 7052 6174  ): flatten_lpRat
│ │ │ │ -0008da90: 696f 6e61 6c4d 6170 5f72 702c 202d 2d20  ionalMap_rp, -- 
│ │ │ │ -0008daa0: 7772 6974 6520 736f 7572 6365 2061 6e64  write source and
│ │ │ │ -0008dab0: 0a20 2020 2074 6172 6765 7420 6173 206e  .    target as n
│ │ │ │ -0008dac0: 6f6e 6465 6765 6e65 7261 7465 2076 6172  ondegenerate var
│ │ │ │ -0008dad0: 6965 7469 6573 0a20 202a 2022 666f 7263  ieties.  * "forc
│ │ │ │ -0008dae0: 6549 6d61 6765 2852 6174 696f 6e61 6c4d  eImage(RationalM
│ │ │ │ -0008daf0: 6170 2c49 6465 616c 2922 202d 2d20 7365  ap,Ideal)" -- se
│ │ │ │ -0008db00: 6520 2a6e 6f74 6520 666f 7263 6549 6d61  e *note forceIma
│ │ │ │ -0008db10: 6765 3a20 666f 7263 6549 6d61 6765 2c20  ge: forceImage, 
│ │ │ │ -0008db20: 2d2d 0a20 2020 2064 6563 6c61 7265 2077  --.    declare w
│ │ │ │ -0008db30: 6869 6368 2069 7320 7468 6520 696d 6167  hich is the imag
│ │ │ │ -0008db40: 6520 6f66 2061 2072 6174 696f 6e61 6c20  e of a rational 
│ │ │ │ -0008db50: 6d61 700a 2020 2a20 2266 6f72 6365 496e  map.  * "forceIn
│ │ │ │ -0008db60: 7665 7273 654d 6170 2852 6174 696f 6e61  verseMap(Rationa
│ │ │ │ -0008db70: 6c4d 6170 2c52 6174 696f 6e61 6c4d 6170  lMap,RationalMap
│ │ │ │ -0008db80: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -0008db90: 666f 7263 6549 6e76 6572 7365 4d61 703a  forceInverseMap:
│ │ │ │ -0008dba0: 0a20 2020 2066 6f72 6365 496e 7665 7273  .    forceInvers
│ │ │ │ -0008dbb0: 654d 6170 2c20 2d2d 2064 6563 6c61 7265  eMap, -- declare
│ │ │ │ -0008dbc0: 2074 6861 7420 7477 6f20 7261 7469 6f6e   that two ration
│ │ │ │ -0008dbd0: 616c 206d 6170 7320 6172 6520 6f6e 6520  al maps are one 
│ │ │ │ -0008dbe0: 7468 6520 696e 7665 7273 6520 6f66 0a20  the inverse of. 
│ │ │ │ -0008dbf0: 2020 2074 6865 206f 7468 6572 0a20 202a     the other.  *
│ │ │ │ -0008dc00: 2022 6772 6170 6828 5261 7469 6f6e 616c   "graph(Rational
│ │ │ │ -0008dc10: 4d61 7029 2220 2d2d 2073 6565 202a 6e6f  Map)" -- see *no
│ │ │ │ -0008dc20: 7465 2067 7261 7068 3a20 6772 6170 682c  te graph: graph,
│ │ │ │ -0008dc30: 202d 2d20 636c 6f73 7572 6520 6f66 2074   -- closure of t
│ │ │ │ -0008dc40: 6865 2067 7261 7068 206f 660a 2020 2020  he graph of.    
│ │ │ │ -0008dc50: 6120 7261 7469 6f6e 616c 206d 6170 0a20  a rational map. 
│ │ │ │ -0008dc60: 202a 202a 6e6f 7465 2069 6465 616c 2852   * *note ideal(R
│ │ │ │ -0008dc70: 6174 696f 6e61 6c4d 6170 293a 2069 6465  ationalMap): ide
│ │ │ │ -0008dc80: 616c 5f6c 7052 6174 696f 6e61 6c4d 6170  al_lpRationalMap
│ │ │ │ -0008dc90: 5f72 702c 202d 2d20 6261 7365 206c 6f63  _rp, -- base loc
│ │ │ │ -0008dca0: 7573 206f 6620 610a 2020 2020 7261 7469  us of a.    rati
│ │ │ │ -0008dcb0: 6f6e 616c 206d 6170 0a20 202a 202a 6e6f  onal map.  * *no
│ │ │ │ -0008dcc0: 7465 2069 6d61 6765 2852 6174 696f 6e61  te image(Rationa
│ │ │ │ -0008dcd0: 6c4d 6170 2c53 7472 696e 6729 3a20 696d  lMap,String): im
│ │ │ │ -0008dce0: 6167 655f 6c70 5261 7469 6f6e 616c 4d61  age_lpRationalMa
│ │ │ │ -0008dcf0: 705f 636d 5374 7269 6e67 5f72 702c 202d  p_cmString_rp, -
│ │ │ │ -0008dd00: 2d0a 2020 2020 636c 6f73 7572 6520 6f66  -.    closure of
│ │ │ │ -0008dd10: 2074 6865 2069 6d61 6765 206f 6620 6120   the image of a 
│ │ │ │ -0008dd20: 7261 7469 6f6e 616c 206d 6170 2075 7369  rational map usi
│ │ │ │ -0008dd30: 6e67 2074 6865 2046 3420 616c 676f 7269  ng the F4 algori
│ │ │ │ -0008dd40: 7468 6d0a 2020 2020 2865 7870 6572 696d  thm.    (experim
│ │ │ │ -0008dd50: 656e 7461 6c29 0a20 202a 2022 696d 6167  ental).  * "imag
│ │ │ │ -0008dd60: 6528 5261 7469 6f6e 616c 4d61 7029 2220  e(RationalMap)" 
│ │ │ │ -0008dd70: 2d2d 2073 6565 202a 6e6f 7465 2069 6d61  -- see *note ima
│ │ │ │ -0008dd80: 6765 2852 6174 696f 6e61 6c4d 6170 2c5a  ge(RationalMap,Z
│ │ │ │ -0008dd90: 5a29 3a0a 2020 2020 696d 6167 655f 6c70  Z):.    image_lp
│ │ │ │ -0008dda0: 5261 7469 6f6e 616c 4d61 705f 636d 5a5a  RationalMap_cmZZ
│ │ │ │ -0008ddb0: 5f72 702c 202d 2d20 636c 6f73 7572 6520  _rp, -- closure 
│ │ │ │ -0008ddc0: 6f66 2074 6865 2069 6d61 6765 206f 6620  of the image of 
│ │ │ │ -0008ddd0: 6120 7261 7469 6f6e 616c 206d 6170 0a20  a rational map. 
│ │ │ │ -0008dde0: 202a 202a 6e6f 7465 2069 6d61 6765 2852   * *note image(R
│ │ │ │ -0008ddf0: 6174 696f 6e61 6c4d 6170 2c5a 5a29 3a20  ationalMap,ZZ): 
│ │ │ │ -0008de00: 696d 6167 655f 6c70 5261 7469 6f6e 616c  image_lpRational
│ │ │ │ -0008de10: 4d61 705f 636d 5a5a 5f72 702c 202d 2d20  Map_cmZZ_rp, -- 
│ │ │ │ -0008de20: 636c 6f73 7572 6520 6f66 2074 6865 0a20  closure of the. 
│ │ │ │ -0008de30: 2020 2069 6d61 6765 206f 6620 6120 7261     image of a ra
│ │ │ │ -0008de40: 7469 6f6e 616c 206d 6170 0a20 202a 202a  tional map.  * *
│ │ │ │ -0008de50: 6e6f 7465 2069 6e76 6572 7365 2852 6174  note inverse(Rat
│ │ │ │ -0008de60: 696f 6e61 6c4d 6170 293a 2069 6e76 6572  ionalMap): inver
│ │ │ │ -0008de70: 7365 5f6c 7052 6174 696f 6e61 6c4d 6170  se_lpRationalMap
│ │ │ │ -0008de80: 5f72 702c 202d 2d20 696e 7665 7273 6520  _rp, -- inverse 
│ │ │ │ -0008de90: 6f66 2061 0a20 2020 2062 6972 6174 696f  of a.    biratio
│ │ │ │ -0008dea0: 6e61 6c20 6d61 700a 2020 2a20 2269 6e76  nal map.  * "inv
│ │ │ │ -0008deb0: 6572 7365 2852 6174 696f 6e61 6c4d 6170  erse(RationalMap
│ │ │ │ -0008dec0: 2c4f 7074 696f 6e29 2220 2d2d 2073 6565  ,Option)" -- see
│ │ │ │ -0008ded0: 202a 6e6f 7465 2069 6e76 6572 7365 2852   *note inverse(R
│ │ │ │ -0008dee0: 6174 696f 6e61 6c4d 6170 293a 0a20 2020  ationalMap):.   
│ │ │ │ -0008def0: 2069 6e76 6572 7365 5f6c 7052 6174 696f   inverse_lpRatio
│ │ │ │ -0008df00: 6e61 6c4d 6170 5f72 702c 202d 2d20 696e  nalMap_rp, -- in
│ │ │ │ -0008df10: 7665 7273 6520 6f66 2061 2062 6972 6174  verse of a birat
│ │ │ │ -0008df20: 696f 6e61 6c20 6d61 700a 2020 2a20 2269  ional map.  * "i
│ │ │ │ -0008df30: 6e76 6572 7365 4d61 7028 5261 7469 6f6e  nverseMap(Ration
│ │ │ │ -0008df40: 616c 4d61 7029 2220 2d2d 2073 6565 202a  alMap)" -- see *
│ │ │ │ -0008df50: 6e6f 7465 2069 6e76 6572 7365 4d61 703a  note inverseMap:
│ │ │ │ -0008df60: 2069 6e76 6572 7365 4d61 702c 202d 2d20   inverseMap, -- 
│ │ │ │ -0008df70: 696e 7665 7273 650a 2020 2020 6f66 2061  inverse.    of a
│ │ │ │ -0008df80: 2062 6972 6174 696f 6e61 6c20 6d61 700a   birational map.
│ │ │ │ -0008df90: 2020 2a20 2269 7342 6972 6174 696f 6e61    * "isBirationa
│ │ │ │ -0008dfa0: 6c28 5261 7469 6f6e 616c 4d61 7029 2220  l(RationalMap)" 
│ │ │ │ -0008dfb0: 2d2d 2073 6565 202a 6e6f 7465 2069 7342  -- see *note isB
│ │ │ │ -0008dfc0: 6972 6174 696f 6e61 6c3a 2069 7342 6972  irational: isBir
│ │ │ │ -0008dfd0: 6174 696f 6e61 6c2c 202d 2d0a 2020 2020  ational, --.    
│ │ │ │ -0008dfe0: 7768 6574 6865 7220 6120 7261 7469 6f6e  whether a ration
│ │ │ │ -0008dff0: 616c 206d 6170 2069 7320 6269 7261 7469  al map is birati
│ │ │ │ -0008e000: 6f6e 616c 0a20 202a 2022 6973 446f 6d69  onal.  * "isDomi
│ │ │ │ -0008e010: 6e61 6e74 2852 6174 696f 6e61 6c4d 6170  nant(RationalMap
│ │ │ │ -0008e020: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -0008e030: 6973 446f 6d69 6e61 6e74 3a20 6973 446f  isDominant: isDo
│ │ │ │ -0008e040: 6d69 6e61 6e74 2c20 2d2d 2077 6865 7468  minant, -- wheth
│ │ │ │ -0008e050: 6572 2061 0a20 2020 2072 6174 696f 6e61  er a.    rationa
│ │ │ │ -0008e060: 6c20 6d61 7020 6973 2064 6f6d 696e 616e  l map is dominan
│ │ │ │ -0008e070: 740a 2020 2a20 2a6e 6f74 6520 6973 496e  t.  * *note isIn
│ │ │ │ -0008e080: 7665 7273 654d 6170 2852 6174 696f 6e61  verseMap(Rationa
│ │ │ │ -0008e090: 6c4d 6170 2c52 6174 696f 6e61 6c4d 6170  lMap,RationalMap
│ │ │ │ -0008e0a0: 293a 0a20 2020 2069 7349 6e76 6572 7365  ):.    isInverse
│ │ │ │ -0008e0b0: 4d61 705f 6c70 5261 7469 6f6e 616c 4d61  Map_lpRationalMa
│ │ │ │ -0008e0c0: 705f 636d 5261 7469 6f6e 616c 4d61 705f  p_cmRationalMap_
│ │ │ │ -0008e0d0: 7270 2c20 2d2d 2063 6865 636b 7320 7768  rp, -- checks wh
│ │ │ │ -0008e0e0: 6574 6865 7220 7477 6f20 7261 7469 6f6e  ether two ration
│ │ │ │ -0008e0f0: 616c 0a20 2020 206d 6170 7320 6172 6520  al.    maps are 
│ │ │ │ -0008e100: 6f6e 6520 7468 6520 696e 7665 7273 6520  one the inverse 
│ │ │ │ -0008e110: 6f66 2074 6865 206f 7468 6572 0a20 202a  of the other.  *
│ │ │ │ -0008e120: 202a 6e6f 7465 2069 7349 736f 6d6f 7270   *note isIsomorp
│ │ │ │ -0008e130: 6869 736d 2852 6174 696f 6e61 6c4d 6170  hism(RationalMap
│ │ │ │ -0008e140: 293a 2069 7349 736f 6d6f 7270 6869 736d  ): isIsomorphism
│ │ │ │ -0008e150: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ -0008e160: 702c 202d 2d0a 2020 2020 7768 6574 6865  p, --.    whethe
│ │ │ │ -0008e170: 7220 6120 6269 7261 7469 6f6e 616c 206d  r a birational m
│ │ │ │ -0008e180: 6170 2069 7320 616e 2069 736f 6d6f 7270  ap is an isomorp
│ │ │ │ -0008e190: 6869 736d 0a20 202a 2022 6973 4d6f 7270  hism.  * "isMorp
│ │ │ │ -0008e1a0: 6869 736d 2852 6174 696f 6e61 6c4d 6170  hism(RationalMap
│ │ │ │ -0008e1b0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -0008e1c0: 6973 4d6f 7270 6869 736d 3a20 6973 4d6f  isMorphism: isMo
│ │ │ │ -0008e1d0: 7270 6869 736d 2c20 2d2d 2077 6865 7468  rphism, -- wheth
│ │ │ │ -0008e1e0: 6572 2061 0a20 2020 2072 6174 696f 6e61  er a.    rationa
│ │ │ │ -0008e1f0: 6c20 6d61 7020 6973 2061 206d 6f72 7068  l map is a morph
│ │ │ │ -0008e200: 6973 6d0a 2020 2a20 2a6e 6f74 6520 6d61  ism.  * *note ma
│ │ │ │ -0008e210: 7028 5261 7469 6f6e 616c 4d61 7029 3a20  p(RationalMap): 
│ │ │ │ -0008e220: 6d61 705f 6c70 5261 7469 6f6e 616c 4d61  map_lpRationalMa
│ │ │ │ -0008e230: 705f 7270 2c20 2d2d 2067 6574 2074 6865  p_rp, -- get the
│ │ │ │ -0008e240: 2072 696e 6720 6d61 7020 6465 6669 6e69   ring map defini
│ │ │ │ -0008e250: 6e67 0a20 2020 2061 2072 6174 696f 6e61  ng.    a rationa
│ │ │ │ -0008e260: 6c20 6d61 700a 2020 2a20 226d 6170 285a  l map.  * "map(Z
│ │ │ │ -0008e270: 5a2c 5261 7469 6f6e 616c 4d61 7029 2220  Z,RationalMap)" 
│ │ │ │ -0008e280: 2d2d 2073 6565 202a 6e6f 7465 206d 6170  -- see *note map
│ │ │ │ -0008e290: 2852 6174 696f 6e61 6c4d 6170 293a 206d  (RationalMap): m
│ │ │ │ -0008e2a0: 6170 5f6c 7052 6174 696f 6e61 6c4d 6170  ap_lpRationalMap
│ │ │ │ -0008e2b0: 5f72 702c 0a20 2020 202d 2d20 6765 7420  _rp,.    -- get 
│ │ │ │ -0008e2c0: 7468 6520 7269 6e67 206d 6170 2064 6566  the ring map def
│ │ │ │ -0008e2d0: 696e 696e 6720 6120 7261 7469 6f6e 616c  ining a rational
│ │ │ │ -0008e2e0: 206d 6170 0a20 202a 202a 6e6f 7465 206d   map.  * *note m
│ │ │ │ -0008e2f0: 6174 7269 7828 5261 7469 6f6e 616c 4d61  atrix(RationalMa
│ │ │ │ -0008e300: 7029 3a20 6d61 7472 6978 5f6c 7052 6174  p): matrix_lpRat
│ │ │ │ -0008e310: 696f 6e61 6c4d 6170 5f72 702c 202d 2d20  ionalMap_rp, -- 
│ │ │ │ -0008e320: 7468 6520 6d61 7472 6978 0a20 2020 2061  the matrix.    a
│ │ │ │ -0008e330: 7373 6f63 6961 7465 6420 746f 2061 2072  ssociated to a r
│ │ │ │ -0008e340: 6174 696f 6e61 6c20 6d61 700a 2020 2a20  ational map.  * 
│ │ │ │ -0008e350: 226d 6174 7269 7828 5a5a 2c52 6174 696f  "matrix(ZZ,Ratio
│ │ │ │ -0008e360: 6e61 6c4d 6170 2922 202d 2d20 7365 6520  nalMap)" -- see 
│ │ │ │ -0008e370: 2a6e 6f74 6520 6d61 7472 6978 2852 6174  *note matrix(Rat
│ │ │ │ -0008e380: 696f 6e61 6c4d 6170 293a 0a20 2020 206d  ionalMap):.    m
│ │ │ │ -0008e390: 6174 7269 785f 6c70 5261 7469 6f6e 616c  atrix_lpRational
│ │ │ │ -0008e3a0: 4d61 705f 7270 2c20 2d2d 2074 6865 206d  Map_rp, -- the m
│ │ │ │ -0008e3b0: 6174 7269 7820 6173 736f 6369 6174 6564  atrix associated
│ │ │ │ -0008e3c0: 2074 6f20 6120 7261 7469 6f6e 616c 206d   to a rational m
│ │ │ │ -0008e3d0: 6170 0a20 202a 202a 6e6f 7465 2070 726f  ap.  * *note pro
│ │ │ │ -0008e3e0: 6a65 6374 6976 6544 6567 7265 6573 2852  jectiveDegrees(R
│ │ │ │ -0008e3f0: 6174 696f 6e61 6c4d 6170 293a 2070 726f  ationalMap): pro
│ │ │ │ -0008e400: 6a65 6374 6976 6544 6567 7265 6573 5f6c  jectiveDegrees_l
│ │ │ │ -0008e410: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ -0008e420: 0a20 2020 202d 2d20 7072 6f6a 6563 7469  .    -- projecti
│ │ │ │ -0008e430: 7665 2064 6567 7265 6573 206f 6620 6120  ve degrees of a 
│ │ │ │ -0008e440: 7261 7469 6f6e 616c 206d 6170 0a20 202a  rational map.  *
│ │ │ │ -0008e450: 202a 6e6f 7465 2052 6174 696f 6e61 6c4d   *note RationalM
│ │ │ │ -0008e460: 6170 2021 3a20 5261 7469 6f6e 616c 4d61  ap !: RationalMa
│ │ │ │ -0008e470: 7020 212c 202d 2d20 6361 6c63 756c 6174  p !, -- calculat
│ │ │ │ -0008e480: 6573 2065 7665 7279 2070 6f73 7369 626c  es every possibl
│ │ │ │ -0008e490: 6520 7468 696e 670a 2020 2a20 2263 6f6d  e thing.  * "com
│ │ │ │ -0008e4a0: 706f 7365 2852 6174 696f 6e61 6c4d 6170  pose(RationalMap
│ │ │ │ -0008e4b0: 2c52 6174 696f 6e61 6c4d 6170 2922 202d  ,RationalMap)" -
│ │ │ │ -0008e4c0: 2d20 7365 6520 2a6e 6f74 6520 5261 7469  - see *note Rati
│ │ │ │ -0008e4d0: 6f6e 616c 4d61 7020 2a20 5261 7469 6f6e  onalMap * Ration
│ │ │ │ -0008e4e0: 616c 4d61 703a 0a20 2020 2052 6174 696f  alMap:.    Ratio
│ │ │ │ -0008e4f0: 6e61 6c4d 6170 205f 7374 2052 6174 696f  nalMap _st Ratio
│ │ │ │ -0008e500: 6e61 6c4d 6170 2c20 2d2d 2063 6f6d 706f  nalMap, -- compo
│ │ │ │ -0008e510: 7369 7469 6f6e 206f 6620 7261 7469 6f6e  sition of ration
│ │ │ │ -0008e520: 616c 206d 6170 730a 2020 2a20 2a6e 6f74  al maps.  * *not
│ │ │ │ -0008e530: 6520 5261 7469 6f6e 616c 4d61 7020 2a20  e RationalMap * 
│ │ │ │ -0008e540: 5261 7469 6f6e 616c 4d61 703a 2052 6174  RationalMap: Rat
│ │ │ │ -0008e550: 696f 6e61 6c4d 6170 205f 7374 2052 6174  ionalMap _st Rat
│ │ │ │ -0008e560: 696f 6e61 6c4d 6170 2c20 2d2d 0a20 2020  ionalMap, --.   
│ │ │ │ -0008e570: 2063 6f6d 706f 7369 7469 6f6e 206f 6620   composition of 
│ │ │ │ -0008e580: 7261 7469 6f6e 616c 206d 6170 730a 2020  rational maps.  
│ │ │ │ -0008e590: 2a20 2a6e 6f74 6520 5261 7469 6f6e 616c  * *note Rational
│ │ │ │ -0008e5a0: 4d61 7020 2a2a 2052 696e 673a 2052 6174  Map ** Ring: Rat
│ │ │ │ -0008e5b0: 696f 6e61 6c4d 6170 205f 7374 5f73 7420  ionalMap _st_st 
│ │ │ │ -0008e5c0: 5269 6e67 2c20 2d2d 2063 6861 6e67 6520  Ring, -- change 
│ │ │ │ -0008e5d0: 7468 650a 2020 2020 636f 6566 6669 6369  the.    coeffici
│ │ │ │ -0008e5e0: 656e 7420 7269 6e67 206f 6620 6120 7261  ent ring of a ra
│ │ │ │ -0008e5f0: 7469 6f6e 616c 206d 6170 0a20 202a 202a  tional map.  * *
│ │ │ │ -0008e600: 6e6f 7465 2052 6174 696f 6e61 6c4d 6170  note RationalMap
│ │ │ │ -0008e610: 203d 3d20 5261 7469 6f6e 616c 4d61 703a   == RationalMap:
│ │ │ │ -0008e620: 2052 6174 696f 6e61 6c4d 6170 203d 3d20   RationalMap == 
│ │ │ │ -0008e630: 5261 7469 6f6e 616c 4d61 702c 202d 2d20  RationalMap, -- 
│ │ │ │ -0008e640: 6571 7561 6c69 7479 0a20 2020 206f 6620  equality.    of 
│ │ │ │ -0008e650: 7261 7469 6f6e 616c 206d 6170 730a 2020  rational maps.  
│ │ │ │ -0008e660: 2a20 2252 6174 696f 6e61 6c4d 6170 203d  * "RationalMap =
│ │ │ │ -0008e670: 3d20 5a5a 2220 2d2d 2073 6565 202a 6e6f  = ZZ" -- see *no
│ │ │ │ -0008e680: 7465 2052 6174 696f 6e61 6c4d 6170 203d  te RationalMap =
│ │ │ │ -0008e690: 3d20 5261 7469 6f6e 616c 4d61 703a 2052  = RationalMap: R
│ │ │ │ -0008e6a0: 6174 696f 6e61 6c4d 6170 203d 3d0a 2020  ationalMap ==.  
│ │ │ │ -0008e6b0: 2020 5261 7469 6f6e 616c 4d61 702c 202d    RationalMap, -
│ │ │ │ -0008e6c0: 2d20 6571 7561 6c69 7479 206f 6620 7261  - equality of ra
│ │ │ │ -0008e6d0: 7469 6f6e 616c 206d 6170 730a 2020 2a20  tional maps.  * 
│ │ │ │ -0008e6e0: 225a 5a20 3d3d 2052 6174 696f 6e61 6c4d  "ZZ == RationalM
│ │ │ │ -0008e6f0: 6170 2220 2d2d 2073 6565 202a 6e6f 7465  ap" -- see *note
│ │ │ │ -0008e700: 2052 6174 696f 6e61 6c4d 6170 203d 3d20   RationalMap == 
│ │ │ │ -0008e710: 5261 7469 6f6e 616c 4d61 703a 2052 6174  RationalMap: Rat
│ │ │ │ -0008e720: 696f 6e61 6c4d 6170 203d 3d0a 2020 2020  ionalMap ==.    
│ │ │ │ -0008e730: 5261 7469 6f6e 616c 4d61 702c 202d 2d20  RationalMap, -- 
│ │ │ │ -0008e740: 6571 7561 6c69 7479 206f 6620 7261 7469  equality of rati
│ │ │ │ -0008e750: 6f6e 616c 206d 6170 730a 2020 2a20 2a6e  onal maps.  * *n
│ │ │ │ -0008e760: 6f74 6520 5261 7469 6f6e 616c 4d61 7020  ote RationalMap 
│ │ │ │ -0008e770: 5e20 5a5a 3a20 5261 7469 6f6e 616c 4d61  ^ ZZ: RationalMa
│ │ │ │ -0008e780: 7020 5e20 5a5a 2c20 2d2d 2070 6f77 6572  p ^ ZZ, -- power
│ │ │ │ -0008e790: 0a20 202a 2022 5261 7469 6f6e 616c 4d61  .  * "RationalMa
│ │ │ │ -0008e7a0: 7020 5e2a 2220 2d2d 2073 6565 202a 6e6f  p ^*" -- see *no
│ │ │ │ -0008e7b0: 7465 2052 6174 696f 6e61 6c4d 6170 205e  te RationalMap ^
│ │ │ │ -0008e7c0: 2a2a 2049 6465 616c 3a20 5261 7469 6f6e  ** Ideal: Ration
│ │ │ │ -0008e7d0: 616c 4d61 7020 5e5f 7374 5f73 740a 2020  alMap ^_st_st.  
│ │ │ │ -0008e7e0: 2020 4964 6561 6c2c 202d 2d20 696e 7665    Ideal, -- inve
│ │ │ │ -0008e7f0: 7273 6520 696d 6167 6520 7669 6120 6120  rse image via a 
│ │ │ │ -0008e800: 7261 7469 6f6e 616c 206d 6170 0a20 202a  rational map.  *
│ │ │ │ -0008e810: 202a 6e6f 7465 2052 6174 696f 6e61 6c4d   *note RationalM
│ │ │ │ -0008e820: 6170 205e 2a2a 2049 6465 616c 3a20 5261  ap ^** Ideal: Ra
│ │ │ │ -0008e830: 7469 6f6e 616c 4d61 7020 5e5f 7374 5f73  tionalMap ^_st_s
│ │ │ │ -0008e840: 7420 4964 6561 6c2c 202d 2d20 696e 7665  t Ideal, -- inve
│ │ │ │ -0008e850: 7273 6520 696d 6167 650a 2020 2020 7669  rse image.    vi
│ │ │ │ -0008e860: 6120 6120 7261 7469 6f6e 616c 206d 6170  a a rational map
│ │ │ │ -0008e870: 0a20 202a 202a 6e6f 7465 2052 6174 696f  .  * *note Ratio
│ │ │ │ -0008e880: 6e61 6c4d 6170 205f 2a3a 2052 6174 696f  nalMap _*: Ratio
│ │ │ │ -0008e890: 6e61 6c4d 6170 205f 7573 5f73 742c 202d  nalMap _us_st, -
│ │ │ │ -0008e8a0: 2d20 6469 7265 6374 2069 6d61 6765 2076  - direct image v
│ │ │ │ -0008e8b0: 6961 2061 2072 6174 696f 6e61 6c0a 2020  ia a rational.  
│ │ │ │ -0008e8c0: 2020 6d61 700a 2020 2a20 2252 6174 696f    map.  * "Ratio
│ │ │ │ -0008e8d0: 6e61 6c4d 6170 2049 6465 616c 2220 2d2d  nalMap Ideal" --
│ │ │ │ -0008e8e0: 2073 6565 202a 6e6f 7465 2052 6174 696f   see *note Ratio
│ │ │ │ -0008e8f0: 6e61 6c4d 6170 205f 2a3a 2052 6174 696f  nalMap _*: Ratio
│ │ │ │ -0008e900: 6e61 6c4d 6170 205f 7573 5f73 742c 202d  nalMap _us_st, -
│ │ │ │ -0008e910: 2d0a 2020 2020 6469 7265 6374 2069 6d61  -.    direct ima
│ │ │ │ -0008e920: 6765 2076 6961 2061 2072 6174 696f 6e61  ge via a rationa
│ │ │ │ -0008e930: 6c20 6d61 700a 2020 2a20 2a6e 6f74 6520  l map.  * *note 
│ │ │ │ -0008e940: 5261 7469 6f6e 616c 4d61 7020 7c20 4964  RationalMap | Id
│ │ │ │ -0008e950: 6561 6c3a 2052 6174 696f 6e61 6c4d 6170  eal: RationalMap
│ │ │ │ -0008e960: 207c 2049 6465 616c 2c20 2d2d 2072 6573   | Ideal, -- res
│ │ │ │ -0008e970: 7472 6963 7469 6f6e 206f 6620 610a 2020  triction of a.  
│ │ │ │ -0008e980: 2020 7261 7469 6f6e 616c 206d 6170 0a20    rational map. 
│ │ │ │ -0008e990: 202a 2022 5261 7469 6f6e 616c 4d61 7020   * "RationalMap 
│ │ │ │ -0008e9a0: 7c20 5269 6e67 2220 2d2d 2073 6565 202a  | Ring" -- see *
│ │ │ │ -0008e9b0: 6e6f 7465 2052 6174 696f 6e61 6c4d 6170  note RationalMap
│ │ │ │ -0008e9c0: 207c 2049 6465 616c 3a20 5261 7469 6f6e   | Ideal: Ration
│ │ │ │ -0008e9d0: 616c 4d61 7020 7c20 4964 6561 6c2c 0a20  alMap | Ideal,. 
│ │ │ │ -0008e9e0: 2020 202d 2d20 7265 7374 7269 6374 696f     -- restrictio
│ │ │ │ -0008e9f0: 6e20 6f66 2061 2072 6174 696f 6e61 6c20  n of a rational 
│ │ │ │ -0008ea00: 6d61 700a 2020 2a20 2252 6174 696f 6e61  map.  * "Rationa
│ │ │ │ -0008ea10: 6c4d 6170 207c 2052 696e 6745 6c65 6d65  lMap | RingEleme
│ │ │ │ -0008ea20: 6e74 2220 2d2d 2073 6565 202a 6e6f 7465  nt" -- see *note
│ │ │ │ -0008ea30: 2052 6174 696f 6e61 6c4d 6170 207c 2049   RationalMap | I
│ │ │ │ -0008ea40: 6465 616c 3a20 5261 7469 6f6e 616c 4d61  deal: RationalMa
│ │ │ │ -0008ea50: 7020 7c0a 2020 2020 4964 6561 6c2c 202d  p |.    Ideal, -
│ │ │ │ -0008ea60: 2d20 7265 7374 7269 6374 696f 6e20 6f66  - restriction of
│ │ │ │ -0008ea70: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ -0008ea80: 2020 2a20 2a6e 6f74 6520 5261 7469 6f6e    * *note Ration
│ │ │ │ -0008ea90: 616c 4d61 7020 7c7c 2049 6465 616c 3a20  alMap || Ideal: 
│ │ │ │ -0008eaa0: 5261 7469 6f6e 616c 4d61 7020 7c7c 2049  RationalMap || I
│ │ │ │ -0008eab0: 6465 616c 2c20 2d2d 2072 6573 7472 6963  deal, -- restric
│ │ │ │ -0008eac0: 7469 6f6e 206f 6620 610a 2020 2020 7261  tion of a.    ra
│ │ │ │ -0008ead0: 7469 6f6e 616c 206d 6170 0a20 202a 2022  tional map.  * "
│ │ │ │ -0008eae0: 5261 7469 6f6e 616c 4d61 7020 7c7c 2052  RationalMap || R
│ │ │ │ -0008eaf0: 696e 6722 202d 2d20 7365 6520 2a6e 6f74  ing" -- see *not
│ │ │ │ -0008eb00: 6520 5261 7469 6f6e 616c 4d61 7020 7c7c  e RationalMap ||
│ │ │ │ -0008eb10: 2049 6465 616c 3a20 5261 7469 6f6e 616c   Ideal: Rational
│ │ │ │ -0008eb20: 4d61 7020 7c7c 0a20 2020 2049 6465 616c  Map ||.    Ideal
│ │ │ │ -0008eb30: 2c20 2d2d 2072 6573 7472 6963 7469 6f6e  , -- restriction
│ │ │ │ -0008eb40: 206f 6620 6120 7261 7469 6f6e 616c 206d   of a rational m
│ │ │ │ -0008eb50: 6170 0a20 202a 2022 5261 7469 6f6e 616c  ap.  * "Rational
│ │ │ │ -0008eb60: 4d61 7020 7c7c 2052 696e 6745 6c65 6d65  Map || RingEleme
│ │ │ │ -0008eb70: 6e74 2220 2d2d 2073 6565 202a 6e6f 7465  nt" -- see *note
│ │ │ │ -0008eb80: 2052 6174 696f 6e61 6c4d 6170 207c 7c20   RationalMap || 
│ │ │ │ -0008eb90: 4964 6561 6c3a 2052 6174 696f 6e61 6c4d  Ideal: RationalM
│ │ │ │ -0008eba0: 6170 0a20 2020 207c 7c20 4964 6561 6c2c  ap.    || Ideal,
│ │ │ │ -0008ebb0: 202d 2d20 7265 7374 7269 6374 696f 6e20   -- restriction 
│ │ │ │ -0008ebc0: 6f66 2061 2072 6174 696f 6e61 6c20 6d61  of a rational ma
│ │ │ │ -0008ebd0: 700a 2020 2a20 2273 6567 7265 2852 6174  p.  * "segre(Rat
│ │ │ │ -0008ebe0: 696f 6e61 6c4d 6170 2922 202d 2d20 7365  ionalMap)" -- se
│ │ │ │ -0008ebf0: 6520 2a6e 6f74 6520 7365 6772 653a 2073  e *note segre: s
│ │ │ │ -0008ec00: 6567 7265 2c20 2d2d 2053 6567 7265 2065  egre, -- Segre e
│ │ │ │ -0008ec10: 6d62 6564 6469 6e67 0a20 202a 2022 5365  mbedding.  * "Se
│ │ │ │ -0008ec20: 6772 6543 6c61 7373 2852 6174 696f 6e61  greClass(Rationa
│ │ │ │ -0008ec30: 6c4d 6170 2922 202d 2d20 7365 6520 2a6e  lMap)" -- see *n
│ │ │ │ -0008ec40: 6f74 6520 5365 6772 6543 6c61 7373 3a20  ote SegreClass: 
│ │ │ │ -0008ec50: 5365 6772 6543 6c61 7373 2c20 2d2d 2053  SegreClass, -- S
│ │ │ │ -0008ec60: 6567 7265 0a20 2020 2063 6c61 7373 206f  egre.    class o
│ │ │ │ -0008ec70: 6620 6120 636c 6f73 6564 2073 7562 7363  f a closed subsc
│ │ │ │ -0008ec80: 6865 6d65 206f 6620 6120 7072 6f6a 6563  heme of a projec
│ │ │ │ -0008ec90: 7469 7665 2076 6172 6965 7479 0a20 202a  tive variety.  *
│ │ │ │ -0008eca0: 202a 6e6f 7465 2073 6f75 7263 6528 5261   *note source(Ra
│ │ │ │ -0008ecb0: 7469 6f6e 616c 4d61 7029 3a20 736f 7572  tionalMap): sour
│ │ │ │ -0008ecc0: 6365 5f6c 7052 6174 696f 6e61 6c4d 6170  ce_lpRationalMap
│ │ │ │ -0008ecd0: 5f72 702c 202d 2d20 636f 6f72 6469 6e61  _rp, -- coordina
│ │ │ │ -0008ece0: 7465 2072 696e 6720 6f66 0a20 2020 2074  te ring of.    t
│ │ │ │ -0008ecf0: 6865 2073 6f75 7263 6520 666f 7220 6120  he source for a 
│ │ │ │ -0008ed00: 7261 7469 6f6e 616c 206d 6170 0a20 202a  rational map.  *
│ │ │ │ -0008ed10: 202a 6e6f 7465 2073 7562 7374 6974 7574   *note substitut
│ │ │ │ -0008ed20: 6528 5261 7469 6f6e 616c 4d61 702c 506f  e(RationalMap,Po
│ │ │ │ -0008ed30: 6c79 6e6f 6d69 616c 5269 6e67 2c50 6f6c  lynomialRing,Pol
│ │ │ │ -0008ed40: 796e 6f6d 6961 6c52 696e 6729 3a0a 2020  ynomialRing):.  
│ │ │ │ -0008ed50: 2020 7375 6273 7469 7475 7465 5f6c 7052    substitute_lpR
│ │ │ │ -0008ed60: 6174 696f 6e61 6c4d 6170 5f63 6d50 6f6c  ationalMap_cmPol
│ │ │ │ -0008ed70: 796e 6f6d 6961 6c52 696e 675f 636d 506f  ynomialRing_cmPo
│ │ │ │ -0008ed80: 6c79 6e6f 6d69 616c 5269 6e67 5f72 702c  lynomialRing_rp,
│ │ │ │ -0008ed90: 202d 2d0a 2020 2020 7375 6273 7469 7475   --.    substitu
│ │ │ │ -0008eda0: 7465 2074 6865 2061 6d62 6965 6e74 2070  te the ambient p
│ │ │ │ -0008edb0: 726f 6a65 6374 6976 6520 7370 6163 6573  rojective spaces
│ │ │ │ -0008edc0: 206f 6620 736f 7572 6365 2061 6e64 2074   of source and t
│ │ │ │ -0008edd0: 6172 6765 740a 2020 2a20 2272 6174 696f  arget.  * "ratio
│ │ │ │ -0008ede0: 6e61 6c4d 6170 2852 6174 696f 6e61 6c4d  nalMap(RationalM
│ │ │ │ -0008edf0: 6170 2922 202d 2d20 7365 6520 2a6e 6f74  ap)" -- see *not
│ │ │ │ -0008ee00: 6520 7375 7065 7228 5261 7469 6f6e 616c  e super(Rational
│ │ │ │ -0008ee10: 4d61 7029 3a0a 2020 2020 7375 7065 725f  Map):.    super_
│ │ │ │ -0008ee20: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ -0008ee30: 2c20 2d2d 2067 6574 2074 6865 2072 6174  , -- get the rat
│ │ │ │ -0008ee40: 696f 6e61 6c20 6d61 7020 7768 6f73 6520  ional map whose 
│ │ │ │ -0008ee50: 7461 7267 6574 2069 7320 610a 2020 2020  target is a.    
│ │ │ │ -0008ee60: 7072 6f6a 6563 7469 7665 2073 7061 6365  projective space
│ │ │ │ -0008ee70: 0a20 202a 202a 6e6f 7465 2073 7570 6572  .  * *note super
│ │ │ │ -0008ee80: 2852 6174 696f 6e61 6c4d 6170 293a 2073  (RationalMap): s
│ │ │ │ -0008ee90: 7570 6572 5f6c 7052 6174 696f 6e61 6c4d  uper_lpRationalM
│ │ │ │ -0008eea0: 6170 5f72 702c 202d 2d20 6765 7420 7468  ap_rp, -- get th
│ │ │ │ -0008eeb0: 6520 7261 7469 6f6e 616c 206d 6170 0a20  e rational map. 
│ │ │ │ -0008eec0: 2020 2077 686f 7365 2074 6172 6765 7420     whose target 
│ │ │ │ -0008eed0: 6973 2061 2070 726f 6a65 6374 6976 6520  is a projective 
│ │ │ │ -0008eee0: 7370 6163 650a 2020 2a20 2a6e 6f74 6520  space.  * *note 
│ │ │ │ -0008eef0: 7461 7267 6574 2852 6174 696f 6e61 6c4d  target(RationalM
│ │ │ │ -0008ef00: 6170 293a 2074 6172 6765 745f 6c70 5261  ap): target_lpRa
│ │ │ │ -0008ef10: 7469 6f6e 616c 4d61 705f 7270 2c20 2d2d  tionalMap_rp, --
│ │ │ │ -0008ef20: 2063 6f6f 7264 696e 6174 6520 7269 6e67   coordinate ring
│ │ │ │ -0008ef30: 206f 660a 2020 2020 7468 6520 7461 7267   of.    the targ
│ │ │ │ -0008ef40: 6574 2066 6f72 2061 2072 6174 696f 6e61  et for a rationa
│ │ │ │ -0008ef50: 6c20 6d61 700a 2020 2a20 2a6e 6f74 6520  l map.  * *note 
│ │ │ │ -0008ef60: 746f 4578 7465 726e 616c 5374 7269 6e67  toExternalString
│ │ │ │ -0008ef70: 2852 6174 696f 6e61 6c4d 6170 293a 2074  (RationalMap): t
│ │ │ │ -0008ef80: 6f45 7874 6572 6e61 6c53 7472 696e 675f  oExternalString_
│ │ │ │ -0008ef90: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ -0008efa0: 2c20 2d2d 0a20 2020 2063 6f6e 7665 7274  , --.    convert
│ │ │ │ -0008efb0: 2074 6f20 6120 7265 6164 6162 6c65 2073   to a readable s
│ │ │ │ -0008efc0: 7472 696e 670a 0a46 6f72 2074 6865 2070  tring..For the p
│ │ │ │ -0008efd0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ -0008efe0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ -0008eff0: 6520 6f62 6a65 6374 202a 6e6f 7465 2052  e object *note R
│ │ │ │ -0008f000: 6174 696f 6e61 6c4d 6170 3a20 5261 7469  ationalMap: Rati
│ │ │ │ -0008f010: 6f6e 616c 4d61 702c 2069 7320 6120 2a6e  onalMap, is a *n
│ │ │ │ -0008f020: 6f74 6520 7479 7065 3a0a 284d 6163 6175  ote type:.(Macau
│ │ │ │ -0008f030: 6c61 7932 446f 6329 5479 7065 2c2c 2077  lay2Doc)Type,, w
│ │ │ │ -0008f040: 6974 6820 616e 6365 7374 6f72 2063 6c61  ith ancestor cla
│ │ │ │ -0008f050: 7373 6573 202a 6e6f 7465 204d 7574 6162  sses *note Mutab
│ │ │ │ -0008f060: 6c65 4861 7368 5461 626c 653a 0a28 4d61  leHashTable:.(Ma
│ │ │ │ -0008f070: 6361 756c 6179 3244 6f63 294d 7574 6162  caulay2Doc)Mutab
│ │ │ │ -0008f080: 6c65 4861 7368 5461 626c 652c 203c 202a  leHashTable, < *
│ │ │ │ -0008f090: 6e6f 7465 2048 6173 6854 6162 6c65 3a20  note HashTable: 
│ │ │ │ -0008f0a0: 284d 6163 6175 6c61 7932 446f 6329 4861  (Macaulay2Doc)Ha
│ │ │ │ -0008f0b0: 7368 5461 626c 652c 203c 0a2a 6e6f 7465  shTable, <.*note
│ │ │ │ -0008f0c0: 2054 6869 6e67 3a20 284d 6163 6175 6c61   Thing: (Macaula
│ │ │ │ -0008f0d0: 7932 446f 6329 5468 696e 672c 2e0a 0a2d  y2Doc)Thing,...-
│ │ │ │ +0008d1f0: 3d0a 0a20 202a 202a 6e6f 7465 2071 7561  =..  * *note qua
│ │ │ │ +0008d200: 6472 6f51 7561 6472 6963 4372 656d 6f6e  droQuadricCremon
│ │ │ │ +0008d210: 6154 7261 6e73 666f 726d 6174 696f 6e3a  aTransformation:
│ │ │ │ +0008d220: 0a20 2020 2071 7561 6472 6f51 7561 6472  .    quadroQuadr
│ │ │ │ +0008d230: 6963 4372 656d 6f6e 6154 7261 6e73 666f  icCremonaTransfo
│ │ │ │ +0008d240: 726d 6174 696f 6e2c 202d 2d20 7175 6164  rmation, -- quad
│ │ │ │ +0008d250: 726f 2d71 7561 6472 6963 2043 7265 6d6f  ro-quadric Cremo
│ │ │ │ +0008d260: 6e61 0a20 2020 2074 7261 6e73 666f 726d  na.    transform
│ │ │ │ +0008d270: 6174 696f 6e73 0a20 202a 202a 6e6f 7465  ations.  * *note
│ │ │ │ +0008d280: 2073 6567 7265 3a20 7365 6772 652c 202d   segre: segre, -
│ │ │ │ +0008d290: 2d20 5365 6772 6520 656d 6265 6464 696e  - Segre embeddin
│ │ │ │ +0008d2a0: 670a 2020 2a20 2a6e 6f74 6520 7370 6563  g.  * *note spec
│ │ │ │ +0008d2b0: 6961 6c43 7265 6d6f 6e61 5472 616e 7366  ialCremonaTransf
│ │ │ │ +0008d2c0: 6f72 6d61 7469 6f6e 3a20 7370 6563 6961  ormation: specia
│ │ │ │ +0008d2d0: 6c43 7265 6d6f 6e61 5472 616e 7366 6f72  lCremonaTransfor
│ │ │ │ +0008d2e0: 6d61 7469 6f6e 2c20 2d2d 0a20 2020 2073  mation, --.    s
│ │ │ │ +0008d2f0: 7065 6369 616c 2043 7265 6d6f 6e61 2074  pecial Cremona t
│ │ │ │ +0008d300: 7261 6e73 666f 726d 6174 696f 6e73 2077  ransformations w
│ │ │ │ +0008d310: 686f 7365 2062 6173 6520 6c6f 6375 7320  hose base locus 
│ │ │ │ +0008d320: 6861 7320 6469 6d65 6e73 696f 6e20 6174  has dimension at
│ │ │ │ +0008d330: 206d 6f73 740a 2020 2020 7468 7265 650a   most.    three.
│ │ │ │ +0008d340: 2020 2a20 2a6e 6f74 6520 7370 6563 6961    * *note specia
│ │ │ │ +0008d350: 6c43 7562 6963 5472 616e 7366 6f72 6d61  lCubicTransforma
│ │ │ │ +0008d360: 7469 6f6e 3a20 7370 6563 6961 6c43 7562  tion: specialCub
│ │ │ │ +0008d370: 6963 5472 616e 7366 6f72 6d61 7469 6f6e  icTransformation
│ │ │ │ +0008d380: 2c20 2d2d 2073 7065 6369 616c 0a20 2020  , -- special.   
│ │ │ │ +0008d390: 2063 7562 6963 2074 7261 6e73 666f 726d   cubic transform
│ │ │ │ +0008d3a0: 6174 696f 6e73 2077 686f 7365 2062 6173  ations whose bas
│ │ │ │ +0008d3b0: 6520 6c6f 6375 7320 6861 7320 6469 6d65  e locus has dime
│ │ │ │ +0008d3c0: 6e73 696f 6e20 6174 206d 6f73 7420 7468  nsion at most th
│ │ │ │ +0008d3d0: 7265 650a 2020 2a20 2a6e 6f74 6520 7370  ree.  * *note sp
│ │ │ │ +0008d3e0: 6563 6961 6c51 7561 6472 6174 6963 5472  ecialQuadraticTr
│ │ │ │ +0008d3f0: 616e 7366 6f72 6d61 7469 6f6e 3a20 7370  ansformation: sp
│ │ │ │ +0008d400: 6563 6961 6c51 7561 6472 6174 6963 5472  ecialQuadraticTr
│ │ │ │ +0008d410: 616e 7366 6f72 6d61 7469 6f6e 2c20 2d2d  ansformation, --
│ │ │ │ +0008d420: 0a20 2020 2073 7065 6369 616c 2071 7561  .    special qua
│ │ │ │ +0008d430: 6472 6174 6963 2074 7261 6e73 666f 726d  dratic transform
│ │ │ │ +0008d440: 6174 696f 6e73 2077 686f 7365 2062 6173  ations whose bas
│ │ │ │ +0008d450: 6520 6c6f 6375 7320 6861 7320 6469 6d65  e locus has dime
│ │ │ │ +0008d460: 6e73 696f 6e20 7468 7265 650a 0a4d 6574  nsion three..Met
│ │ │ │ +0008d470: 686f 6473 2074 6861 7420 7573 6520 6120  hods that use a 
│ │ │ │ +0008d480: 7261 7469 6f6e 616c 206d 6170 3a0a 3d3d  rational map:.==
│ │ │ │ +0008d490: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0008d4a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +0008d4b0: 2020 2a20 2261 6273 7472 6163 7452 6174    * "abstractRat
│ │ │ │ +0008d4c0: 696f 6e61 6c4d 6170 2852 6174 696f 6e61  ionalMap(Rationa
│ │ │ │ +0008d4d0: 6c4d 6170 2922 202d 2d20 7365 6520 2a6e  lMap)" -- see *n
│ │ │ │ +0008d4e0: 6f74 6520 6162 7374 7261 6374 5261 7469  ote abstractRati
│ │ │ │ +0008d4f0: 6f6e 616c 4d61 703a 0a20 2020 2061 6273  onalMap:.    abs
│ │ │ │ +0008d500: 7472 6163 7452 6174 696f 6e61 6c4d 6170  tractRationalMap
│ │ │ │ +0008d510: 2c20 2d2d 206d 616b 6520 616e 2061 6273  , -- make an abs
│ │ │ │ +0008d520: 7472 6163 7420 7261 7469 6f6e 616c 206d  tract rational m
│ │ │ │ +0008d530: 6170 0a20 202a 2022 6170 7072 6f78 696d  ap.  * "approxim
│ │ │ │ +0008d540: 6174 6549 6e76 6572 7365 4d61 7028 5261  ateInverseMap(Ra
│ │ │ │ +0008d550: 7469 6f6e 616c 4d61 7029 2220 2d2d 2073  tionalMap)" -- s
│ │ │ │ +0008d560: 6565 202a 6e6f 7465 2061 7070 726f 7869  ee *note approxi
│ │ │ │ +0008d570: 6d61 7465 496e 7665 7273 654d 6170 3a0a  mateInverseMap:.
│ │ │ │ +0008d580: 2020 2020 6170 7072 6f78 696d 6174 6549      approximateI
│ │ │ │ +0008d590: 6e76 6572 7365 4d61 702c 202d 2d20 7261  nverseMap, -- ra
│ │ │ │ +0008d5a0: 6e64 6f6d 206d 6170 2072 656c 6174 6564  ndom map related
│ │ │ │ +0008d5b0: 2074 6f20 7468 6520 696e 7665 7273 6520   to the inverse 
│ │ │ │ +0008d5c0: 6f66 2061 2062 6972 6174 696f 6e61 6c0a  of a birational.
│ │ │ │ +0008d5d0: 2020 2020 6d61 700a 2020 2a20 2261 7070      map.  * "app
│ │ │ │ +0008d5e0: 726f 7869 6d61 7465 496e 7665 7273 654d  roximateInverseM
│ │ │ │ +0008d5f0: 6170 2852 6174 696f 6e61 6c4d 6170 2c5a  ap(RationalMap,Z
│ │ │ │ +0008d600: 5a29 2220 2d2d 2073 6565 202a 6e6f 7465  Z)" -- see *note
│ │ │ │ +0008d610: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
│ │ │ │ +0008d620: 7273 654d 6170 3a0a 2020 2020 6170 7072  rseMap:.    appr
│ │ │ │ +0008d630: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +0008d640: 702c 202d 2d20 7261 6e64 6f6d 206d 6170  p, -- random map
│ │ │ │ +0008d650: 2072 656c 6174 6564 2074 6f20 7468 6520   related to the 
│ │ │ │ +0008d660: 696e 7665 7273 6520 6f66 2061 2062 6972  inverse of a bir
│ │ │ │ +0008d670: 6174 696f 6e61 6c0a 2020 2020 6d61 700a  ational.    map.
│ │ │ │ +0008d680: 2020 2a20 2a6e 6f74 6520 636f 6566 6669    * *note coeffi
│ │ │ │ +0008d690: 6369 656e 7452 696e 6728 5261 7469 6f6e  cientRing(Ration
│ │ │ │ +0008d6a0: 616c 4d61 7029 3a20 636f 6566 6669 6369  alMap): coeffici
│ │ │ │ +0008d6b0: 656e 7452 696e 675f 6c70 5261 7469 6f6e  entRing_lpRation
│ │ │ │ +0008d6c0: 616c 4d61 705f 7270 2c20 2d2d 0a20 2020  alMap_rp, --.   
│ │ │ │ +0008d6d0: 2063 6f65 6666 6963 6965 6e74 2072 696e   coefficient rin
│ │ │ │ +0008d6e0: 6720 6f66 2061 2072 6174 696f 6e61 6c20  g of a rational 
│ │ │ │ +0008d6f0: 6d61 700a 2020 2a20 2a6e 6f74 6520 636f  map.  * *note co
│ │ │ │ +0008d700: 6566 6669 6369 656e 7473 2852 6174 696f  efficients(Ratio
│ │ │ │ +0008d710: 6e61 6c4d 6170 293a 2063 6f65 6666 6963  nalMap): coeffic
│ │ │ │ +0008d720: 6965 6e74 735f 6c70 5261 7469 6f6e 616c  ients_lpRational
│ │ │ │ +0008d730: 4d61 705f 7270 2c20 2d2d 0a20 2020 2063  Map_rp, --.    c
│ │ │ │ +0008d740: 6f65 6666 6963 6965 6e74 206d 6174 7269  oefficient matri
│ │ │ │ +0008d750: 7820 6f66 2061 2072 6174 696f 6e61 6c20  x of a rational 
│ │ │ │ +0008d760: 6d61 700a 2020 2a20 2a6e 6f74 6520 6465  map.  * *note de
│ │ │ │ +0008d770: 6772 6565 2852 6174 696f 6e61 6c4d 6170  gree(RationalMap
│ │ │ │ +0008d780: 293a 2064 6567 7265 655f 6c70 5261 7469  ): degree_lpRati
│ │ │ │ +0008d790: 6f6e 616c 4d61 705f 7270 2c20 2d2d 2064  onalMap_rp, -- d
│ │ │ │ +0008d7a0: 6567 7265 6520 6f66 2061 2072 6174 696f  egree of a ratio
│ │ │ │ +0008d7b0: 6e61 6c0a 2020 2020 6d61 700a 2020 2a20  nal.    map.  * 
│ │ │ │ +0008d7c0: 2a6e 6f74 6520 6465 6772 6565 4d61 7028  *note degreeMap(
│ │ │ │ +0008d7d0: 5261 7469 6f6e 616c 4d61 7029 3a20 6465  RationalMap): de
│ │ │ │ +0008d7e0: 6772 6565 4d61 705f 6c70 5261 7469 6f6e  greeMap_lpRation
│ │ │ │ +0008d7f0: 616c 4d61 705f 7270 2c20 2d2d 2064 6567  alMap_rp, -- deg
│ │ │ │ +0008d800: 7265 6520 6f66 2061 0a20 2020 2072 6174  ree of a.    rat
│ │ │ │ +0008d810: 696f 6e61 6c20 6d61 700a 2020 2a20 2a6e  ional map.  * *n
│ │ │ │ +0008d820: 6f74 6520 6465 6772 6565 7328 5261 7469  ote degrees(Rati
│ │ │ │ +0008d830: 6f6e 616c 4d61 7029 3a20 6465 6772 6565  onalMap): degree
│ │ │ │ +0008d840: 735f 6c70 5261 7469 6f6e 616c 4d61 705f  s_lpRationalMap_
│ │ │ │ +0008d850: 7270 2c20 2d2d 2070 726f 6a65 6374 6976  rp, -- projectiv
│ │ │ │ +0008d860: 6520 6465 6772 6565 730a 2020 2020 6f66  e degrees.    of
│ │ │ │ +0008d870: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ +0008d880: 2020 2a20 226d 756c 7469 6465 6772 6565    * "multidegree
│ │ │ │ +0008d890: 2852 6174 696f 6e61 6c4d 6170 2922 202d  (RationalMap)" -
│ │ │ │ +0008d8a0: 2d20 7365 6520 2a6e 6f74 6520 6465 6772  - see *note degr
│ │ │ │ +0008d8b0: 6565 7328 5261 7469 6f6e 616c 4d61 7029  ees(RationalMap)
│ │ │ │ +0008d8c0: 3a0a 2020 2020 6465 6772 6565 735f 6c70  :.    degrees_lp
│ │ │ │ +0008d8d0: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ +0008d8e0: 2d2d 2070 726f 6a65 6374 6976 6520 6465  -- projective de
│ │ │ │ +0008d8f0: 6772 6565 7320 6f66 2061 2072 6174 696f  grees of a ratio
│ │ │ │ +0008d900: 6e61 6c20 6d61 700a 2020 2a20 2a6e 6f74  nal map.  * *not
│ │ │ │ +0008d910: 6520 6465 7363 7269 6265 2852 6174 696f  e describe(Ratio
│ │ │ │ +0008d920: 6e61 6c4d 6170 293a 2064 6573 6372 6962  nalMap): describ
│ │ │ │ +0008d930: 655f 6c70 5261 7469 6f6e 616c 4d61 705f  e_lpRationalMap_
│ │ │ │ +0008d940: 7270 2c20 2d2d 2064 6573 6372 6962 6520  rp, -- describe 
│ │ │ │ +0008d950: 610a 2020 2020 7261 7469 6f6e 616c 206d  a.    rational m
│ │ │ │ +0008d960: 6170 0a20 202a 202a 6e6f 7465 2065 6e74  ap.  * *note ent
│ │ │ │ +0008d970: 7269 6573 2852 6174 696f 6e61 6c4d 6170  ries(RationalMap
│ │ │ │ +0008d980: 293a 2065 6e74 7269 6573 5f6c 7052 6174  ): entries_lpRat
│ │ │ │ +0008d990: 696f 6e61 6c4d 6170 5f72 702c 202d 2d20  ionalMap_rp, -- 
│ │ │ │ +0008d9a0: 7468 6520 656e 7472 6965 7320 6f66 2074  the entries of t
│ │ │ │ +0008d9b0: 6865 0a20 2020 206d 6174 7269 7820 6173  he.    matrix as
│ │ │ │ +0008d9c0: 736f 6369 6174 6564 2074 6f20 6120 7261  sociated to a ra
│ │ │ │ +0008d9d0: 7469 6f6e 616c 206d 6170 0a20 202a 2022  tional map.  * "
│ │ │ │ +0008d9e0: 6578 6365 7074 696f 6e61 6c4c 6f63 7573  exceptionalLocus
│ │ │ │ +0008d9f0: 2852 6174 696f 6e61 6c4d 6170 2922 202d  (RationalMap)" -
│ │ │ │ +0008da00: 2d20 7365 6520 2a6e 6f74 6520 6578 6365  - see *note exce
│ │ │ │ +0008da10: 7074 696f 6e61 6c4c 6f63 7573 3a0a 2020  ptionalLocus:.  
│ │ │ │ +0008da20: 2020 6578 6365 7074 696f 6e61 6c4c 6f63    exceptionalLoc
│ │ │ │ +0008da30: 7573 2c20 2d2d 2065 7863 6570 7469 6f6e  us, -- exception
│ │ │ │ +0008da40: 616c 206c 6f63 7573 206f 6620 6120 6269  al locus of a bi
│ │ │ │ +0008da50: 7261 7469 6f6e 616c 206d 6170 0a20 202a  rational map.  *
│ │ │ │ +0008da60: 202a 6e6f 7465 2066 6c61 7474 656e 2852   *note flatten(R
│ │ │ │ +0008da70: 6174 696f 6e61 6c4d 6170 293a 2066 6c61  ationalMap): fla
│ │ │ │ +0008da80: 7474 656e 5f6c 7052 6174 696f 6e61 6c4d  tten_lpRationalM
│ │ │ │ +0008da90: 6170 5f72 702c 202d 2d20 7772 6974 6520  ap_rp, -- write 
│ │ │ │ +0008daa0: 736f 7572 6365 2061 6e64 0a20 2020 2074  source and.    t
│ │ │ │ +0008dab0: 6172 6765 7420 6173 206e 6f6e 6465 6765  arget as nondege
│ │ │ │ +0008dac0: 6e65 7261 7465 2076 6172 6965 7469 6573  nerate varieties
│ │ │ │ +0008dad0: 0a20 202a 2022 666f 7263 6549 6d61 6765  .  * "forceImage
│ │ │ │ +0008dae0: 2852 6174 696f 6e61 6c4d 6170 2c49 6465  (RationalMap,Ide
│ │ │ │ +0008daf0: 616c 2922 202d 2d20 7365 6520 2a6e 6f74  al)" -- see *not
│ │ │ │ +0008db00: 6520 666f 7263 6549 6d61 6765 3a20 666f  e forceImage: fo
│ │ │ │ +0008db10: 7263 6549 6d61 6765 2c20 2d2d 0a20 2020  rceImage, --.   
│ │ │ │ +0008db20: 2064 6563 6c61 7265 2077 6869 6368 2069   declare which i
│ │ │ │ +0008db30: 7320 7468 6520 696d 6167 6520 6f66 2061  s the image of a
│ │ │ │ +0008db40: 2072 6174 696f 6e61 6c20 6d61 700a 2020   rational map.  
│ │ │ │ +0008db50: 2a20 2266 6f72 6365 496e 7665 7273 654d  * "forceInverseM
│ │ │ │ +0008db60: 6170 2852 6174 696f 6e61 6c4d 6170 2c52  ap(RationalMap,R
│ │ │ │ +0008db70: 6174 696f 6e61 6c4d 6170 2922 202d 2d20  ationalMap)" -- 
│ │ │ │ +0008db80: 7365 6520 2a6e 6f74 6520 666f 7263 6549  see *note forceI
│ │ │ │ +0008db90: 6e76 6572 7365 4d61 703a 0a20 2020 2066  nverseMap:.    f
│ │ │ │ +0008dba0: 6f72 6365 496e 7665 7273 654d 6170 2c20  orceInverseMap, 
│ │ │ │ +0008dbb0: 2d2d 2064 6563 6c61 7265 2074 6861 7420  -- declare that 
│ │ │ │ +0008dbc0: 7477 6f20 7261 7469 6f6e 616c 206d 6170  two rational map
│ │ │ │ +0008dbd0: 7320 6172 6520 6f6e 6520 7468 6520 696e  s are one the in
│ │ │ │ +0008dbe0: 7665 7273 6520 6f66 0a20 2020 2074 6865  verse of.    the
│ │ │ │ +0008dbf0: 206f 7468 6572 0a20 202a 2022 6772 6170   other.  * "grap
│ │ │ │ +0008dc00: 6828 5261 7469 6f6e 616c 4d61 7029 2220  h(RationalMap)" 
│ │ │ │ +0008dc10: 2d2d 2073 6565 202a 6e6f 7465 2067 7261  -- see *note gra
│ │ │ │ +0008dc20: 7068 3a20 6772 6170 682c 202d 2d20 636c  ph: graph, -- cl
│ │ │ │ +0008dc30: 6f73 7572 6520 6f66 2074 6865 2067 7261  osure of the gra
│ │ │ │ +0008dc40: 7068 206f 660a 2020 2020 6120 7261 7469  ph of.    a rati
│ │ │ │ +0008dc50: 6f6e 616c 206d 6170 0a20 202a 202a 6e6f  onal map.  * *no
│ │ │ │ +0008dc60: 7465 2069 6465 616c 2852 6174 696f 6e61  te ideal(Rationa
│ │ │ │ +0008dc70: 6c4d 6170 293a 2069 6465 616c 5f6c 7052  lMap): ideal_lpR
│ │ │ │ +0008dc80: 6174 696f 6e61 6c4d 6170 5f72 702c 202d  ationalMap_rp, -
│ │ │ │ +0008dc90: 2d20 6261 7365 206c 6f63 7573 206f 6620  - base locus of 
│ │ │ │ +0008dca0: 610a 2020 2020 7261 7469 6f6e 616c 206d  a.    rational m
│ │ │ │ +0008dcb0: 6170 0a20 202a 202a 6e6f 7465 2069 6d61  ap.  * *note ima
│ │ │ │ +0008dcc0: 6765 2852 6174 696f 6e61 6c4d 6170 2c53  ge(RationalMap,S
│ │ │ │ +0008dcd0: 7472 696e 6729 3a20 696d 6167 655f 6c70  tring): image_lp
│ │ │ │ +0008dce0: 5261 7469 6f6e 616c 4d61 705f 636d 5374  RationalMap_cmSt
│ │ │ │ +0008dcf0: 7269 6e67 5f72 702c 202d 2d0a 2020 2020  ring_rp, --.    
│ │ │ │ +0008dd00: 636c 6f73 7572 6520 6f66 2074 6865 2069  closure of the i
│ │ │ │ +0008dd10: 6d61 6765 206f 6620 6120 7261 7469 6f6e  mage of a ration
│ │ │ │ +0008dd20: 616c 206d 6170 2075 7369 6e67 2074 6865  al map using the
│ │ │ │ +0008dd30: 2046 3420 616c 676f 7269 7468 6d0a 2020   F4 algorithm.  
│ │ │ │ +0008dd40: 2020 2865 7870 6572 696d 656e 7461 6c29    (experimental)
│ │ │ │ +0008dd50: 0a20 202a 2022 696d 6167 6528 5261 7469  .  * "image(Rati
│ │ │ │ +0008dd60: 6f6e 616c 4d61 7029 2220 2d2d 2073 6565  onalMap)" -- see
│ │ │ │ +0008dd70: 202a 6e6f 7465 2069 6d61 6765 2852 6174   *note image(Rat
│ │ │ │ +0008dd80: 696f 6e61 6c4d 6170 2c5a 5a29 3a0a 2020  ionalMap,ZZ):.  
│ │ │ │ +0008dd90: 2020 696d 6167 655f 6c70 5261 7469 6f6e    image_lpRation
│ │ │ │ +0008dda0: 616c 4d61 705f 636d 5a5a 5f72 702c 202d  alMap_cmZZ_rp, -
│ │ │ │ +0008ddb0: 2d20 636c 6f73 7572 6520 6f66 2074 6865  - closure of the
│ │ │ │ +0008ddc0: 2069 6d61 6765 206f 6620 6120 7261 7469   image of a rati
│ │ │ │ +0008ddd0: 6f6e 616c 206d 6170 0a20 202a 202a 6e6f  onal map.  * *no
│ │ │ │ +0008dde0: 7465 2069 6d61 6765 2852 6174 696f 6e61  te image(Rationa
│ │ │ │ +0008ddf0: 6c4d 6170 2c5a 5a29 3a20 696d 6167 655f  lMap,ZZ): image_
│ │ │ │ +0008de00: 6c70 5261 7469 6f6e 616c 4d61 705f 636d  lpRationalMap_cm
│ │ │ │ +0008de10: 5a5a 5f72 702c 202d 2d20 636c 6f73 7572  ZZ_rp, -- closur
│ │ │ │ +0008de20: 6520 6f66 2074 6865 0a20 2020 2069 6d61  e of the.    ima
│ │ │ │ +0008de30: 6765 206f 6620 6120 7261 7469 6f6e 616c  ge of a rational
│ │ │ │ +0008de40: 206d 6170 0a20 202a 202a 6e6f 7465 2069   map.  * *note i
│ │ │ │ +0008de50: 6e76 6572 7365 2852 6174 696f 6e61 6c4d  nverse(RationalM
│ │ │ │ +0008de60: 6170 293a 2069 6e76 6572 7365 5f6c 7052  ap): inverse_lpR
│ │ │ │ +0008de70: 6174 696f 6e61 6c4d 6170 5f72 702c 202d  ationalMap_rp, -
│ │ │ │ +0008de80: 2d20 696e 7665 7273 6520 6f66 2061 0a20  - inverse of a. 
│ │ │ │ +0008de90: 2020 2062 6972 6174 696f 6e61 6c20 6d61     birational ma
│ │ │ │ +0008dea0: 700a 2020 2a20 2269 6e76 6572 7365 2852  p.  * "inverse(R
│ │ │ │ +0008deb0: 6174 696f 6e61 6c4d 6170 2c4f 7074 696f  ationalMap,Optio
│ │ │ │ +0008dec0: 6e29 2220 2d2d 2073 6565 202a 6e6f 7465  n)" -- see *note
│ │ │ │ +0008ded0: 2069 6e76 6572 7365 2852 6174 696f 6e61   inverse(Rationa
│ │ │ │ +0008dee0: 6c4d 6170 293a 0a20 2020 2069 6e76 6572  lMap):.    inver
│ │ │ │ +0008def0: 7365 5f6c 7052 6174 696f 6e61 6c4d 6170  se_lpRationalMap
│ │ │ │ +0008df00: 5f72 702c 202d 2d20 696e 7665 7273 6520  _rp, -- inverse 
│ │ │ │ +0008df10: 6f66 2061 2062 6972 6174 696f 6e61 6c20  of a birational 
│ │ │ │ +0008df20: 6d61 700a 2020 2a20 2269 6e76 6572 7365  map.  * "inverse
│ │ │ │ +0008df30: 4d61 7028 5261 7469 6f6e 616c 4d61 7029  Map(RationalMap)
│ │ │ │ +0008df40: 2220 2d2d 2073 6565 202a 6e6f 7465 2069  " -- see *note i
│ │ │ │ +0008df50: 6e76 6572 7365 4d61 703a 2069 6e76 6572  nverseMap: inver
│ │ │ │ +0008df60: 7365 4d61 702c 202d 2d20 696e 7665 7273  seMap, -- invers
│ │ │ │ +0008df70: 650a 2020 2020 6f66 2061 2062 6972 6174  e.    of a birat
│ │ │ │ +0008df80: 696f 6e61 6c20 6d61 700a 2020 2a20 2269  ional map.  * "i
│ │ │ │ +0008df90: 7342 6972 6174 696f 6e61 6c28 5261 7469  sBirational(Rati
│ │ │ │ +0008dfa0: 6f6e 616c 4d61 7029 2220 2d2d 2073 6565  onalMap)" -- see
│ │ │ │ +0008dfb0: 202a 6e6f 7465 2069 7342 6972 6174 696f   *note isBiratio
│ │ │ │ +0008dfc0: 6e61 6c3a 2069 7342 6972 6174 696f 6e61  nal: isBirationa
│ │ │ │ +0008dfd0: 6c2c 202d 2d0a 2020 2020 7768 6574 6865  l, --.    whethe
│ │ │ │ +0008dfe0: 7220 6120 7261 7469 6f6e 616c 206d 6170  r a rational map
│ │ │ │ +0008dff0: 2069 7320 6269 7261 7469 6f6e 616c 0a20   is birational. 
│ │ │ │ +0008e000: 202a 2022 6973 446f 6d69 6e61 6e74 2852   * "isDominant(R
│ │ │ │ +0008e010: 6174 696f 6e61 6c4d 6170 2922 202d 2d20  ationalMap)" -- 
│ │ │ │ +0008e020: 7365 6520 2a6e 6f74 6520 6973 446f 6d69  see *note isDomi
│ │ │ │ +0008e030: 6e61 6e74 3a20 6973 446f 6d69 6e61 6e74  nant: isDominant
│ │ │ │ +0008e040: 2c20 2d2d 2077 6865 7468 6572 2061 0a20  , -- whether a. 
│ │ │ │ +0008e050: 2020 2072 6174 696f 6e61 6c20 6d61 7020     rational map 
│ │ │ │ +0008e060: 6973 2064 6f6d 696e 616e 740a 2020 2a20  is dominant.  * 
│ │ │ │ +0008e070: 2a6e 6f74 6520 6973 496e 7665 7273 654d  *note isInverseM
│ │ │ │ +0008e080: 6170 2852 6174 696f 6e61 6c4d 6170 2c52  ap(RationalMap,R
│ │ │ │ +0008e090: 6174 696f 6e61 6c4d 6170 293a 0a20 2020  ationalMap):.   
│ │ │ │ +0008e0a0: 2069 7349 6e76 6572 7365 4d61 705f 6c70   isInverseMap_lp
│ │ │ │ +0008e0b0: 5261 7469 6f6e 616c 4d61 705f 636d 5261  RationalMap_cmRa
│ │ │ │ +0008e0c0: 7469 6f6e 616c 4d61 705f 7270 2c20 2d2d  tionalMap_rp, --
│ │ │ │ +0008e0d0: 2063 6865 636b 7320 7768 6574 6865 7220   checks whether 
│ │ │ │ +0008e0e0: 7477 6f20 7261 7469 6f6e 616c 0a20 2020  two rational.   
│ │ │ │ +0008e0f0: 206d 6170 7320 6172 6520 6f6e 6520 7468   maps are one th
│ │ │ │ +0008e100: 6520 696e 7665 7273 6520 6f66 2074 6865  e inverse of the
│ │ │ │ +0008e110: 206f 7468 6572 0a20 202a 202a 6e6f 7465   other.  * *note
│ │ │ │ +0008e120: 2069 7349 736f 6d6f 7270 6869 736d 2852   isIsomorphism(R
│ │ │ │ +0008e130: 6174 696f 6e61 6c4d 6170 293a 2069 7349  ationalMap): isI
│ │ │ │ +0008e140: 736f 6d6f 7270 6869 736d 5f6c 7052 6174  somorphism_lpRat
│ │ │ │ +0008e150: 696f 6e61 6c4d 6170 5f72 702c 202d 2d0a  ionalMap_rp, --.
│ │ │ │ +0008e160: 2020 2020 7768 6574 6865 7220 6120 6269      whether a bi
│ │ │ │ +0008e170: 7261 7469 6f6e 616c 206d 6170 2069 7320  rational map is 
│ │ │ │ +0008e180: 616e 2069 736f 6d6f 7270 6869 736d 0a20  an isomorphism. 
│ │ │ │ +0008e190: 202a 2022 6973 4d6f 7270 6869 736d 2852   * "isMorphism(R
│ │ │ │ +0008e1a0: 6174 696f 6e61 6c4d 6170 2922 202d 2d20  ationalMap)" -- 
│ │ │ │ +0008e1b0: 7365 6520 2a6e 6f74 6520 6973 4d6f 7270  see *note isMorp
│ │ │ │ +0008e1c0: 6869 736d 3a20 6973 4d6f 7270 6869 736d  hism: isMorphism
│ │ │ │ +0008e1d0: 2c20 2d2d 2077 6865 7468 6572 2061 0a20  , -- whether a. 
│ │ │ │ +0008e1e0: 2020 2072 6174 696f 6e61 6c20 6d61 7020     rational map 
│ │ │ │ +0008e1f0: 6973 2061 206d 6f72 7068 6973 6d0a 2020  is a morphism.  
│ │ │ │ +0008e200: 2a20 2a6e 6f74 6520 6d61 7028 5261 7469  * *note map(Rati
│ │ │ │ +0008e210: 6f6e 616c 4d61 7029 3a20 6d61 705f 6c70  onalMap): map_lp
│ │ │ │ +0008e220: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ +0008e230: 2d2d 2067 6574 2074 6865 2072 696e 6720  -- get the ring 
│ │ │ │ +0008e240: 6d61 7020 6465 6669 6e69 6e67 0a20 2020  map defining.   
│ │ │ │ +0008e250: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ +0008e260: 2020 2a20 226d 6170 285a 5a2c 5261 7469    * "map(ZZ,Rati
│ │ │ │ +0008e270: 6f6e 616c 4d61 7029 2220 2d2d 2073 6565  onalMap)" -- see
│ │ │ │ +0008e280: 202a 6e6f 7465 206d 6170 2852 6174 696f   *note map(Ratio
│ │ │ │ +0008e290: 6e61 6c4d 6170 293a 206d 6170 5f6c 7052  nalMap): map_lpR
│ │ │ │ +0008e2a0: 6174 696f 6e61 6c4d 6170 5f72 702c 0a20  ationalMap_rp,. 
│ │ │ │ +0008e2b0: 2020 202d 2d20 6765 7420 7468 6520 7269     -- get the ri
│ │ │ │ +0008e2c0: 6e67 206d 6170 2064 6566 696e 696e 6720  ng map defining 
│ │ │ │ +0008e2d0: 6120 7261 7469 6f6e 616c 206d 6170 0a20  a rational map. 
│ │ │ │ +0008e2e0: 202a 202a 6e6f 7465 206d 6174 7269 7828   * *note matrix(
│ │ │ │ +0008e2f0: 5261 7469 6f6e 616c 4d61 7029 3a20 6d61  RationalMap): ma
│ │ │ │ +0008e300: 7472 6978 5f6c 7052 6174 696f 6e61 6c4d  trix_lpRationalM
│ │ │ │ +0008e310: 6170 5f72 702c 202d 2d20 7468 6520 6d61  ap_rp, -- the ma
│ │ │ │ +0008e320: 7472 6978 0a20 2020 2061 7373 6f63 6961  trix.    associa
│ │ │ │ +0008e330: 7465 6420 746f 2061 2072 6174 696f 6e61  ted to a rationa
│ │ │ │ +0008e340: 6c20 6d61 700a 2020 2a20 226d 6174 7269  l map.  * "matri
│ │ │ │ +0008e350: 7828 5a5a 2c52 6174 696f 6e61 6c4d 6170  x(ZZ,RationalMap
│ │ │ │ +0008e360: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ +0008e370: 6d61 7472 6978 2852 6174 696f 6e61 6c4d  matrix(RationalM
│ │ │ │ +0008e380: 6170 293a 0a20 2020 206d 6174 7269 785f  ap):.    matrix_
│ │ │ │ +0008e390: 6c70 5261 7469 6f6e 616c 4d61 705f 7270  lpRationalMap_rp
│ │ │ │ +0008e3a0: 2c20 2d2d 2074 6865 206d 6174 7269 7820  , -- the matrix 
│ │ │ │ +0008e3b0: 6173 736f 6369 6174 6564 2074 6f20 6120  associated to a 
│ │ │ │ +0008e3c0: 7261 7469 6f6e 616c 206d 6170 0a20 202a  rational map.  *
│ │ │ │ +0008e3d0: 202a 6e6f 7465 2070 726f 6a65 6374 6976   *note projectiv
│ │ │ │ +0008e3e0: 6544 6567 7265 6573 2852 6174 696f 6e61  eDegrees(Rationa
│ │ │ │ +0008e3f0: 6c4d 6170 293a 2070 726f 6a65 6374 6976  lMap): projectiv
│ │ │ │ +0008e400: 6544 6567 7265 6573 5f6c 7052 6174 696f  eDegrees_lpRatio
│ │ │ │ +0008e410: 6e61 6c4d 6170 5f72 702c 0a20 2020 202d  nalMap_rp,.    -
│ │ │ │ +0008e420: 2d20 7072 6f6a 6563 7469 7665 2064 6567  - projective deg
│ │ │ │ +0008e430: 7265 6573 206f 6620 6120 7261 7469 6f6e  rees of a ration
│ │ │ │ +0008e440: 616c 206d 6170 0a20 202a 202a 6e6f 7465  al map.  * *note
│ │ │ │ +0008e450: 2052 6174 696f 6e61 6c4d 6170 2021 3a20   RationalMap !: 
│ │ │ │ +0008e460: 5261 7469 6f6e 616c 4d61 7020 212c 202d  RationalMap !, -
│ │ │ │ +0008e470: 2d20 6361 6c63 756c 6174 6573 2065 7665  - calculates eve
│ │ │ │ +0008e480: 7279 2070 6f73 7369 626c 6520 7468 696e  ry possible thin
│ │ │ │ +0008e490: 670a 2020 2a20 2263 6f6d 706f 7365 2852  g.  * "compose(R
│ │ │ │ +0008e4a0: 6174 696f 6e61 6c4d 6170 2c52 6174 696f  ationalMap,Ratio
│ │ │ │ +0008e4b0: 6e61 6c4d 6170 2922 202d 2d20 7365 6520  nalMap)" -- see 
│ │ │ │ +0008e4c0: 2a6e 6f74 6520 5261 7469 6f6e 616c 4d61  *note RationalMa
│ │ │ │ +0008e4d0: 7020 2a20 5261 7469 6f6e 616c 4d61 703a  p * RationalMap:
│ │ │ │ +0008e4e0: 0a20 2020 2052 6174 696f 6e61 6c4d 6170  .    RationalMap
│ │ │ │ +0008e4f0: 205f 7374 2052 6174 696f 6e61 6c4d 6170   _st RationalMap
│ │ │ │ +0008e500: 2c20 2d2d 2063 6f6d 706f 7369 7469 6f6e  , -- composition
│ │ │ │ +0008e510: 206f 6620 7261 7469 6f6e 616c 206d 6170   of rational map
│ │ │ │ +0008e520: 730a 2020 2a20 2a6e 6f74 6520 5261 7469  s.  * *note Rati
│ │ │ │ +0008e530: 6f6e 616c 4d61 7020 2a20 5261 7469 6f6e  onalMap * Ration
│ │ │ │ +0008e540: 616c 4d61 703a 2052 6174 696f 6e61 6c4d  alMap: RationalM
│ │ │ │ +0008e550: 6170 205f 7374 2052 6174 696f 6e61 6c4d  ap _st RationalM
│ │ │ │ +0008e560: 6170 2c20 2d2d 0a20 2020 2063 6f6d 706f  ap, --.    compo
│ │ │ │ +0008e570: 7369 7469 6f6e 206f 6620 7261 7469 6f6e  sition of ration
│ │ │ │ +0008e580: 616c 206d 6170 730a 2020 2a20 2a6e 6f74  al maps.  * *not
│ │ │ │ +0008e590: 6520 5261 7469 6f6e 616c 4d61 7020 2a2a  e RationalMap **
│ │ │ │ +0008e5a0: 2052 696e 673a 2052 6174 696f 6e61 6c4d   Ring: RationalM
│ │ │ │ +0008e5b0: 6170 205f 7374 5f73 7420 5269 6e67 2c20  ap _st_st Ring, 
│ │ │ │ +0008e5c0: 2d2d 2063 6861 6e67 6520 7468 650a 2020  -- change the.  
│ │ │ │ +0008e5d0: 2020 636f 6566 6669 6369 656e 7420 7269    coefficient ri
│ │ │ │ +0008e5e0: 6e67 206f 6620 6120 7261 7469 6f6e 616c  ng of a rational
│ │ │ │ +0008e5f0: 206d 6170 0a20 202a 202a 6e6f 7465 2052   map.  * *note R
│ │ │ │ +0008e600: 6174 696f 6e61 6c4d 6170 203d 3d20 5261  ationalMap == Ra
│ │ │ │ +0008e610: 7469 6f6e 616c 4d61 703a 2052 6174 696f  tionalMap: Ratio
│ │ │ │ +0008e620: 6e61 6c4d 6170 203d 3d20 5261 7469 6f6e  nalMap == Ration
│ │ │ │ +0008e630: 616c 4d61 702c 202d 2d20 6571 7561 6c69  alMap, -- equali
│ │ │ │ +0008e640: 7479 0a20 2020 206f 6620 7261 7469 6f6e  ty.    of ration
│ │ │ │ +0008e650: 616c 206d 6170 730a 2020 2a20 2252 6174  al maps.  * "Rat
│ │ │ │ +0008e660: 696f 6e61 6c4d 6170 203d 3d20 5a5a 2220  ionalMap == ZZ" 
│ │ │ │ +0008e670: 2d2d 2073 6565 202a 6e6f 7465 2052 6174  -- see *note Rat
│ │ │ │ +0008e680: 696f 6e61 6c4d 6170 203d 3d20 5261 7469  ionalMap == Rati
│ │ │ │ +0008e690: 6f6e 616c 4d61 703a 2052 6174 696f 6e61  onalMap: Rationa
│ │ │ │ +0008e6a0: 6c4d 6170 203d 3d0a 2020 2020 5261 7469  lMap ==.    Rati
│ │ │ │ +0008e6b0: 6f6e 616c 4d61 702c 202d 2d20 6571 7561  onalMap, -- equa
│ │ │ │ +0008e6c0: 6c69 7479 206f 6620 7261 7469 6f6e 616c  lity of rational
│ │ │ │ +0008e6d0: 206d 6170 730a 2020 2a20 225a 5a20 3d3d   maps.  * "ZZ ==
│ │ │ │ +0008e6e0: 2052 6174 696f 6e61 6c4d 6170 2220 2d2d   RationalMap" --
│ │ │ │ +0008e6f0: 2073 6565 202a 6e6f 7465 2052 6174 696f   see *note Ratio
│ │ │ │ +0008e700: 6e61 6c4d 6170 203d 3d20 5261 7469 6f6e  nalMap == Ration
│ │ │ │ +0008e710: 616c 4d61 703a 2052 6174 696f 6e61 6c4d  alMap: RationalM
│ │ │ │ +0008e720: 6170 203d 3d0a 2020 2020 5261 7469 6f6e  ap ==.    Ration
│ │ │ │ +0008e730: 616c 4d61 702c 202d 2d20 6571 7561 6c69  alMap, -- equali
│ │ │ │ +0008e740: 7479 206f 6620 7261 7469 6f6e 616c 206d  ty of rational m
│ │ │ │ +0008e750: 6170 730a 2020 2a20 2a6e 6f74 6520 5261  aps.  * *note Ra
│ │ │ │ +0008e760: 7469 6f6e 616c 4d61 7020 5e20 5a5a 3a20  tionalMap ^ ZZ: 
│ │ │ │ +0008e770: 5261 7469 6f6e 616c 4d61 7020 5e20 5a5a  RationalMap ^ ZZ
│ │ │ │ +0008e780: 2c20 2d2d 2070 6f77 6572 0a20 202a 2022  , -- power.  * "
│ │ │ │ +0008e790: 5261 7469 6f6e 616c 4d61 7020 5e2a 2220  RationalMap ^*" 
│ │ │ │ +0008e7a0: 2d2d 2073 6565 202a 6e6f 7465 2052 6174  -- see *note Rat
│ │ │ │ +0008e7b0: 696f 6e61 6c4d 6170 205e 2a2a 2049 6465  ionalMap ^** Ide
│ │ │ │ +0008e7c0: 616c 3a20 5261 7469 6f6e 616c 4d61 7020  al: RationalMap 
│ │ │ │ +0008e7d0: 5e5f 7374 5f73 740a 2020 2020 4964 6561  ^_st_st.    Idea
│ │ │ │ +0008e7e0: 6c2c 202d 2d20 696e 7665 7273 6520 696d  l, -- inverse im
│ │ │ │ +0008e7f0: 6167 6520 7669 6120 6120 7261 7469 6f6e  age via a ration
│ │ │ │ +0008e800: 616c 206d 6170 0a20 202a 202a 6e6f 7465  al map.  * *note
│ │ │ │ +0008e810: 2052 6174 696f 6e61 6c4d 6170 205e 2a2a   RationalMap ^**
│ │ │ │ +0008e820: 2049 6465 616c 3a20 5261 7469 6f6e 616c   Ideal: Rational
│ │ │ │ +0008e830: 4d61 7020 5e5f 7374 5f73 7420 4964 6561  Map ^_st_st Idea
│ │ │ │ +0008e840: 6c2c 202d 2d20 696e 7665 7273 6520 696d  l, -- inverse im
│ │ │ │ +0008e850: 6167 650a 2020 2020 7669 6120 6120 7261  age.    via a ra
│ │ │ │ +0008e860: 7469 6f6e 616c 206d 6170 0a20 202a 202a  tional map.  * *
│ │ │ │ +0008e870: 6e6f 7465 2052 6174 696f 6e61 6c4d 6170  note RationalMap
│ │ │ │ +0008e880: 205f 2a3a 2052 6174 696f 6e61 6c4d 6170   _*: RationalMap
│ │ │ │ +0008e890: 205f 7573 5f73 742c 202d 2d20 6469 7265   _us_st, -- dire
│ │ │ │ +0008e8a0: 6374 2069 6d61 6765 2076 6961 2061 2072  ct image via a r
│ │ │ │ +0008e8b0: 6174 696f 6e61 6c0a 2020 2020 6d61 700a  ational.    map.
│ │ │ │ +0008e8c0: 2020 2a20 2252 6174 696f 6e61 6c4d 6170    * "RationalMap
│ │ │ │ +0008e8d0: 2049 6465 616c 2220 2d2d 2073 6565 202a   Ideal" -- see *
│ │ │ │ +0008e8e0: 6e6f 7465 2052 6174 696f 6e61 6c4d 6170  note RationalMap
│ │ │ │ +0008e8f0: 205f 2a3a 2052 6174 696f 6e61 6c4d 6170   _*: RationalMap
│ │ │ │ +0008e900: 205f 7573 5f73 742c 202d 2d0a 2020 2020   _us_st, --.    
│ │ │ │ +0008e910: 6469 7265 6374 2069 6d61 6765 2076 6961  direct image via
│ │ │ │ +0008e920: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ +0008e930: 2020 2a20 2a6e 6f74 6520 5261 7469 6f6e    * *note Ration
│ │ │ │ +0008e940: 616c 4d61 7020 7c20 4964 6561 6c3a 2052  alMap | Ideal: R
│ │ │ │ +0008e950: 6174 696f 6e61 6c4d 6170 207c 2049 6465  ationalMap | Ide
│ │ │ │ +0008e960: 616c 2c20 2d2d 2072 6573 7472 6963 7469  al, -- restricti
│ │ │ │ +0008e970: 6f6e 206f 6620 610a 2020 2020 7261 7469  on of a.    rati
│ │ │ │ +0008e980: 6f6e 616c 206d 6170 0a20 202a 2022 5261  onal map.  * "Ra
│ │ │ │ +0008e990: 7469 6f6e 616c 4d61 7020 7c20 5269 6e67  tionalMap | Ring
│ │ │ │ +0008e9a0: 2220 2d2d 2073 6565 202a 6e6f 7465 2052  " -- see *note R
│ │ │ │ +0008e9b0: 6174 696f 6e61 6c4d 6170 207c 2049 6465  ationalMap | Ide
│ │ │ │ +0008e9c0: 616c 3a20 5261 7469 6f6e 616c 4d61 7020  al: RationalMap 
│ │ │ │ +0008e9d0: 7c20 4964 6561 6c2c 0a20 2020 202d 2d20  | Ideal,.    -- 
│ │ │ │ +0008e9e0: 7265 7374 7269 6374 696f 6e20 6f66 2061  restriction of a
│ │ │ │ +0008e9f0: 2072 6174 696f 6e61 6c20 6d61 700a 2020   rational map.  
│ │ │ │ +0008ea00: 2a20 2252 6174 696f 6e61 6c4d 6170 207c  * "RationalMap |
│ │ │ │ +0008ea10: 2052 696e 6745 6c65 6d65 6e74 2220 2d2d   RingElement" --
│ │ │ │ +0008ea20: 2073 6565 202a 6e6f 7465 2052 6174 696f   see *note Ratio
│ │ │ │ +0008ea30: 6e61 6c4d 6170 207c 2049 6465 616c 3a20  nalMap | Ideal: 
│ │ │ │ +0008ea40: 5261 7469 6f6e 616c 4d61 7020 7c0a 2020  RationalMap |.  
│ │ │ │ +0008ea50: 2020 4964 6561 6c2c 202d 2d20 7265 7374    Ideal, -- rest
│ │ │ │ +0008ea60: 7269 6374 696f 6e20 6f66 2061 2072 6174  riction of a rat
│ │ │ │ +0008ea70: 696f 6e61 6c20 6d61 700a 2020 2a20 2a6e  ional map.  * *n
│ │ │ │ +0008ea80: 6f74 6520 5261 7469 6f6e 616c 4d61 7020  ote RationalMap 
│ │ │ │ +0008ea90: 7c7c 2049 6465 616c 3a20 5261 7469 6f6e  || Ideal: Ration
│ │ │ │ +0008eaa0: 616c 4d61 7020 7c7c 2049 6465 616c 2c20  alMap || Ideal, 
│ │ │ │ +0008eab0: 2d2d 2072 6573 7472 6963 7469 6f6e 206f  -- restriction o
│ │ │ │ +0008eac0: 6620 610a 2020 2020 7261 7469 6f6e 616c  f a.    rational
│ │ │ │ +0008ead0: 206d 6170 0a20 202a 2022 5261 7469 6f6e   map.  * "Ration
│ │ │ │ +0008eae0: 616c 4d61 7020 7c7c 2052 696e 6722 202d  alMap || Ring" -
│ │ │ │ +0008eaf0: 2d20 7365 6520 2a6e 6f74 6520 5261 7469  - see *note Rati
│ │ │ │ +0008eb00: 6f6e 616c 4d61 7020 7c7c 2049 6465 616c  onalMap || Ideal
│ │ │ │ +0008eb10: 3a20 5261 7469 6f6e 616c 4d61 7020 7c7c  : RationalMap ||
│ │ │ │ +0008eb20: 0a20 2020 2049 6465 616c 2c20 2d2d 2072  .    Ideal, -- r
│ │ │ │ +0008eb30: 6573 7472 6963 7469 6f6e 206f 6620 6120  estriction of a 
│ │ │ │ +0008eb40: 7261 7469 6f6e 616c 206d 6170 0a20 202a  rational map.  *
│ │ │ │ +0008eb50: 2022 5261 7469 6f6e 616c 4d61 7020 7c7c   "RationalMap ||
│ │ │ │ +0008eb60: 2052 696e 6745 6c65 6d65 6e74 2220 2d2d   RingElement" --
│ │ │ │ +0008eb70: 2073 6565 202a 6e6f 7465 2052 6174 696f   see *note Ratio
│ │ │ │ +0008eb80: 6e61 6c4d 6170 207c 7c20 4964 6561 6c3a  nalMap || Ideal:
│ │ │ │ +0008eb90: 2052 6174 696f 6e61 6c4d 6170 0a20 2020   RationalMap.   
│ │ │ │ +0008eba0: 207c 7c20 4964 6561 6c2c 202d 2d20 7265   || Ideal, -- re
│ │ │ │ +0008ebb0: 7374 7269 6374 696f 6e20 6f66 2061 2072  striction of a r
│ │ │ │ +0008ebc0: 6174 696f 6e61 6c20 6d61 700a 2020 2a20  ational map.  * 
│ │ │ │ +0008ebd0: 2273 6567 7265 2852 6174 696f 6e61 6c4d  "segre(RationalM
│ │ │ │ +0008ebe0: 6170 2922 202d 2d20 7365 6520 2a6e 6f74  ap)" -- see *not
│ │ │ │ +0008ebf0: 6520 7365 6772 653a 2073 6567 7265 2c20  e segre: segre, 
│ │ │ │ +0008ec00: 2d2d 2053 6567 7265 2065 6d62 6564 6469  -- Segre embeddi
│ │ │ │ +0008ec10: 6e67 0a20 202a 2022 5365 6772 6543 6c61  ng.  * "SegreCla
│ │ │ │ +0008ec20: 7373 2852 6174 696f 6e61 6c4d 6170 2922  ss(RationalMap)"
│ │ │ │ +0008ec30: 202d 2d20 7365 6520 2a6e 6f74 6520 5365   -- see *note Se
│ │ │ │ +0008ec40: 6772 6543 6c61 7373 3a20 5365 6772 6543  greClass: SegreC
│ │ │ │ +0008ec50: 6c61 7373 2c20 2d2d 2053 6567 7265 0a20  lass, -- Segre. 
│ │ │ │ +0008ec60: 2020 2063 6c61 7373 206f 6620 6120 636c     class of a cl
│ │ │ │ +0008ec70: 6f73 6564 2073 7562 7363 6865 6d65 206f  osed subscheme o
│ │ │ │ +0008ec80: 6620 6120 7072 6f6a 6563 7469 7665 2076  f a projective v
│ │ │ │ +0008ec90: 6172 6965 7479 0a20 202a 202a 6e6f 7465  ariety.  * *note
│ │ │ │ +0008eca0: 2073 6f75 7263 6528 5261 7469 6f6e 616c   source(Rational
│ │ │ │ +0008ecb0: 4d61 7029 3a20 736f 7572 6365 5f6c 7052  Map): source_lpR
│ │ │ │ +0008ecc0: 6174 696f 6e61 6c4d 6170 5f72 702c 202d  ationalMap_rp, -
│ │ │ │ +0008ecd0: 2d20 636f 6f72 6469 6e61 7465 2072 696e  - coordinate rin
│ │ │ │ +0008ece0: 6720 6f66 0a20 2020 2074 6865 2073 6f75  g of.    the sou
│ │ │ │ +0008ecf0: 7263 6520 666f 7220 6120 7261 7469 6f6e  rce for a ration
│ │ │ │ +0008ed00: 616c 206d 6170 0a20 202a 202a 6e6f 7465  al map.  * *note
│ │ │ │ +0008ed10: 2073 7562 7374 6974 7574 6528 5261 7469   substitute(Rati
│ │ │ │ +0008ed20: 6f6e 616c 4d61 702c 506f 6c79 6e6f 6d69  onalMap,Polynomi
│ │ │ │ +0008ed30: 616c 5269 6e67 2c50 6f6c 796e 6f6d 6961  alRing,Polynomia
│ │ │ │ +0008ed40: 6c52 696e 6729 3a0a 2020 2020 7375 6273  lRing):.    subs
│ │ │ │ +0008ed50: 7469 7475 7465 5f6c 7052 6174 696f 6e61  titute_lpRationa
│ │ │ │ +0008ed60: 6c4d 6170 5f63 6d50 6f6c 796e 6f6d 6961  lMap_cmPolynomia
│ │ │ │ +0008ed70: 6c52 696e 675f 636d 506f 6c79 6e6f 6d69  lRing_cmPolynomi
│ │ │ │ +0008ed80: 616c 5269 6e67 5f72 702c 202d 2d0a 2020  alRing_rp, --.  
│ │ │ │ +0008ed90: 2020 7375 6273 7469 7475 7465 2074 6865    substitute the
│ │ │ │ +0008eda0: 2061 6d62 6965 6e74 2070 726f 6a65 6374   ambient project
│ │ │ │ +0008edb0: 6976 6520 7370 6163 6573 206f 6620 736f  ive spaces of so
│ │ │ │ +0008edc0: 7572 6365 2061 6e64 2074 6172 6765 740a  urce and target.
│ │ │ │ +0008edd0: 2020 2a20 2272 6174 696f 6e61 6c4d 6170    * "rationalMap
│ │ │ │ +0008ede0: 2852 6174 696f 6e61 6c4d 6170 2922 202d  (RationalMap)" -
│ │ │ │ +0008edf0: 2d20 7365 6520 2a6e 6f74 6520 7375 7065  - see *note supe
│ │ │ │ +0008ee00: 7228 5261 7469 6f6e 616c 4d61 7029 3a0a  r(RationalMap):.
│ │ │ │ +0008ee10: 2020 2020 7375 7065 725f 6c70 5261 7469      super_lpRati
│ │ │ │ +0008ee20: 6f6e 616c 4d61 705f 7270 2c20 2d2d 2067  onalMap_rp, -- g
│ │ │ │ +0008ee30: 6574 2074 6865 2072 6174 696f 6e61 6c20  et the rational 
│ │ │ │ +0008ee40: 6d61 7020 7768 6f73 6520 7461 7267 6574  map whose target
│ │ │ │ +0008ee50: 2069 7320 610a 2020 2020 7072 6f6a 6563   is a.    projec
│ │ │ │ +0008ee60: 7469 7665 2073 7061 6365 0a20 202a 202a  tive space.  * *
│ │ │ │ +0008ee70: 6e6f 7465 2073 7570 6572 2852 6174 696f  note super(Ratio
│ │ │ │ +0008ee80: 6e61 6c4d 6170 293a 2073 7570 6572 5f6c  nalMap): super_l
│ │ │ │ +0008ee90: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ +0008eea0: 202d 2d20 6765 7420 7468 6520 7261 7469   -- get the rati
│ │ │ │ +0008eeb0: 6f6e 616c 206d 6170 0a20 2020 2077 686f  onal map.    who
│ │ │ │ +0008eec0: 7365 2074 6172 6765 7420 6973 2061 2070  se target is a p
│ │ │ │ +0008eed0: 726f 6a65 6374 6976 6520 7370 6163 650a  rojective space.
│ │ │ │ +0008eee0: 2020 2a20 2a6e 6f74 6520 7461 7267 6574    * *note target
│ │ │ │ +0008eef0: 2852 6174 696f 6e61 6c4d 6170 293a 2074  (RationalMap): t
│ │ │ │ +0008ef00: 6172 6765 745f 6c70 5261 7469 6f6e 616c  arget_lpRational
│ │ │ │ +0008ef10: 4d61 705f 7270 2c20 2d2d 2063 6f6f 7264  Map_rp, -- coord
│ │ │ │ +0008ef20: 696e 6174 6520 7269 6e67 206f 660a 2020  inate ring of.  
│ │ │ │ +0008ef30: 2020 7468 6520 7461 7267 6574 2066 6f72    the target for
│ │ │ │ +0008ef40: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ +0008ef50: 2020 2a20 2a6e 6f74 6520 746f 4578 7465    * *note toExte
│ │ │ │ +0008ef60: 726e 616c 5374 7269 6e67 2852 6174 696f  rnalString(Ratio
│ │ │ │ +0008ef70: 6e61 6c4d 6170 293a 2074 6f45 7874 6572  nalMap): toExter
│ │ │ │ +0008ef80: 6e61 6c53 7472 696e 675f 6c70 5261 7469  nalString_lpRati
│ │ │ │ +0008ef90: 6f6e 616c 4d61 705f 7270 2c20 2d2d 0a20  onalMap_rp, --. 
│ │ │ │ +0008efa0: 2020 2063 6f6e 7665 7274 2074 6f20 6120     convert to a 
│ │ │ │ +0008efb0: 7265 6164 6162 6c65 2073 7472 696e 670a  readable string.
│ │ │ │ +0008efc0: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ +0008efd0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ +0008efe0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ +0008eff0: 6374 202a 6e6f 7465 2052 6174 696f 6e61  ct *note Rationa
│ │ │ │ +0008f000: 6c4d 6170 3a20 5261 7469 6f6e 616c 4d61  lMap: RationalMa
│ │ │ │ +0008f010: 702c 2069 7320 6120 2a6e 6f74 6520 7479  p, is a *note ty
│ │ │ │ +0008f020: 7065 3a0a 284d 6163 6175 6c61 7932 446f  pe:.(Macaulay2Do
│ │ │ │ +0008f030: 6329 5479 7065 2c2c 2077 6974 6820 616e  c)Type,, with an
│ │ │ │ +0008f040: 6365 7374 6f72 2063 6c61 7373 6573 202a  cestor classes *
│ │ │ │ +0008f050: 6e6f 7465 204d 7574 6162 6c65 4861 7368  note MutableHash
│ │ │ │ +0008f060: 5461 626c 653a 0a28 4d61 6361 756c 6179  Table:.(Macaulay
│ │ │ │ +0008f070: 3244 6f63 294d 7574 6162 6c65 4861 7368  2Doc)MutableHash
│ │ │ │ +0008f080: 5461 626c 652c 203c 202a 6e6f 7465 2048  Table, < *note H
│ │ │ │ +0008f090: 6173 6854 6162 6c65 3a20 284d 6163 6175  ashTable: (Macau
│ │ │ │ +0008f0a0: 6c61 7932 446f 6329 4861 7368 5461 626c  lay2Doc)HashTabl
│ │ │ │ +0008f0b0: 652c 203c 0a2a 6e6f 7465 2054 6869 6e67  e, <.*note Thing
│ │ │ │ +0008f0c0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0008f0d0: 5468 696e 672c 2e0a 0a2d 2d2d 2d2d 2d2d  Thing,...-------
│ │ │ │  0008f0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008f120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ -0008f130: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ -0008f140: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ -0008f150: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ -0008f160: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ -0008f170: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ -0008f180: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ -0008f190: 6b61 6765 732f 4372 656d 6f6e 612f 0a64  kages/Cremona/.d
│ │ │ │ -0008f1a0: 6f63 756d 656e 7461 7469 6f6e 2e6d 323a  ocumentation.m2:
│ │ │ │ -0008f1b0: 3335 323a 302e 0a1f 0a46 696c 653a 2043  352:0....File: C
│ │ │ │ -0008f1c0: 7265 6d6f 6e61 2e69 6e66 6f2c 204e 6f64  remona.info, Nod
│ │ │ │ -0008f1d0: 653a 2072 6174 696f 6e61 6c4d 6170 2c20  e: rationalMap, 
│ │ │ │ -0008f1e0: 4e65 7874 3a20 5261 7469 6f6e 616c 4d61  Next: RationalMa
│ │ │ │ -0008f1f0: 7020 212c 2050 7265 763a 2052 6174 696f  p !, Prev: Ratio
│ │ │ │ -0008f200: 6e61 6c4d 6170 2c20 5570 3a20 546f 700a  nalMap, Up: Top.
│ │ │ │ -0008f210: 0a72 6174 696f 6e61 6c4d 6170 202d 2d20  .rationalMap -- 
│ │ │ │ -0008f220: 6d61 6b65 7320 6120 7261 7469 6f6e 616c  makes a rational
│ │ │ │ -0008f230: 206d 6170 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a   map.***********
│ │ │ │ +0008f120: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ +0008f130: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ +0008f140: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ +0008f150: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ +0008f160: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ +0008f170: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163  .25.11+ds/M2/Mac
│ │ │ │ +0008f180: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ +0008f190: 4372 656d 6f6e 612f 0a64 6f63 756d 656e  Cremona/.documen
│ │ │ │ +0008f1a0: 7461 7469 6f6e 2e6d 323a 3335 323a 302e  tation.m2:352:0.
│ │ │ │ +0008f1b0: 0a1f 0a46 696c 653a 2043 7265 6d6f 6e61  ...File: Cremona
│ │ │ │ +0008f1c0: 2e69 6e66 6f2c 204e 6f64 653a 2072 6174  .info, Node: rat
│ │ │ │ +0008f1d0: 696f 6e61 6c4d 6170 2c20 4e65 7874 3a20  ionalMap, Next: 
│ │ │ │ +0008f1e0: 5261 7469 6f6e 616c 4d61 7020 212c 2050  RationalMap !, P
│ │ │ │ +0008f1f0: 7265 763a 2052 6174 696f 6e61 6c4d 6170  rev: RationalMap
│ │ │ │ +0008f200: 2c20 5570 3a20 546f 700a 0a72 6174 696f  , Up: Top..ratio
│ │ │ │ +0008f210: 6e61 6c4d 6170 202d 2d20 6d61 6b65 7320  nalMap -- makes 
│ │ │ │ +0008f220: 6120 7261 7469 6f6e 616c 206d 6170 0a2a  a rational map.*
│ │ │ │ +0008f230: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0008f240: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0008f250: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ -0008f260: 6167 653a 200a 2020 2020 2020 2020 7261  age: .        ra
│ │ │ │ -0008f270: 7469 6f6e 616c 4d61 7020 7068 6920 0a20  tionalMap phi . 
│ │ │ │ -0008f280: 2020 2020 2020 2072 6174 696f 6e61 6c4d         rationalM
│ │ │ │ -0008f290: 6170 2046 0a20 202a 2049 6e70 7574 733a  ap F.  * Inputs:
│ │ │ │ -0008f2a0: 0a20 2020 2020 202a 2061 202a 6e6f 7465  .      * a *note
│ │ │ │ -0008f2b0: 2072 696e 6720 6d61 703a 2028 4d61 6361   ring map: (Maca
│ │ │ │ -0008f2c0: 756c 6179 3244 6f63 2952 696e 674d 6170  ulay2Doc)RingMap
│ │ │ │ -0008f2d0: 2c2c 206f 7220 6120 2a6e 6f74 6520 6d61  ,, or a *note ma
│ │ │ │ -0008f2e0: 7472 6978 3a0a 2020 2020 2020 2020 284d  trix:.        (M
│ │ │ │ -0008f2f0: 6163 6175 6c61 7932 446f 6329 4d61 7472  acaulay2Doc)Matr
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│ │ │ │ -0008f320: 6f63 294c 6973 742c 2c20 6574 632e 0a20  oc)List,, etc.. 
│ │ │ │ -0008f330: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
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│ │ │ │ -0008f380: 2020 2020 202a 202a 6e6f 7465 2044 6f6d       * *note Dom
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│ │ │ │ -0008f3c0: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
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│ │ │ │ -0008f450: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
│ │ │ │ -0008f460: 7320 6973 2074 6865 2062 6173 6963 2063  s is the basic c
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│ │ │ │ -0008f4a0: 702c 2e20 5468 650a 6d65 7468 6f64 2069  p,. The.method i
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│ │ │ │ -0008f4c0: 746f 202a 6e6f 7465 2074 6f4d 6170 3a20  to *note toMap: 
│ │ │ │ -0008f4d0: 746f 4d61 702c 2c20 6578 6365 7074 2074  toMap,, except t
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│ │ │ │ -0008f500: 6170 3a20 5261 7469 6f6e 616c 4d61 702c  ap: RationalMap,
│ │ │ │ -0008f510: 2069 6e73 7465 6164 206f 6620 6120 2a6e   instead of a *n
│ │ │ │ -0008f520: 6f74 6520 5269 6e67 4d61 703a 2028 4d61  ote RingMap: (Ma
│ │ │ │ -0008f530: 6361 756c 6179 3244 6f63 2952 696e 674d  caulay2Doc)RingM
│ │ │ │ -0008f540: 6170 2c2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ap,...+---------
│ │ │ │ +0008f250: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a  **..  * Usage: .
│ │ │ │ +0008f260: 2020 2020 2020 2020 7261 7469 6f6e 616c          rational
│ │ │ │ +0008f270: 4d61 7020 7068 6920 0a20 2020 2020 2020  Map phi .       
│ │ │ │ +0008f280: 2072 6174 696f 6e61 6c4d 6170 2046 0a20   rationalMap F. 
│ │ │ │ +0008f290: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +0008f2a0: 202a 2061 202a 6e6f 7465 2072 696e 6720   * a *note ring 
│ │ │ │ +0008f2b0: 6d61 703a 2028 4d61 6361 756c 6179 3244  map: (Macaulay2D
│ │ │ │ +0008f2c0: 6f63 2952 696e 674d 6170 2c2c 206f 7220  oc)RingMap,, or 
│ │ │ │ +0008f2d0: 6120 2a6e 6f74 6520 6d61 7472 6978 3a0a  a *note matrix:.
│ │ │ │ +0008f2e0: 2020 2020 2020 2020 284d 6163 6175 6c61          (Macaula
│ │ │ │ +0008f2f0: 7932 446f 6329 4d61 7472 6978 2c20 6f72  y2Doc)Matrix, or
│ │ │ │ +0008f300: 2061 202a 6e6f 7465 206c 6973 743a 2028   a *note list: (
│ │ │ │ +0008f310: 4d61 6361 756c 6179 3244 6f63 294c 6973  Macaulay2Doc)Lis
│ │ │ │ +0008f320: 742c 2c20 6574 632e 0a20 202a 202a 6e6f  t,, etc..  * *no
│ │ │ │ +0008f330: 7465 204f 7074 696f 6e61 6c20 696e 7075  te Optional inpu
│ │ │ │ +0008f340: 7473 3a20 284d 6163 6175 6c61 7932 446f  ts: (Macaulay2Do
│ │ │ │ +0008f350: 6329 7573 696e 6720 6675 6e63 7469 6f6e  c)using function
│ │ │ │ +0008f360: 7320 7769 7468 206f 7074 696f 6e61 6c20  s with optional 
│ │ │ │ +0008f370: 696e 7075 7473 2c3a 0a20 2020 2020 202a  inputs,:.      *
│ │ │ │ +0008f380: 202a 6e6f 7465 2044 6f6d 696e 616e 743a   *note Dominant:
│ │ │ │ +0008f390: 2044 6f6d 696e 616e 742c 203d 3e20 2e2e   Dominant, => ..
│ │ │ │ +0008f3a0: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ +0008f3b0: 206e 756c 6c2c 200a 2020 2a20 4f75 7470   null, .  * Outp
│ │ │ │ +0008f3c0: 7574 733a 0a20 2020 2020 202a 2061 202a  uts:.      * a *
│ │ │ │ +0008f3d0: 6e6f 7465 2072 6174 696f 6e61 6c20 6d61  note rational ma
│ │ │ │ +0008f3e0: 703a 2052 6174 696f 6e61 6c4d 6170 2c2c  p: RationalMap,,
│ │ │ │ +0008f3f0: 2074 6865 2072 6174 696f 6e61 6c20 6d61   the rational ma
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│ │ │ │ +0008f410: 2070 6869 0a20 2020 2020 2020 206f 7220   phi.        or 
│ │ │ │ +0008f420: 6279 2074 6865 2072 696e 6720 6d61 7020  by the ring map 
│ │ │ │ +0008f430: 6465 6669 6e65 6420 6279 2046 0a0a 4465  defined by F..De
│ │ │ │ +0008f440: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +0008f450: 3d3d 3d3d 3d0a 0a54 6869 7320 6973 2074  =====..This is t
│ │ │ │ +0008f460: 6865 2062 6173 6963 2063 6f6e 7374 7275  he basic constru
│ │ │ │ +0008f470: 6374 696f 6e20 666f 7220 6120 2a6e 6f74  ction for a *not
│ │ │ │ +0008f480: 6520 7261 7469 6f6e 616c 206d 6170 3a20  e rational map: 
│ │ │ │ +0008f490: 5261 7469 6f6e 616c 4d61 702c 2e20 5468  RationalMap,. Th
│ │ │ │ +0008f4a0: 650a 6d65 7468 6f64 2069 7320 7175 6974  e.method is quit
│ │ │ │ +0008f4b0: 6520 7369 6d69 6c61 7220 746f 202a 6e6f  e similar to *no
│ │ │ │ +0008f4c0: 7465 2074 6f4d 6170 3a20 746f 4d61 702c  te toMap: toMap,
│ │ │ │ +0008f4d0: 2c20 6578 6365 7074 2074 6861 7420 6974  , except that it
│ │ │ │ +0008f4e0: 2072 6574 7572 6e73 2061 202a 6e6f 7465   returns a *note
│ │ │ │ +0008f4f0: 0a52 6174 696f 6e61 6c4d 6170 3a20 5261  .RationalMap: Ra
│ │ │ │ +0008f500: 7469 6f6e 616c 4d61 702c 2069 6e73 7465  tionalMap, inste
│ │ │ │ +0008f510: 6164 206f 6620 6120 2a6e 6f74 6520 5269  ad of a *note Ri
│ │ │ │ +0008f520: 6e67 4d61 703a 2028 4d61 6361 756c 6179  ngMap: (Macaulay
│ │ │ │ +0008f530: 3244 6f63 2952 696e 674d 6170 2c2e 0a0a  2Doc)RingMap,...
│ │ │ │ +0008f540: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0008f550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008f580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0008f590: 3120 3a20 5220 3a3d 2051 515b 745f 302e  1 : R := QQ[t_0.
│ │ │ │ -0008f5a0: 2e74 5f38 5d20 2020 2020 2020 2020 2020  .t_8]           
│ │ │ │ +0008f580: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5220  ------+.|i1 : R 
│ │ │ │ +0008f590: 3a3d 2051 515b 745f 302e 2e74 5f38 5d20  := QQ[t_0..t_8] 
│ │ │ │ +0008f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f5d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0008f5c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008f5d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0008f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f610: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0008f620: 3120 3d20 5151 5b74 202e 2e74 205d 2020  1 = QQ[t ..t ]  
│ │ │ │ +0008f610: 2020 2020 2020 7c0a 7c6f 3120 3d20 5151        |.|o1 = QQ
│ │ │ │ +0008f620: 5b74 202e 2e74 205d 2020 2020 2020 2020  [t ..t ]        
│ │ │ │  0008f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f660: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0008f670: 3020 2020 3820 2020 2020 2020 2020 2020  0   8           
│ │ │ │ +0008f650: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008f660: 7c20 2020 2020 2020 2020 3020 2020 3820  |         0   8 
│ │ │ │ +0008f670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f6a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0008f6a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0008f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f6f0: 2020 2020 7c0a 7c6f 3120 3a20 506f 6c79      |.|o1 : Poly
│ │ │ │ -0008f700: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │ +0008f6e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008f6f0: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c  |o1 : Polynomial
│ │ │ │ +0008f700: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │  0008f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f730: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0008f730: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  0008f740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008f770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008f780: 2d2d 2d2d 2b0a 7c69 3220 3a20 4620 3d20  ----+.|i2 : F = 
│ │ │ │ -0008f790: 6d61 7472 6978 7b7b 745f 302a 745f 332a  matrix{{t_0*t_3*
│ │ │ │ -0008f7a0: 745f 352c 2074 5f31 2a74 5f33 2a74 5f36  t_5, t_1*t_3*t_6
│ │ │ │ -0008f7b0: 2c20 745f 322a 745f 342a 745f 372c 2074  , t_2*t_4*t_7, t
│ │ │ │ -0008f7c0: 5f32 2a74 5f34 2a74 5f38 7d7d 7c0a 7c20  _2*t_4*t_8}}|.| 
│ │ │ │ +0008f770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0008f780: 7c69 3220 3a20 4620 3d20 6d61 7472 6978  |i2 : F = matrix
│ │ │ │ +0008f790: 7b7b 745f 302a 745f 332a 745f 352c 2074  {{t_0*t_3*t_5, t
│ │ │ │ +0008f7a0: 5f31 2a74 5f33 2a74 5f36 2c20 745f 322a  _1*t_3*t_6, t_2*
│ │ │ │ +0008f7b0: 745f 342a 745f 372c 2074 5f32 2a74 5f34  t_4*t_7, t_2*t_4
│ │ │ │ +0008f7c0: 2a74 5f38 7d7d 7c0a 7c20 2020 2020 2020  *t_8}}|.|       
│ │ │ │  0008f7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f810: 2020 2020 7c0a 7c6f 3220 3d20 7c20 745f      |.|o2 = | t_
│ │ │ │ -0008f820: 3074 5f33 745f 3520 745f 3174 5f33 745f  0t_3t_5 t_1t_3t_
│ │ │ │ -0008f830: 3620 745f 3274 5f34 745f 3720 745f 3274  6 t_2t_4t_7 t_2t
│ │ │ │ -0008f840: 5f34 745f 3820 7c20 2020 2020 2020 2020  _4t_8 |         
│ │ │ │ -0008f850: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0008f800: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008f810: 7c6f 3220 3d20 7c20 745f 3074 5f33 745f  |o2 = | t_0t_3t_
│ │ │ │ +0008f820: 3520 745f 3174 5f33 745f 3620 745f 3274  5 t_1t_3t_6 t_2t
│ │ │ │ +0008f830: 5f34 745f 3720 745f 3274 5f34 745f 3820  _4t_7 t_2t_4t_8 
│ │ │ │ +0008f840: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008f850: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0008f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f8a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0008f8b0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -0008f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f8d0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -0008f8e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0008f8f0: 3220 3a20 4d61 7472 6978 2028 5151 5b74  2 : Matrix (QQ[t
│ │ │ │ -0008f900: 202e 2e74 205d 2920 203c 2d2d 2028 5151   ..t ])  <-- (QQ
│ │ │ │ -0008f910: 5b74 202e 2e74 205d 2920 2020 2020 2020  [t ..t ])       
│ │ │ │ -0008f920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f930: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0008f940: 2020 2020 2020 2020 3020 2020 3820 2020          0   8   
│ │ │ │ -0008f950: 2020 2020 2020 2020 2020 3020 2020 3820            0   8 
│ │ │ │ +0008f890: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008f8a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008f8b0: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ +0008f8c0: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ +0008f8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008f8e0: 2020 2020 2020 7c0a 7c6f 3220 3a20 4d61        |.|o2 : Ma
│ │ │ │ +0008f8f0: 7472 6978 2028 5151 5b74 202e 2e74 205d  trix (QQ[t ..t ]
│ │ │ │ +0008f900: 2920 203c 2d2d 2028 5151 5b74 202e 2e74  )  <-- (QQ[t ..t
│ │ │ │ +0008f910: 205d 2920 2020 2020 2020 2020 2020 2020   ])             
│ │ │ │ +0008f920: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008f930: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008f940: 2020 3020 2020 3820 2020 2020 2020 2020    0   8         
│ │ │ │ +0008f950: 2020 2020 3020 2020 3820 2020 2020 2020      0   8       
│ │ │ │  0008f960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008f970: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0008f970: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  0008f980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008f9c0: 2d2d 2d2d 2b0a 7c69 3320 3a20 7068 6920  ----+.|i3 : phi 
│ │ │ │ -0008f9d0: 3d20 746f 4d61 7020 4620 2020 2020 2020  = toMap F       
│ │ │ │ +0008f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0008f9c0: 7c69 3320 3a20 7068 6920 3d20 746f 4d61  |i3 : phi = toMa
│ │ │ │ +0008f9d0: 7020 4620 2020 2020 2020 2020 2020 2020  p F             
│ │ │ │  0008f9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008f9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fa00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0008fa00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0008fa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fa50: 2020 2020 7c0a 7c6f 3320 3d20 6d61 7020      |.|o3 = map 
│ │ │ │ -0008fa60: 2851 515b 7420 2e2e 7420 5d2c 2051 515b  (QQ[t ..t ], QQ[
│ │ │ │ -0008fa70: 7820 2e2e 7820 5d2c 207b 7420 7420 7420  x ..x ], {t t t 
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│ │ │ │ -0008fa90: 2c20 7420 7420 7420 7d29 2020 7c0a 7c20  , t t t })  |.| 
│ │ │ │ -0008faa0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -0008fab0: 2038 2020 2020 2020 2030 2020 2033 2020   8       0   3  
│ │ │ │ -0008fac0: 2020 2030 2033 2035 2020 2031 2033 2036     0 3 5   1 3 6
│ │ │ │ -0008fad0: 2020 2032 2034 2037 2020 2032 2034 2038     2 4 7   2 4 8
│ │ │ │ -0008fae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0008fa40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008fa50: 7c6f 3320 3d20 6d61 7020 2851 515b 7420  |o3 = map (QQ[t 
│ │ │ │ +0008fa60: 2e2e 7420 5d2c 2051 515b 7820 2e2e 7820  ..t ], QQ[x ..x 
│ │ │ │ +0008fa70: 5d2c 207b 7420 7420 7420 2c20 7420 7420  ], {t t t , t t 
│ │ │ │ +0008fa80: 7420 2c20 7420 7420 7420 2c20 7420 7420  t , t t t , t t 
│ │ │ │ +0008fa90: 7420 7d29 2020 7c0a 7c20 2020 2020 2020  t })  |.|       
│ │ │ │ +0008faa0: 2020 2020 2020 2030 2020 2038 2020 2020         0   8    
│ │ │ │ +0008fab0: 2020 2030 2020 2033 2020 2020 2030 2033     0   3     0 3
│ │ │ │ +0008fac0: 2035 2020 2031 2033 2036 2020 2032 2034   5   1 3 6   2 4
│ │ │ │ +0008fad0: 2037 2020 2032 2034 2038 2020 2020 7c0a   7   2 4 8    |.
│ │ │ │ +0008fae0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0008faf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fb20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0008fb30: 3320 3a20 5269 6e67 4d61 7020 5151 5b74  3 : RingMap QQ[t
│ │ │ │ -0008fb40: 202e 2e74 205d 203c 2d2d 2051 515b 7820   ..t ] <-- QQ[x 
│ │ │ │ -0008fb50: 2e2e 7820 5d20 2020 2020 2020 2020 2020  ..x ]           
│ │ │ │ -0008fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fb70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0008fb80: 2020 2020 2020 2020 3020 2020 3820 2020          0   8   
│ │ │ │ -0008fb90: 2020 2020 2020 2030 2020 2033 2020 2020         0   3    
│ │ │ │ +0008fb20: 2020 2020 2020 7c0a 7c6f 3320 3a20 5269        |.|o3 : Ri
│ │ │ │ +0008fb30: 6e67 4d61 7020 5151 5b74 202e 2e74 205d  ngMap QQ[t ..t ]
│ │ │ │ +0008fb40: 203c 2d2d 2051 515b 7820 2e2e 7820 5d20   <-- QQ[x ..x ] 
│ │ │ │ +0008fb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008fb60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008fb70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008fb80: 2020 3020 2020 3820 2020 2020 2020 2020    0   8         
│ │ │ │ +0008fb90: 2030 2020 2033 2020 2020 2020 2020 2020   0   3          
│ │ │ │  0008fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fbb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0008fbb0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  0008fbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008fbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008fbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008fbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0008fc00: 2d2d 2d2d 2b0a 7c69 3420 3a20 7261 7469  ----+.|i4 : rati
│ │ │ │ -0008fc10: 6f6e 616c 4d61 7020 7068 6920 2020 2020  onalMap phi     
│ │ │ │ +0008fbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0008fc00: 7c69 3420 3a20 7261 7469 6f6e 616c 4d61  |i4 : rationalMa
│ │ │ │ +0008fc10: 7020 7068 6920 2020 2020 2020 2020 2020  p phi           
│ │ │ │  0008fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fc40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0008fc40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0008fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fc90: 2020 2020 7c0a 7c6f 3420 3d20 2d2d 2072      |.|o4 = -- r
│ │ │ │ -0008fca0: 6174 696f 6e61 6c20 6d61 7020 2d2d 2020  ational map --  
│ │ │ │ +0008fc80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008fc90: 7c6f 3420 3d20 2d2d 2072 6174 696f 6e61  |o4 = -- rationa
│ │ │ │ +0008fca0: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │  0008fcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fcd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0008fce0: 2020 2020 736f 7572 6365 3a20 5072 6f6a      source: Proj
│ │ │ │ -0008fcf0: 2851 515b 7420 2c20 7420 2c20 7420 2c20  (QQ[t , t , t , 
│ │ │ │ -0008fd00: 7420 2c20 7420 2c20 7420 2c20 7420 2c20  t , t , t , t , 
│ │ │ │ -0008fd10: 7420 2c20 7420 5d29 2020 2020 2020 2020  t , t ])        
│ │ │ │ -0008fd20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0008fd30: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -0008fd40: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -0008fd50: 2035 2020 2036 2020 2037 2020 2038 2020   5   6   7   8  
│ │ │ │ -0008fd60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0008fd70: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
│ │ │ │ -0008fd80: 2851 515b 7820 2c20 7820 2c20 7820 2c20  (QQ[x , x , x , 
│ │ │ │ -0008fd90: 7820 5d29 2020 2020 2020 2020 2020 2020  x ])            
│ │ │ │ -0008fda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fdb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0008fdc0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -0008fdd0: 2031 2020 2032 2020 2033 2020 2020 2020   1   2   3      
│ │ │ │ +0008fcd0: 2020 2020 2020 7c0a 7c20 2020 2020 736f        |.|     so
│ │ │ │ +0008fce0: 7572 6365 3a20 5072 6f6a 2851 515b 7420  urce: Proj(QQ[t 
│ │ │ │ +0008fcf0: 2c20 7420 2c20 7420 2c20 7420 2c20 7420  , t , t , t , t 
│ │ │ │ +0008fd00: 2c20 7420 2c20 7420 2c20 7420 2c20 7420  , t , t , t , t 
│ │ │ │ +0008fd10: 5d29 2020 2020 2020 2020 2020 2020 7c0a  ])            |.
│ │ │ │ +0008fd20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008fd30: 2020 2020 2020 2030 2020 2031 2020 2032         0   1   2
│ │ │ │ +0008fd40: 2020 2033 2020 2034 2020 2035 2020 2036     3   4   5   6
│ │ │ │ +0008fd50: 2020 2037 2020 2038 2020 2020 2020 2020     7   8        
│ │ │ │ +0008fd60: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ +0008fd70: 7267 6574 3a20 5072 6f6a 2851 515b 7820  rget: Proj(QQ[x 
│ │ │ │ +0008fd80: 2c20 7820 2c20 7820 2c20 7820 5d29 2020  , x , x , x ])  
│ │ │ │ +0008fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0008fda0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008fdb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008fdc0: 2020 2020 2020 2030 2020 2031 2020 2032         0   1   2
│ │ │ │ +0008fdd0: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │  0008fde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fdf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0008fe00: 2020 2020 6465 6669 6e69 6e67 2066 6f72      defining for
│ │ │ │ -0008fe10: 6d73 3a20 7b20 2020 2020 2020 2020 2020  ms: {           
│ │ │ │ +0008fdf0: 2020 2020 2020 7c0a 7c20 2020 2020 6465        |.|     de
│ │ │ │ +0008fe00: 6669 6e69 6e67 2066 6f72 6d73 3a20 7b20  fining forms: { 
│ │ │ │ +0008fe10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fe20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fe30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fe40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0008fe50: 2020 2020 2020 2020 2020 2020 2074 2074               t t
│ │ │ │ -0008fe60: 2074 202c 2020 2020 2020 2020 2020 2020   t ,            
│ │ │ │ +0008fe30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008fe40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008fe50: 2020 2020 2020 2074 2074 2074 202c 2020         t t t ,  
│ │ │ │ +0008fe60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fe70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fe80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0008fe80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0008fe90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fea0: 2020 2020 2020 3020 3320 3520 2020 2020        0 3 5     
│ │ │ │ +0008fea0: 3020 3320 3520 2020 2020 2020 2020 2020  0 3 5           
│ │ │ │  0008feb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fed0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0008fec0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008fed0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0008fee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008fef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008ff00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008ff10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0008ff20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008ff30: 2020 2020 2074 2074 2074 202c 2020 2020       t t t ,    
│ │ │ │ +0008ff10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0008ff20: 2020 2020 2020 2020 2020 2020 2020 2074                 t
│ │ │ │ +0008ff30: 2074 2074 202c 2020 2020 2020 2020 2020   t t ,          
│ │ │ │  0008ff40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008ff50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008ff60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0008ff70: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -0008ff80: 3320 3620 2020 2020 2020 2020 2020 2020  3 6             
│ │ │ │ +0008ff50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008ff60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0008ff70: 2020 2020 2020 2020 3120 3320 3620 2020          1 3 6   
│ │ │ │ +0008ff80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008ff90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008ffa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0008ffa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0008ffb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008ffc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008ffd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008ffe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008fff0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00090000: 2020 2020 2020 2020 2020 2020 2074 2074               t t
│ │ │ │ -00090010: 2074 202c 2020 2020 2020 2020 2020 2020   t ,            
│ │ │ │ +0008ffe0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0008fff0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00090000: 2020 2020 2020 2074 2074 2074 202c 2020         t t t ,  
│ │ │ │ +00090010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090030: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00090030: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00090040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090050: 2020 2020 2020 3220 3420 3720 2020 2020        2 4 7     
│ │ │ │ +00090050: 3220 3420 3720 2020 2020 2020 2020 2020  2 4 7           
│ │ │ │  00090060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090080: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00090070: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00090080: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00090090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000900a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000900b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000900c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000900d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000900e0: 2020 2020 2074 2074 2074 2020 2020 2020       t t t      
│ │ │ │ +000900c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000900d0: 2020 2020 2020 2020 2020 2020 2020 2074                 t
│ │ │ │ +000900e0: 2074 2074 2020 2020 2020 2020 2020 2020   t t            
│ │ │ │  000900f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090110: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00090120: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00090130: 3420 3820 2020 2020 2020 2020 2020 2020  4 8             
│ │ │ │ +00090100: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00090110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00090120: 2020 2020 2020 2020 3220 3420 3820 2020          2 4 8   
│ │ │ │ +00090130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090150: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00090160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090170: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │ +00090150: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00090160: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
│ │ │ │ +00090170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000901a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00090190: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000901a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000901b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000901c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000901d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000901e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000901f0: 3420 3a20 5261 7469 6f6e 616c 4d61 7020  4 : RationalMap 
│ │ │ │ -00090200: 2863 7562 6963 2072 6174 696f 6e61 6c20  (cubic rational 
│ │ │ │ -00090210: 6d61 7020 6672 6f6d 2050 505e 3820 746f  map from PP^8 to
│ │ │ │ -00090220: 2050 505e 3329 2020 2020 2020 2020 2020   PP^3)          
│ │ │ │ -00090230: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000901e0: 2020 2020 2020 7c0a 7c6f 3420 3a20 5261        |.|o4 : Ra
│ │ │ │ +000901f0: 7469 6f6e 616c 4d61 7020 2863 7562 6963  tionalMap (cubic
│ │ │ │ +00090200: 2072 6174 696f 6e61 6c20 6d61 7020 6672   rational map fr
│ │ │ │ +00090210: 6f6d 2050 505e 3820 746f 2050 505e 3329  om PP^8 to PP^3)
│ │ │ │ +00090220: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00090230: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00090240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00090250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00090260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00090270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00090280: 3520 3a20 7261 7469 6f6e 616c 4d61 7020  5 : rationalMap 
│ │ │ │ -00090290: 4620 2020 2020 2020 2020 2020 2020 2020  F               
│ │ │ │ +00090270: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7261  ------+.|i5 : ra
│ │ │ │ +00090280: 7469 6f6e 616c 4d61 7020 4620 2020 2020  tionalMap F     
│ │ │ │ +00090290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000902a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000902b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000902c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000902b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000902c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000902d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000902e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000902f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090300: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00090310: 3520 3d20 2d2d 2072 6174 696f 6e61 6c20  5 = -- rational 
│ │ │ │ -00090320: 6d61 7020 2d2d 2020 2020 2020 2020 2020  map --          
│ │ │ │ +00090300: 2020 2020 2020 7c0a 7c6f 3520 3d20 2d2d        |.|o5 = --
│ │ │ │ +00090310: 2072 6174 696f 6e61 6c20 6d61 7020 2d2d   rational map --
│ │ │ │ +00090320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090350: 2020 2020 7c0a 7c20 2020 2020 736f 7572      |.|     sour
│ │ │ │ -00090360: 6365 3a20 5072 6f6a 2851 515b 7420 2c20  ce: Proj(QQ[t , 
│ │ │ │ -00090370: 7420 2c20 7420 2c20 7420 2c20 7420 2c20  t , t , t , t , 
│ │ │ │ -00090380: 7420 2c20 7420 2c20 7420 2c20 7420 5d29  t , t , t , t ])
│ │ │ │ -00090390: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000903a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000903b0: 2020 2020 2030 2020 2031 2020 2032 2020       0   1   2  
│ │ │ │ -000903c0: 2033 2020 2034 2020 2035 2020 2036 2020   3   4   5   6  
│ │ │ │ -000903d0: 2037 2020 2038 2020 2020 2020 2020 2020   7   8          
│ │ │ │ -000903e0: 2020 2020 7c0a 7c20 2020 2020 7461 7267      |.|     targ
│ │ │ │ -000903f0: 6574 3a20 5072 6f6a 2851 515b 7820 2c20  et: Proj(QQ[x , 
│ │ │ │ -00090400: 7820 2c20 7820 2c20 7820 5d29 2020 2020  x , x , x ])    
│ │ │ │ +00090340: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00090350: 7c20 2020 2020 736f 7572 6365 3a20 5072  |     source: Pr
│ │ │ │ +00090360: 6f6a 2851 515b 7420 2c20 7420 2c20 7420  oj(QQ[t , t , t 
│ │ │ │ +00090370: 2c20 7420 2c20 7420 2c20 7420 2c20 7420  , t , t , t , t 
│ │ │ │ +00090380: 2c20 7420 2c20 7420 5d29 2020 2020 2020  , t , t ])      
│ │ │ │ +00090390: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000903a0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +000903b0: 2020 2031 2020 2032 2020 2033 2020 2034     1   2   3   4
│ │ │ │ +000903c0: 2020 2035 2020 2036 2020 2037 2020 2038     5   6   7   8
│ │ │ │ +000903d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000903e0: 7c20 2020 2020 7461 7267 6574 3a20 5072  |     target: Pr
│ │ │ │ +000903f0: 6f6a 2851 515b 7820 2c20 7820 2c20 7820  oj(QQ[x , x , x 
│ │ │ │ +00090400: 2c20 7820 5d29 2020 2020 2020 2020 2020  , x ])          
│ │ │ │  00090410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090420: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00090430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090440: 2020 2020 2030 2020 2031 2020 2032 2020       0   1   2  
│ │ │ │ -00090450: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -00090460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090470: 2020 2020 7c0a 7c20 2020 2020 6465 6669      |.|     defi
│ │ │ │ -00090480: 6e69 6e67 2066 6f72 6d73 3a20 7b20 2020  ning forms: {   
│ │ │ │ +00090420: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00090430: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00090440: 2020 2031 2020 2032 2020 2033 2020 2020     1   2   3    
│ │ │ │ +00090450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090460: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00090470: 7c20 2020 2020 6465 6669 6e69 6e67 2066  |     defining f
│ │ │ │ +00090480: 6f72 6d73 3a20 7b20 2020 2020 2020 2020  orms: {         
│ │ │ │  00090490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000904a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000904b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000904c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000904d0: 2020 2020 2074 2074 2074 202c 2020 2020       t t t ,    
│ │ │ │ +000904b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000904c0: 2020 2020 2020 2020 2020 2020 2020 2074                 t
│ │ │ │ +000904d0: 2074 2074 202c 2020 2020 2020 2020 2020   t t ,          
│ │ │ │  000904e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000904f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090500: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00090510: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00090520: 3320 3520 2020 2020 2020 2020 2020 2020  3 5             
│ │ │ │ +000904f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00090500: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00090510: 2020 2020 2020 2020 3020 3320 3520 2020          0 3 5   
│ │ │ │ +00090520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090540: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00090540: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00090550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090590: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000905a0: 2020 2020 2020 2020 2020 2020 2074 2074               t t
│ │ │ │ -000905b0: 2074 202c 2020 2020 2020 2020 2020 2020   t ,            
│ │ │ │ +00090580: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00090590: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000905a0: 2020 2020 2020 2074 2074 2074 202c 2020         t t t ,  
│ │ │ │ +000905b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000905c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000905d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000905d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000905e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000905f0: 2020 2020 2020 3120 3320 3620 2020 2020        1 3 6     
│ │ │ │ +000905f0: 3120 3320 3620 2020 2020 2020 2020 2020  1 3 6           
│ │ │ │  00090600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090620: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00090610: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00090620: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00090630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090660: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00090670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090680: 2020 2020 2074 2074 2074 202c 2020 2020       t t t ,    
│ │ │ │ +00090660: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00090670: 2020 2020 2020 2020 2020 2020 2020 2074                 t
│ │ │ │ +00090680: 2074 2074 202c 2020 2020 2020 2020 2020   t t ,          
│ │ │ │  00090690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000906a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000906b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000906c0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -000906d0: 3420 3720 2020 2020 2020 2020 2020 2020  4 7             
│ │ │ │ +000906a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000906b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000906c0: 2020 2020 2020 2020 3220 3420 3720 2020          2 4 7   
│ │ │ │ +000906d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000906e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000906f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000906f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00090700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090740: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00090750: 2020 2020 2020 2020 2020 2020 2074 2074               t t
│ │ │ │ -00090760: 2074 2020 2020 2020 2020 2020 2020 2020   t              
│ │ │ │ +00090730: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00090740: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00090750: 2020 2020 2020 2074 2074 2074 2020 2020         t t t    
│ │ │ │ +00090760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090780: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00090780: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00090790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000907a0: 2020 2020 2020 3220 3420 3820 2020 2020        2 4 8     
│ │ │ │ +000907a0: 3220 3420 3820 2020 2020 2020 2020 2020  2 4 8           
│ │ │ │  000907b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000907c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000907d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000907e0: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ +000907c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000907d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000907e0: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │  000907f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090810: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00090810: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00090820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090860: 2020 2020 7c0a 7c6f 3520 3a20 5261 7469      |.|o5 : Rati
│ │ │ │ -00090870: 6f6e 616c 4d61 7020 2863 7562 6963 2072  onalMap (cubic r
│ │ │ │ -00090880: 6174 696f 6e61 6c20 6d61 7020 6672 6f6d  ational map from
│ │ │ │ -00090890: 2050 505e 3820 746f 2050 505e 3329 2020   PP^8 to PP^3)  
│ │ │ │ -000908a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00090850: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00090860: 7c6f 3520 3a20 5261 7469 6f6e 616c 4d61  |o5 : RationalMa
│ │ │ │ +00090870: 7020 2863 7562 6963 2072 6174 696f 6e61  p (cubic rationa
│ │ │ │ +00090880: 6c20 6d61 7020 6672 6f6d 2050 505e 3820  l map from PP^8 
│ │ │ │ +00090890: 746f 2050 505e 3329 2020 2020 2020 2020  to PP^3)        
│ │ │ │ +000908a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  000908b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000908c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000908d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000908e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000908f0: 2d2d 2d2d 2b0a 0a4d 756c 7469 6772 6164  ----+..Multigrad
│ │ │ │ -00090900: 6564 2072 696e 6773 2061 7265 2061 6c73  ed rings are als
│ │ │ │ -00090910: 6f20 7065 726d 6974 7465 6420 6275 7420  o permitted but 
│ │ │ │ -00090920: 696e 2074 6869 7320 6361 7365 2074 6865  in this case the
│ │ │ │ -00090930: 206d 6574 686f 6420 7265 7475 726e 7320   method returns 
│ │ │ │ -00090940: 616e 0a6f 626a 6563 7420 6f66 2074 6865  an.object of the
│ │ │ │ -00090950: 2063 6c61 7373 204d 756c 7469 686f 6d6f   class Multihomo
│ │ │ │ -00090960: 6765 6e65 6f75 7352 6174 696f 6e61 6c4d  geneousRationalM
│ │ │ │ -00090970: 6170 2c20 7768 6963 6820 6361 6e20 6265  ap, which can be
│ │ │ │ -00090980: 2063 6f6e 7369 6465 7265 6420 6173 2061   considered as a
│ │ │ │ -00090990: 6e0a 6578 7465 6e73 696f 6e20 6f66 2074  n.extension of t
│ │ │ │ -000909a0: 6865 2063 6c61 7373 2052 6174 696f 6e61  he class Rationa
│ │ │ │ -000909b0: 6c4d 6170 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  lMap...+--------
│ │ │ │ +000908e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000908f0: 0a4d 756c 7469 6772 6164 6564 2072 696e  .Multigraded rin
│ │ │ │ +00090900: 6773 2061 7265 2061 6c73 6f20 7065 726d  gs are also perm
│ │ │ │ +00090910: 6974 7465 6420 6275 7420 696e 2074 6869  itted but in thi
│ │ │ │ +00090920: 7320 6361 7365 2074 6865 206d 6574 686f  s case the metho
│ │ │ │ +00090930: 6420 7265 7475 726e 7320 616e 0a6f 626a  d returns an.obj
│ │ │ │ +00090940: 6563 7420 6f66 2074 6865 2063 6c61 7373  ect of the class
│ │ │ │ +00090950: 204d 756c 7469 686f 6d6f 6765 6e65 6f75   Multihomogeneou
│ │ │ │ +00090960: 7352 6174 696f 6e61 6c4d 6170 2c20 7768  sRationalMap, wh
│ │ │ │ +00090970: 6963 6820 6361 6e20 6265 2063 6f6e 7369  ich can be consi
│ │ │ │ +00090980: 6465 7265 6420 6173 2061 6e0a 6578 7465  dered as an.exte
│ │ │ │ +00090990: 6e73 696f 6e20 6f66 2074 6865 2063 6c61  nsion of the cla
│ │ │ │ +000909a0: 7373 2052 6174 696f 6e61 6c4d 6170 2e0a  ss RationalMap..
│ │ │ │ +000909b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000909c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000909d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000909e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000909f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00090a00: 2d2d 2d2d 2d2b 0a7c 6936 203a 2052 2720  -----+.|i6 : R' 
│ │ │ │ -00090a10: 3a3d 206e 6577 5269 6e67 2852 2c44 6567  := newRing(R,Deg
│ │ │ │ -00090a20: 7265 6573 3d3e 7b33 3a7b 312c 302c 307d  rees=>{3:{1,0,0}
│ │ │ │ -00090a30: 2c32 3a7b 302c 312c 307d 2c34 3a7b 302c  ,2:{0,1,0},4:{0,
│ │ │ │ -00090a40: 302c 317d 7d29 2020 2020 2020 2020 2020  0,1}})          
│ │ │ │ -00090a50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000909f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00090a00: 0a7c 6936 203a 2052 2720 3a3d 206e 6577  .|i6 : R' := new
│ │ │ │ +00090a10: 5269 6e67 2852 2c44 6567 7265 6573 3d3e  Ring(R,Degrees=>
│ │ │ │ +00090a20: 7b33 3a7b 312c 302c 307d 2c32 3a7b 302c  {3:{1,0,0},2:{0,
│ │ │ │ +00090a30: 312c 307d 2c34 3a7b 302c 302c 317d 7d29  1,0},4:{0,0,1}})
│ │ │ │ +00090a40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00090a50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00090a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090aa0: 2020 2020 207c 0a7c 6f36 203d 2051 515b       |.|o6 = QQ[
│ │ │ │ -00090ab0: 7420 2e2e 7420 5d20 2020 2020 2020 2020  t ..t ]         
│ │ │ │ +00090a90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00090aa0: 0a7c 6f36 203d 2051 515b 7420 2e2e 7420  .|o6 = QQ[t ..t 
│ │ │ │ +00090ab0: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │  00090ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090af0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00090b00: 2030 2020 2038 2020 2020 2020 2020 2020   0   8          
│ │ │ │ +00090ae0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00090af0: 0a7c 2020 2020 2020 2020 2030 2020 2038  .|         0   8
│ │ │ │ +00090b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090b40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00090b30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00090b40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00090b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090b90: 2020 2020 207c 0a7c 6f36 203a 2050 6f6c       |.|o6 : Pol
│ │ │ │ -00090ba0: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ +00090b80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00090b90: 0a7c 6f36 203a 2050 6f6c 796e 6f6d 6961  .|o6 : Polynomia
│ │ │ │ +00090ba0: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │  00090bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090be0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00090bd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00090be0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00090bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00090c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00090c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00090c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00090c30: 2d2d 2d2d 2d2b 0a7c 6937 203a 2046 2720  -----+.|i7 : F' 
│ │ │ │ -00090c40: 3d20 7375 6228 462c 5227 2920 2020 2020  = sub(F,R')     
│ │ │ │ +00090c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00090c30: 0a7c 6937 203a 2046 2720 3d20 7375 6228  .|i7 : F' = sub(
│ │ │ │ +00090c40: 462c 5227 2920 2020 2020 2020 2020 2020  F,R')           
│ │ │ │  00090c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090c80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00090c70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00090c80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00090c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090cd0: 2020 2020 207c 0a7c 6f37 203d 207c 2074       |.|o7 = | t
│ │ │ │ -00090ce0: 5f30 745f 3374 5f35 2074 5f31 745f 3374  _0t_3t_5 t_1t_3t
│ │ │ │ -00090cf0: 5f36 2074 5f32 745f 3474 5f37 2074 5f32  _6 t_2t_4t_7 t_2
│ │ │ │ -00090d00: 745f 3474 5f38 207c 2020 2020 2020 2020  t_4t_8 |        
│ │ │ │ -00090d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090d20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00090cc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00090cd0: 0a7c 6f37 203d 207c 2074 5f30 745f 3374  .|o7 = | t_0t_3t
│ │ │ │ +00090ce0: 5f35 2074 5f31 745f 3374 5f36 2074 5f32  _5 t_1t_3t_6 t_2
│ │ │ │ +00090cf0: 745f 3474 5f37 2074 5f32 745f 3474 5f38  t_4t_7 t_2t_4t_8
│ │ │ │ +00090d00: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +00090d10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00090d20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00090d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090d70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00090d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090d90: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -00090da0: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ -00090db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090dc0: 2020 2020 207c 0a7c 6f37 203a 204d 6174       |.|o7 : Mat
│ │ │ │ -00090dd0: 7269 7820 2851 515b 7420 2e2e 7420 5d29  rix (QQ[t ..t ])
│ │ │ │ -00090de0: 2020 3c2d 2d20 2851 515b 7420 2e2e 7420    <-- (QQ[t ..t 
│ │ │ │ -00090df0: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ -00090e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090e10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00090e20: 2020 2020 2020 2020 2030 2020 2038 2020           0   8  
│ │ │ │ -00090e30: 2020 2020 2020 2020 2020 2030 2020 2038             0   8
│ │ │ │ +00090d60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00090d70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00090d80: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ +00090d90: 2020 2020 2020 2020 2020 2020 3420 2020              4   
│ │ │ │ +00090da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090db0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00090dc0: 0a7c 6f37 203a 204d 6174 7269 7820 2851  .|o7 : Matrix (Q
│ │ │ │ +00090dd0: 515b 7420 2e2e 7420 5d29 2020 3c2d 2d20  Q[t ..t ])  <-- 
│ │ │ │ +00090de0: 2851 515b 7420 2e2e 7420 5d29 2020 2020  (QQ[t ..t ])    
│ │ │ │ +00090df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00090e00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00090e10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00090e20: 2020 2030 2020 2038 2020 2020 2020 2020     0   8        
│ │ │ │ +00090e30: 2020 2020 2030 2020 2038 2020 2020 2020       0   8      
│ │ │ │  00090e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090e60: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00090e50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00090e60: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00090e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00090e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00090e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00090ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00090eb0: 2d2d 2d2d 2d2b 0a7c 6938 203a 2070 6869  -----+.|i8 : phi
│ │ │ │ -00090ec0: 2720 3d20 746f 4d61 7020 4627 2020 2020  ' = toMap F'    
│ │ │ │ +00090ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00090eb0: 0a7c 6938 203a 2070 6869 2720 3d20 746f  .|i8 : phi' = to
│ │ │ │ +00090ec0: 4d61 7020 4627 2020 2020 2020 2020 2020  Map F'          
│ │ │ │  00090ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090f00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00090ef0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00090f00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00090f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00090f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00090f50: 2020 2020 207c 0a7c 6f38 203d 206d 6170       |.|o8 = map
│ │ │ │ -00090f60: 2028 5151 5b74 202e 2e74 205d 2c20 5151   (QQ[t ..t ], QQ
│ │ │ │ -00090f70: 5b78 202e 2e78 205d 2c20 7b74 2074 2074  [x ..x ], {t t t
│ │ │ │ -00090f80: 202c 2074 2074 2074 202c 2074 2074 2074   , t t t , t t t
│ │ │ │ -00090f90: 202c 2074 2074 2074 207d 2920 2020 2020   , t t t })     
│ │ │ │ -00090fa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00090fb0: 2020 2020 2020 3020 2020 3820 2020 2020        0   8     
│ │ │ │ -00090fc0: 2020 3020 2020 3320 2020 2020 3020 3320    0   3     0 3 
│ │ │ │ -00090fd0: 3520 2020 3120 3320 3620 2020 3220 3420  5   1 3 6   2 4 
│ │ │ │ -00090fe0: 3720 2020 3220 3420 3820 2020 2020 2020  7   2 4 8       
│ │ │ │ -00090ff0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00090f40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00090f50: 0a7c 6f38 203d 206d 6170 2028 5151 5b74  .|o8 = map (QQ[t
│ │ │ │ +00090f60: 202e 2e74 205d 2c20 5151 5b78 202e 2e78   ..t ], QQ[x ..x
│ │ │ │ +00090f70: 205d 2c20 7b74 2074 2074 202c 2074 2074   ], {t t t , t t
│ │ │ │ +00090f80: 2074 202c 2074 2074 2074 202c 2074 2074   t , t t t , t t
│ │ │ │ +00090f90: 2074 207d 2920 2020 2020 2020 2020 207c   t })          |
│ │ │ │ +00090fa0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00090fb0: 3020 2020 3820 2020 2020 2020 3020 2020  0   8       0   
│ │ │ │ +00090fc0: 3320 2020 2020 3020 3320 3520 2020 3120  3     0 3 5   1 
│ │ │ │ +00090fd0: 3320 3620 2020 3220 3420 3720 2020 3220  3 6   2 4 7   2 
│ │ │ │ +00090fe0: 3420 3820 2020 2020 2020 2020 2020 207c  4 8            |
│ │ │ │ +00090ff0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091040: 2020 2020 207c 0a7c 6f38 203a 2052 696e       |.|o8 : Rin
│ │ │ │ -00091050: 674d 6170 2051 515b 7420 2e2e 7420 5d20  gMap QQ[t ..t ] 
│ │ │ │ -00091060: 3c2d 2d20 5151 5b78 202e 2e78 205d 2020  <-- QQ[x ..x ]  
│ │ │ │ +00091030: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091040: 0a7c 6f38 203a 2052 696e 674d 6170 2051  .|o8 : RingMap Q
│ │ │ │ +00091050: 515b 7420 2e2e 7420 5d20 3c2d 2d20 5151  Q[t ..t ] <-- QQ
│ │ │ │ +00091060: 5b78 202e 2e78 205d 2020 2020 2020 2020  [x ..x ]        
│ │ │ │  00091070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091090: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000910a0: 2020 2020 2020 2020 2030 2020 2038 2020           0   8  
│ │ │ │ -000910b0: 2020 2020 2020 2020 3020 2020 3320 2020          0   3   
│ │ │ │ +00091080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091090: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000910a0: 2020 2030 2020 2038 2020 2020 2020 2020     0   8        
│ │ │ │ +000910b0: 2020 3020 2020 3320 2020 2020 2020 2020    0   3         
│ │ │ │  000910c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000910d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000910e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000910d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000910e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000910f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00091100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00091110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00091120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00091130: 2d2d 2d2d 2d2b 0a7c 6939 203a 2072 6174  -----+.|i9 : rat
│ │ │ │ -00091140: 696f 6e61 6c4d 6170 2070 6869 2720 2020  ionalMap phi'   
│ │ │ │ +00091120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00091130: 0a7c 6939 203a 2072 6174 696f 6e61 6c4d  .|i9 : rationalM
│ │ │ │ +00091140: 6170 2070 6869 2720 2020 2020 2020 2020  ap phi'         
│ │ │ │  00091150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091180: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091170: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091180: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000911a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000911b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000911c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000911d0: 2020 2020 207c 0a7c 6f39 203d 202d 2d20       |.|o9 = -- 
│ │ │ │ -000911e0: 7261 7469 6f6e 616c 206d 6170 202d 2d20  rational map -- 
│ │ │ │ +000911c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000911d0: 0a7c 6f39 203d 202d 2d20 7261 7469 6f6e  .|o9 = -- ration
│ │ │ │ +000911e0: 616c 206d 6170 202d 2d20 2020 2020 2020  al map --       
│ │ │ │  000911f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091220: 2020 2020 207c 0a7c 2020 2020 2073 6f75       |.|     sou
│ │ │ │ -00091230: 7263 653a 2050 726f 6a28 5151 5b74 202c  rce: Proj(QQ[t ,
│ │ │ │ -00091240: 2074 202c 2074 205d 2920 7820 5072 6f6a   t , t ]) x Proj
│ │ │ │ -00091250: 2851 515b 7420 2c20 7420 5d29 2078 2050  (QQ[t , t ]) x P
│ │ │ │ -00091260: 726f 6a28 5151 5b74 202c 2074 202c 2020  roj(QQ[t , t ,  
│ │ │ │ -00091270: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00091280: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00091290: 2020 3120 2020 3220 2020 2020 2020 2020    1   2         
│ │ │ │ -000912a0: 2020 2020 2033 2020 2034 2020 2020 2020       3   4      
│ │ │ │ -000912b0: 2020 2020 2020 2020 3520 2020 3620 2020          5   6   
│ │ │ │ -000912c0: 2020 2020 207c 0a7c 2020 2020 2074 6172       |.|     tar
│ │ │ │ -000912d0: 6765 743a 2050 726f 6a28 5151 5b78 202c  get: Proj(QQ[x ,
│ │ │ │ -000912e0: 2078 202c 2078 202c 2078 205d 2920 2020   x , x , x ])   
│ │ │ │ +00091210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091220: 0a7c 2020 2020 2073 6f75 7263 653a 2050  .|     source: P
│ │ │ │ +00091230: 726f 6a28 5151 5b74 202c 2074 202c 2074  roj(QQ[t , t , t
│ │ │ │ +00091240: 205d 2920 7820 5072 6f6a 2851 515b 7420   ]) x Proj(QQ[t 
│ │ │ │ +00091250: 2c20 7420 5d29 2078 2050 726f 6a28 5151  , t ]) x Proj(QQ
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│ │ │ │ +00091270: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00091280: 2020 2020 2020 2020 3020 2020 3120 2020          0   1   
│ │ │ │ +00091290: 3220 2020 2020 2020 2020 2020 2020 2033  2              3
│ │ │ │ +000912a0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +000912b0: 2020 3520 2020 3620 2020 2020 2020 207c    5   6        |
│ │ │ │ +000912c0: 0a7c 2020 2020 2074 6172 6765 743a 2050  .|     target: P
│ │ │ │ +000912d0: 726f 6a28 5151 5b78 202c 2078 202c 2078  roj(QQ[x , x , x
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│ │ │ │  000912f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091310: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00091320: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00091330: 2020 3120 2020 3220 2020 3320 2020 2020    1   2   3     
│ │ │ │ +00091300: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091310: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00091320: 2020 2020 2020 2020 3020 2020 3120 2020          0   1   
│ │ │ │ +00091330: 3220 2020 3320 2020 2020 2020 2020 2020  2   3           
│ │ │ │  00091340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091360: 2020 2020 207c 0a7c 2020 2020 2064 6566       |.|     def
│ │ │ │ -00091370: 696e 696e 6720 666f 726d 733a 207b 2020  ining forms: {  
│ │ │ │ +00091350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091360: 0a7c 2020 2020 2064 6566 696e 696e 6720  .|     defining 
│ │ │ │ +00091370: 666f 726d 733a 207b 2020 2020 2020 2020  forms: {        
│ │ │ │  00091380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000913a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000913b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000913c0: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ -000913d0: 7420 7420 2c20 2020 2020 2020 2020 2020  t t ,           
│ │ │ │ +000913a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000913b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000913c0: 2020 2020 2020 2020 7420 7420 7420 2c20          t t t , 
│ │ │ │ +000913d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000913e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000913f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091400: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00091410: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00091420: 2033 2035 2020 2020 2020 2020 2020 2020   3 5            
│ │ │ │ +000913f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091400: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00091410: 2020 2020 2020 2020 2030 2033 2035 2020           0 3 5  
│ │ │ │ +00091420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091450: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091440: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091450: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000914a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000914b0: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ -000914c0: 7420 7420 2c20 2020 2020 2020 2020 2020  t t ,           
│ │ │ │ +00091490: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000914a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000914b0: 2020 2020 2020 2020 7420 7420 7420 2c20          t t t , 
│ │ │ │ +000914c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000914d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000914e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000914f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00091500: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00091510: 2033 2036 2020 2020 2020 2020 2020 2020   3 6            
│ │ │ │ +000914e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000914f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00091500: 2020 2020 2020 2020 2031 2033 2036 2020           1 3 6  
│ │ │ │ +00091510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091540: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091530: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091540: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091590: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000915a0: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ -000915b0: 7420 7420 2c20 2020 2020 2020 2020 2020  t t ,           
│ │ │ │ +00091580: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091590: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000915a0: 2020 2020 2020 2020 7420 7420 7420 2c20          t t t , 
│ │ │ │ +000915b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000915c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000915d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000915e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000915f0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00091600: 2034 2037 2020 2020 2020 2020 2020 2020   4 7            
│ │ │ │ +000915d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000915e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000915f0: 2020 2020 2020 2020 2032 2034 2037 2020           2 4 7  
│ │ │ │ +00091600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091630: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091620: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091630: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091680: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00091690: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ -000916a0: 7420 7420 2020 2020 2020 2020 2020 2020  t t             
│ │ │ │ +00091670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091680: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00091690: 2020 2020 2020 2020 7420 7420 7420 2020          t t t   
│ │ │ │ +000916a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000916b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000916c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000916d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000916e0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -000916f0: 2034 2038 2020 2020 2020 2020 2020 2020   4 8            
│ │ │ │ +000916c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000916d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000916e0: 2020 2020 2020 2020 2032 2034 2038 2020           2 4 8  
│ │ │ │ +000916f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091720: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00091730: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ +00091710: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091720: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00091730: 2020 2020 2020 207d 2020 2020 2020 2020         }        
│ │ │ │  00091740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091770: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091760: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091770: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000917a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000917b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000917c0: 2020 2020 207c 0a7c 6f39 203a 204d 756c       |.|o9 : Mul
│ │ │ │ -000917d0: 7469 686f 6d6f 6765 6e65 6f75 7352 6174  tihomogeneousRat
│ │ │ │ -000917e0: 696f 6e61 6c4d 6170 2028 7261 7469 6f6e  ionalMap (ration
│ │ │ │ -000917f0: 616c 206d 6170 2066 726f 6d20 5050 5e32  al map from PP^2
│ │ │ │ -00091800: 2078 2050 505e 3120 7820 5050 5e33 2020   x PP^1 x PP^3  
│ │ │ │ -00091810: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +000917b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000917c0: 0a7c 6f39 203a 204d 756c 7469 686f 6d6f  .|o9 : Multihomo
│ │ │ │ +000917d0: 6765 6e65 6f75 7352 6174 696f 6e61 6c4d  geneousRationalM
│ │ │ │ +000917e0: 6170 2028 7261 7469 6f6e 616c 206d 6170  ap (rational map
│ │ │ │ +000917f0: 2066 726f 6d20 5050 5e32 2078 2050 505e   from PP^2 x PP^
│ │ │ │ +00091800: 3120 7820 5050 5e33 2020 2020 2020 207c  1 x PP^3       |
│ │ │ │ +00091810: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00091820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00091830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00091840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00091850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00091860: 2d2d 2d2d 2d7c 0a7c 7420 2c20 7420 5d29  -----|.|t , t ])
│ │ │ │ +00091850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00091860: 0a7c 7420 2c20 7420 5d29 2020 2020 2020  .|t , t ])      
│ │ │ │  00091870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000918a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000918b0: 2020 2020 207c 0a7c 2037 2020 2038 2020       |.| 7   8  
│ │ │ │ +000918a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000918b0: 0a7c 2037 2020 2038 2020 2020 2020 2020  .| 7   8        
│ │ │ │  000918c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000918d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000918e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000918f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091900: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000918f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091900: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091950: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091940: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091950: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000919a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091990: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000919a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000919b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000919c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000919d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000919e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000919f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000919e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000919f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091a40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091a30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091a40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091a90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091a80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091a90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091ae0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091ad0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091ae0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091b30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091b20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091b30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091b80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091b70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091b80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091bd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091bc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091bd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091c20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091c10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091c20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091c70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091c60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091c70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091cc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091cb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091cc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091d10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091d00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091d10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091d60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091d50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091d60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091db0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091db0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091e00: 2020 2020 207c 0a7c 746f 2050 505e 3329       |.|to PP^3)
│ │ │ │ +00091df0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091e00: 0a7c 746f 2050 505e 3329 2020 2020 2020  .|to PP^3)      
│ │ │ │  00091e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091e50: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00091e40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091e50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00091e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00091e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00091e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00091e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00091ea0: 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 7261  -----+.|i10 : ra
│ │ │ │ -00091eb0: 7469 6f6e 616c 4d61 7020 4627 2020 2020  tionalMap F'    
│ │ │ │ +00091e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00091ea0: 0a7c 6931 3020 3a20 7261 7469 6f6e 616c  .|i10 : rational
│ │ │ │ +00091eb0: 4d61 7020 4627 2020 2020 2020 2020 2020  Map F'          
│ │ │ │  00091ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091ef0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00091ee0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091ef0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00091f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00091f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091f40: 2020 2020 207c 0a7c 6f31 3020 3d20 2d2d       |.|o10 = --
│ │ │ │ -00091f50: 2072 6174 696f 6e61 6c20 6d61 7020 2d2d   rational map --
│ │ │ │ +00091f30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00091f40: 0a7c 6f31 3020 3d20 2d2d 2072 6174 696f  .|o10 = -- ratio
│ │ │ │ +00091f50: 6e61 6c20 6d61 7020 2d2d 2020 2020 2020  nal map --      
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│ │ │ │  00091f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00091f90: 2020 2020 207c 0a7c 2020 2020 2020 736f       |.|      so
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│ │ │ │  000920b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000920c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000920d0: 2020 2020 207c 0a7c 2020 2020 2020 6465       |.|      de
│ │ │ │ -000920e0: 6669 6e69 6e67 2066 6f72 6d73 3a20 7b20  fining forms: { 
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│ │ │ │ -00092360: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000923b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00092450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092460: 3220 3420 3820 2020 2020 2020 2020 2020  2 4 8           
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│ │ │ │ -000924a0: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
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│ │ │ │ -00092530: 2020 2020 207c 0a7c 6f31 3020 3a20 4d75       |.|o10 : Mu
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│ │ │ │ +00092530: 0a7c 6f31 3020 3a20 4d75 6c74 6968 6f6d  .|o10 : Multihom
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│ │ │ │  000925b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  000925e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00092640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00092650: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00092b70: 2020 2020 207c 0a7c 746f 2050 505e 3329       |.|to PP^3)
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│ │ │ │ +00092b70: 0a7c 746f 2050 505e 3329 2020 2020 2020  .|to PP^3)      
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│ │ │ │ -00092bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00092bc0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00092bb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00092bc0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00092bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00092be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00092bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00092c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00092c10: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f  -----+..See also
│ │ │ │ -00092c20: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ -00092c30: 6e6f 7465 2074 6f4d 6170 3a20 746f 4d61  note toMap: toMa
│ │ │ │ -00092c40: 702c 202d 2d20 7261 7469 6f6e 616c 206d  p, -- rational m
│ │ │ │ -00092c50: 6170 2064 6566 696e 6564 2062 7920 6120  ap defined by a 
│ │ │ │ -00092c60: 6c69 6e65 6172 2073 7973 7465 6d0a 2020  linear system.  
│ │ │ │ -00092c70: 2a20 2a6e 6f74 6520 7261 7469 6f6e 616c  * *note rational
│ │ │ │ -00092c80: 4d61 7028 4964 6561 6c29 3a20 7261 7469  Map(Ideal): rati
│ │ │ │ -00092c90: 6f6e 616c 4d61 705f 6c70 4964 6561 6c5f  onalMap_lpIdeal_
│ │ │ │ -00092ca0: 636d 5a5a 5f63 6d5a 5a5f 7270 2c20 2d2d  cmZZ_cmZZ_rp, --
│ │ │ │ -00092cb0: 206d 616b 6573 2061 0a20 2020 2072 6174   makes a.    rat
│ │ │ │ -00092cc0: 696f 6e61 6c20 6d61 7020 6672 6f6d 2061  ional map from a
│ │ │ │ -00092cd0: 6e20 6964 6561 6c0a 2020 2a20 2a6e 6f74  n ideal.  * *not
│ │ │ │ -00092ce0: 6520 7261 7469 6f6e 616c 4d61 7028 5461  e rationalMap(Ta
│ │ │ │ -00092cf0: 6c6c 7929 3a20 7261 7469 6f6e 616c 4d61  lly): rationalMa
│ │ │ │ -00092d00: 705f 6c70 5269 6e67 5f63 6d54 616c 6c79  p_lpRing_cmTally
│ │ │ │ -00092d10: 5f72 702c 202d 2d20 7261 7469 6f6e 616c  _rp, -- rational
│ │ │ │ -00092d20: 206d 6170 0a20 2020 2064 6566 696e 6564   map.    defined
│ │ │ │ -00092d30: 2062 7920 616e 2065 6666 6563 7469 7665   by an effective
│ │ │ │ -00092d40: 2064 6976 6973 6f72 0a0a 5761 7973 2074   divisor..Ways t
│ │ │ │ -00092d50: 6f20 7573 6520 7261 7469 6f6e 616c 4d61  o use rationalMa
│ │ │ │ -00092d60: 703a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  p:.=============
│ │ │ │ -00092d70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -00092d80: 2022 7261 7469 6f6e 616c 4d61 7028 4c69   "rationalMap(Li
│ │ │ │ -00092d90: 7374 2922 0a20 202a 2022 7261 7469 6f6e  st)".  * "ration
│ │ │ │ -00092da0: 616c 4d61 7028 4d61 7472 6978 2922 0a20  alMap(Matrix)". 
│ │ │ │ -00092db0: 202a 2022 7261 7469 6f6e 616c 4d61 7028   * "rationalMap(
│ │ │ │ -00092dc0: 5269 6e67 2922 0a20 202a 2022 7261 7469  Ring)".  * "rati
│ │ │ │ -00092dd0: 6f6e 616c 4d61 7028 5269 6e67 2c52 696e  onalMap(Ring,Rin
│ │ │ │ -00092de0: 6729 220a 2020 2a20 2272 6174 696f 6e61  g)".  * "rationa
│ │ │ │ -00092df0: 6c4d 6170 2852 696e 672c 5269 6e67 2c4c  lMap(Ring,Ring,L
│ │ │ │ -00092e00: 6973 7429 220a 2020 2a20 2272 6174 696f  ist)".  * "ratio
│ │ │ │ -00092e10: 6e61 6c4d 6170 2852 696e 672c 5269 6e67  nalMap(Ring,Ring
│ │ │ │ -00092e20: 2c4d 6174 7269 7829 220a 2020 2a20 2272  ,Matrix)".  * "r
│ │ │ │ -00092e30: 6174 696f 6e61 6c4d 6170 2852 696e 674d  ationalMap(RingM
│ │ │ │ -00092e40: 6170 2922 0a20 202a 2022 7261 7469 6f6e  ap)".  * "ration
│ │ │ │ -00092e50: 616c 4d61 7028 4964 6561 6c29 2220 2d2d  alMap(Ideal)" --
│ │ │ │ -00092e60: 2073 6565 202a 6e6f 7465 2072 6174 696f   see *note ratio
│ │ │ │ -00092e70: 6e61 6c4d 6170 2849 6465 616c 2c5a 5a2c  nalMap(Ideal,ZZ,
│ │ │ │ -00092e80: 5a5a 293a 0a20 2020 2072 6174 696f 6e61  ZZ):.    rationa
│ │ │ │ -00092e90: 6c4d 6170 5f6c 7049 6465 616c 5f63 6d5a  lMap_lpIdeal_cmZ
│ │ │ │ -00092ea0: 5a5f 636d 5a5a 5f72 702c 202d 2d20 6d61  Z_cmZZ_rp, -- ma
│ │ │ │ -00092eb0: 6b65 7320 6120 7261 7469 6f6e 616c 206d  kes a rational m
│ │ │ │ -00092ec0: 6170 2066 726f 6d20 616e 2069 6465 616c  ap from an ideal
│ │ │ │ -00092ed0: 0a20 202a 2022 7261 7469 6f6e 616c 4d61  .  * "rationalMa
│ │ │ │ -00092ee0: 7028 4964 6561 6c2c 4c69 7374 2922 202d  p(Ideal,List)" -
│ │ │ │ -00092ef0: 2d20 7365 6520 2a6e 6f74 6520 7261 7469  - see *note rati
│ │ │ │ -00092f00: 6f6e 616c 4d61 7028 4964 6561 6c2c 5a5a  onalMap(Ideal,ZZ
│ │ │ │ -00092f10: 2c5a 5a29 3a0a 2020 2020 7261 7469 6f6e  ,ZZ):.    ration
│ │ │ │ -00092f20: 616c 4d61 705f 6c70 4964 6561 6c5f 636d  alMap_lpIdeal_cm
│ │ │ │ -00092f30: 5a5a 5f63 6d5a 5a5f 7270 2c20 2d2d 206d  ZZ_cmZZ_rp, -- m
│ │ │ │ -00092f40: 616b 6573 2061 2072 6174 696f 6e61 6c20  akes a rational 
│ │ │ │ -00092f50: 6d61 7020 6672 6f6d 2061 6e20 6964 6561  map from an idea
│ │ │ │ -00092f60: 6c0a 2020 2a20 2272 6174 696f 6e61 6c4d  l.  * "rationalM
│ │ │ │ -00092f70: 6170 2849 6465 616c 2c5a 5a29 2220 2d2d  ap(Ideal,ZZ)" --
│ │ │ │ -00092f80: 2073 6565 202a 6e6f 7465 2072 6174 696f   see *note ratio
│ │ │ │ -00092f90: 6e61 6c4d 6170 2849 6465 616c 2c5a 5a2c  nalMap(Ideal,ZZ,
│ │ │ │ -00092fa0: 5a5a 293a 0a20 2020 2072 6174 696f 6e61  ZZ):.    rationa
│ │ │ │ -00092fb0: 6c4d 6170 5f6c 7049 6465 616c 5f63 6d5a  lMap_lpIdeal_cmZ
│ │ │ │ -00092fc0: 5a5f 636d 5a5a 5f72 702c 202d 2d20 6d61  Z_cmZZ_rp, -- ma
│ │ │ │ -00092fd0: 6b65 7320 6120 7261 7469 6f6e 616c 206d  kes a rational m
│ │ │ │ -00092fe0: 6170 2066 726f 6d20 616e 2069 6465 616c  ap from an ideal
│ │ │ │ -00092ff0: 0a20 202a 202a 6e6f 7465 2072 6174 696f  .  * *note ratio
│ │ │ │ -00093000: 6e61 6c4d 6170 2849 6465 616c 2c5a 5a2c  nalMap(Ideal,ZZ,
│ │ │ │ -00093010: 5a5a 293a 2072 6174 696f 6e61 6c4d 6170  ZZ): rationalMap
│ │ │ │ -00093020: 5f6c 7049 6465 616c 5f63 6d5a 5a5f 636d  _lpIdeal_cmZZ_cm
│ │ │ │ -00093030: 5a5a 5f72 702c 202d 2d20 6d61 6b65 730a  ZZ_rp, -- makes.
│ │ │ │ -00093040: 2020 2020 6120 7261 7469 6f6e 616c 206d      a rational m
│ │ │ │ -00093050: 6170 2066 726f 6d20 616e 2069 6465 616c  ap from an ideal
│ │ │ │ -00093060: 0a20 202a 202a 6e6f 7465 2072 6174 696f  .  * *note ratio
│ │ │ │ -00093070: 6e61 6c4d 6170 2850 6f6c 796e 6f6d 6961  nalMap(Polynomia
│ │ │ │ -00093080: 6c52 696e 672c 4c69 7374 293a 0a20 2020  lRing,List):.   
│ │ │ │ -00093090: 2072 6174 696f 6e61 6c4d 6170 5f6c 7050   rationalMap_lpP
│ │ │ │ -000930a0: 6f6c 796e 6f6d 6961 6c52 696e 675f 636d  olynomialRing_cm
│ │ │ │ -000930b0: 4c69 7374 5f72 702c 202d 2d20 7261 7469  List_rp, -- rati
│ │ │ │ -000930c0: 6f6e 616c 206d 6170 2064 6566 696e 6564  onal map defined
│ │ │ │ -000930d0: 2062 7920 7468 650a 2020 2020 6c69 6e65   by the.    line
│ │ │ │ -000930e0: 6172 2073 7973 7465 6d20 6f66 2068 7970  ar system of hyp
│ │ │ │ -000930f0: 6572 7375 7266 6163 6573 2070 6173 7369  ersurfaces passi
│ │ │ │ -00093100: 6e67 2074 6872 6f75 6768 2072 616e 646f  ng through rando
│ │ │ │ -00093110: 6d20 706f 696e 7473 2077 6974 680a 2020  m points with.  
│ │ │ │ -00093120: 2020 6d75 6c74 6970 6c69 6369 7479 0a20    multiplicity. 
│ │ │ │ -00093130: 202a 202a 6e6f 7465 2072 6174 696f 6e61   * *note rationa
│ │ │ │ -00093140: 6c4d 6170 2852 696e 672c 5461 6c6c 7929  lMap(Ring,Tally)
│ │ │ │ -00093150: 3a20 7261 7469 6f6e 616c 4d61 705f 6c70  : rationalMap_lp
│ │ │ │ -00093160: 5269 6e67 5f63 6d54 616c 6c79 5f72 702c  Ring_cmTally_rp,
│ │ │ │ -00093170: 202d 2d20 7261 7469 6f6e 616c 0a20 2020   -- rational.   
│ │ │ │ -00093180: 206d 6170 2064 6566 696e 6564 2062 7920   map defined by 
│ │ │ │ -00093190: 616e 2065 6666 6563 7469 7665 2064 6976  an effective div
│ │ │ │ -000931a0: 6973 6f72 0a20 202a 2022 7261 7469 6f6e  isor.  * "ration
│ │ │ │ -000931b0: 616c 4d61 7028 5461 6c6c 7929 2220 2d2d  alMap(Tally)" --
│ │ │ │ -000931c0: 2073 6565 202a 6e6f 7465 2072 6174 696f   see *note ratio
│ │ │ │ -000931d0: 6e61 6c4d 6170 2852 696e 672c 5461 6c6c  nalMap(Ring,Tall
│ │ │ │ -000931e0: 7929 3a0a 2020 2020 7261 7469 6f6e 616c  y):.    rational
│ │ │ │ -000931f0: 4d61 705f 6c70 5269 6e67 5f63 6d54 616c  Map_lpRing_cmTal
│ │ │ │ -00093200: 6c79 5f72 702c 202d 2d20 7261 7469 6f6e  ly_rp, -- ration
│ │ │ │ -00093210: 616c 206d 6170 2064 6566 696e 6564 2062  al map defined b
│ │ │ │ -00093220: 7920 616e 2065 6666 6563 7469 7665 0a20  y an effective. 
│ │ │ │ -00093230: 2020 2064 6976 6973 6f72 0a20 202a 2022     divisor.  * "
│ │ │ │ -00093240: 7261 7469 6f6e 616c 4d61 7028 5261 7469  rationalMap(Rati
│ │ │ │ -00093250: 6f6e 616c 4d61 7029 2220 2d2d 2073 6565  onalMap)" -- see
│ │ │ │ -00093260: 202a 6e6f 7465 2073 7570 6572 2852 6174   *note super(Rat
│ │ │ │ -00093270: 696f 6e61 6c4d 6170 293a 0a20 2020 2073  ionalMap):.    s
│ │ │ │ -00093280: 7570 6572 5f6c 7052 6174 696f 6e61 6c4d  uper_lpRationalM
│ │ │ │ -00093290: 6170 5f72 702c 202d 2d20 6765 7420 7468  ap_rp, -- get th
│ │ │ │ -000932a0: 6520 7261 7469 6f6e 616c 206d 6170 2077  e rational map w
│ │ │ │ -000932b0: 686f 7365 2074 6172 6765 7420 6973 2061  hose target is a
│ │ │ │ -000932c0: 0a20 2020 2070 726f 6a65 6374 6976 6520  .    projective 
│ │ │ │ -000932d0: 7370 6163 650a 0a46 6f72 2074 6865 2070  space..For the p
│ │ │ │ -000932e0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ -000932f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ -00093300: 6520 6f62 6a65 6374 202a 6e6f 7465 2072  e object *note r
│ │ │ │ -00093310: 6174 696f 6e61 6c4d 6170 3a20 7261 7469  ationalMap: rati
│ │ │ │ -00093320: 6f6e 616c 4d61 702c 2069 7320 6120 2a6e  onalMap, is a *n
│ │ │ │ -00093330: 6f74 6520 6d65 7468 6f64 2066 756e 6374  ote method funct
│ │ │ │ -00093340: 696f 6e20 7769 7468 0a6f 7074 696f 6e73  ion with.options
│ │ │ │ -00093350: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00093360: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769  MethodFunctionWi
│ │ │ │ -00093370: 7468 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d  thOptions,...---
│ │ │ │ +00092c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00092c10: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ +00092c20: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2074  ===..  * *note t
│ │ │ │ +00092c30: 6f4d 6170 3a20 746f 4d61 702c 202d 2d20  oMap: toMap, -- 
│ │ │ │ +00092c40: 7261 7469 6f6e 616c 206d 6170 2064 6566  rational map def
│ │ │ │ +00092c50: 696e 6564 2062 7920 6120 6c69 6e65 6172  ined by a linear
│ │ │ │ +00092c60: 2073 7973 7465 6d0a 2020 2a20 2a6e 6f74   system.  * *not
│ │ │ │ +00092c70: 6520 7261 7469 6f6e 616c 4d61 7028 4964  e rationalMap(Id
│ │ │ │ +00092c80: 6561 6c29 3a20 7261 7469 6f6e 616c 4d61  eal): rationalMa
│ │ │ │ +00092c90: 705f 6c70 4964 6561 6c5f 636d 5a5a 5f63  p_lpIdeal_cmZZ_c
│ │ │ │ +00092ca0: 6d5a 5a5f 7270 2c20 2d2d 206d 616b 6573  mZZ_rp, -- makes
│ │ │ │ +00092cb0: 2061 0a20 2020 2072 6174 696f 6e61 6c20   a.    rational 
│ │ │ │ +00092cc0: 6d61 7020 6672 6f6d 2061 6e20 6964 6561  map from an idea
│ │ │ │ +00092cd0: 6c0a 2020 2a20 2a6e 6f74 6520 7261 7469  l.  * *note rati
│ │ │ │ +00092ce0: 6f6e 616c 4d61 7028 5461 6c6c 7929 3a20  onalMap(Tally): 
│ │ │ │ +00092cf0: 7261 7469 6f6e 616c 4d61 705f 6c70 5269  rationalMap_lpRi
│ │ │ │ +00092d00: 6e67 5f63 6d54 616c 6c79 5f72 702c 202d  ng_cmTally_rp, -
│ │ │ │ +00092d10: 2d20 7261 7469 6f6e 616c 206d 6170 0a20  - rational map. 
│ │ │ │ +00092d20: 2020 2064 6566 696e 6564 2062 7920 616e     defined by an
│ │ │ │ +00092d30: 2065 6666 6563 7469 7665 2064 6976 6973   effective divis
│ │ │ │ +00092d40: 6f72 0a0a 5761 7973 2074 6f20 7573 6520  or..Ways to use 
│ │ │ │ +00092d50: 7261 7469 6f6e 616c 4d61 703a 0a3d 3d3d  rationalMap:.===
│ │ │ │ +00092d60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00092d70: 3d3d 3d3d 3d0a 0a20 202a 2022 7261 7469  =====..  * "rati
│ │ │ │ +00092d80: 6f6e 616c 4d61 7028 4c69 7374 2922 0a20  onalMap(List)". 
│ │ │ │ +00092d90: 202a 2022 7261 7469 6f6e 616c 4d61 7028   * "rationalMap(
│ │ │ │ +00092da0: 4d61 7472 6978 2922 0a20 202a 2022 7261  Matrix)".  * "ra
│ │ │ │ +00092db0: 7469 6f6e 616c 4d61 7028 5269 6e67 2922  tionalMap(Ring)"
│ │ │ │ +00092dc0: 0a20 202a 2022 7261 7469 6f6e 616c 4d61  .  * "rationalMa
│ │ │ │ +00092dd0: 7028 5269 6e67 2c52 696e 6729 220a 2020  p(Ring,Ring)".  
│ │ │ │ +00092de0: 2a20 2272 6174 696f 6e61 6c4d 6170 2852  * "rationalMap(R
│ │ │ │ +00092df0: 696e 672c 5269 6e67 2c4c 6973 7429 220a  ing,Ring,List)".
│ │ │ │ +00092e00: 2020 2a20 2272 6174 696f 6e61 6c4d 6170    * "rationalMap
│ │ │ │ +00092e10: 2852 696e 672c 5269 6e67 2c4d 6174 7269  (Ring,Ring,Matri
│ │ │ │ +00092e20: 7829 220a 2020 2a20 2272 6174 696f 6e61  x)".  * "rationa
│ │ │ │ +00092e30: 6c4d 6170 2852 696e 674d 6170 2922 0a20  lMap(RingMap)". 
│ │ │ │ +00092e40: 202a 2022 7261 7469 6f6e 616c 4d61 7028   * "rationalMap(
│ │ │ │ +00092e50: 4964 6561 6c29 2220 2d2d 2073 6565 202a  Ideal)" -- see *
│ │ │ │ +00092e60: 6e6f 7465 2072 6174 696f 6e61 6c4d 6170  note rationalMap
│ │ │ │ +00092e70: 2849 6465 616c 2c5a 5a2c 5a5a 293a 0a20  (Ideal,ZZ,ZZ):. 
│ │ │ │ +00092e80: 2020 2072 6174 696f 6e61 6c4d 6170 5f6c     rationalMap_l
│ │ │ │ +00092e90: 7049 6465 616c 5f63 6d5a 5a5f 636d 5a5a  pIdeal_cmZZ_cmZZ
│ │ │ │ +00092ea0: 5f72 702c 202d 2d20 6d61 6b65 7320 6120  _rp, -- makes a 
│ │ │ │ +00092eb0: 7261 7469 6f6e 616c 206d 6170 2066 726f  rational map fro
│ │ │ │ +00092ec0: 6d20 616e 2069 6465 616c 0a20 202a 2022  m an ideal.  * "
│ │ │ │ +00092ed0: 7261 7469 6f6e 616c 4d61 7028 4964 6561  rationalMap(Idea
│ │ │ │ +00092ee0: 6c2c 4c69 7374 2922 202d 2d20 7365 6520  l,List)" -- see 
│ │ │ │ +00092ef0: 2a6e 6f74 6520 7261 7469 6f6e 616c 4d61  *note rationalMa
│ │ │ │ +00092f00: 7028 4964 6561 6c2c 5a5a 2c5a 5a29 3a0a  p(Ideal,ZZ,ZZ):.
│ │ │ │ +00092f10: 2020 2020 7261 7469 6f6e 616c 4d61 705f      rationalMap_
│ │ │ │ +00092f20: 6c70 4964 6561 6c5f 636d 5a5a 5f63 6d5a  lpIdeal_cmZZ_cmZ
│ │ │ │ +00092f30: 5a5f 7270 2c20 2d2d 206d 616b 6573 2061  Z_rp, -- makes a
│ │ │ │ +00092f40: 2072 6174 696f 6e61 6c20 6d61 7020 6672   rational map fr
│ │ │ │ +00092f50: 6f6d 2061 6e20 6964 6561 6c0a 2020 2a20  om an ideal.  * 
│ │ │ │ +00092f60: 2272 6174 696f 6e61 6c4d 6170 2849 6465  "rationalMap(Ide
│ │ │ │ +00092f70: 616c 2c5a 5a29 2220 2d2d 2073 6565 202a  al,ZZ)" -- see *
│ │ │ │ +00092f80: 6e6f 7465 2072 6174 696f 6e61 6c4d 6170  note rationalMap
│ │ │ │ +00092f90: 2849 6465 616c 2c5a 5a2c 5a5a 293a 0a20  (Ideal,ZZ,ZZ):. 
│ │ │ │ +00092fa0: 2020 2072 6174 696f 6e61 6c4d 6170 5f6c     rationalMap_l
│ │ │ │ +00092fb0: 7049 6465 616c 5f63 6d5a 5a5f 636d 5a5a  pIdeal_cmZZ_cmZZ
│ │ │ │ +00092fc0: 5f72 702c 202d 2d20 6d61 6b65 7320 6120  _rp, -- makes a 
│ │ │ │ +00092fd0: 7261 7469 6f6e 616c 206d 6170 2066 726f  rational map fro
│ │ │ │ +00092fe0: 6d20 616e 2069 6465 616c 0a20 202a 202a  m an ideal.  * *
│ │ │ │ +00092ff0: 6e6f 7465 2072 6174 696f 6e61 6c4d 6170  note rationalMap
│ │ │ │ +00093000: 2849 6465 616c 2c5a 5a2c 5a5a 293a 2072  (Ideal,ZZ,ZZ): r
│ │ │ │ +00093010: 6174 696f 6e61 6c4d 6170 5f6c 7049 6465  ationalMap_lpIde
│ │ │ │ +00093020: 616c 5f63 6d5a 5a5f 636d 5a5a 5f72 702c  al_cmZZ_cmZZ_rp,
│ │ │ │ +00093030: 202d 2d20 6d61 6b65 730a 2020 2020 6120   -- makes.    a 
│ │ │ │ +00093040: 7261 7469 6f6e 616c 206d 6170 2066 726f  rational map fro
│ │ │ │ +00093050: 6d20 616e 2069 6465 616c 0a20 202a 202a  m an ideal.  * *
│ │ │ │ +00093060: 6e6f 7465 2072 6174 696f 6e61 6c4d 6170  note rationalMap
│ │ │ │ +00093070: 2850 6f6c 796e 6f6d 6961 6c52 696e 672c  (PolynomialRing,
│ │ │ │ +00093080: 4c69 7374 293a 0a20 2020 2072 6174 696f  List):.    ratio
│ │ │ │ +00093090: 6e61 6c4d 6170 5f6c 7050 6f6c 796e 6f6d  nalMap_lpPolynom
│ │ │ │ +000930a0: 6961 6c52 696e 675f 636d 4c69 7374 5f72  ialRing_cmList_r
│ │ │ │ +000930b0: 702c 202d 2d20 7261 7469 6f6e 616c 206d  p, -- rational m
│ │ │ │ +000930c0: 6170 2064 6566 696e 6564 2062 7920 7468  ap defined by th
│ │ │ │ +000930d0: 650a 2020 2020 6c69 6e65 6172 2073 7973  e.    linear sys
│ │ │ │ +000930e0: 7465 6d20 6f66 2068 7970 6572 7375 7266  tem of hypersurf
│ │ │ │ +000930f0: 6163 6573 2070 6173 7369 6e67 2074 6872  aces passing thr
│ │ │ │ +00093100: 6f75 6768 2072 616e 646f 6d20 706f 696e  ough random poin
│ │ │ │ +00093110: 7473 2077 6974 680a 2020 2020 6d75 6c74  ts with.    mult
│ │ │ │ +00093120: 6970 6c69 6369 7479 0a20 202a 202a 6e6f  iplicity.  * *no
│ │ │ │ +00093130: 7465 2072 6174 696f 6e61 6c4d 6170 2852  te rationalMap(R
│ │ │ │ +00093140: 696e 672c 5461 6c6c 7929 3a20 7261 7469  ing,Tally): rati
│ │ │ │ +00093150: 6f6e 616c 4d61 705f 6c70 5269 6e67 5f63  onalMap_lpRing_c
│ │ │ │ +00093160: 6d54 616c 6c79 5f72 702c 202d 2d20 7261  mTally_rp, -- ra
│ │ │ │ +00093170: 7469 6f6e 616c 0a20 2020 206d 6170 2064  tional.    map d
│ │ │ │ +00093180: 6566 696e 6564 2062 7920 616e 2065 6666  efined by an eff
│ │ │ │ +00093190: 6563 7469 7665 2064 6976 6973 6f72 0a20  ective divisor. 
│ │ │ │ +000931a0: 202a 2022 7261 7469 6f6e 616c 4d61 7028   * "rationalMap(
│ │ │ │ +000931b0: 5461 6c6c 7929 2220 2d2d 2073 6565 202a  Tally)" -- see *
│ │ │ │ +000931c0: 6e6f 7465 2072 6174 696f 6e61 6c4d 6170  note rationalMap
│ │ │ │ +000931d0: 2852 696e 672c 5461 6c6c 7929 3a0a 2020  (Ring,Tally):.  
│ │ │ │ +000931e0: 2020 7261 7469 6f6e 616c 4d61 705f 6c70    rationalMap_lp
│ │ │ │ +000931f0: 5269 6e67 5f63 6d54 616c 6c79 5f72 702c  Ring_cmTally_rp,
│ │ │ │ +00093200: 202d 2d20 7261 7469 6f6e 616c 206d 6170   -- rational map
│ │ │ │ +00093210: 2064 6566 696e 6564 2062 7920 616e 2065   defined by an e
│ │ │ │ +00093220: 6666 6563 7469 7665 0a20 2020 2064 6976  ffective.    div
│ │ │ │ +00093230: 6973 6f72 0a20 202a 2022 7261 7469 6f6e  isor.  * "ration
│ │ │ │ +00093240: 616c 4d61 7028 5261 7469 6f6e 616c 4d61  alMap(RationalMa
│ │ │ │ +00093250: 7029 2220 2d2d 2073 6565 202a 6e6f 7465  p)" -- see *note
│ │ │ │ +00093260: 2073 7570 6572 2852 6174 696f 6e61 6c4d   super(RationalM
│ │ │ │ +00093270: 6170 293a 0a20 2020 2073 7570 6572 5f6c  ap):.    super_l
│ │ │ │ +00093280: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ +00093290: 202d 2d20 6765 7420 7468 6520 7261 7469   -- get the rati
│ │ │ │ +000932a0: 6f6e 616c 206d 6170 2077 686f 7365 2074  onal map whose t
│ │ │ │ +000932b0: 6172 6765 7420 6973 2061 0a20 2020 2070  arget is a.    p
│ │ │ │ +000932c0: 726f 6a65 6374 6976 6520 7370 6163 650a  rojective space.
│ │ │ │ +000932d0: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ +000932e0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ +000932f0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ +00093300: 6374 202a 6e6f 7465 2072 6174 696f 6e61  ct *note rationa
│ │ │ │ +00093310: 6c4d 6170 3a20 7261 7469 6f6e 616c 4d61  lMap: rationalMa
│ │ │ │ +00093320: 702c 2069 7320 6120 2a6e 6f74 6520 6d65  p, is a *note me
│ │ │ │ +00093330: 7468 6f64 2066 756e 6374 696f 6e20 7769  thod function wi
│ │ │ │ +00093340: 7468 0a6f 7074 696f 6e73 3a20 284d 6163  th.options: (Mac
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│ │ │ │ +00093360: 4675 6e63 7469 6f6e 5769 7468 4f70 7469  FunctionWithOpti
│ │ │ │ +00093370: 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d  ons,...---------
│ │ │ │  00093380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000933a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000933b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000933c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
│ │ │ │ -000933d0: 6520 736f 7572 6365 206f 6620 7468 6973  e source of this
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│ │ │ │ -000933f0: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ -00093400: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ -00093410: 7932 2d31 2e32 352e 3131 2b64 732f 4d32  y2-1.25.11+ds/M2
│ │ │ │ -00093420: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ -00093430: 6765 732f 4372 656d 6f6e 612f 0a64 6f63  ges/Cremona/.doc
│ │ │ │ -00093440: 756d 656e 7461 7469 6f6e 2e6d 323a 3830  umentation.m2:80
│ │ │ │ -00093450: 303a 302e 0a1f 0a46 696c 653a 2043 7265  0:0....File: Cre
│ │ │ │ -00093460: 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 653a  mona.info, Node:
│ │ │ │ -00093470: 2052 6174 696f 6e61 6c4d 6170 2021 2c20   RationalMap !, 
│ │ │ │ -00093480: 4e65 7874 3a20 5261 7469 6f6e 616c 4d61  Next: RationalMa
│ │ │ │ -00093490: 7020 5f73 7420 5261 7469 6f6e 616c 4d61  p _st RationalMa
│ │ │ │ -000934a0: 702c 2050 7265 763a 2072 6174 696f 6e61  p, Prev: rationa
│ │ │ │ -000934b0: 6c4d 6170 2c20 5570 3a20 546f 700a 0a52  lMap, Up: Top..R
│ │ │ │ -000934c0: 6174 696f 6e61 6c4d 6170 2021 202d 2d20  ationalMap ! -- 
│ │ │ │ -000934d0: 6361 6c63 756c 6174 6573 2065 7665 7279  calculates every
│ │ │ │ -000934e0: 2070 6f73 7369 626c 6520 7468 696e 670a   possible thing.
│ │ │ │ +000933c0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ +000933d0: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ +000933e0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ +000933f0: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ +00093400: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ +00093410: 352e 3131 2b64 732f 4d32 2f4d 6163 6175  5.11+ds/M2/Macau
│ │ │ │ +00093420: 6c61 7932 2f70 6163 6b61 6765 732f 4372  lay2/packages/Cr
│ │ │ │ +00093430: 656d 6f6e 612f 0a64 6f63 756d 656e 7461  emona/.documenta
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│ │ │ │ +00093460: 6e66 6f2c 204e 6f64 653a 2052 6174 696f  nfo, Node: Ratio
│ │ │ │ +00093470: 6e61 6c4d 6170 2021 2c20 4e65 7874 3a20  nalMap !, Next: 
│ │ │ │ +00093480: 5261 7469 6f6e 616c 4d61 7020 5f73 7420  RationalMap _st 
│ │ │ │ +00093490: 5261 7469 6f6e 616c 4d61 702c 2050 7265  RationalMap, Pre
│ │ │ │ +000934a0: 763a 2072 6174 696f 6e61 6c4d 6170 2c20  v: rationalMap, 
│ │ │ │ +000934b0: 5570 3a20 546f 700a 0a52 6174 696f 6e61  Up: Top..Rationa
│ │ │ │ +000934c0: 6c4d 6170 2021 202d 2d20 6361 6c63 756c  lMap ! -- calcul
│ │ │ │ +000934d0: 6174 6573 2065 7665 7279 2070 6f73 7369  ates every possi
│ │ │ │ +000934e0: 626c 6520 7468 696e 670a 2a2a 2a2a 2a2a  ble thing.******
│ │ │ │  000934f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00093500: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00093510: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00093520: 0a0a 2020 2a20 4f70 6572 6174 6f72 3a20  ..  * Operator: 
│ │ │ │ -00093530: 2a6e 6f74 6520 213a 2028 4d61 6361 756c  *note !: (Macaul
│ │ │ │ -00093540: 6179 3244 6f63 2921 2c0a 2020 2a20 5573  ay2Doc)!,.  * Us
│ │ │ │ -00093550: 6167 653a 200a 2020 2020 2020 2020 7068  age: .        ph
│ │ │ │ -00093560: 6921 0a20 202a 2049 6e70 7574 733a 0a20  i!.  * Inputs:. 
│ │ │ │ -00093570: 2020 2020 202a 2070 6869 2c20 6120 2a6e       * phi, a *n
│ │ │ │ -00093580: 6f74 6520 7261 7469 6f6e 616c 206d 6170  ote rational map
│ │ │ │ -00093590: 3a20 5261 7469 6f6e 616c 4d61 702c 0a20  : RationalMap,. 
│ │ │ │ -000935a0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -000935b0: 2020 2a20 6120 2a6e 6f74 6520 7261 7469    * a *note rati
│ │ │ │ -000935c0: 6f6e 616c 206d 6170 3a20 5261 7469 6f6e  onal map: Ration
│ │ │ │ -000935d0: 616c 4d61 702c 2c20 7468 6520 7361 6d65  alMap,, the same
│ │ │ │ -000935e0: 2072 6174 696f 6e61 6c20 6d61 7020 7068   rational map ph
│ │ │ │ -000935f0: 690a 0a44 6573 6372 6970 7469 6f6e 0a3d  i..Description.=
│ │ │ │ -00093600: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973  ==========..This
│ │ │ │ -00093610: 206d 6574 686f 6420 286d 6169 6e6c 7920   method (mainly 
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│ │ │ │ -00093630: 6170 706c 6965 7320 616c 6d6f 7374 2061  applies almost a
│ │ │ │ -00093640: 6c6c 2074 6865 2064 6574 6572 6d69 6e69  ll the determini
│ │ │ │ -00093650: 7374 6963 0a6d 6574 686f 6473 2074 6861  stic.methods tha
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│ │ │ │ -00093670: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
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│ │ │ │ +00093520: 4f70 6572 6174 6f72 3a20 2a6e 6f74 6520  Operator: *note 
│ │ │ │ +00093530: 213a 2028 4d61 6361 756c 6179 3244 6f63  !: (Macaulay2Doc
│ │ │ │ +00093540: 2921 2c0a 2020 2a20 5573 6167 653a 200a  )!,.  * Usage: .
│ │ │ │ +00093550: 2020 2020 2020 2020 7068 6921 0a20 202a          phi!.  *
│ │ │ │ +00093560: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +00093570: 2070 6869 2c20 6120 2a6e 6f74 6520 7261   phi, a *note ra
│ │ │ │ +00093580: 7469 6f6e 616c 206d 6170 3a20 5261 7469  tional map: Rati
│ │ │ │ +00093590: 6f6e 616c 4d61 702c 0a20 202a 204f 7574  onalMap,.  * Out
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│ │ │ │ +000935b0: 2a6e 6f74 6520 7261 7469 6f6e 616c 206d  *note rational m
│ │ │ │ +000935c0: 6170 3a20 5261 7469 6f6e 616c 4d61 702c  ap: RationalMap,
│ │ │ │ +000935d0: 2c20 7468 6520 7361 6d65 2072 6174 696f  , the same ratio
│ │ │ │ +000935e0: 6e61 6c20 6d61 7020 7068 690a 0a44 6573  nal map phi..Des
│ │ │ │ +000935f0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +00093600: 3d3d 3d3d 0a0a 5468 6973 206d 6574 686f  ====..This metho
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│ │ │ │ +00093620: 6f72 2074 6573 7473 2920 6170 706c 6965  or tests) applie
│ │ │ │ +00093630: 7320 616c 6d6f 7374 2061 6c6c 2074 6865  s almost all the
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│ │ │ │ +00093660: 6176 6169 6c61 626c 652e 0a0a 2b2d 2d2d  available...+---
│ │ │ │ +00093670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000936a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000936b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000936c0: 2b0a 7c69 3120 3a20 5151 5b78 5f30 2e2e  +.|i1 : QQ[x_0..
│ │ │ │ -000936d0: 785f 355d 3b20 7068 6920 3d20 7261 7469  x_5]; phi = rati
│ │ │ │ -000936e0: 6f6e 616c 4d61 7020 2020 2020 2020 2020  onalMap         
│ │ │ │ +000936b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ +000936c0: 3a20 5151 5b78 5f30 2e2e 785f 355d 3b20  : QQ[x_0..x_5]; 
│ │ │ │ +000936d0: 7068 6920 3d20 7261 7469 6f6e 616c 4d61  phi = rationalMa
│ │ │ │ +000936e0: 7020 2020 2020 2020 2020 2020 2020 2020  p               
│ │ │ │  000936f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093710: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00093700: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00093710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093760: 7c0a 7c6f 3220 3a20 5261 7469 6f6e 616c  |.|o2 : Rational
│ │ │ │ -00093770: 4d61 7020 2871 7561 6472 6174 6963 2072  Map (quadratic r
│ │ │ │ -00093780: 6174 696f 6e61 6c20 2020 2020 2020 2020  ational         
│ │ │ │ +00093750: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +00093760: 3a20 5261 7469 6f6e 616c 4d61 7020 2871  : RationalMap (q
│ │ │ │ +00093770: 7561 6472 6174 6963 2072 6174 696f 6e61  uadratic rationa
│ │ │ │ +00093780: 6c20 2020 2020 2020 2020 2020 2020 2020  l               
│ │ │ │  00093790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000937a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000937b0: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │ +000937a0: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │ +000937b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000937c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000937d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000937e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000937f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00093800: 7c0a 7c7b 785f 345e 322d 785f 332a 785f  |.|{x_4^2-x_3*x_
│ │ │ │ -00093810: 352c 785f 322a 785f 342d 785f 312a 785f  5,x_2*x_4-x_1*x_
│ │ │ │ -00093820: 352c 785f 322a 785f 332d 785f 312a 785f  5,x_2*x_3-x_1*x_
│ │ │ │ -00093830: 342c 785f 325e 322d 785f 302a 785f 352c  4,x_2^2-x_0*x_5,
│ │ │ │ -00093840: 785f 312a 785f 322d 785f 302a 785f 342c  x_1*x_2-x_0*x_4,
│ │ │ │ -00093850: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000937f0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c7b 785f  ----------|.|{x_
│ │ │ │ +00093800: 345e 322d 785f 332a 785f 352c 785f 322a  4^2-x_3*x_5,x_2*
│ │ │ │ +00093810: 785f 342d 785f 312a 785f 352c 785f 322a  x_4-x_1*x_5,x_2*
│ │ │ │ +00093820: 785f 332d 785f 312a 785f 342c 785f 325e  x_3-x_1*x_4,x_2^
│ │ │ │ +00093830: 322d 785f 302a 785f 352c 785f 312a 785f  2-x_0*x_5,x_1*x_
│ │ │ │ +00093840: 322d 785f 302a 785f 342c 7c0a 7c20 2020  2-x_0*x_4,|.|   
│ │ │ │ +00093850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000938a0: 7c0a 7c6d 6170 2066 726f 6d20 5050 5e35  |.|map from PP^5
│ │ │ │ -000938b0: 2074 6f20 5050 5e35 2920 2020 2020 2020   to PP^5)       
│ │ │ │ +00093890: 2020 2020 2020 2020 2020 7c0a 7c6d 6170            |.|map
│ │ │ │ +000938a0: 2066 726f 6d20 5050 5e35 2074 6f20 5050   from PP^5 to PP
│ │ │ │ +000938b0: 5e35 2920 2020 2020 2020 2020 2020 2020  ^5)             
│ │ │ │  000938c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000938d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000938e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000938f0: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │ +000938e0: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │ +000938f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00093930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00093940: 7c0a 7c78 5f31 5e32 2d78 5f30 2a78 5f33  |.|x_1^2-x_0*x_3
│ │ │ │ -00093950: 7d3b 2020 2020 2020 2020 2020 2020 2020  };              
│ │ │ │ +00093930: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c78 5f31  ----------|.|x_1
│ │ │ │ +00093940: 5e32 2d78 5f30 2a78 5f33 7d3b 2020 2020  ^2-x_0*x_3};    
│ │ │ │ +00093950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093990: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00093980: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00093990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000939a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000939b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000939c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000939d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000939e0: 2b0a 7c69 3320 3a20 6465 7363 7269 6265  +.|i3 : describe
│ │ │ │ -000939f0: 2070 6869 2020 2020 2020 2020 2020 2020   phi            
│ │ │ │ +000939d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ +000939e0: 3a20 6465 7363 7269 6265 2070 6869 2020  : describe phi  
│ │ │ │ +000939f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093a30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00093a20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00093a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093a80: 7c0a 7c6f 3320 3d20 7261 7469 6f6e 616c  |.|o3 = rational
│ │ │ │ -00093a90: 206d 6170 2064 6566 696e 6564 2062 7920   map defined by 
│ │ │ │ -00093aa0: 666f 726d 7320 6f66 2064 6567 7265 6520  forms of degree 
│ │ │ │ -00093ab0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00093ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093ad0: 7c0a 7c20 2020 2020 736f 7572 6365 2076  |.|     source v
│ │ │ │ -00093ae0: 6172 6965 7479 3a20 5050 5e35 2020 2020  ariety: PP^5    
│ │ │ │ +00093a70: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ +00093a80: 3d20 7261 7469 6f6e 616c 206d 6170 2064  = rational map d
│ │ │ │ +00093a90: 6566 696e 6564 2062 7920 666f 726d 7320  efined by forms 
│ │ │ │ +00093aa0: 6f66 2064 6567 7265 6520 3220 2020 2020  of degree 2     
│ │ │ │ +00093ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093ac0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00093ad0: 2020 736f 7572 6365 2076 6172 6965 7479    source variety
│ │ │ │ +00093ae0: 3a20 5050 5e35 2020 2020 2020 2020 2020  : PP^5          
│ │ │ │  00093af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093b20: 7c0a 7c20 2020 2020 7461 7267 6574 2076  |.|     target v
│ │ │ │ -00093b30: 6172 6965 7479 3a20 5050 5e35 2020 2020  ariety: PP^5    
│ │ │ │ +00093b10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00093b20: 2020 7461 7267 6574 2076 6172 6965 7479    target variety
│ │ │ │ +00093b30: 3a20 5050 5e35 2020 2020 2020 2020 2020  : PP^5          
│ │ │ │  00093b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093b70: 7c0a 7c20 2020 2020 636f 6566 6669 6369  |.|     coeffici
│ │ │ │ -00093b80: 656e 7420 7269 6e67 3a20 5151 2020 2020  ent ring: QQ    
│ │ │ │ +00093b60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00093b70: 2020 636f 6566 6669 6369 656e 7420 7269    coefficient ri
│ │ │ │ +00093b80: 6e67 3a20 5151 2020 2020 2020 2020 2020  ng: QQ          
│ │ │ │  00093b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093bc0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00093bb0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00093bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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  ! ;             
│ │ │ │ +00093c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ +00093c10: 3a20 7469 6d65 2070 6869 2120 3b20 2020  : time phi! ;   
│ │ │ │ +00093c20: 2020 2020 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: 3332 3432 3973 2028 6370 7529 3b20 302e  32429s (cpu); 0.
│ │ │ │ -00093c80: 3035 3238 3939 3273 2028 7468 7265 6164  0528992s (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  |.|             
│ │ │ │ +00093c50: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ +00093c60: 2075 7365 6420 302e 3131 3836 3338 7320   used 0.118638s 
│ │ │ │ +00093c70: 2863 7075 293b 2030 2e30 3730 3939 3037  (cpu); 0.0709907
│ │ │ │ +00093c80: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +00093c90: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +00093ca0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00093cb0: 2020 2020 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                  
│ │ │ │ -00093d00: 7c0a 7c6f 3420 3a20 5261 7469 6f6e 616c  |.|o4 : Rational
│ │ │ │ -00093d10: 4d61 7020 2843 7265 6d6f 6e61 2074 7261  Map (Cremona tra
│ │ │ │ -00093d20: 6e73 666f 726d 6174 696f 6e20 6f66 2050  nsformation of P
│ │ │ │ -00093d30: 505e 3520 6f66 2074 7970 6520 2832 2c32  P^5 of type (2,2
│ │ │ │ -00093d40: 2929 2020 2020 2020 2020 2020 2020 2020  ))              
│ │ │ │ -00093d50: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00093cf0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ +00093d00: 3a20 5261 7469 6f6e 616c 4d61 7020 2843  : RationalMap (C
│ │ │ │ +00093d10: 7265 6d6f 6e61 2074 7261 6e73 666f 726d  remona transform
│ │ │ │ +00093d20: 6174 696f 6e20 6f66 2050 505e 3520 6f66  ation of PP^5 of
│ │ │ │ +00093d30: 2074 7970 6520 2832 2c32 2929 2020 2020   type (2,2))    
│ │ │ │ +00093d40: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00093d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00093d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00093da0: 2b0a 7c69 3520 3a20 6465 7363 7269 6265  +.|i5 : describe
│ │ │ │ -00093db0: 2070 6869 2020 2020 2020 2020 2020 2020   phi            
│ │ │ │ +00093d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ +00093da0: 3a20 6465 7363 7269 6265 2070 6869 2020  : describe phi  
│ │ │ │ +00093db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093df0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00093de0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00093df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093e40: 7c0a 7c6f 3520 3d20 7261 7469 6f6e 616c  |.|o5 = rational
│ │ │ │ -00093e50: 206d 6170 2064 6566 696e 6564 2062 7920   map defined by 
│ │ │ │ -00093e60: 666f 726d 7320 6f66 2064 6567 7265 6520  forms of degree 
│ │ │ │ -00093e70: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00093e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093e90: 7c0a 7c20 2020 2020 736f 7572 6365 2076  |.|     source v
│ │ │ │ -00093ea0: 6172 6965 7479 3a20 5050 5e35 2020 2020  ariety: PP^5    
│ │ │ │ +00093e30: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ +00093e40: 3d20 7261 7469 6f6e 616c 206d 6170 2064  = rational map d
│ │ │ │ +00093e50: 6566 696e 6564 2062 7920 666f 726d 7320  efined by forms 
│ │ │ │ +00093e60: 6f66 2064 6567 7265 6520 3220 2020 2020  of degree 2     
│ │ │ │ +00093e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00093e80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00093e90: 2020 736f 7572 6365 2076 6172 6965 7479    source variety
│ │ │ │ +00093ea0: 3a20 5050 5e35 2020 2020 2020 2020 2020  : PP^5          
│ │ │ │  00093eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093ee0: 7c0a 7c20 2020 2020 7461 7267 6574 2076  |.|     target v
│ │ │ │ -00093ef0: 6172 6965 7479 3a20 5050 5e35 2020 2020  ariety: PP^5    
│ │ │ │ +00093ed0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00093ee0: 2020 7461 7267 6574 2076 6172 6965 7479    target variety
│ │ │ │ +00093ef0: 3a20 5050 5e35 2020 2020 2020 2020 2020  : PP^5          
│ │ │ │  00093f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093f30: 7c0a 7c20 2020 2020 646f 6d69 6e61 6e63  |.|     dominanc
│ │ │ │ -00093f40: 653a 2074 7275 6520 2020 2020 2020 2020  e: true         
│ │ │ │ +00093f20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00093f30: 2020 646f 6d69 6e61 6e63 653a 2074 7275    dominance: tru
│ │ │ │ +00093f40: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │  00093f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093f80: 7c0a 7c20 2020 2020 6269 7261 7469 6f6e  |.|     biration
│ │ │ │ -00093f90: 616c 6974 793a 2074 7275 6520 2874 6865  ality: true (the
│ │ │ │ -00093fa0: 2069 6e76 6572 7365 206d 6170 2069 7320   inverse map is 
│ │ │ │ -00093fb0: 616c 7265 6164 7920 6361 6c63 756c 6174  already calculat
│ │ │ │ -00093fc0: 6564 2920 2020 2020 2020 2020 2020 2020  ed)             
│ │ │ │ -00093fd0: 7c0a 7c20 2020 2020 7072 6f6a 6563 7469  |.|     projecti
│ │ │ │ -00093fe0: 7665 2064 6567 7265 6573 3a20 7b31 2c20  ve degrees: {1, 
│ │ │ │ -00093ff0: 322c 2034 2c20 342c 2032 2c20 317d 2020  2, 4, 4, 2, 1}  
│ │ │ │ +00093f70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00093f80: 2020 6269 7261 7469 6f6e 616c 6974 793a    birationality:
│ │ │ │ +00093f90: 2074 7275 6520 2874 6865 2069 6e76 6572   true (the inver
│ │ │ │ +00093fa0: 7365 206d 6170 2069 7320 616c 7265 6164  se map is alread
│ │ │ │ +00093fb0: 7920 6361 6c63 756c 6174 6564 2920 2020  y calculated)   
│ │ │ │ +00093fc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00093fd0: 2020 7072 6f6a 6563 7469 7665 2064 6567    projective deg
│ │ │ │ +00093fe0: 7265 6573 3a20 7b31 2c20 322c 2034 2c20  rees: {1, 2, 4, 
│ │ │ │ +00093ff0: 342c 2032 2c20 317d 2020 2020 2020 2020  4, 2, 1}        
│ │ │ │  00094000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094020: 7c0a 7c20 2020 2020 6e75 6d62 6572 206f  |.|     number o
│ │ │ │ -00094030: 6620 6d69 6e69 6d61 6c20 7265 7072 6573  f minimal repres
│ │ │ │ -00094040: 656e 7461 7469 7665 733a 2031 2020 2020  entatives: 1    
│ │ │ │ +00094010: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00094020: 2020 6e75 6d62 6572 206f 6620 6d69 6e69    number of mini
│ │ │ │ +00094030: 6d61 6c20 7265 7072 6573 656e 7461 7469  mal representati
│ │ │ │ +00094040: 7665 733a 2031 2020 2020 2020 2020 2020  ves: 1          
│ │ │ │  00094050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094070: 7c0a 7c20 2020 2020 6469 6d65 6e73 696f  |.|     dimensio
│ │ │ │ -00094080: 6e20 6261 7365 206c 6f63 7573 3a20 3220  n base locus: 2 
│ │ │ │ +00094060: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00094070: 2020 6469 6d65 6e73 696f 6e20 6261 7365    dimension base
│ │ │ │ +00094080: 206c 6f63 7573 3a20 3220 2020 2020 2020   locus: 2       
│ │ │ │  00094090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000940a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000940b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000940c0: 7c0a 7c20 2020 2020 6465 6772 6565 2062  |.|     degree b
│ │ │ │ -000940d0: 6173 6520 6c6f 6375 733a 2034 2020 2020  ase locus: 4    
│ │ │ │ +000940b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000940c0: 2020 6465 6772 6565 2062 6173 6520 6c6f    degree base lo
│ │ │ │ +000940d0: 6375 733a 2034 2020 2020 2020 2020 2020  cus: 4          
│ │ │ │  000940e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000940f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094110: 7c0a 7c20 2020 2020 636f 6566 6669 6369  |.|     coeffici
│ │ │ │ -00094120: 656e 7420 7269 6e67 3a20 5151 2020 2020  ent ring: QQ    
│ │ │ │ +00094100: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00094110: 2020 636f 6566 6669 6369 656e 7420 7269    coefficient ri
│ │ │ │ +00094120: 6e67 3a20 5151 2020 2020 2020 2020 2020  ng: QQ          
│ │ │ │  00094130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094160: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00094150: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00094160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000941a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000941b0: 2b0a 7c69 3620 3a20 5151 5b78 5f30 2e2e  +.|i6 : QQ[x_0..
│ │ │ │ -000941c0: 785f 345d 3b20 7068 6920 3d20 7261 7469  x_4]; phi = rati
│ │ │ │ -000941d0: 6f6e 616c 4d61 7020 2020 2020 2020 2020  onalMap         
│ │ │ │ +000941a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ +000941b0: 3a20 5151 5b78 5f30 2e2e 785f 345d 3b20  : QQ[x_0..x_4]; 
│ │ │ │ +000941c0: 7068 6920 3d20 7261 7469 6f6e 616c 4d61  phi = rationalMa
│ │ │ │ +000941d0: 7020 2020 2020 2020 2020 2020 2020 2020  p               
│ │ │ │  000941e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000941f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094200: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000941f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00094200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094250: 7c0a 7c6f 3720 3a20 5261 7469 6f6e 616c  |.|o7 : Rational
│ │ │ │ -00094260: 4d61 7020 2871 7561 6472 6174 6963 2072  Map (quadratic r
│ │ │ │ -00094270: 6174 696f 6e61 6c20 2020 2020 2020 2020  ational         
│ │ │ │ +00094240: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ +00094250: 3a20 5261 7469 6f6e 616c 4d61 7020 2871  : RationalMap (q
│ │ │ │ +00094260: 7561 6472 6174 6963 2072 6174 696f 6e61  uadratic rationa
│ │ │ │ +00094270: 6c20 2020 2020 2020 2020 2020 2020 2020  l               
│ │ │ │  00094280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000942a0: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │ +00094290: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │ +000942a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000942b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000942c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000942d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000942e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000942f0: 7c0a 7c7b 2d78 5f31 5e32 2b78 5f30 2a78  |.|{-x_1^2+x_0*x
│ │ │ │ -00094300: 5f32 2c2d 785f 312a 785f 322b 785f 302a  _2,-x_1*x_2+x_0*
│ │ │ │ -00094310: 785f 332c 2d78 5f32 5e32 2b78 5f31 2a78  x_3,-x_2^2+x_1*x
│ │ │ │ -00094320: 5f33 2c2d 785f 312a 785f 332b 785f 302a  _3,-x_1*x_3+x_0*
│ │ │ │ -00094330: 785f 342c 2d78 5f32 2a78 5f33 2b78 5f31  x_4,-x_2*x_3+x_1
│ │ │ │ -00094340: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000942e0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c7b 2d78  ----------|.|{-x
│ │ │ │ +000942f0: 5f31 5e32 2b78 5f30 2a78 5f32 2c2d 785f  _1^2+x_0*x_2,-x_
│ │ │ │ +00094300: 312a 785f 322b 785f 302a 785f 332c 2d78  1*x_2+x_0*x_3,-x
│ │ │ │ +00094310: 5f32 5e32 2b78 5f31 2a78 5f33 2c2d 785f  _2^2+x_1*x_3,-x_
│ │ │ │ +00094320: 312a 785f 332b 785f 302a 785f 342c 2d78  1*x_3+x_0*x_4,-x
│ │ │ │ +00094330: 5f32 2a78 5f33 2b78 5f31 7c0a 7c20 2020  _2*x_3+x_1|.|   
│ │ │ │ +00094340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094390: 7c0a 7c6d 6170 2066 726f 6d20 5050 5e34  |.|map from PP^4
│ │ │ │ -000943a0: 2074 6f20 5050 5e35 2920 2020 2020 2020   to PP^5)       
│ │ │ │ +00094380: 2020 2020 2020 2020 2020 7c0a 7c6d 6170            |.|map
│ │ │ │ +00094390: 2066 726f 6d20 5050 5e34 2074 6f20 5050   from PP^4 to PP
│ │ │ │ +000943a0: 5e35 2920 2020 2020 2020 2020 2020 2020  ^5)             
│ │ │ │  000943b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000943c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000943d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000943e0: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │ +000943d0: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │ +000943e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000943f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00094420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00094430: 7c0a 7c2a 785f 342c 2d78 5f33 5e32 2b78  |.|*x_4,-x_3^2+x
│ │ │ │ -00094440: 5f32 2a78 5f34 7d3b 2020 2020 2020 2020  _2*x_4};        
│ │ │ │ +00094420: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a 785f  ----------|.|*x_
│ │ │ │ +00094430: 342c 2d78 5f33 5e32 2b78 5f32 2a78 5f34  4,-x_3^2+x_2*x_4
│ │ │ │ +00094440: 7d3b 2020 2020 2020 2020 2020 2020 2020  };              
│ │ │ │  00094450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094480: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00094470: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00094480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000944a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000944b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000944c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000944d0: 2b0a 7c69 3820 3a20 6465 7363 7269 6265  +.|i8 : describe
│ │ │ │ -000944e0: 2070 6869 2020 2020 2020 2020 2020 2020   phi            
│ │ │ │ +000944c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ +000944d0: 3a20 6465 7363 7269 6265 2070 6869 2020  : describe phi  
│ │ │ │ +000944e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000944f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094520: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00094510: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00094520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094570: 7c0a 7c6f 3820 3d20 7261 7469 6f6e 616c  |.|o8 = rational
│ │ │ │ -00094580: 206d 6170 2064 6566 696e 6564 2062 7920   map defined by 
│ │ │ │ -00094590: 666f 726d 7320 6f66 2064 6567 7265 6520  forms of degree 
│ │ │ │ -000945a0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -000945b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000945c0: 7c0a 7c20 2020 2020 736f 7572 6365 2076  |.|     source v
│ │ │ │ -000945d0: 6172 6965 7479 3a20 5050 5e34 2020 2020  ariety: PP^4    
│ │ │ │ +00094560: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ +00094570: 3d20 7261 7469 6f6e 616c 206d 6170 2064  = rational map d
│ │ │ │ +00094580: 6566 696e 6564 2062 7920 666f 726d 7320  efined by forms 
│ │ │ │ +00094590: 6f66 2064 6567 7265 6520 3220 2020 2020  of degree 2     
│ │ │ │ +000945a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000945b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000945c0: 2020 736f 7572 6365 2076 6172 6965 7479    source variety
│ │ │ │ +000945d0: 3a20 5050 5e34 2020 2020 2020 2020 2020  : PP^4          
│ │ │ │  000945e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000945f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094610: 7c0a 7c20 2020 2020 7461 7267 6574 2076  |.|     target v
│ │ │ │ -00094620: 6172 6965 7479 3a20 5050 5e35 2020 2020  ariety: PP^5    
│ │ │ │ +00094600: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00094610: 2020 7461 7267 6574 2076 6172 6965 7479    target variety
│ │ │ │ +00094620: 3a20 5050 5e35 2020 2020 2020 2020 2020  : PP^5          
│ │ │ │  00094630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094660: 7c0a 7c20 2020 2020 636f 6566 6669 6369  |.|     coeffici
│ │ │ │ -00094670: 656e 7420 7269 6e67 3a20 5151 2020 2020  ent ring: QQ    
│ │ │ │ +00094650: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00094660: 2020 636f 6566 6669 6369 656e 7420 7269    coefficient ri
│ │ │ │ +00094670: 6e67 3a20 5151 2020 2020 2020 2020 2020  ng: QQ          
│ │ │ │  00094680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000946a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000946b0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +000946a0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000946b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000946c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000946d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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  ! ;             
│ │ │ │ +000946f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920  ----------+.|i9 
│ │ │ │ +00094700: 3a20 7469 6d65 2070 6869 2120 3b20 2020  : time phi! ;   
│ │ │ │ +00094710: 2020 2020 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 3033  |.| -- used 0.03
│ │ │ │ -00094760: 3630 3632 3773 2028 6370 7529 3b20 302e  60627s (cpu); 0.
│ │ │ │ -00094770: 3033 3537 3034 3973 2028 7468 7265 6164  0357049s (thread
│ │ │ │ -00094780: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ -00094790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000947a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00094740: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ +00094750: 2075 7365 6420 302e 3035 3830 3535 3373   used 0.0580553s
│ │ │ │ +00094760: 2028 6370 7529 3b20 302e 3034 3439 3735   (cpu); 0.044975
│ │ │ │ +00094770: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +00094780: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +00094790: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000947a0: 2020 2020 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                  
│ │ │ │ -000947f0: 7c0a 7c6f 3920 3a20 5261 7469 6f6e 616c  |.|o9 : Rational
│ │ │ │ -00094800: 4d61 7020 2871 7561 6472 6174 6963 2072  Map (quadratic r
│ │ │ │ -00094810: 6174 696f 6e61 6c20 6d61 7020 6672 6f6d  ational map from
│ │ │ │ -00094820: 2050 505e 3420 746f 2050 505e 3529 2020   PP^4 to PP^5)  
│ │ │ │ -00094830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094840: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +000947e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3920            |.|o9 
│ │ │ │ +000947f0: 3a20 5261 7469 6f6e 616c 4d61 7020 2871  : RationalMap (q
│ │ │ │ +00094800: 7561 6472 6174 6963 2072 6174 696f 6e61  uadratic rationa
│ │ │ │ +00094810: 6c20 6d61 7020 6672 6f6d 2050 505e 3420  l map from PP^4 
│ │ │ │ +00094820: 746f 2050 505e 3529 2020 2020 2020 2020  to PP^5)        
│ │ │ │ +00094830: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00094840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00094880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00094890: 2b0a 7c69 3130 203a 2064 6573 6372 6962  +.|i10 : describ
│ │ │ │ -000948a0: 6520 7068 6920 2020 2020 2020 2020 2020  e phi           
│ │ │ │ +00094880: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130  ----------+.|i10
│ │ │ │ +00094890: 203a 2064 6573 6372 6962 6520 7068 6920   : describe phi 
│ │ │ │ +000948a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000948b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000948c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000948d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000948e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000948d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000948e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000948f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094930: 7c0a 7c6f 3130 203d 2072 6174 696f 6e61  |.|o10 = rationa
│ │ │ │ -00094940: 6c20 6d61 7020 6465 6669 6e65 6420 6279  l map defined by
│ │ │ │ -00094950: 2066 6f72 6d73 206f 6620 6465 6772 6565   forms of degree
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│ │ │ │ -00094b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00094b50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │  00094c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00094c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00094c90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │  00094cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00094d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00094d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00094d40: 2b0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  +..See also.====
│ │ │ │ -00094d50: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
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│ │ │ │ -00094da0: 2020 2020 7261 7469 6f6e 616c 206d 6170      rational map
│ │ │ │ -00094db0: 0a0a 5761 7973 2074 6f20 7573 6520 7468  ..Ways to use th
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│ │ │ │ -00094dd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00094de0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2052  ===..  * *note R
│ │ │ │ -00094df0: 6174 696f 6e61 6c4d 6170 2021 3a20 5261  ationalMap !: Ra
│ │ │ │ -00094e00: 7469 6f6e 616c 4d61 7020 212c 202d 2d20  tionalMap !, -- 
│ │ │ │ -00094e10: 6361 6c63 756c 6174 6573 2065 7665 7279  calculates every
│ │ │ │ -00094e20: 2070 6f73 7369 626c 6520 7468 696e 670a   possible thing.
│ │ │ │ +00094d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565  ----------+..See
│ │ │ │ +00094d40: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a   also.========..
│ │ │ │ +00094d50: 2020 2a20 2a6e 6f74 6520 6465 7363 7269    * *note descri
│ │ │ │ +00094d60: 6265 2852 6174 696f 6e61 6c4d 6170 293a  be(RationalMap):
│ │ │ │ +00094d70: 2064 6573 6372 6962 655f 6c70 5261 7469   describe_lpRati
│ │ │ │ +00094d80: 6f6e 616c 4d61 705f 7270 2c20 2d2d 2064  onalMap_rp, -- d
│ │ │ │ +00094d90: 6573 6372 6962 6520 610a 2020 2020 7261  escribe a.    ra
│ │ │ │ +00094da0: 7469 6f6e 616c 206d 6170 0a0a 5761 7973  tional map..Ways
│ │ │ │ +00094db0: 2074 6f20 7573 6520 7468 6973 206d 6574   to use this met
│ │ │ │ +00094dc0: 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  hod:.===========
│ │ │ │ +00094dd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +00094de0: 202a 202a 6e6f 7465 2052 6174 696f 6e61   * *note Rationa
│ │ │ │ +00094df0: 6c4d 6170 2021 3a20 5261 7469 6f6e 616c  lMap !: Rational
│ │ │ │ +00094e00: 4d61 7020 212c 202d 2d20 6361 6c63 756c  Map !, -- calcul
│ │ │ │ +00094e10: 6174 6573 2065 7665 7279 2070 6f73 7369  ates every possi
│ │ │ │ +00094e20: 626c 6520 7468 696e 670a 2d2d 2d2d 2d2d  ble thing.------
│ │ │ │  00094e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00094e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ -00094e80: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
│ │ │ │ -00094e90: 6869 7320 646f 6375 6d65 6e74 2069 7320  his document is 
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│ │ │ │ -00094eb0: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ -00094ec0: 756c 6179 322d 312e 3235 2e31 312b 6473  ulay2-1.25.11+ds
│ │ │ │ -00094ed0: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ -00094ee0: 636b 6167 6573 2f43 7265 6d6f 6e61 2f0a  ckages/Cremona/.
│ │ │ │ -00094ef0: 646f 6375 6d65 6e74 6174 696f 6e2e 6d32  documentation.m2
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│ │ │ │ -00094f20: 6465 3a20 5261 7469 6f6e 616c 4d61 7020  de: RationalMap 
│ │ │ │ -00094f30: 5f73 7420 5261 7469 6f6e 616c 4d61 702c  _st RationalMap,
│ │ │ │ -00094f40: 204e 6578 743a 2052 6174 696f 6e61 6c4d   Next: RationalM
│ │ │ │ -00094f50: 6170 205f 7374 5f73 7420 5269 6e67 2c20  ap _st_st Ring, 
│ │ │ │ -00094f60: 5072 6576 3a20 5261 7469 6f6e 616c 4d61  Prev: RationalMa
│ │ │ │ -00094f70: 7020 212c 2055 703a 2054 6f70 0a0a 5261  p !, Up: Top..Ra
│ │ │ │ -00094f80: 7469 6f6e 616c 4d61 7020 2a20 5261 7469  tionalMap * Rati
│ │ │ │ -00094f90: 6f6e 616c 4d61 7020 2d2d 2063 6f6d 706f  onalMap -- compo
│ │ │ │ -00094fa0: 7369 7469 6f6e 206f 6620 7261 7469 6f6e  sition of ration
│ │ │ │ -00094fb0: 616c 206d 6170 730a 2a2a 2a2a 2a2a 2a2a  al maps.********
│ │ │ │ +00094e70: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ +00094e80: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ +00094e90: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +00094ea0: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +00094eb0: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +00094ec0: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
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│ │ │ │ +00094ef0: 6e74 6174 696f 6e2e 6d32 3a35 3132 3a30  ntation.m2:512:0
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│ │ │ │ +00094f20: 7469 6f6e 616c 4d61 7020 5f73 7420 5261  tionalMap _st Ra
│ │ │ │ +00094f30: 7469 6f6e 616c 4d61 702c 204e 6578 743a  tionalMap, Next:
│ │ │ │ +00094f40: 2052 6174 696f 6e61 6c4d 6170 205f 7374   RationalMap _st
│ │ │ │ +00094f50: 5f73 7420 5269 6e67 2c20 5072 6576 3a20  _st Ring, Prev: 
│ │ │ │ +00094f60: 5261 7469 6f6e 616c 4d61 7020 212c 2055  RationalMap !, U
│ │ │ │ +00094f70: 703a 2054 6f70 0a0a 5261 7469 6f6e 616c  p: Top..Rational
│ │ │ │ +00094f80: 4d61 7020 2a20 5261 7469 6f6e 616c 4d61  Map * RationalMa
│ │ │ │ +00094f90: 7020 2d2d 2063 6f6d 706f 7369 7469 6f6e  p -- composition
│ │ │ │ +00094fa0: 206f 6620 7261 7469 6f6e 616c 206d 6170   of rational map
│ │ │ │ +00094fb0: 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  s.**************
│ │ │ │  00094fc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00094fd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00094fe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00094ff0: 2a0a 0a20 202a 204f 7065 7261 746f 723a  *..  * Operator:
│ │ │ │ -00095000: 202a 6e6f 7465 202a 3a20 284d 6163 6175   *note *: (Macau
│ │ │ │ -00095010: 6c61 7932 446f 6329 5f73 742c 0a20 202a  lay2Doc)_st,.  *
│ │ │ │ -00095020: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -00095030: 2070 6869 202a 2070 7369 200a 2020 2020   phi * psi .    
│ │ │ │ -00095040: 2020 2020 636f 6d70 6f73 6528 7068 692c      compose(phi,
│ │ │ │ -00095050: 7073 6929 0a20 202a 2049 6e70 7574 733a  psi).  * Inputs:
│ │ │ │ -00095060: 0a20 2020 2020 202a 2070 6869 2c20 6120  .      * phi, a 
│ │ │ │ -00095070: 2a6e 6f74 6520 7261 7469 6f6e 616c 206d  *note rational m
│ │ │ │ -00095080: 6170 3a20 5261 7469 6f6e 616c 4d61 702c  ap: RationalMap,
│ │ │ │ -00095090: 2c20 2458 205c 6461 7368 7269 6768 7461  , $X \dashrighta
│ │ │ │ -000950a0: 7272 6f77 2059 240a 2020 2020 2020 2a20  rrow Y$.      * 
│ │ │ │ -000950b0: 7073 692c 2061 202a 6e6f 7465 2072 6174  psi, a *note rat
│ │ │ │ -000950c0: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ -000950d0: 6e61 6c4d 6170 2c2c 2024 5920 5c64 6173  nalMap,, $Y \das
│ │ │ │ -000950e0: 6872 6967 6874 6172 726f 7720 5a24 0a20  hrightarrow Z$. 
│ │ │ │ -000950f0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -00095100: 2020 2a20 6120 2a6e 6f74 6520 7261 7469    * a *note rati
│ │ │ │ -00095110: 6f6e 616c 206d 6170 3a20 5261 7469 6f6e  onal map: Ration
│ │ │ │ -00095120: 616c 4d61 702c 2c20 2458 205c 6461 7368  alMap,, $X \dash
│ │ │ │ -00095130: 7269 6768 7461 7272 6f77 205a 242c 2074  rightarrow Z$, t
│ │ │ │ -00095140: 6865 0a20 2020 2020 2020 2063 6f6d 706f  he.        compo
│ │ │ │ -00095150: 7369 7469 6f6e 206f 6620 7068 6920 616e  sition of phi an
│ │ │ │ -00095160: 6420 7073 690a 0a44 6573 6372 6970 7469  d psi..Descripti
│ │ │ │ -00095170: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -00095180: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00094fe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ +00094ff0: 204f 7065 7261 746f 723a 202a 6e6f 7465   Operator: *note
│ │ │ │ +00095000: 202a 3a20 284d 6163 6175 6c61 7932 446f   *: (Macaulay2Do
│ │ │ │ +00095010: 6329 5f73 742c 0a20 202a 2055 7361 6765  c)_st,.  * Usage
│ │ │ │ +00095020: 3a20 0a20 2020 2020 2020 2070 6869 202a  : .        phi *
│ │ │ │ +00095030: 2070 7369 200a 2020 2020 2020 2020 636f   psi .        co
│ │ │ │ +00095040: 6d70 6f73 6528 7068 692c 7073 6929 0a20  mpose(phi,psi). 
│ │ │ │ +00095050: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +00095060: 202a 2070 6869 2c20 6120 2a6e 6f74 6520   * phi, a *note 
│ │ │ │ +00095070: 7261 7469 6f6e 616c 206d 6170 3a20 5261  rational map: Ra
│ │ │ │ +00095080: 7469 6f6e 616c 4d61 702c 2c20 2458 205c  tionalMap,, $X \
│ │ │ │ +00095090: 6461 7368 7269 6768 7461 7272 6f77 2059  dashrightarrow Y
│ │ │ │ +000950a0: 240a 2020 2020 2020 2a20 7073 692c 2061  $.      * psi, a
│ │ │ │ +000950b0: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ +000950c0: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ +000950d0: 2c2c 2024 5920 5c64 6173 6872 6967 6874  ,, $Y \dashright
│ │ │ │ +000950e0: 6172 726f 7720 5a24 0a20 202a 204f 7574  arrow Z$.  * Out
│ │ │ │ +000950f0: 7075 7473 3a0a 2020 2020 2020 2a20 6120  puts:.      * a 
│ │ │ │ +00095100: 2a6e 6f74 6520 7261 7469 6f6e 616c 206d  *note rational m
│ │ │ │ +00095110: 6170 3a20 5261 7469 6f6e 616c 4d61 702c  ap: RationalMap,
│ │ │ │ +00095120: 2c20 2458 205c 6461 7368 7269 6768 7461  , $X \dashrighta
│ │ │ │ +00095130: 7272 6f77 205a 242c 2074 6865 0a20 2020  rrow Z$, the.   
│ │ │ │ +00095140: 2020 2020 2063 6f6d 706f 7369 7469 6f6e       composition
│ │ │ │ +00095150: 206f 6620 7068 6920 616e 6420 7073 690a   of phi and psi.
│ │ │ │ +00095160: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +00095170: 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d  ========..+-----
│ │ │ │ +00095180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00095190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000951a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000951b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000951c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -000951d0: 5220 3d20 5151 5b78 5f30 2e2e 785f 335d  R = QQ[x_0..x_3]
│ │ │ │ -000951e0: 3b20 5320 3d20 5151 5b79 5f30 2e2e 795f  ; S = QQ[y_0..y_
│ │ │ │ -000951f0: 345d 3b20 5420 3d20 5151 5b7a 5f30 2e2e  4]; T = QQ[z_0..
│ │ │ │ -00095200: 7a5f 345d 3b20 2020 2020 2020 2020 2020  z_4];           
│ │ │ │ -00095210: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000951c0: 2d2d 2b0a 7c69 3120 3a20 5220 3d20 5151  --+.|i1 : R = QQ
│ │ │ │ +000951d0: 5b78 5f30 2e2e 785f 335d 3b20 5320 3d20  [x_0..x_3]; S = 
│ │ │ │ +000951e0: 5151 5b79 5f30 2e2e 795f 345d 3b20 5420  QQ[y_0..y_4]; T 
│ │ │ │ +000951f0: 3d20 5151 5b7a 5f30 2e2e 7a5f 345d 3b20  = QQ[z_0..z_4]; 
│ │ │ │ +00095200: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00095210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00095220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00095230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00095240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00095250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00095260: 3420 3a20 7068 6920 3d20 7261 7469 6f6e  4 : phi = ration
│ │ │ │ -00095270: 616c 4d61 7028 522c 532c 7b78 5f30 2a78  alMap(R,S,{x_0*x
│ │ │ │ -00095280: 5f32 2c78 5f30 2a78 5f33 2c78 5f31 2a78  _2,x_0*x_3,x_1*x
│ │ │ │ -00095290: 5f32 2c78 5f31 2a78 5f33 2c78 5f32 2a78  _2,x_1*x_3,x_2*x
│ │ │ │ -000952a0: 5f33 7d29 2020 7c0a 7c20 2020 2020 2020  _3})  |.|       
│ │ │ │ +00095250: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 7068  ------+.|i4 : ph
│ │ │ │ +00095260: 6920 3d20 7261 7469 6f6e 616c 4d61 7028  i = rationalMap(
│ │ │ │ +00095270: 522c 532c 7b78 5f30 2a78 5f32 2c78 5f30  R,S,{x_0*x_2,x_0
│ │ │ │ +00095280: 2a78 5f33 2c78 5f31 2a78 5f32 2c78 5f31  *x_3,x_1*x_2,x_1
│ │ │ │ +00095290: 2a78 5f33 2c78 5f32 2a78 5f33 7d29 2020  *x_3,x_2*x_3})  
│ │ │ │ +000952a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000952b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000952c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000952d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000952e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000952f0: 7c0a 7c6f 3420 3d20 2d2d 2072 6174 696f  |.|o4 = -- ratio
│ │ │ │ -00095300: 6e61 6c20 6d61 7020 2d2d 2020 2020 2020  nal map --      
│ │ │ │ +000952e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ +000952f0: 3d20 2d2d 2072 6174 696f 6e61 6c20 6d61  = -- rational ma
│ │ │ │ +00095300: 7020 2d2d 2020 2020 2020 2020 2020 2020  p --            
│ │ │ │  00095310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095330: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00095340: 2020 736f 7572 6365 3a20 5072 6f6a 2851    source: Proj(Q
│ │ │ │ -00095350: 515b 7820 2c20 7820 2c20 7820 2c20 7820  Q[x , x , x , x 
│ │ │ │ -00095360: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ -00095370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095380: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00095390: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000953a0: 2031 2020 2032 2020 2033 2020 2020 2020   1   2   3      
│ │ │ │ +00095330: 2020 2020 7c0a 7c20 2020 2020 736f 7572      |.|     sour
│ │ │ │ +00095340: 6365 3a20 5072 6f6a 2851 515b 7820 2c20  ce: Proj(QQ[x , 
│ │ │ │ +00095350: 7820 2c20 7820 2c20 7820 5d29 2020 2020  x , x , x ])    
│ │ │ │ +00095360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095370: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00095380: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00095390: 2020 2020 2020 2030 2020 2031 2020 2032         0   1   2
│ │ │ │ +000953a0: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │  000953b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000953c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000953d0: 7c20 2020 2020 7461 7267 6574 3a20 5072  |     target: Pr
│ │ │ │ -000953e0: 6f6a 2851 515b 7920 2c20 7920 2c20 7920  oj(QQ[y , y , y 
│ │ │ │ -000953f0: 2c20 7920 2c20 7920 5d29 2020 2020 2020  , y , y ])      
│ │ │ │ +000953c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000953d0: 7461 7267 6574 3a20 5072 6f6a 2851 515b  target: Proj(QQ[
│ │ │ │ +000953e0: 7920 2c20 7920 2c20 7920 2c20 7920 2c20  y , y , y , y , 
│ │ │ │ +000953f0: 7920 5d29 2020 2020 2020 2020 2020 2020  y ])            
│ │ │ │  00095400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095410: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00095420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095430: 2030 2020 2031 2020 2032 2020 2033 2020   0   1   2   3  
│ │ │ │ -00095440: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -00095450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095460: 2020 7c0a 7c20 2020 2020 6465 6669 6e69    |.|     defini
│ │ │ │ -00095470: 6e67 2066 6f72 6d73 3a20 7b20 2020 2020  ng forms: {     
│ │ │ │ +00095410: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00095420: 2020 2020 2020 2020 2020 2030 2020 2031             0   1
│ │ │ │ +00095430: 2020 2032 2020 2033 2020 2034 2020 2020     2   3   4    
│ │ │ │ +00095440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095450: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00095460: 2020 2020 6465 6669 6e69 6e67 2066 6f72      defining for
│ │ │ │ +00095470: 6d73 3a20 7b20 2020 2020 2020 2020 2020  ms: {           
│ │ │ │  00095480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000954a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000954b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000954c0: 2020 2020 2078 2078 202c 2020 2020 2020       x x ,      
│ │ │ │ +000954a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000954b0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +000954c0: 2078 202c 2020 2020 2020 2020 2020 2020   x ,            
│ │ │ │  000954d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000954e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000954f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00095500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095510: 3020 3220 2020 2020 2020 2020 2020 2020  0 2             
│ │ │ │ +000954f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00095500: 2020 2020 2020 2020 2020 3020 3220 2020            0 2   
│ │ │ │ +00095510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095540: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00095530: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00095540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095580: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00095590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000955a0: 2020 2078 2078 202c 2020 2020 2020 2020     x x ,        
│ │ │ │ +00095580: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00095590: 2020 2020 2020 2020 2020 2020 2078 2078               x x
│ │ │ │ +000955a0: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  000955b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000955c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000955d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000955e0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -000955f0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +000955c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000955d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000955e0: 2020 2020 2020 2020 3020 3320 2020 2020          0 3     
│ │ │ │ +000955f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095610: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00095620: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00095610: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00095620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095660: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00095670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095680: 2078 2078 202c 2020 2020 2020 2020 2020   x x ,          
│ │ │ │ +00095660: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00095670: 2020 2020 2020 2020 2020 2078 2078 202c             x x ,
│ │ │ │ +00095680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000956a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000956b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -000956c0: 2020 2020 2020 2020 2020 2020 3120 3220              1 2 
│ │ │ │ +000956a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000956b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000956c0: 2020 2020 2020 3120 3220 2020 2020 2020        1 2       
│ │ │ │  000956d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000956e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000956f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000956f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00095700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095740: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00095750: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ -00095760: 2078 202c 2020 2020 2020 2020 2020 2020   x ,            
│ │ │ │ +00095740: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00095750: 2020 2020 2020 2020 2078 2078 202c 2020           x x ,  
│ │ │ │ +00095760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095790: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000957a0: 2020 2020 2020 2020 2020 3120 3320 2020            1 3   
│ │ │ │ +00095780: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00095790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000957a0: 2020 2020 3120 3320 2020 2020 2020 2020      1 3         
│ │ │ │  000957b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000957c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000957d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000957d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000957e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000957f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095820: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00095830: 2020 2020 2020 2020 2020 2020 2078 2078               x x
│ │ │ │ +00095810: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00095820: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00095830: 2020 2020 2020 2078 2078 2020 2020 2020         x x      
│ │ │ │  00095840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095860: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00095870: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00095880: 2020 2020 2020 2020 3220 3320 2020 2020          2 3     
│ │ │ │ +00095860: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00095870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095880: 2020 3220 3320 2020 2020 2020 2020 2020    2 3           
│ │ │ │  00095890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000958a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000958b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000958c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000958d0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +000958b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000958c0: 2020 2020 2020 2020 2020 7d20 2020 2020            }     
│ │ │ │ +000958d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000958e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000958f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095900: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000958f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00095900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095940: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00095950: 3420 3a20 5261 7469 6f6e 616c 4d61 7020  4 : RationalMap 
│ │ │ │ -00095960: 2871 7561 6472 6174 6963 2072 6174 696f  (quadratic ratio
│ │ │ │ -00095970: 6e61 6c20 6d61 7020 6672 6f6d 2050 505e  nal map from PP^
│ │ │ │ -00095980: 3320 746f 2050 505e 3429 2020 2020 2020  3 to PP^4)      
│ │ │ │ -00095990: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00095940: 2020 2020 2020 7c0a 7c6f 3420 3a20 5261        |.|o4 : Ra
│ │ │ │ +00095950: 7469 6f6e 616c 4d61 7020 2871 7561 6472  tionalMap (quadr
│ │ │ │ +00095960: 6174 6963 2072 6174 696f 6e61 6c20 6d61  atic rational ma
│ │ │ │ +00095970: 7020 6672 6f6d 2050 505e 3320 746f 2050  p from PP^3 to P
│ │ │ │ +00095980: 505e 3429 2020 2020 2020 2020 2020 2020  P^4)            
│ │ │ │ +00095990: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  000959a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000959b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000959c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000959d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000959e0: 2b0a 7c69 3520 3a20 7073 6920 3d20 7261  +.|i5 : psi = ra
│ │ │ │ -000959f0: 7469 6f6e 616c 4d61 7028 532c 542c 7b79  tionalMap(S,T,{y
│ │ │ │ -00095a00: 5f30 2a79 5f33 2c2d 795f 322a 795f 332c  _0*y_3,-y_2*y_3,
│ │ │ │ -00095a10: 795f 312a 795f 322c 795f 322a 795f 342c  y_1*y_2,y_2*y_4,
│ │ │ │ -00095a20: 2d79 5f33 2a79 5f34 7d29 7c0a 7c20 2020  -y_3*y_4})|.|   
│ │ │ │ +000959d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ +000959e0: 3a20 7073 6920 3d20 7261 7469 6f6e 616c  : psi = rational
│ │ │ │ +000959f0: 4d61 7028 532c 542c 7b79 5f30 2a79 5f33  Map(S,T,{y_0*y_3
│ │ │ │ +00095a00: 2c2d 795f 322a 795f 332c 795f 312a 795f  ,-y_2*y_3,y_1*y_
│ │ │ │ +00095a10: 322c 795f 322a 795f 342c 2d79 5f33 2a79  2,y_2*y_4,-y_3*y
│ │ │ │ +00095a20: 5f34 7d29 7c0a 7c20 2020 2020 2020 2020  _4})|.|         
│ │ │ │  00095a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095a70: 2020 2020 7c0a 7c6f 3520 3d20 2d2d 2072      |.|o5 = -- r
│ │ │ │ -00095a80: 6174 696f 6e61 6c20 6d61 7020 2d2d 2020  ational map --  
│ │ │ │ +00095a60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00095a70: 7c6f 3520 3d20 2d2d 2072 6174 696f 6e61  |o5 = -- rationa
│ │ │ │ +00095a80: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │  00095a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00095aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095ab0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00095ac0: 7c20 2020 2020 736f 7572 6365 3a20 5072  |     source: Pr
│ │ │ │ -00095ad0: 6f6a 2851 515b 7920 2c20 7920 2c20 7920  oj(QQ[y , y , y 
│ │ │ │ -00095ae0: 2c20 7920 2c20 7920 5d29 2020 2020 2020  , y , y ])      
│ │ │ │ +00095ab0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00095ac0: 736f 7572 6365 3a20 5072 6f6a 2851 515b  source: Proj(QQ[
│ │ │ │ +00095ad0: 7920 2c20 7920 2c20 7920 2c20 7920 2c20  y , y , y , y , 
│ │ │ │ +00095ae0: 7920 5d29 2020 2020 2020 2020 2020 2020  y ])            
│ │ │ │  00095af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095b00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │ -00095b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00095b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00095b40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00095b50: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
│ │ │ │ +00095b60: 2851 515b 7a20 2c20 7a20 2c20 7a20 2c20  (QQ[z , z , z , 
│ │ │ │ +00095b70: 7a20 2c20 7a20 5d29 2020 2020 2020 2020  z , z ])        
│ │ │ │  00095b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095b90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00095ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095bb0: 2020 2020 2030 2020 2031 2020 2032 2020       0   1   2  
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│ │ │ │ -00095be0: 2020 2020 2020 7c0a 7c20 2020 2020 6465        |.|     de
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│ │ │ │  00095c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00095c30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
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│ │ │ │ -00095de0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00095df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095e00: 2020 2020 2079 2079 202c 2020 2020 2020       y y ,      
│ │ │ │ +00095de0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
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│ │ │ │  00095e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00095e30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
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│ │ │ │ -00095e50: 3120 3220 2020 2020 2020 2020 2020 2020  1 2             
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│ │ │ │  00095e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00095e80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
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│ │ │ │  00095eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00095ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00095ee0: 2020 2079 2079 202c 2020 2020 2020 2020     y y ,        
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│ │ │ │ -00095fa0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
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│ │ │ │  00096010: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00096030: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │ -00096050: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
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│ │ │ │ +00096050: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00096070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096080: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00096080: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00096090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000960a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000960b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000960c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000960d0: 7c0a 7c6f 3520 3a20 5261 7469 6f6e 616c  |.|o5 : Rational
│ │ │ │ -000960e0: 4d61 7020 2871 7561 6472 6174 6963 2072  Map (quadratic r
│ │ │ │ -000960f0: 6174 696f 6e61 6c20 6d61 7020 6672 6f6d  ational map from
│ │ │ │ -00096100: 2050 505e 3420 746f 2050 505e 3429 2020   PP^4 to PP^4)  
│ │ │ │ -00096110: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000960c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ +000960d0: 3a20 5261 7469 6f6e 616c 4d61 7020 2871  : RationalMap (q
│ │ │ │ +000960e0: 7561 6472 6174 6963 2072 6174 696f 6e61  uadratic rationa
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│ │ │ │ +00096100: 746f 2050 505e 3429 2020 2020 2020 2020  to PP^4)        
│ │ │ │ +00096110: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00096120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00096150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00096160: 2d2d 2d2d 2b0a 7c69 3620 3a20 7068 6920  ----+.|i6 : phi 
│ │ │ │ -00096170: 2a20 7073 6920 2020 2020 2020 2020 2020  * psi           
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│ │ │ │  00096180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00096190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000961a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000961b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000961a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000961b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000961c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000961d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000961e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000961f0: 2020 7c0a 7c6f 3620 3d20 2d2d 2072 6174    |.|o6 = -- rat
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│ │ │ │ +000962b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000962c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000962d0: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ -000962e0: 7267 6574 3a20 5072 6f6a 2851 515b 7a20  rget: Proj(QQ[z 
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│ │ │ │ -00096310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00096320: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
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│ │ │ │ -00096340: 2032 2020 2033 2020 2034 2020 2020 2020   2   3   4      
│ │ │ │ +000962d0: 7c0a 7c20 2020 2020 7461 7267 6574 3a20  |.|     target: 
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│ │ │ │ +00096300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00096310: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00096320: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00096470: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000964e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000964f0: 2020 2020 2020 2031 2020 2020 2020 2020         1        
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│ │ │ │  00096680: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00096ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00096b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00096bd0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ -00096be0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
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│ │ │ │ -00096c00: 7020 2a20 5261 7469 6f6e 616c 4d61 703a  p * RationalMap:
│ │ │ │ -00096c10: 2052 6174 696f 6e61 6c4d 6170 205f 7374   RationalMap _st
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│ │ │ │ +00096b40: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ +00096b50: 6520 5269 6e67 4d61 7020 2a20 5269 6e67  e RingMap * Ring
│ │ │ │ +00096b60: 4d61 703a 2028 4d61 6361 756c 6179 3244  Map: (Macaulay2D
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│ │ │ │ +00096bb0: 6e0a 0a57 6179 7320 746f 2075 7365 2074  n..Ways to use t
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│ │ │ │ +00096bd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00096be0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ +00096bf0: 5261 7469 6f6e 616c 4d61 7020 2a20 5261  RationalMap * Ra
│ │ │ │ +00096c00: 7469 6f6e 616c 4d61 703a 2052 6174 696f  tionalMap: Ratio
│ │ │ │ +00096c10: 6e61 6c4d 6170 205f 7374 2052 6174 696f  nalMap _st Ratio
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│ │ │ │ +00096c30: 6f6d 706f 7369 7469 6f6e 206f 6620 7261  omposition of ra
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│ │ │ │  00096c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00096c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00096c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00096c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00096ca0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66  -..The source of
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│ │ │ │ -00096d80: 2050 7265 763a 2052 6174 696f 6e61 6c4d   Prev: RationalM
│ │ │ │ -00096d90: 6170 205f 7374 2052 6174 696f 6e61 6c4d  ap _st RationalM
│ │ │ │ -00096da0: 6170 2c20 5570 3a20 546f 700a 0a52 6174  ap, Up: Top..Rat
│ │ │ │ -00096db0: 696f 6e61 6c4d 6170 202a 2a20 5269 6e67  ionalMap ** Ring
│ │ │ │ -00096dc0: 202d 2d20 6368 616e 6765 2074 6865 2063   -- change the c
│ │ │ │ -00096dd0: 6f65 6666 6963 6965 6e74 2072 696e 6720  oefficient ring 
│ │ │ │ -00096de0: 6f66 2061 2072 6174 696f 6e61 6c20 6d61  of a rational ma
│ │ │ │ -00096df0: 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  p.**************
│ │ │ │ +00096c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
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│ │ │ │ +00096ce0: 322d 312e 3235 2e31 312b 6473 2f4d 322f  2-1.25.11+ds/M2/
│ │ │ │ +00096cf0: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ +00096d00: 6573 2f43 7265 6d6f 6e61 2f0a 646f 6375  es/Cremona/.docu
│ │ │ │ +00096d10: 6d65 6e74 6174 696f 6e2e 6d32 3a36 3330  mentation.m2:630
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│ │ │ │ +00096d40: 5261 7469 6f6e 616c 4d61 7020 5f73 745f  RationalMap _st_
│ │ │ │ +00096d50: 7374 2052 696e 672c 204e 6578 743a 2052  st Ring, Next: R
│ │ │ │ +00096d60: 6174 696f 6e61 6c4d 6170 203d 3d20 5261  ationalMap == Ra
│ │ │ │ +00096d70: 7469 6f6e 616c 4d61 702c 2050 7265 763a  tionalMap, Prev:
│ │ │ │ +00096d80: 2052 6174 696f 6e61 6c4d 6170 205f 7374   RationalMap _st
│ │ │ │ +00096d90: 2052 6174 696f 6e61 6c4d 6170 2c20 5570   RationalMap, Up
│ │ │ │ +00096da0: 3a20 546f 700a 0a52 6174 696f 6e61 6c4d  : Top..RationalM
│ │ │ │ +00096db0: 6170 202a 2a20 5269 6e67 202d 2d20 6368  ap ** Ring -- ch
│ │ │ │ +00096dc0: 616e 6765 2074 6865 2063 6f65 6666 6963  ange the coeffic
│ │ │ │ +00096dd0: 6965 6e74 2072 696e 6720 6f66 2061 2072  ient ring of a r
│ │ │ │ +00096de0: 6174 696f 6e61 6c20 6d61 700a 2a2a 2a2a  ational map.****
│ │ │ │ +00096df0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00096e00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00096e10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00096e20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00096e30: 2a2a 2a2a 2a2a 0a0a 2020 2a20 4f70 6572  ******..  * Oper
│ │ │ │ -00096e40: 6174 6f72 3a20 2a6e 6f74 6520 2a2a 3a20  ator: *note **: 
│ │ │ │ -00096e50: 284d 6163 6175 6c61 7932 446f 6329 5f73  (Macaulay2Doc)_s
│ │ │ │ -00096e60: 745f 7374 2c0a 2020 2a20 5573 6167 653a  t_st,.  * Usage:
│ │ │ │ -00096e70: 200a 2020 2020 2020 2020 7068 6920 2a2a   .        phi **
│ │ │ │ -00096e80: 204b 0a20 202a 2049 6e70 7574 733a 0a20   K.  * Inputs:. 
│ │ │ │ -00096e90: 2020 2020 202a 2070 6869 2c20 6120 2a6e       * phi, a *n
│ │ │ │ -00096ea0: 6f74 6520 7261 7469 6f6e 616c 206d 6170  ote rational map
│ │ │ │ -00096eb0: 3a20 5261 7469 6f6e 616c 4d61 702c 2c20  : RationalMap,, 
│ │ │ │ -00096ec0: 6465 6669 6e65 6420 6f76 6572 2061 2063  defined over a c
│ │ │ │ -00096ed0: 6f65 6666 6963 6965 6e74 0a20 2020 2020  oefficient.     
│ │ │ │ -00096ee0: 2020 2072 696e 6720 460a 2020 2020 2020     ring F.      
│ │ │ │ -00096ef0: 2a20 4b2c 2061 202a 6e6f 7465 2072 696e  * K, a *note rin
│ │ │ │ -00096f00: 673a 2028 4d61 6361 756c 6179 3244 6f63  g: (Macaulay2Doc
│ │ │ │ -00096f10: 2952 696e 672c 2c20 7468 6520 6e65 7720  )Ring,, the new 
│ │ │ │ -00096f20: 636f 6566 6669 6369 656e 7420 7269 6e67  coefficient ring
│ │ │ │ -00096f30: 2028 7768 6963 680a 2020 2020 2020 2020   (which.        
│ │ │ │ -00096f40: 6d75 7374 2062 6520 6120 6669 656c 6429  must be a field)
│ │ │ │ -00096f50: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -00096f60: 2020 2020 2a20 6120 2a6e 6f74 6520 7261      * a *note ra
│ │ │ │ -00096f70: 7469 6f6e 616c 206d 6170 3a20 5261 7469  tional map: Rati
│ │ │ │ -00096f80: 6f6e 616c 4d61 702c 2c20 6120 7261 7469  onalMap,, a rati
│ │ │ │ -00096f90: 6f6e 616c 206d 6170 2064 6566 696e 6564  onal map defined
│ │ │ │ -00096fa0: 206f 7665 7220 4b2c 0a20 2020 2020 2020   over K,.       
│ │ │ │ -00096fb0: 206f 6274 6169 6e65 6420 6279 2063 6f65   obtained by coe
│ │ │ │ -00096fc0: 7263 696e 6720 7468 6520 636f 6566 6669  rcing the coeffi
│ │ │ │ -00096fd0: 6369 656e 7473 206f 6620 7468 6520 666f  cients of the fo
│ │ │ │ -00096fe0: 726d 7320 6465 6669 6e69 6e67 2070 6869  rms defining phi
│ │ │ │ -00096ff0: 2069 6e74 6f20 4b0a 0a44 6573 6372 6970   into K..Descrip
│ │ │ │ -00097000: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -00097010: 0a0a 4974 2069 7320 6e65 6365 7373 6172  ..It is necessar
│ │ │ │ -00097020: 7920 7468 6174 2061 6c6c 2066 6f72 6d73  y that all forms
│ │ │ │ -00097030: 2069 6e20 7468 6520 6f6c 6420 636f 6566   in the old coef
│ │ │ │ -00097040: 6669 6369 656e 7420 7269 6e67 2046 2063  ficient ring F c
│ │ │ │ -00097050: 616e 2062 650a 6175 746f 6d61 7469 6361  an be.automatica
│ │ │ │ -00097060: 6c6c 7920 636f 6572 6365 6420 696e 746f  lly coerced into
│ │ │ │ -00097070: 2074 6865 206e 6577 2063 6f65 6666 6963   the new coeffic
│ │ │ │ -00097080: 6965 6e74 2072 696e 6720 4b2e 0a0a 2b2d  ient ring K...+-
│ │ │ │ +00096e30: 0a0a 2020 2a20 4f70 6572 6174 6f72 3a20  ..  * Operator: 
│ │ │ │ +00096e40: 2a6e 6f74 6520 2a2a 3a20 284d 6163 6175  *note **: (Macau
│ │ │ │ +00096e50: 6c61 7932 446f 6329 5f73 745f 7374 2c0a  lay2Doc)_st_st,.
│ │ │ │ +00096e60: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +00096e70: 2020 2020 7068 6920 2a2a 204b 0a20 202a      phi ** K.  *
│ │ │ │ +00096e80: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +00096e90: 2070 6869 2c20 6120 2a6e 6f74 6520 7261   phi, a *note ra
│ │ │ │ +00096ea0: 7469 6f6e 616c 206d 6170 3a20 5261 7469  tional map: Rati
│ │ │ │ +00096eb0: 6f6e 616c 4d61 702c 2c20 6465 6669 6e65  onalMap,, define
│ │ │ │ +00096ec0: 6420 6f76 6572 2061 2063 6f65 6666 6963  d over a coeffic
│ │ │ │ +00096ed0: 6965 6e74 0a20 2020 2020 2020 2072 696e  ient.        rin
│ │ │ │ +00096ee0: 6720 460a 2020 2020 2020 2a20 4b2c 2061  g F.      * K, a
│ │ │ │ +00096ef0: 202a 6e6f 7465 2072 696e 673a 2028 4d61   *note ring: (Ma
│ │ │ │ +00096f00: 6361 756c 6179 3244 6f63 2952 696e 672c  caulay2Doc)Ring,
│ │ │ │ +00096f10: 2c20 7468 6520 6e65 7720 636f 6566 6669  , the new coeffi
│ │ │ │ +00096f20: 6369 656e 7420 7269 6e67 2028 7768 6963  cient ring (whic
│ │ │ │ +00096f30: 680a 2020 2020 2020 2020 6d75 7374 2062  h.        must b
│ │ │ │ +00096f40: 6520 6120 6669 656c 6429 0a20 202a 204f  e a field).  * O
│ │ │ │ +00096f50: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ +00096f60: 6120 2a6e 6f74 6520 7261 7469 6f6e 616c  a *note rational
│ │ │ │ +00096f70: 206d 6170 3a20 5261 7469 6f6e 616c 4d61   map: RationalMa
│ │ │ │ +00096f80: 702c 2c20 6120 7261 7469 6f6e 616c 206d  p,, a rational m
│ │ │ │ +00096f90: 6170 2064 6566 696e 6564 206f 7665 7220  ap defined over 
│ │ │ │ +00096fa0: 4b2c 0a20 2020 2020 2020 206f 6274 6169  K,.        obtai
│ │ │ │ +00096fb0: 6e65 6420 6279 2063 6f65 7263 696e 6720  ned by coercing 
│ │ │ │ +00096fc0: 7468 6520 636f 6566 6669 6369 656e 7473  the coefficients
│ │ │ │ +00096fd0: 206f 6620 7468 6520 666f 726d 7320 6465   of the forms de
│ │ │ │ +00096fe0: 6669 6e69 6e67 2070 6869 2069 6e74 6f20  fining phi into 
│ │ │ │ +00096ff0: 4b0a 0a44 6573 6372 6970 7469 6f6e 0a3d  K..Description.=
│ │ │ │ +00097000: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4974 2069  ==========..It i
│ │ │ │ +00097010: 7320 6e65 6365 7373 6172 7920 7468 6174  s necessary that
│ │ │ │ +00097020: 2061 6c6c 2066 6f72 6d73 2069 6e20 7468   all forms in th
│ │ │ │ +00097030: 6520 6f6c 6420 636f 6566 6669 6369 656e  e old coefficien
│ │ │ │ +00097040: 7420 7269 6e67 2046 2063 616e 2062 650a  t ring F can be.
│ │ │ │ +00097050: 6175 746f 6d61 7469 6361 6c6c 7920 636f  automatically co
│ │ │ │ +00097060: 6572 6365 6420 696e 746f 2074 6865 206e  erced into the n
│ │ │ │ +00097070: 6577 2063 6f65 6666 6963 6965 6e74 2072  ew coefficient r
│ │ │ │ +00097080: 696e 6720 4b2e 0a0a 2b2d 2d2d 2d2d 2d2d  ing K...+-------
│ │ │ │  00097090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000970a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000970b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000970c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000970d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000970e0: 3120 3a20 5151 5b76 6172 7328 302e 2e35  1 : QQ[vars(0..5
│ │ │ │ -000970f0: 295d 2020 2020 2020 2020 2020 2020 2020  )]              
│ │ │ │ +000970d0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5151  ------+.|i1 : QQ
│ │ │ │ +000970e0: 5b76 6172 7328 302e 2e35 295d 2020 2020  [vars(0..5)]    
│ │ │ │ +000970f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097120: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00097120: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00097130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097170: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00097180: 3120 3d20 5151 5b61 2e2e 665d 2020 2020  1 = QQ[a..f]    
│ │ │ │ +00097170: 2020 2020 2020 7c0a 7c6f 3120 3d20 5151        |.|o1 = QQ
│ │ │ │ +00097180: 5b61 2e2e 665d 2020 2020 2020 2020 2020  [a..f]          
│ │ │ │  00097190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000971a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000971b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000971c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000971c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000971d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000971e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000971f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097210: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00097220: 3120 3a20 506f 6c79 6e6f 6d69 616c 5269  1 : PolynomialRi
│ │ │ │ -00097230: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +00097210: 2020 2020 2020 7c0a 7c6f 3120 3a20 506f        |.|o1 : Po
│ │ │ │ +00097220: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020  lynomialRing    
│ │ │ │ +00097230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097260: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00097260: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  00097270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00097280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00097290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000972a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000972b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000972c0: 3220 3a20 7068 6920 3d20 7261 7469 6f6e  2 : phi = ration
│ │ │ │ -000972d0: 616c 4d61 7020 7b65 5e32 2d64 2a66 2c20  alMap {e^2-d*f, 
│ │ │ │ -000972e0: 632a 652d 622a 662c 2063 2a64 2d62 2a65  c*e-b*f, c*d-b*e
│ │ │ │ -000972f0: 2c20 635e 322d 612a 662c 2062 2a63 2d61  , c^2-a*f, b*c-a
│ │ │ │ -00097300: 2a65 2c20 625e 322d 612a 647d 7c0a 7c20  *e, b^2-a*d}|.| 
│ │ │ │ +000972b0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 7068  ------+.|i2 : ph
│ │ │ │ +000972c0: 6920 3d20 7261 7469 6f6e 616c 4d61 7020  i = rationalMap 
│ │ │ │ +000972d0: 7b65 5e32 2d64 2a66 2c20 632a 652d 622a  {e^2-d*f, c*e-b*
│ │ │ │ +000972e0: 662c 2063 2a64 2d62 2a65 2c20 635e 322d  f, c*d-b*e, c^2-
│ │ │ │ +000972f0: 612a 662c 2062 2a63 2d61 2a65 2c20 625e  a*f, b*c-a*e, b^
│ │ │ │ +00097300: 322d 612a 647d 7c0a 7c20 2020 2020 2020  2-a*d}|.|       
│ │ │ │  00097310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097350: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00097360: 3220 3d20 2d2d 2072 6174 696f 6e61 6c20  2 = -- rational 
│ │ │ │ -00097370: 6d61 7020 2d2d 2020 2020 2020 2020 2020  map --          
│ │ │ │ +00097350: 2020 2020 2020 7c0a 7c6f 3220 3d20 2d2d        |.|o2 = --
│ │ │ │ +00097360: 2072 6174 696f 6e61 6c20 6d61 7020 2d2d   rational map --
│ │ │ │ +00097370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000973a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000973b0: 2020 2020 736f 7572 6365 3a20 5072 6f6a      source: Proj
│ │ │ │ -000973c0: 2851 515b 612c 2062 2c20 632c 2064 2c20  (QQ[a, b, c, d, 
│ │ │ │ -000973d0: 652c 2066 5d29 2020 2020 2020 2020 2020  e, f])          
│ │ │ │ +000973a0: 2020 2020 2020 7c0a 7c20 2020 2020 736f        |.|     so
│ │ │ │ +000973b0: 7572 6365 3a20 5072 6f6a 2851 515b 612c  urce: Proj(QQ[a,
│ │ │ │ +000973c0: 2062 2c20 632c 2064 2c20 652c 2066 5d29   b, c, d, e, f])
│ │ │ │ +000973d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000973e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000973f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00097400: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
│ │ │ │ -00097410: 2851 515b 612c 2062 2c20 632c 2064 2c20  (QQ[a, b, c, d, 
│ │ │ │ -00097420: 652c 2066 5d29 2020 2020 2020 2020 2020  e, f])          
│ │ │ │ +000973f0: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ +00097400: 7267 6574 3a20 5072 6f6a 2851 515b 612c  rget: Proj(QQ[a,
│ │ │ │ +00097410: 2062 2c20 632c 2064 2c20 652c 2066 5d29   b, c, d, e, f])
│ │ │ │ +00097420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097440: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00097450: 2020 2020 6465 6669 6e69 6e67 2066 6f72      defining for
│ │ │ │ -00097460: 6d73 3a20 7b20 2020 2020 2020 2020 2020  ms: {           
│ │ │ │ +00097440: 2020 2020 2020 7c0a 7c20 2020 2020 6465        |.|     de
│ │ │ │ +00097450: 6669 6e69 6e67 2066 6f72 6d73 3a20 7b20  fining forms: { 
│ │ │ │ +00097460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097490: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00097490: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000974a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000974b0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +000974b0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  000974c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000974d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000974e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000974f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097500: 2020 2020 2065 2020 2d20 642a 662c 2020       e  - d*f,  
│ │ │ │ +000974e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000974f0: 2020 2020 2020 2020 2020 2020 2020 2065                 e
│ │ │ │ +00097500: 2020 2d20 642a 662c 2020 2020 2020 2020    - d*f,        
│ │ │ │  00097510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097530: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00097530: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00097540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097580: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00097590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000975a0: 2020 2020 2063 2a65 202d 2062 2a66 2c20       c*e - b*f, 
│ │ │ │ +00097580: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00097590: 2020 2020 2020 2020 2020 2020 2020 2063                 c
│ │ │ │ +000975a0: 2a65 202d 2062 2a66 2c20 2020 2020 2020  *e - b*f,       
│ │ │ │  000975b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000975c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000975d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000975d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000975e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000975f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097620: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00097630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097640: 2020 2020 2063 2a64 202d 2062 2a65 2c20       c*d - b*e, 
│ │ │ │ +00097620: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00097630: 2020 2020 2020 2020 2020 2020 2020 2063                 c
│ │ │ │ +00097640: 2a64 202d 2062 2a65 2c20 2020 2020 2020  *d - b*e,       
│ │ │ │  00097650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097670: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00097670: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00097680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000976a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000976b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000976c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000976c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000976d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000976e0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +000976e0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  000976f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097710: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00097720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097730: 2020 2020 2063 2020 2d20 612a 662c 2020       c  - a*f,  
│ │ │ │ +00097710: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00097720: 2020 2020 2020 2020 2020 2020 2020 2063                 c
│ │ │ │ +00097730: 2020 2d20 612a 662c 2020 2020 2020 2020    - a*f,        
│ │ │ │  00097740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097760: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00097760: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00097770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000977a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000977b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000977c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000977d0: 2020 2020 2062 2a63 202d 2061 2a65 2c20       b*c - a*e, 
│ │ │ │ +000977b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000977c0: 2020 2020 2020 2020 2020 2020 2020 2062                 b
│ │ │ │ +000977d0: 2a63 202d 2061 2a65 2c20 2020 2020 2020  *c - a*e,       
│ │ │ │  000977e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000977f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097800: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00097800: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00097810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097850: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00097850: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00097860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097870: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00097870: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  00097880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000978a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000978b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000978c0: 2020 2020 2062 2020 2d20 612a 6420 2020       b  - a*d   
│ │ │ │ +000978a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000978b0: 2020 2020 2020 2020 2020 2020 2020 2062                 b
│ │ │ │ +000978c0: 2020 2d20 612a 6420 2020 2020 2020 2020    - a*d         
│ │ │ │  000978d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000978e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000978f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00097900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097910: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │ +000978f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00097900: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
│ │ │ │ +00097910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097940: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00097940: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00097950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097990: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000979a0: 3220 3a20 5261 7469 6f6e 616c 4d61 7020  2 : RationalMap 
│ │ │ │ -000979b0: 2871 7561 6472 6174 6963 2072 6174 696f  (quadratic ratio
│ │ │ │ -000979c0: 6e61 6c20 6d61 7020 6672 6f6d 2050 505e  nal map from PP^
│ │ │ │ -000979d0: 3520 746f 2050 505e 3529 2020 2020 2020  5 to PP^5)      
│ │ │ │ -000979e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00097990: 2020 2020 2020 7c0a 7c6f 3220 3a20 5261        |.|o2 : Ra
│ │ │ │ +000979a0: 7469 6f6e 616c 4d61 7020 2871 7561 6472  tionalMap (quadr
│ │ │ │ +000979b0: 6174 6963 2072 6174 696f 6e61 6c20 6d61  atic rational ma
│ │ │ │ +000979c0: 7020 6672 6f6d 2050 505e 3520 746f 2050  p from PP^5 to P
│ │ │ │ +000979d0: 505e 3529 2020 2020 2020 2020 2020 2020  P^5)            
│ │ │ │ +000979e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  000979f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00097a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00097a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00097a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00097a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00097a40: 3320 3a20 4b20 3d20 5a5a 2f36 3535 3231  3 : K = ZZ/65521
│ │ │ │ -00097a50: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00097a30: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 4b20  ------+.|i3 : K 
│ │ │ │ +00097a40: 3d20 5a5a 2f36 3535 3231 3b20 2020 2020  = ZZ/65521;     
│ │ │ │ +00097a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097a80: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00097a80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  00097a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00097aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00097ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00097ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00097ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00097ae0: 3420 3a20 7068 6920 2a2a 204b 2020 2020  4 : phi ** K    
│ │ │ │ +00097ad0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 7068  ------+.|i4 : ph
│ │ │ │ +00097ae0: 6920 2a2a 204b 2020 2020 2020 2020 2020  i ** K          
│ │ │ │  00097af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097b20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00097b20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00097b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097b70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00097b80: 3420 3d20 2d2d 2072 6174 696f 6e61 6c20  4 = -- rational 
│ │ │ │ -00097b90: 6d61 7020 2d2d 2020 2020 2020 2020 2020  map --          
│ │ │ │ +00097b70: 2020 2020 2020 7c0a 7c6f 3420 3d20 2d2d        |.|o4 = --
│ │ │ │ +00097b80: 2072 6174 696f 6e61 6c20 6d61 7020 2d2d   rational map --
│ │ │ │ +00097b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097bc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00097bd0: 2020 2020 736f 7572 6365 3a20 5072 6f6a      source: Proj
│ │ │ │ -00097be0: 284b 5b61 2c20 622c 2063 2c20 642c 2065  (K[a, b, c, d, e
│ │ │ │ -00097bf0: 2c20 665d 2920 2020 2020 2020 2020 2020  , f])           
│ │ │ │ +00097bc0: 2020 2020 2020 7c0a 7c20 2020 2020 736f        |.|     so
│ │ │ │ +00097bd0: 7572 6365 3a20 5072 6f6a 284b 5b61 2c20  urce: Proj(K[a, 
│ │ │ │ +00097be0: 622c 2063 2c20 642c 2065 2c20 665d 2920  b, c, d, e, f]) 
│ │ │ │ +00097bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097c10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00097c20: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
│ │ │ │ -00097c30: 284b 5b61 2c20 622c 2063 2c20 642c 2065  (K[a, b, c, d, e
│ │ │ │ -00097c40: 2c20 665d 2920 2020 2020 2020 2020 2020  , f])           
│ │ │ │ +00097c10: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ +00097c20: 7267 6574 3a20 5072 6f6a 284b 5b61 2c20  rget: Proj(K[a, 
│ │ │ │ +00097c30: 622c 2063 2c20 642c 2065 2c20 665d 2920  b, c, d, e, f]) 
│ │ │ │ +00097c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097c60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00097c70: 2020 2020 6465 6669 6e69 6e67 2066 6f72      defining for
│ │ │ │ -00097c80: 6d73 3a20 7b20 2020 2020 2020 2020 2020  ms: {           
│ │ │ │ +00097c60: 2020 2020 2020 7c0a 7c20 2020 2020 6465        |.|     de
│ │ │ │ +00097c70: 6669 6e69 6e67 2066 6f72 6d73 3a20 7b20  fining forms: { 
│ │ │ │ +00097c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097cb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00097cb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00097cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097cd0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00097cd0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  00097ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097d00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00097d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097d20: 2020 2020 2065 2020 2d20 642a 662c 2020       e  - d*f,  
│ │ │ │ +00097d00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00097d10: 2020 2020 2020 2020 2020 2020 2020 2065                 e
│ │ │ │ +00097d20: 2020 2d20 642a 662c 2020 2020 2020 2020    - d*f,        
│ │ │ │  00097d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097d50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00097d50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00097d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00097db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097dc0: 2020 2020 2063 2a65 202d 2062 2a66 2c20       c*e - b*f, 
│ │ │ │ +00097da0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00097db0: 2020 2020 2020 2020 2020 2020 2020 2063                 c
│ │ │ │ +00097dc0: 2a65 202d 2062 2a66 2c20 2020 2020 2020  *e - b*f,       
│ │ │ │  00097dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097df0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00097df0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00097e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097e40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00097e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097e60: 2020 2020 2063 2a64 202d 2062 2a65 2c20       c*d - b*e, 
│ │ │ │ +00097e40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00097e50: 2020 2020 2020 2020 2020 2020 2020 2063                 c
│ │ │ │ +00097e60: 2a64 202d 2062 2a65 2c20 2020 2020 2020  *d - b*e,       
│ │ │ │  00097e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097e90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00097e90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00097ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097ee0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00097ee0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00097ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097f00: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00097f00: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  00097f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097f30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00097f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097f50: 2020 2020 2063 2020 2d20 612a 662c 2020       c  - a*f,  
│ │ │ │ +00097f30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00097f40: 2020 2020 2020 2020 2020 2020 2020 2063                 c
│ │ │ │ +00097f50: 2020 2d20 612a 662c 2020 2020 2020 2020    - a*f,        
│ │ │ │  00097f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097f80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00097f80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00097f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00097fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00097fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00097ff0: 2020 2020 2062 2a63 202d 2061 2a65 2c20       b*c - a*e, 
│ │ │ │ +00097fd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00097fe0: 2020 2020 2020 2020 2020 2020 2020 2062                 b
│ │ │ │ +00097ff0: 2a63 202d 2061 2a65 2c20 2020 2020 2020  *c - a*e,       
│ │ │ │  00098000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098020: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00098020: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00098030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098070: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00098070: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00098080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098090: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00098090: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  000980a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000980b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000980c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000980d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000980e0: 2020 2020 2062 2020 2d20 612a 6420 2020       b  - a*d   
│ │ │ │ +000980c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000980d0: 2020 2020 2020 2020 2020 2020 2020 2062                 b
│ │ │ │ +000980e0: 2020 2d20 612a 6420 2020 2020 2020 2020    - a*d         
│ │ │ │  000980f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098110: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00098120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098130: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │ +00098110: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00098120: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
│ │ │ │ +00098130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098160: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00098160: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00098170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000981a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000981b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000981c0: 3420 3a20 5261 7469 6f6e 616c 4d61 7020  4 : RationalMap 
│ │ │ │ -000981d0: 2871 7561 6472 6174 6963 2072 6174 696f  (quadratic ratio
│ │ │ │ -000981e0: 6e61 6c20 6d61 7020 6672 6f6d 2050 505e  nal map from PP^
│ │ │ │ -000981f0: 3520 746f 2050 505e 3529 2020 2020 2020  5 to PP^5)      
│ │ │ │ -00098200: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000981b0: 2020 2020 2020 7c0a 7c6f 3420 3a20 5261        |.|o4 : Ra
│ │ │ │ +000981c0: 7469 6f6e 616c 4d61 7020 2871 7561 6472  tionalMap (quadr
│ │ │ │ +000981d0: 6174 6963 2072 6174 696f 6e61 6c20 6d61  atic rational ma
│ │ │ │ +000981e0: 7020 6672 6f6d 2050 505e 3520 746f 2050  p from PP^5 to P
│ │ │ │ +000981f0: 505e 3529 2020 2020 2020 2020 2020 2020  P^5)            
│ │ │ │ +00098200: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  00098210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00098220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00098230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00098240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00098250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00098260: 3520 3a20 7068 6920 2a2a 2066 7261 6328  5 : phi ** frac(
│ │ │ │ -00098270: 4b5b 745d 2920 2020 2020 2020 2020 2020  K[t])           
│ │ │ │ +00098250: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7068  ------+.|i5 : ph
│ │ │ │ +00098260: 6920 2a2a 2066 7261 6328 4b5b 745d 2920  i ** frac(K[t]) 
│ │ │ │ +00098270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000982a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000982a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000982b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000982c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000982d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000982e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000982f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00098300: 3520 3d20 2d2d 2072 6174 696f 6e61 6c20  5 = -- rational 
│ │ │ │ -00098310: 6d61 7020 2d2d 2020 2020 2020 2020 2020  map --          
│ │ │ │ +000982f0: 2020 2020 2020 7c0a 7c6f 3520 3d20 2d2d        |.|o5 = --
│ │ │ │ +00098300: 2072 6174 696f 6e61 6c20 6d61 7020 2d2d   rational map --
│ │ │ │ +00098310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098340: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00098350: 2020 2020 736f 7572 6365 3a20 5072 6f6a      source: Proj
│ │ │ │ -00098360: 2866 7261 6328 4b5b 745d 295b 612c 2062  (frac(K[t])[a, b
│ │ │ │ -00098370: 2c20 632c 2064 2c20 652c 2066 5d29 2020  , c, d, e, f])  
│ │ │ │ +00098340: 2020 2020 2020 7c0a 7c20 2020 2020 736f        |.|     so
│ │ │ │ +00098350: 7572 6365 3a20 5072 6f6a 2866 7261 6328  urce: Proj(frac(
│ │ │ │ +00098360: 4b5b 745d 295b 612c 2062 2c20 632c 2064  K[t])[a, b, c, d
│ │ │ │ +00098370: 2c20 652c 2066 5d29 2020 2020 2020 2020  , e, f])        
│ │ │ │  00098380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098390: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000983a0: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
│ │ │ │ -000983b0: 2866 7261 6328 4b5b 745d 295b 612c 2062  (frac(K[t])[a, b
│ │ │ │ -000983c0: 2c20 632c 2064 2c20 652c 2066 5d29 2020  , c, d, e, f])  
│ │ │ │ +00098390: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ +000983a0: 7267 6574 3a20 5072 6f6a 2866 7261 6328  rget: Proj(frac(
│ │ │ │ +000983b0: 4b5b 745d 295b 612c 2062 2c20 632c 2064  K[t])[a, b, c, d
│ │ │ │ +000983c0: 2c20 652c 2066 5d29 2020 2020 2020 2020  , e, f])        
│ │ │ │  000983d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000983e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000983f0: 2020 2020 6465 6669 6e69 6e67 2066 6f72      defining for
│ │ │ │ -00098400: 6d73 3a20 7b20 2020 2020 2020 2020 2020  ms: {           
│ │ │ │ +000983e0: 2020 2020 2020 7c0a 7c20 2020 2020 6465        |.|     de
│ │ │ │ +000983f0: 6669 6e69 6e67 2066 6f72 6d73 3a20 7b20  fining forms: { 
│ │ │ │ +00098400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098430: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00098430: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00098440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098450: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00098450: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  00098460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098480: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00098490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000984a0: 2020 2020 2065 2020 2d20 642a 662c 2020       e  - d*f,  
│ │ │ │ +00098480: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00098490: 2020 2020 2020 2020 2020 2020 2020 2065                 e
│ │ │ │ +000984a0: 2020 2d20 642a 662c 2020 2020 2020 2020    - d*f,        
│ │ │ │  000984b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000984c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000984d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000984d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
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│ │ │ │ +00098980: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
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│ │ │ │  000989a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000989b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000989c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00098aa0: 7469 6f6e 616c 4d61 7020 2a2a 2052 696e  tionalMap ** Rin
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│ │ │ │  00098b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00098b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00098b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00098b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00098c70: 7020 2d2d 2065 7175 616c 6974 7920 6f66  p -- equality of
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│ │ │ │ +00098bb0: 6163 6b61 6765 732f 4372 656d 6f6e 612f  ackages/Cremona/
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│ │ │ │ +00098c00: 203d 3d20 5261 7469 6f6e 616c 4d61 702c   == RationalMap,
│ │ │ │ +00098c10: 204e 6578 743a 2052 6174 696f 6e61 6c4d   Next: RationalM
│ │ │ │ +00098c20: 6170 205e 205a 5a2c 2050 7265 763a 2052  ap ^ ZZ, Prev: R
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│ │ │ │ +00098c70: 7175 616c 6974 7920 6f66 2072 6174 696f  quality of ratio
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│ │ │ │  00098c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00098ca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00098cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00098cf0: 2c0a 2020 2a20 5573 6167 653a 200a 2020  ,.  * Usage: .  
│ │ │ │ -00098d00: 2020 2020 2020 7068 6920 3d3d 2070 7369        phi == psi
│ │ │ │ -00098d10: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ -00098d20: 2020 202a 2070 6869 2c20 6120 2a6e 6f74     * phi, a *not
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│ │ │ │ -00098d40: 5261 7469 6f6e 616c 4d61 702c 0a20 2020  RationalMap,.   
│ │ │ │ -00098d50: 2020 202a 2070 7369 2c20 6120 2a6e 6f74     * psi, a *not
│ │ │ │ -00098d60: 6520 7261 7469 6f6e 616c 206d 6170 3a20  e rational map: 
│ │ │ │ -00098d70: 5261 7469 6f6e 616c 4d61 702c 0a20 202a  RationalMap,.  *
│ │ │ │ -00098d80: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
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│ │ │ │ -00098da0: 6e20 7661 6c75 653a 2028 4d61 6361 756c  n value: (Macaul
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│ │ │ │ -00098dc0: 2020 7768 6574 6865 7220 7068 6920 616e    whether phi an
│ │ │ │ -00098dd0: 6420 7073 6920 6172 650a 2020 2020 2020  d psi are.      
│ │ │ │ -00098de0: 2020 7468 6520 7361 6d65 2072 6174 696f    the same ratio
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│ │ │ │ +00098cf0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +00098d00: 7068 6920 3d3d 2070 7369 0a20 202a 2049  phi == psi.  * I
│ │ │ │ +00098d10: 6e70 7574 733a 0a20 2020 2020 202a 2070  nputs:.      * p
│ │ │ │ +00098d20: 6869 2c20 6120 2a6e 6f74 6520 7261 7469  hi, a *note rati
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│ │ │ │ +00098d40: 616c 4d61 702c 0a20 2020 2020 202a 2070  alMap,.      * p
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│ │ │ │ +00098da0: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ +00098db0: 2942 6f6f 6c65 616e 2c2c 2020 7768 6574  )Boolean,,  whet
│ │ │ │ +00098dc0: 6865 7220 7068 6920 616e 6420 7073 6920  her phi and psi 
│ │ │ │ +00098dd0: 6172 650a 2020 2020 2020 2020 7468 6520  are.        the 
│ │ │ │ +00098de0: 7361 6d65 2072 6174 696f 6e61 6c20 6d61  same rational ma
│ │ │ │ +00098df0: 700a 0a44 6573 6372 6970 7469 6f6e 0a3d  p..Description.=
│ │ │ │ +00098e00: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d  ==========..+---
│ │ │ │ +00098e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00098e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00098e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00098e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00098e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00098e60: 2b0a 7c69 3120 3a20 5151 5b78 5f30 2e2e  +.|i1 : QQ[x_0..
│ │ │ │ -00098e70: 785f 355d 2020 2020 2020 2020 2020 2020  x_5]            
│ │ │ │ +00098e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ +00098e60: 3a20 5151 5b78 5f30 2e2e 785f 355d 2020  : QQ[x_0..x_5]  
│ │ │ │ +00098e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098eb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00098ea0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00098eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098f00: 7c0a 7c6f 3120 3d20 5151 5b78 202e 2e78  |.|o1 = QQ[x ..x
│ │ │ │ -00098f10: 205d 2020 2020 2020 2020 2020 2020 2020   ]              
│ │ │ │ +00098ef0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +00098f00: 3d20 5151 5b78 202e 2e78 205d 2020 2020  = QQ[x ..x ]    
│ │ │ │ +00098f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098f50: 7c0a 7c20 2020 2020 2020 2020 3020 2020  |.|         0   
│ │ │ │ -00098f60: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │ +00098f40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00098f50: 2020 2020 2020 3020 2020 3520 2020 2020        0   5     
│ │ │ │ +00098f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098fa0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00098f90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00098fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00098fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00098ff0: 7c0a 7c6f 3120 3a20 506f 6c79 6e6f 6d69  |.|o1 : Polynomi
│ │ │ │ -00099000: 616c 5269 6e67 2020 2020 2020 2020 2020  alRing          
│ │ │ │ +00098fe0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +00098ff0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │ +00099000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099040: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00099030: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00099040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00099080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00099090: 2b0a 7c69 3220 3a20 7068 6920 3d20 7261  +.|i2 : phi = ra
│ │ │ │ -000990a0: 7469 6f6e 616c 4d61 7020 7b78 5f30 2a78  tionalMap {x_0*x
│ │ │ │ -000990b0: 5f34 5e32 2d78 5f30 2a78 5f33 2a78 5f35  _4^2-x_0*x_3*x_5
│ │ │ │ -000990c0: 2c78 5f30 2a78 5f32 2a78 5f34 2d78 5f30  ,x_0*x_2*x_4-x_0
│ │ │ │ -000990d0: 2a78 5f31 2a78 5f35 2c78 5f30 2a78 5f32  *x_1*x_5,x_0*x_2
│ │ │ │ -000990e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00099080: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +00099090: 3a20 7068 6920 3d20 7261 7469 6f6e 616c  : phi = rational
│ │ │ │ +000990a0: 4d61 7020 7b78 5f30 2a78 5f34 5e32 2d78  Map {x_0*x_4^2-x
│ │ │ │ +000990b0: 5f30 2a78 5f33 2a78 5f35 2c78 5f30 2a78  _0*x_3*x_5,x_0*x
│ │ │ │ +000990c0: 5f32 2a78 5f34 2d78 5f30 2a78 5f31 2a78  _2*x_4-x_0*x_1*x
│ │ │ │ +000990d0: 5f35 2c78 5f30 2a78 5f32 7c0a 7c20 2020  _5,x_0*x_2|.|   
│ │ │ │ +000990e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000990f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099130: 7c0a 7c6f 3220 3d20 2d2d 2072 6174 696f  |.|o2 = -- ratio
│ │ │ │ -00099140: 6e61 6c20 6d61 7020 2d2d 2020 2020 2020  nal map --      
│ │ │ │ +00099120: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +00099130: 3d20 2d2d 2072 6174 696f 6e61 6c20 6d61  = -- rational ma
│ │ │ │ +00099140: 7020 2d2d 2020 2020 2020 2020 2020 2020  p --            
│ │ │ │  00099150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099180: 7c0a 7c20 2020 2020 736f 7572 6365 3a20  |.|     source: 
│ │ │ │ -00099190: 5072 6f6a 2851 515b 7820 2c20 7820 2c20  Proj(QQ[x , x , 
│ │ │ │ -000991a0: 7820 2c20 7820 2c20 7820 2c20 7820 5d29  x , x , x , x ])
│ │ │ │ +00099170: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00099180: 2020 736f 7572 6365 3a20 5072 6f6a 2851    source: Proj(Q
│ │ │ │ +00099190: 515b 7820 2c20 7820 2c20 7820 2c20 7820  Q[x , x , x , x 
│ │ │ │ +000991a0: 2c20 7820 2c20 7820 5d29 2020 2020 2020  , x , x ])      
│ │ │ │  000991b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000991c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000991d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000991e0: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
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│ │ │ │ +000991c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000991d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000991e0: 2020 2030 2020 2031 2020 2032 2020 2033     0   1   2   3
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│ │ │ │  00099200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099220: 7c0a 7c20 2020 2020 7461 7267 6574 3a20  |.|     target: 
│ │ │ │ -00099230: 5072 6f6a 2851 515b 7820 2c20 7820 2c20  Proj(QQ[x , x , 
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│ │ │ │ +00099210: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00099220: 2020 7461 7267 6574 3a20 5072 6f6a 2851    target: Proj(Q
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│ │ │ │  00099250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099270: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
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│ │ │ │ +00099260: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00099270: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000992a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000992c0: 7c0a 7c20 2020 2020 6465 6669 6e69 6e67  |.|     defining
│ │ │ │ -000992d0: 2066 6f72 6d73 3a20 7b20 2020 2020 2020   forms: {       
│ │ │ │ +000992b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000992c0: 2020 6465 6669 6e69 6e67 2066 6f72 6d73    defining forms
│ │ │ │ +000992d0: 3a20 7b20 2020 2020 2020 2020 2020 2020  : {             
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│ │ │ │  00099890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000998a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000998b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000998c0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -000998d0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000998a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000998b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000998c0: 2020 2020 2020 3220 2020 2032 2020 2020        2    2    
│ │ │ │ +000998d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000998e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000998f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099900: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00099910: 2020 2020 2020 2020 2078 2078 2020 2d20           x x  - 
│ │ │ │ -00099920: 7820 7820 2020 2020 2020 2020 2020 2020  x x             
│ │ │ │ +000998f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00099900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099910: 2020 2078 2078 2020 2d20 7820 7820 2020     x x  - x x   
│ │ │ │ +00099920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099950: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00099960: 2020 2020 2020 2020 2020 3020 3120 2020            0 1   
│ │ │ │ -00099970: 2030 2033 2020 2020 2020 2020 2020 2020   0 3            
│ │ │ │ +00099940: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00099950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099960: 2020 2020 3020 3120 2020 2030 2033 2020      0 1    0 3  
│ │ │ │ +00099970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000999a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000999b0: 2020 2020 2020 2020 7d20 2020 2020 2020          }       
│ │ │ │ +00099990: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000999a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000999b0: 2020 7d20 2020 2020 2020 2020 2020 2020    }             
│ │ │ │  000999c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000999d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000999e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000999f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000999e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000999f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099a40: 7c0a 7c6f 3220 3a20 5261 7469 6f6e 616c  |.|o2 : Rational
│ │ │ │ -00099a50: 4d61 7020 2863 7562 6963 2072 6174 696f  Map (cubic ratio
│ │ │ │ -00099a60: 6e61 6c20 6d61 7020 6672 6f6d 2050 505e  nal map from PP^
│ │ │ │ -00099a70: 3520 746f 2050 505e 3529 2020 2020 2020  5 to PP^5)      
│ │ │ │ -00099a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099a90: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │ +00099a30: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +00099a40: 3a20 5261 7469 6f6e 616c 4d61 7020 2863  : RationalMap (c
│ │ │ │ +00099a50: 7562 6963 2072 6174 696f 6e61 6c20 6d61  ubic rational ma
│ │ │ │ +00099a60: 7020 6672 6f6d 2050 505e 3520 746f 2050  p from PP^5 to P
│ │ │ │ +00099a70: 505e 3529 2020 2020 2020 2020 2020 2020  P^5)            
│ │ │ │ +00099a80: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │ +00099a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00099ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00099ae0: 7c0a 7c2a 785f 332d 785f 302a 785f 312a  |.|*x_3-x_0*x_1*
│ │ │ │ -00099af0: 785f 342c 785f 302a 785f 325e 322d 785f  x_4,x_0*x_2^2-x_
│ │ │ │ -00099b00: 305e 322a 785f 352c 785f 302a 785f 312a  0^2*x_5,x_0*x_1*
│ │ │ │ -00099b10: 785f 322d 785f 305e 322a 785f 342c 785f  x_2-x_0^2*x_4,x_
│ │ │ │ -00099b20: 302a 785f 315e 322d 785f 305e 322a 785f  0*x_1^2-x_0^2*x_
│ │ │ │ -00099b30: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │ +00099ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2a 785f  ----------|.|*x_
│ │ │ │ +00099ae0: 332d 785f 302a 785f 312a 785f 342c 785f  3-x_0*x_1*x_4,x_
│ │ │ │ +00099af0: 302a 785f 325e 322d 785f 305e 322a 785f  0*x_2^2-x_0^2*x_
│ │ │ │ +00099b00: 352c 785f 302a 785f 312a 785f 322d 785f  5,x_0*x_1*x_2-x_
│ │ │ │ +00099b10: 305e 322a 785f 342c 785f 302a 785f 315e  0^2*x_4,x_0*x_1^
│ │ │ │ +00099b20: 322d 785f 305e 322a 785f 7c0a 7c2d 2d2d  2-x_0^2*x_|.|---
│ │ │ │ +00099b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00099b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00099b80: 7c0a 7c33 7d20 2020 2020 2020 2020 2020  |.|3}           
│ │ │ │ +00099b70: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c33 7d20  ----------|.|3} 
│ │ │ │ +00099b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099bd0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00099bc0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00099bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00099c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00099c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00099c20: 2b0a 7c69 3320 3a20 7073 6920 3d20 7261  +.|i3 : psi = ra
│ │ │ │ -00099c30: 7469 6f6e 616c 4d61 7020 7b78 5f34 5e32  tionalMap {x_4^2
│ │ │ │ -00099c40: 2d78 5f33 2a78 5f35 2c78 5f32 2a78 5f34  -x_3*x_5,x_2*x_4
│ │ │ │ -00099c50: 2d78 5f31 2a78 5f35 2c78 5f32 2a78 5f33  -x_1*x_5,x_2*x_3
│ │ │ │ -00099c60: 2d78 5f31 2a78 5f34 2c78 5f32 5e32 2d78  -x_1*x_4,x_2^2-x
│ │ │ │ -00099c70: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00099c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ +00099c20: 3a20 7073 6920 3d20 7261 7469 6f6e 616c  : psi = rational
│ │ │ │ +00099c30: 4d61 7020 7b78 5f34 5e32 2d78 5f33 2a78  Map {x_4^2-x_3*x
│ │ │ │ +00099c40: 5f35 2c78 5f32 2a78 5f34 2d78 5f31 2a78  _5,x_2*x_4-x_1*x
│ │ │ │ +00099c50: 5f35 2c78 5f32 2a78 5f33 2d78 5f31 2a78  _5,x_2*x_3-x_1*x
│ │ │ │ +00099c60: 5f34 2c78 5f32 5e32 2d78 7c0a 7c20 2020  _4,x_2^2-x|.|   
│ │ │ │ +00099c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099cc0: 7c0a 7c6f 3320 3d20 2d2d 2072 6174 696f  |.|o3 = -- ratio
│ │ │ │ -00099cd0: 6e61 6c20 6d61 7020 2d2d 2020 2020 2020  nal map --      
│ │ │ │ +00099cb0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ +00099cc0: 3d20 2d2d 2072 6174 696f 6e61 6c20 6d61  = -- rational ma
│ │ │ │ +00099cd0: 7020 2d2d 2020 2020 2020 2020 2020 2020  p --            
│ │ │ │  00099ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099d10: 7c0a 7c20 2020 2020 736f 7572 6365 3a20  |.|     source: 
│ │ │ │ -00099d20: 5072 6f6a 2851 515b 7820 2c20 7820 2c20  Proj(QQ[x , x , 
│ │ │ │ -00099d30: 7820 2c20 7820 2c20 7820 2c20 7820 5d29  x , x , x , x ])
│ │ │ │ +00099d00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00099d10: 2020 736f 7572 6365 3a20 5072 6f6a 2851    source: Proj(Q
│ │ │ │ +00099d20: 515b 7820 2c20 7820 2c20 7820 2c20 7820  Q[x , x , x , x 
│ │ │ │ +00099d30: 2c20 7820 2c20 7820 5d29 2020 2020 2020  , x , x ])      
│ │ │ │  00099d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099d60: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00099d70: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ -00099d80: 2032 2020 2033 2020 2034 2020 2035 2020   2   3   4   5  
│ │ │ │ +00099d50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00099d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099d70: 2020 2030 2020 2031 2020 2032 2020 2033     0   1   2   3
│ │ │ │ +00099d80: 2020 2034 2020 2035 2020 2020 2020 2020     4   5        
│ │ │ │  00099d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099db0: 7c0a 7c20 2020 2020 7461 7267 6574 3a20  |.|     target: 
│ │ │ │ -00099dc0: 5072 6f6a 2851 515b 7820 2c20 7820 2c20  Proj(QQ[x , x , 
│ │ │ │ -00099dd0: 7820 2c20 7820 2c20 7820 2c20 7820 5d29  x , x , x , x ])
│ │ │ │ +00099da0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00099db0: 2020 7461 7267 6574 3a20 5072 6f6a 2851    target: Proj(Q
│ │ │ │ +00099dc0: 515b 7820 2c20 7820 2c20 7820 2c20 7820  Q[x , x , x , x 
│ │ │ │ +00099dd0: 2c20 7820 2c20 7820 5d29 2020 2020 2020  , x , x ])      
│ │ │ │  00099de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099e00: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00099e10: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ -00099e20: 2032 2020 2033 2020 2034 2020 2035 2020   2   3   4   5  
│ │ │ │ +00099df0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00099e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099e10: 2020 2030 2020 2031 2020 2032 2020 2033     0   1   2   3
│ │ │ │ +00099e20: 2020 2034 2020 2035 2020 2020 2020 2020     4   5        
│ │ │ │  00099e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099e50: 7c0a 7c20 2020 2020 6465 6669 6e69 6e67  |.|     defining
│ │ │ │ -00099e60: 2066 6f72 6d73 3a20 7b20 2020 2020 2020   forms: {       
│ │ │ │ +00099e40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00099e50: 2020 6465 6669 6e69 6e67 2066 6f72 6d73    defining forms
│ │ │ │ +00099e60: 3a20 7b20 2020 2020 2020 2020 2020 2020  : {             
│ │ │ │  00099e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099ea0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00099eb0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00099e90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00099ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099eb0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  00099ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099ef0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00099f00: 2020 2020 2020 2020 2078 2020 2d20 7820           x  - x 
│ │ │ │ -00099f10: 7820 2c20 2020 2020 2020 2020 2020 2020  x ,             
│ │ │ │ +00099ee0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00099ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099f00: 2020 2078 2020 2d20 7820 7820 2c20 2020     x  - x x ,   
│ │ │ │ +00099f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099f40: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00099f50: 2020 2020 2020 2020 2020 3420 2020 2033            4    3
│ │ │ │ -00099f60: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ +00099f30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00099f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099f50: 2020 2020 3420 2020 2033 2035 2020 2020      4    3 5    
│ │ │ │ +00099f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099f90: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00099f80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00099f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00099fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00099fe0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00099ff0: 2020 2020 2020 2020 2078 2078 2020 2d20           x x  - 
│ │ │ │ -0009a000: 7820 7820 2c20 2020 2020 2020 2020 2020  x x ,           
│ │ │ │ +00099fd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00099fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00099ff0: 2020 2078 2078 2020 2d20 7820 7820 2c20     x x  - x x , 
│ │ │ │ +0009a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a030: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0009a040: 2020 2020 2020 2020 2020 3220 3420 2020            2 4   
│ │ │ │ -0009a050: 2031 2035 2020 2020 2020 2020 2020 2020   1 5            
│ │ │ │ +0009a020: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0009a030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009a040: 2020 2020 3220 3420 2020 2031 2035 2020      2 4    1 5  
│ │ │ │ +0009a050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009a060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a080: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0009a070: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0009a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a0d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0009a0e0: 2020 2020 2020 2020 2078 2078 2020 2d20           x x  - 
│ │ │ │ -0009a0f0: 7820 7820 2c20 2020 2020 2020 2020 2020  x x ,           
│ │ │ │ +0009a0c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0009a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009a0e0: 2020 2078 2078 2020 2d20 7820 7820 2c20     x x  - x x , 
│ │ │ │ +0009a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a120: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0009a130: 2020 2020 2020 2020 2020 3220 3320 2020            2 3   
│ │ │ │ -0009a140: 2031 2034 2020 2020 2020 2020 2020 2020   1 4            
│ │ │ │ +0009a110: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0009a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009a130: 2020 2020 3220 3320 2020 2031 2034 2020      2 3    1 4  
│ │ │ │ +0009a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a170: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0009a160: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0009a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a1c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0009a1d0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +0009a1b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0009a1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009a1d0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  0009a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009a1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a210: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0009a220: 2020 2020 2020 2020 2078 2020 2d20 7820           x  - x 
│ │ │ │ -0009a230: 7820 2c20 2020 2020 2020 2020 2020 2020  x ,             
│ │ │ │ +0009a200: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0009a210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009a220: 2020 2078 2020 2d20 7820 7820 2c20 2020     x  - x x ,   
│ │ │ │ +0009a230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a260: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0009a270: 2020 2020 2020 2020 2020 3220 2020 2030            2    0
│ │ │ │ -0009a280: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ +0009a250: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0009a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009a270: 2020 2020 3220 2020 2030 2035 2020 2020      2    0 5    
│ │ │ │ +0009a280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009a290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009a2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0009a9c0: 6170 205e 205a 5a2c 204e 6578 743a 2052  ap ^ ZZ, Next: R
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│ │ │ │  0009ac60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0009ada0: 7020 5e2a 2a20 4964 6561 6c20 2d2d 2069  p ^** Ideal -- i
│ │ │ │ +0009adb0: 6e76 6572 7365 2069 6d61 6765 2076 6961  nverse image via
│ │ │ │ +0009adc0: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ +0009add0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0009ade0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0009adf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0009ae00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
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│ │ │ │ -0009ae50: 2020 2020 2070 6869 5e2a 2a20 490a 2020       phi^** I.  
│ │ │ │ -0009ae60: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -0009ae70: 2a20 7068 692c 2061 202a 6e6f 7465 2072  * phi, a *note r
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│ │ │ │ -0009af70: 2020 2020 2020 696e 7665 7273 6520 696d        inverse im
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│ │ │ │ -0009b090: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
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│ │ │ │ +0009ae30: 6329 5e5f 7374 5f73 742c 0a20 202a 2055  c)^_st_st,.  * U
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│ │ │ │ +0009ae50: 6869 5e2a 2a20 490a 2020 2a20 496e 7075  hi^** I.  * Inpu
│ │ │ │ +0009ae60: 7473 3a0a 2020 2020 2020 2a20 7068 692c  ts:.      * phi,
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│ │ │ │ +0009ae80: 6c20 6d61 703a 2052 6174 696f 6e61 6c4d  l map: RationalM
│ │ │ │ +0009ae90: 6170 2c0a 2020 2020 2020 2a20 492c 2061  ap,.      * I, a
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│ │ │ │ +0009aee0: 2020 2020 2020 2020 636f 6f72 6469 6e61          coordina
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│ │ │ │ +0009af10: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
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│ │ │ │ +0009af40: 4964 6561 6c2c 2c20 7468 6520 6964 6561  Ideal,, the idea
│ │ │ │ +0009af50: 6c20 6f66 2074 6865 2063 6c6f 7375 7265  l of the closure
│ │ │ │ +0009af60: 206f 6620 7468 650a 2020 2020 2020 2020   of the.        
│ │ │ │ +0009af70: 696e 7665 7273 6520 696d 6167 6520 6f66  inverse image of
│ │ │ │ +0009af80: 2056 2849 2920 7669 6120 7068 690a 0a44   V(I) via phi..D
│ │ │ │ +0009af90: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +0009afa0: 3d3d 3d3d 3d3d 0a0a 496e 206d 6f73 7420  ======..In most 
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│ │ │ │ +0009afc0: 7569 7661 6c65 6e74 2074 6f20 7068 695e  uivalent to phi^
│ │ │ │ +0009afd0: 2a20 492c 2077 6869 6368 2069 7320 6661  * I, which is fa
│ │ │ │ +0009afe0: 7374 6572 2062 7574 206d 6179 206e 6f74  ster but may not
│ │ │ │ +0009aff0: 2074 616b 650a 696e 746f 2061 6363 6f75   take.into accou
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│ │ │ │ +0009b010: 6e74 6174 696f 6e73 206f 6620 7468 6520  ntations of the 
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│ │ │ │ +0009b030: 6d70 6c65 2062 656c 6f77 2c20 7765 2061  mple below, we a
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│ │ │ │ +0009b050: 746f 2063 6865 636b 2074 6865 2062 6972  to check the bir
│ │ │ │ +0009b060: 6174 696f 6e61 6c69 7479 206f 6620 6120  ationality of a 
│ │ │ │ +0009b070: 6d61 700a 2864 6574 6572 6d69 6e69 7374  map.(determinist
│ │ │ │ +0009b080: 6963 616c 6c79 292e 0a0a 2b2d 2d2d 2d2d  ically)...+-----
│ │ │ │ +0009b090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009b0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009b0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009b0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0009b0e0: 7c69 3120 3a20 7068 6920 3d20 7175 6164  |i1 : phi = quad
│ │ │ │ -0009b0f0: 726f 5175 6164 7269 6343 7265 6d6f 6e61  roQuadricCremona
│ │ │ │ -0009b100: 5472 616e 7366 6f72 6d61 7469 6f6e 2835  Transformation(5
│ │ │ │ -0009b110: 2c31 2920 2020 2020 2020 2020 2020 2020  ,1)             
│ │ │ │ -0009b120: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b130: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009b0d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +0009b0e0: 7068 6920 3d20 7175 6164 726f 5175 6164  phi = quadroQuad
│ │ │ │ +0009b0f0: 7269 6343 7265 6d6f 6e61 5472 616e 7366  ricCremonaTransf
│ │ │ │ +0009b100: 6f72 6d61 7469 6f6e 2835 2c31 2920 2020  ormation(5,1)   
│ │ │ │ +0009b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b120: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b170: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b180: 7c6f 3120 3d20 2d2d 2072 6174 696f 6e61  |o1 = -- rationa
│ │ │ │ -0009b190: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │ +0009b170: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ +0009b180: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ +0009b190: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │  0009b1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b1c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b1d0: 7c20 2020 2020 736f 7572 6365 3a20 5072  |     source: Pr
│ │ │ │ -0009b1e0: 6f6a 2851 515b 782c 2079 2c20 7a2c 2074  oj(QQ[x, y, z, t
│ │ │ │ -0009b1f0: 2c20 752c 2076 5d29 2020 2020 2020 2020  , u, v])        
│ │ │ │ +0009b1c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b1d0: 736f 7572 6365 3a20 5072 6f6a 2851 515b  source: Proj(QQ[
│ │ │ │ +0009b1e0: 782c 2079 2c20 7a2c 2074 2c20 752c 2076  x, y, z, t, u, v
│ │ │ │ +0009b1f0: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │  0009b200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b210: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b220: 7c20 2020 2020 7461 7267 6574 3a20 5072  |     target: Pr
│ │ │ │ -0009b230: 6f6a 2851 515b 782c 2079 2c20 7a2c 2074  oj(QQ[x, y, z, t
│ │ │ │ -0009b240: 2c20 752c 2076 5d29 2020 2020 2020 2020  , u, v])        
│ │ │ │ +0009b210: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b220: 7461 7267 6574 3a20 5072 6f6a 2851 515b  target: Proj(QQ[
│ │ │ │ +0009b230: 782c 2079 2c20 7a2c 2074 2c20 752c 2076  x, y, z, t, u, v
│ │ │ │ +0009b240: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │  0009b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b260: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b270: 7c20 2020 2020 6465 6669 6e69 6e67 2066  |     defining f
│ │ │ │ -0009b280: 6f72 6d73 3a20 7b20 2020 2020 2020 2020  orms: {         
│ │ │ │ +0009b260: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b270: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20  defining forms: 
│ │ │ │ +0009b280: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │  0009b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b2b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b2c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009b2d0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0009b2b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b2d0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0009b2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b300: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b310: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009b320: 2020 2020 2020 2079 2a7a 202d 2076 202c         y*z - v ,
│ │ │ │ +0009b300: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b320: 2079 2a7a 202d 2076 202c 2020 2020 2020   y*z - v ,      
│ │ │ │  0009b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b350: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b360: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009b350: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b3a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b3b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009b3c0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0009b3a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b3c0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0009b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b3f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b400: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009b410: 2020 2020 2020 2078 2a7a 202d 2075 202c         x*z - u ,
│ │ │ │ +0009b3f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b410: 2078 2a7a 202d 2075 202c 2020 2020 2020   x*z - u ,      
│ │ │ │  0009b420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b440: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b450: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009b440: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b490: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b4a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009b4b0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0009b490: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b4b0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0009b4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b4e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b4f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009b500: 2020 2020 2020 2078 2a79 202d 2074 202c         x*y - t ,
│ │ │ │ +0009b4e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b500: 2078 2a79 202d 2074 202c 2020 2020 2020   x*y - t ,      
│ │ │ │  0009b510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b530: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b540: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009b530: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b580: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b590: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009b5a0: 2020 2020 2020 202d 207a 2a74 202b 2075         - z*t + u
│ │ │ │ -0009b5b0: 2a76 2c20 2020 2020 2020 2020 2020 2020  *v,             
│ │ │ │ +0009b580: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b5a0: 202d 207a 2a74 202b 2075 2a76 2c20 2020   - z*t + u*v,   
│ │ │ │ +0009b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b5d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b5e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009b5d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b620: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b630: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009b640: 2020 2020 2020 202d 2079 2a75 202b 2074         - y*u + t
│ │ │ │ -0009b650: 2a76 2c20 2020 2020 2020 2020 2020 2020  *v,             
│ │ │ │ +0009b620: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b640: 202d 2079 2a75 202b 2074 2a76 2c20 2020   - y*u + t*v,   
│ │ │ │ +0009b650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b670: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b680: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009b670: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b6c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b6d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009b6e0: 2020 2020 2020 2074 2a75 202d 2078 2a76         t*u - x*v
│ │ │ │ +0009b6c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b6e0: 2074 2a75 202d 2078 2a76 2020 2020 2020   t*u - x*v      
│ │ │ │  0009b6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b710: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b720: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009b730: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │ +0009b710: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b730: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │  0009b740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b760: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b770: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009b760: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b7b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b7c0: 7c6f 3120 3a20 5261 7469 6f6e 616c 4d61  |o1 : RationalMa
│ │ │ │ -0009b7d0: 7020 2843 7265 6d6f 6e61 2074 7261 6e73  p (Cremona trans
│ │ │ │ -0009b7e0: 666f 726d 6174 696f 6e20 6f66 2050 505e  formation of PP^
│ │ │ │ -0009b7f0: 3520 6f66 2074 7970 6520 2832 2c32 2929  5 of type (2,2))
│ │ │ │ -0009b800: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b810: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0009b7b0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ +0009b7c0: 5261 7469 6f6e 616c 4d61 7020 2843 7265  RationalMap (Cre
│ │ │ │ +0009b7d0: 6d6f 6e61 2074 7261 6e73 666f 726d 6174  mona transformat
│ │ │ │ +0009b7e0: 696f 6e20 6f66 2050 505e 3520 6f66 2074  ion of PP^5 of t
│ │ │ │ +0009b7f0: 7970 6520 2832 2c32 2929 2020 2020 2020  ype (2,2))      
│ │ │ │ +0009b800: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0009b810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009b820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009b830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009b840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009b850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0009b860: 7c69 3220 3a20 4b20 3a3d 2066 7261 6328  |i2 : K := frac(
│ │ │ │ -0009b870: 5151 5b76 6172 7328 302e 2e35 295d 293b  QQ[vars(0..5)]);
│ │ │ │ -0009b880: 2070 6869 203d 2070 6869 202a 2a20 4b20   phi = phi ** K 
│ │ │ │ +0009b850: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ +0009b860: 4b20 3a3d 2066 7261 6328 5151 5b76 6172  K := frac(QQ[var
│ │ │ │ +0009b870: 7328 302e 2e35 295d 293b 2070 6869 203d  s(0..5)]); phi =
│ │ │ │ +0009b880: 2070 6869 202a 2a20 4b20 2020 2020 2020   phi ** K       
│ │ │ │  0009b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b8a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b8b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009b8a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b8f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b900: 7c6f 3320 3d20 2d2d 2072 6174 696f 6e61  |o3 = -- rationa
│ │ │ │ -0009b910: 6c20 6d61 7020 2d2d 2020 2020 2020 2020  l map --        
│ │ │ │ +0009b8f0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ +0009b900: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ +0009b910: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │  0009b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009b930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009b940: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b950: 7c20 2020 2020 736f 7572 6365 3a20 5072  |     source: Pr
│ │ │ │ -0009b960: 6f6a 2866 7261 6328 5151 5b61 2e2e 665d  oj(frac(QQ[a..f]
│ │ │ │ -0009b970: 295b 782c 2079 2c20 7a2c 2074 2c20 752c  )[x, y, z, t, u,
│ │ │ │ -0009b980: 2076 5d29 2020 2020 2020 2020 2020 2020   v])            
│ │ │ │ -0009b990: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b9a0: 7c20 2020 2020 7461 7267 6574 3a20 5072  |     target: Pr
│ │ │ │ -0009b9b0: 6f6a 2866 7261 6328 5151 5b61 2e2e 665d  oj(frac(QQ[a..f]
│ │ │ │ -0009b9c0: 295b 782c 2079 2c20 7a2c 2074 2c20 752c  )[x, y, z, t, u,
│ │ │ │ -0009b9d0: 2076 5d29 2020 2020 2020 2020 2020 2020   v])            
│ │ │ │ -0009b9e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009b9f0: 7c20 2020 2020 6465 6669 6e69 6e67 2066  |     defining f
│ │ │ │ -0009ba00: 6f72 6d73 3a20 7b20 2020 2020 2020 2020  orms: {         
│ │ │ │ +0009b940: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b950: 736f 7572 6365 3a20 5072 6f6a 2866 7261  source: Proj(fra
│ │ │ │ +0009b960: 6328 5151 5b61 2e2e 665d 295b 782c 2079  c(QQ[a..f])[x, y
│ │ │ │ +0009b970: 2c20 7a2c 2074 2c20 752c 2076 5d29 2020  , z, t, u, v])  
│ │ │ │ +0009b980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b990: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b9a0: 7461 7267 6574 3a20 5072 6f6a 2866 7261  target: Proj(fra
│ │ │ │ +0009b9b0: 6328 5151 5b61 2e2e 665d 295b 782c 2079  c(QQ[a..f])[x, y
│ │ │ │ +0009b9c0: 2c20 7a2c 2074 2c20 752c 2076 5d29 2020  , z, t, u, v])  
│ │ │ │ +0009b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009b9e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009b9f0: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20  defining forms: 
│ │ │ │ +0009ba00: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │  0009ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ba30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009ba40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009ba50: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0009ba30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009ba50: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0009ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ba80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009ba90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009baa0: 2020 2020 2020 2079 2a7a 202d 2076 202c         y*z - v ,
│ │ │ │ +0009ba80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009baa0: 2079 2a7a 202d 2076 202c 2020 2020 2020   y*z - v ,      
│ │ │ │  0009bab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bad0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009bae0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009bad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009bae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009baf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bb20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009bb30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009bb40: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0009bb20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009bb40: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0009bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bb70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009bb80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009bb90: 2020 2020 2020 2078 2a7a 202d 2075 202c         x*z - u ,
│ │ │ │ +0009bb70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009bb90: 2078 2a7a 202d 2075 202c 2020 2020 2020   x*z - u ,      
│ │ │ │  0009bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bbc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009bbd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009bbc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bc10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009bc20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009bc30: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0009bc10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009bc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009bc30: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0009bc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bc60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009bc70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009bc80: 2020 2020 2020 2078 2a79 202d 2074 202c         x*y - t ,
│ │ │ │ +0009bc60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009bc80: 2078 2a79 202d 2074 202c 2020 2020 2020   x*y - t ,      
│ │ │ │  0009bc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bcb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009bcc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009bcb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bd00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009bd10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009bd20: 2020 2020 2020 202d 207a 2a74 202b 2075         - z*t + u
│ │ │ │ -0009bd30: 2a76 2c20 2020 2020 2020 2020 2020 2020  *v,             
│ │ │ │ +0009bd00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009bd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009bd20: 202d 207a 2a74 202b 2075 2a76 2c20 2020   - z*t + u*v,   
│ │ │ │ +0009bd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bd50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009bd60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009bd50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bda0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009bdb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009bdc0: 2020 2020 2020 202d 2079 2a75 202b 2074         - y*u + t
│ │ │ │ -0009bdd0: 2a76 2c20 2020 2020 2020 2020 2020 2020  *v,             
│ │ │ │ +0009bda0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009bdc0: 202d 2079 2a75 202b 2074 2a76 2c20 2020   - y*u + t*v,   
│ │ │ │ +0009bdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bdf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009be00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009bdf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009be00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009be10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009be20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009be40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009be50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009be60: 2020 2020 2020 2074 2a75 202d 2078 2a76         t*u - x*v
│ │ │ │ +0009be40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009be60: 2074 2a75 202d 2078 2a76 2020 2020 2020   t*u - x*v      
│ │ │ │  0009be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009be80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009be90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009bea0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009beb0: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │ +0009be90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009beb0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │  0009bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bee0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009bef0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009bee0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009bef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009bf30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009bf40: 7c6f 3320 3a20 5261 7469 6f6e 616c 4d61  |o3 : RationalMa
│ │ │ │ -0009bf50: 7020 2871 7561 6472 6174 6963 2072 6174  p (quadratic rat
│ │ │ │ -0009bf60: 696f 6e61 6c20 6d61 7020 6672 6f6d 2050  ional map from P
│ │ │ │ -0009bf70: 505e 3520 746f 2050 505e 3529 2020 2020  P^5 to PP^5)    
│ │ │ │ -0009bf80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009bf90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0009bf30: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ +0009bf40: 5261 7469 6f6e 616c 4d61 7020 2871 7561  RationalMap (qua
│ │ │ │ +0009bf50: 6472 6174 6963 2072 6174 696f 6e61 6c20  dratic rational 
│ │ │ │ +0009bf60: 6d61 7020 6672 6f6d 2050 505e 3520 746f  map from PP^5 to
│ │ │ │ +0009bf70: 2050 505e 3529 2020 2020 2020 2020 2020   PP^5)          
│ │ │ │ +0009bf80: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0009bf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009bfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009bfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009bfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009bfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0009bfe0: 7c69 3420 3a20 7020 3d20 7472 696d 206d  |i4 : p = trim m
│ │ │ │ -0009bff0: 696e 6f72 7328 322c 2876 6172 7320 4b29  inors(2,(vars K)
│ │ │ │ -0009c000: 7c7c 2876 6172 7320 736f 7572 6365 2070  ||(vars source p
│ │ │ │ -0009c010: 6869 2929 2020 2020 2020 2020 2020 2020  hi))            
│ │ │ │ -0009c020: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c030: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009bfd0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ +0009bfe0: 7020 3d20 7472 696d 206d 696e 6f72 7328  p = trim minors(
│ │ │ │ +0009bff0: 322c 2876 6172 7320 4b29 7c7c 2876 6172  2,(vars K)||(var
│ │ │ │ +0009c000: 7320 736f 7572 6365 2070 6869 2929 2020  s source phi))  
│ │ │ │ +0009c010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009c020: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c070: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c080: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009c090: 2065 2020 2020 2020 2020 6420 2020 2020   e        d     
│ │ │ │ -0009c0a0: 2020 2063 2020 2020 2020 2020 6220 2020     c        b   
│ │ │ │ -0009c0b0: 2020 2020 2061 2020 2020 2020 2020 2020       a          
│ │ │ │ -0009c0c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c0d0: 7c6f 3420 3d20 6964 6561 6c20 2875 202d  |o4 = ideal (u -
│ │ │ │ -0009c0e0: 202d 2a76 2c20 7420 2d20 2d2a 762c 207a   -*v, t - -*v, z
│ │ │ │ -0009c0f0: 202d 202d 2a76 2c20 7920 2d20 2d2a 762c   - -*v, y - -*v,
│ │ │ │ -0009c100: 2078 202d 202d 2a76 2920 2020 2020 2020   x - -*v)       
│ │ │ │ -0009c110: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c120: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009c130: 2066 2020 2020 2020 2020 6620 2020 2020   f        f     
│ │ │ │ -0009c140: 2020 2066 2020 2020 2020 2020 6620 2020     f        f   
│ │ │ │ -0009c150: 2020 2020 2066 2020 2020 2020 2020 2020       f          
│ │ │ │ -0009c160: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c170: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009c070: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c080: 2020 2020 2020 2020 2020 2065 2020 2020             e    
│ │ │ │ +0009c090: 2020 2020 6420 2020 2020 2020 2063 2020      d        c  
│ │ │ │ +0009c0a0: 2020 2020 2020 6220 2020 2020 2020 2061        b        a
│ │ │ │ +0009c0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009c0c0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ +0009c0d0: 6964 6561 6c20 2875 202d 202d 2a76 2c20  ideal (u - -*v, 
│ │ │ │ +0009c0e0: 7420 2d20 2d2a 762c 207a 202d 202d 2a76  t - -*v, z - -*v
│ │ │ │ +0009c0f0: 2c20 7920 2d20 2d2a 762c 2078 202d 202d  , y - -*v, x - -
│ │ │ │ +0009c100: 2a76 2920 2020 2020 2020 2020 2020 2020  *v)             
│ │ │ │ +0009c110: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c120: 2020 2020 2020 2020 2020 2066 2020 2020             f    
│ │ │ │ +0009c130: 2020 2020 6620 2020 2020 2020 2066 2020      f        f  
│ │ │ │ +0009c140: 2020 2020 2020 6620 2020 2020 2020 2066        f        f
│ │ │ │ +0009c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009c160: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c1b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c1c0: 7c6f 3420 3a20 4964 6561 6c20 6f66 2066  |o4 : Ideal of f
│ │ │ │ -0009c1d0: 7261 6328 5151 5b61 2e2e 665d 295b 782c  rac(QQ[a..f])[x,
│ │ │ │ -0009c1e0: 2079 2c20 7a2c 2074 2c20 752c 2076 5d20   y, z, t, u, v] 
│ │ │ │ +0009c1b0: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ +0009c1c0: 4964 6561 6c20 6f66 2066 7261 6328 5151  Ideal of frac(QQ
│ │ │ │ +0009c1d0: 5b61 2e2e 665d 295b 782c 2079 2c20 7a2c  [a..f])[x, y, z,
│ │ │ │ +0009c1e0: 2074 2c20 752c 2076 5d20 2020 2020 2020   t, u, v]       
│ │ │ │  0009c1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c200: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c210: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0009c200: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0009c210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009c220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009c230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009c240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009c250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0009c260: 7c69 3520 3a20 7120 3d20 7068 6920 7020  |i5 : q = phi p 
│ │ │ │ +0009c250: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ +0009c260: 7120 3d20 7068 6920 7020 2020 2020 2020  q = phi p       
│ │ │ │  0009c270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c2a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c2b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009c2a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c2f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c300: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009c2f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c330: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -0009c340: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c350: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009c360: 2062 2a65 202d 2064 2a66 2020 2020 2020   b*e - d*f      
│ │ │ │ -0009c370: 2020 632a 6420 2d20 652a 6620 2020 2020    c*d - e*f     
│ │ │ │ -0009c380: 2020 202d 2061 2a62 202b 2064 2020 2020     - a*b + d    
│ │ │ │ -0009c390: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c3a0: 7c6f 3520 3d20 6964 6561 6c20 2875 202b  |o5 = ideal (u +
│ │ │ │ -0009c3b0: 202d 2d2d 2d2d 2d2d 2d2d 2a76 2c20 7420   ---------*v, t 
│ │ │ │ -0009c3c0: 2b20 2d2d 2d2d 2d2d 2d2d 2d2a 762c 207a  + ---------*v, z
│ │ │ │ -0009c3d0: 202b 202d 2d2d 2d2d 2d2d 2d2d 2d2a 762c   + ----------*v,
│ │ │ │ -0009c3e0: 2079 202b 2020 2020 2020 2020 2020 7c0a   y +          |.
│ │ │ │ -0009c3f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009c400: 2064 2a65 202d 2061 2a66 2020 2020 2020   d*e - a*f      
│ │ │ │ -0009c410: 2020 642a 6520 2d20 612a 6620 2020 2020    d*e - a*f     
│ │ │ │ -0009c420: 2020 2020 642a 6520 2d20 612a 6620 2020      d*e - a*f   
│ │ │ │ -0009c430: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c440: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ +0009c330: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0009c340: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c350: 2020 2020 2020 2020 2020 2062 2a65 202d             b*e -
│ │ │ │ +0009c360: 2064 2a66 2020 2020 2020 2020 632a 6420   d*f        c*d 
│ │ │ │ +0009c370: 2d20 652a 6620 2020 2020 2020 202d 2061  - e*f        - a
│ │ │ │ +0009c380: 2a62 202b 2064 2020 2020 2020 2020 2020  *b + d          
│ │ │ │ +0009c390: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ +0009c3a0: 6964 6561 6c20 2875 202b 202d 2d2d 2d2d  ideal (u + -----
│ │ │ │ +0009c3b0: 2d2d 2d2d 2a76 2c20 7420 2b20 2d2d 2d2d  ----*v, t + ----
│ │ │ │ +0009c3c0: 2d2d 2d2d 2d2a 762c 207a 202b 202d 2d2d  -----*v, z + ---
│ │ │ │ +0009c3d0: 2d2d 2d2d 2d2d 2d2a 762c 2079 202b 2020  -------*v, y +  
│ │ │ │ +0009c3e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c3f0: 2020 2020 2020 2020 2020 2064 2a65 202d             d*e -
│ │ │ │ +0009c400: 2061 2a66 2020 2020 2020 2020 642a 6520   a*f        d*e 
│ │ │ │ +0009c410: 2d20 612a 6620 2020 2020 2020 2020 642a  - a*f         d*
│ │ │ │ +0009c420: 6520 2d20 612a 6620 2020 2020 2020 2020  e - a*f         
│ │ │ │ +0009c430: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009c450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009c460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009c470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009c480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -0009c490: 7c20 2020 2020 2020 2020 2020 2020 2032  |              2
│ │ │ │ -0009c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c4b0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0009c480: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ +0009c490: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0009c4a0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0009c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c4d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c4e0: 7c20 2020 2020 2d20 612a 6320 2b20 6520  |     - a*c + e 
│ │ │ │ -0009c4f0: 2020 2020 2020 2020 2d20 622a 6320 2b20          - b*c + 
│ │ │ │ -0009c500: 6620 2020 2020 2020 2020 2020 2020 2020  f               
│ │ │ │ +0009c4d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c4e0: 2d20 612a 6320 2b20 6520 2020 2020 2020  - a*c + e       
│ │ │ │ +0009c4f0: 2020 2d20 622a 6320 2b20 6620 2020 2020    - b*c + f     
│ │ │ │ +0009c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c520: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c530: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d  |     ----------
│ │ │ │ -0009c540: 2a76 2c20 7820 2b20 2d2d 2d2d 2d2d 2d2d  *v, x + --------
│ │ │ │ -0009c550: 2d2d 2a76 2920 2020 2020 2020 2020 2020  --*v)           
│ │ │ │ +0009c520: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c530: 2d2d 2d2d 2d2d 2d2d 2d2d 2a76 2c20 7820  ----------*v, x 
│ │ │ │ +0009c540: 2b20 2d2d 2d2d 2d2d 2d2d 2d2d 2a76 2920  + ----------*v) 
│ │ │ │ +0009c550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c570: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c580: 7c20 2020 2020 2064 2a65 202d 2061 2a66  |      d*e - a*f
│ │ │ │ -0009c590: 2020 2020 2020 2020 2064 2a65 202d 2061           d*e - a
│ │ │ │ -0009c5a0: 2a66 2020 2020 2020 2020 2020 2020 2020  *f              
│ │ │ │ +0009c570: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c580: 2064 2a65 202d 2061 2a66 2020 2020 2020   d*e - a*f      
│ │ │ │ +0009c590: 2020 2064 2a65 202d 2061 2a66 2020 2020     d*e - a*f    
│ │ │ │ +0009c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c5c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c5d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009c5c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c610: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c620: 7c6f 3520 3a20 4964 6561 6c20 6f66 2066  |o5 : Ideal of f
│ │ │ │ -0009c630: 7261 6328 5151 5b61 2e2e 665d 295b 782c  rac(QQ[a..f])[x,
│ │ │ │ -0009c640: 2079 2c20 7a2c 2074 2c20 752c 2076 5d20   y, z, t, u, v] 
│ │ │ │ +0009c610: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20          |.|o5 : 
│ │ │ │ +0009c620: 4964 6561 6c20 6f66 2066 7261 6328 5151  Ideal of frac(QQ
│ │ │ │ +0009c630: 5b61 2e2e 665d 295b 782c 2079 2c20 7a2c  [a..f])[x, y, z,
│ │ │ │ +0009c640: 2074 2c20 752c 2076 5d20 2020 2020 2020   t, u, v]       
│ │ │ │  0009c650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c660: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c670: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0009c660: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0009c670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009c680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009c690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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             
│ │ │ │ +0009c6b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ +0009c6c0: 7469 6d65 2070 6869 5e2a 2a20 7120 2020  time phi^** q   
│ │ │ │ +0009c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 3135 3735  | -- used 0.1575
│ │ │ │ -0009c720: 3331 7320 2863 7075 293b 2030 2e31 3537  31s (cpu); 0.157
│ │ │ │ -0009c730: 3532 3773 2028 7468 7265 6164 293b 2030  527s (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  |               
│ │ │ │ +0009c700: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ +0009c710: 7365 6420 302e 3138 3433 3637 7320 2863  sed 0.184367s (c
│ │ │ │ +0009c720: 7075 293b 2030 2e31 3834 3336 3673 2028  pu); 0.184366s (
│ │ │ │ +0009c730: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ +0009c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009c750: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c760: 2020 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                |.
│ │ │ │ -0009c7b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009c7c0: 2065 2020 2020 2020 2020 6420 2020 2020   e        d     
│ │ │ │ -0009c7d0: 2020 2063 2020 2020 2020 2020 6220 2020     c        b   
│ │ │ │ -0009c7e0: 2020 2020 2061 2020 2020 2020 2020 2020       a          
│ │ │ │ -0009c7f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c800: 7c6f 3620 3d20 6964 6561 6c20 2875 202d  |o6 = ideal (u -
│ │ │ │ -0009c810: 202d 2a76 2c20 7420 2d20 2d2a 762c 207a   -*v, t - -*v, z
│ │ │ │ -0009c820: 202d 202d 2a76 2c20 7920 2d20 2d2a 762c   - -*v, y - -*v,
│ │ │ │ -0009c830: 2078 202d 202d 2a76 2920 2020 2020 2020   x - -*v)       
│ │ │ │ -0009c840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c850: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0009c860: 2066 2020 2020 2020 2020 6620 2020 2020   f        f     
│ │ │ │ -0009c870: 2020 2066 2020 2020 2020 2020 6620 2020     f        f   
│ │ │ │ -0009c880: 2020 2020 2066 2020 2020 2020 2020 2020       f          
│ │ │ │ -0009c890: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c8a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009c7a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c7b0: 2020 2020 2020 2020 2020 2065 2020 2020             e    
│ │ │ │ +0009c7c0: 2020 2020 6420 2020 2020 2020 2063 2020      d        c  
│ │ │ │ +0009c7d0: 2020 2020 2020 6220 2020 2020 2020 2061        b        a
│ │ │ │ +0009c7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009c7f0: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ +0009c800: 6964 6561 6c20 2875 202d 202d 2a76 2c20  ideal (u - -*v, 
│ │ │ │ +0009c810: 7420 2d20 2d2a 762c 207a 202d 202d 2a76  t - -*v, z - -*v
│ │ │ │ +0009c820: 2c20 7920 2d20 2d2a 762c 2078 202d 202d  , y - -*v, x - -
│ │ │ │ +0009c830: 2a76 2920 2020 2020 2020 2020 2020 2020  *v)             
│ │ │ │ +0009c840: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c850: 2020 2020 2020 2020 2020 2066 2020 2020             f    
│ │ │ │ +0009c860: 2020 2020 6620 2020 2020 2020 2066 2020      f        f  
│ │ │ │ +0009c870: 2020 2020 2020 6620 2020 2020 2020 2066        f        f
│ │ │ │ +0009c880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009c890: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c8e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c8f0: 7c6f 3620 3a20 4964 6561 6c20 6f66 2066  |o6 : Ideal of f
│ │ │ │ -0009c900: 7261 6328 5151 5b61 2e2e 665d 295b 782c  rac(QQ[a..f])[x,
│ │ │ │ -0009c910: 2079 2c20 7a2c 2074 2c20 752c 2076 5d20   y, z, t, u, v] 
│ │ │ │ +0009c8e0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ +0009c8f0: 4964 6561 6c20 6f66 2066 7261 6328 5151  Ideal of frac(QQ
│ │ │ │ +0009c900: 5b61 2e2e 665d 295b 782c 2079 2c20 7a2c  [a..f])[x, y, z,
│ │ │ │ +0009c910: 2074 2c20 752c 2076 5d20 2020 2020 2020   t, u, v]       
│ │ │ │  0009c920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c930: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c940: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0009c930: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0009c940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009c950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009c960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009c970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009c980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0009c990: 7c69 3720 3a20 6f6f 203d 3d20 7020 2020  |i7 : oo == p   
│ │ │ │ +0009c980: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20  --------+.|i7 : 
│ │ │ │ +0009c990: 6f6f 203d 3d20 7020 2020 2020 2020 2020  oo == p         
│ │ │ │  0009c9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009c9d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009c9e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0009c9d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0009c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ca00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ca10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ca20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009ca30: 7c6f 3720 3d20 7472 7565 2020 2020 2020  |o7 = true      
│ │ │ │ +0009ca20: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20          |.|o7 = 
│ │ │ │ +0009ca30: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │  0009ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ca60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ca70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0009ca80: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0009ca70: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0009ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009caa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009cab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009cac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0009cad0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -0009cae0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 5261  ==..  * *note Ra
│ │ │ │ -0009caf0: 7469 6f6e 616c 4d61 7020 5f2a 3a20 5261  tionalMap _*: Ra
│ │ │ │ -0009cb00: 7469 6f6e 616c 4d61 7020 5f75 735f 7374  tionalMap _us_st
│ │ │ │ -0009cb10: 2c20 2d2d 2064 6972 6563 7420 696d 6167  , -- direct imag
│ │ │ │ -0009cb20: 6520 7669 6120 6120 7261 7469 6f6e 616c  e via a rational
│ │ │ │ -0009cb30: 0a20 2020 206d 6170 0a20 202a 202a 6e6f  .    map.  * *no
│ │ │ │ -0009cb40: 7465 2052 6174 696f 6e61 6c4d 6170 202a  te RationalMap *
│ │ │ │ -0009cb50: 2a20 5269 6e67 3a20 5261 7469 6f6e 616c  * Ring: Rational
│ │ │ │ -0009cb60: 4d61 7020 5f73 745f 7374 2052 696e 672c  Map _st_st Ring,
│ │ │ │ -0009cb70: 202d 2d20 6368 616e 6765 2074 6865 0a20   -- change the. 
│ │ │ │ -0009cb80: 2020 2063 6f65 6666 6963 6965 6e74 2072     coefficient r
│ │ │ │ -0009cb90: 696e 6720 6f66 2061 2072 6174 696f 6e61  ing of a rationa
│ │ │ │ -0009cba0: 6c20 6d61 700a 2020 2a20 2a6e 6f74 6520  l map.  * *note 
│ │ │ │ -0009cbb0: 7461 7267 6574 2852 6174 696f 6e61 6c4d  target(RationalM
│ │ │ │ -0009cbc0: 6170 293a 2074 6172 6765 745f 6c70 5261  ap): target_lpRa
│ │ │ │ -0009cbd0: 7469 6f6e 616c 4d61 705f 7270 2c20 2d2d  tionalMap_rp, --
│ │ │ │ -0009cbe0: 2063 6f6f 7264 696e 6174 6520 7269 6e67   coordinate ring
│ │ │ │ -0009cbf0: 206f 660a 2020 2020 7468 6520 7461 7267   of.    the targ
│ │ │ │ -0009cc00: 6574 2066 6f72 2061 2072 6174 696f 6e61  et for a rationa
│ │ │ │ -0009cc10: 6c20 6d61 700a 0a57 6179 7320 746f 2075  l map..Ways to u
│ │ │ │ -0009cc20: 7365 2074 6869 7320 6d65 7468 6f64 3a0a  se this method:.
│ │ │ │ +0009cac0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ +0009cad0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ +0009cae0: 2a20 2a6e 6f74 6520 5261 7469 6f6e 616c  * *note Rational
│ │ │ │ +0009caf0: 4d61 7020 5f2a 3a20 5261 7469 6f6e 616c  Map _*: Rational
│ │ │ │ +0009cb00: 4d61 7020 5f75 735f 7374 2c20 2d2d 2064  Map _us_st, -- d
│ │ │ │ +0009cb10: 6972 6563 7420 696d 6167 6520 7669 6120  irect image via 
│ │ │ │ +0009cb20: 6120 7261 7469 6f6e 616c 0a20 2020 206d  a rational.    m
│ │ │ │ +0009cb30: 6170 0a20 202a 202a 6e6f 7465 2052 6174  ap.  * *note Rat
│ │ │ │ +0009cb40: 696f 6e61 6c4d 6170 202a 2a20 5269 6e67  ionalMap ** Ring
│ │ │ │ +0009cb50: 3a20 5261 7469 6f6e 616c 4d61 7020 5f73  : RationalMap _s
│ │ │ │ +0009cb60: 745f 7374 2052 696e 672c 202d 2d20 6368  t_st Ring, -- ch
│ │ │ │ +0009cb70: 616e 6765 2074 6865 0a20 2020 2063 6f65  ange the.    coe
│ │ │ │ +0009cb80: 6666 6963 6965 6e74 2072 696e 6720 6f66  fficient ring of
│ │ │ │ +0009cb90: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ +0009cba0: 2020 2a20 2a6e 6f74 6520 7461 7267 6574    * *note target
│ │ │ │ +0009cbb0: 2852 6174 696f 6e61 6c4d 6170 293a 2074  (RationalMap): t
│ │ │ │ +0009cbc0: 6172 6765 745f 6c70 5261 7469 6f6e 616c  arget_lpRational
│ │ │ │ +0009cbd0: 4d61 705f 7270 2c20 2d2d 2063 6f6f 7264  Map_rp, -- coord
│ │ │ │ +0009cbe0: 696e 6174 6520 7269 6e67 206f 660a 2020  inate ring of.  
│ │ │ │ +0009cbf0: 2020 7468 6520 7461 7267 6574 2066 6f72    the target for
│ │ │ │ +0009cc00: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ +0009cc10: 0a57 6179 7320 746f 2075 7365 2074 6869  .Ways to use thi
│ │ │ │ +0009cc20: 7320 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d  s method:.======
│ │ │ │  0009cc30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0009cc40: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ -0009cc50: 6f74 6520 5261 7469 6f6e 616c 4d61 7020  ote RationalMap 
│ │ │ │ -0009cc60: 5e2a 2a20 4964 6561 6c3a 2052 6174 696f  ^** Ideal: Ratio
│ │ │ │ -0009cc70: 6e61 6c4d 6170 205e 5f73 745f 7374 2049  nalMap ^_st_st I
│ │ │ │ -0009cc80: 6465 616c 2c20 2d2d 2069 6e76 6572 7365  deal, -- inverse
│ │ │ │ -0009cc90: 2069 6d61 6765 0a20 2020 2076 6961 2061   image.    via a
│ │ │ │ -0009cca0: 2072 6174 696f 6e61 6c20 6d61 700a 2d2d   rational map.--
│ │ │ │ +0009cc40: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 5261  ==..  * *note Ra
│ │ │ │ +0009cc50: 7469 6f6e 616c 4d61 7020 5e2a 2a20 4964  tionalMap ^** Id
│ │ │ │ +0009cc60: 6561 6c3a 2052 6174 696f 6e61 6c4d 6170  eal: RationalMap
│ │ │ │ +0009cc70: 205e 5f73 745f 7374 2049 6465 616c 2c20   ^_st_st Ideal, 
│ │ │ │ +0009cc80: 2d2d 2069 6e76 6572 7365 2069 6d61 6765  -- inverse image
│ │ │ │ +0009cc90: 0a20 2020 2076 6961 2061 2072 6174 696f  .    via a ratio
│ │ │ │ +0009cca0: 6e61 6c20 6d61 700a 2d2d 2d2d 2d2d 2d2d  nal map.--------
│ │ │ │  0009ccb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009ccc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009ccd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009cce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009ccf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ -0009cd00: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
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│ │ │ │ -0009cd40: 6179 322d 312e 3235 2e31 312b 6473 2f4d  ay2-1.25.11+ds/M
│ │ │ │ -0009cd50: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
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│ │ │ │ -0009cdb0: 735f 7374 2c20 4e65 7874 3a20 5261 7469  s_st, Next: Rati
│ │ │ │ -0009cdc0: 6f6e 616c 4d61 7020 7c20 4964 6561 6c2c  onalMap | Ideal,
│ │ │ │ -0009cdd0: 2050 7265 763a 2052 6174 696f 6e61 6c4d   Prev: RationalM
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│ │ │ │ -0009cdf0: 2c20 5570 3a20 546f 700a 0a52 6174 696f  , Up: Top..Ratio
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│ │ │ │ -0009ce20: 7261 7469 6f6e 616c 206d 6170 0a2a 2a2a  rational map.***
│ │ │ │ +0009ccf0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
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│ │ │ │ +0009cdb0: 4e65 7874 3a20 5261 7469 6f6e 616c 4d61  Next: RationalMa
│ │ │ │ +0009cdc0: 7020 7c20 4964 6561 6c2c 2050 7265 763a  p | Ideal, Prev:
│ │ │ │ +0009cdd0: 2052 6174 696f 6e61 6c4d 6170 205e 5f73   RationalMap ^_s
│ │ │ │ +0009cde0: 745f 7374 2049 6465 616c 2c20 5570 3a20  t_st Ideal, Up: 
│ │ │ │ +0009cdf0: 546f 700a 0a52 6174 696f 6e61 6c4d 6170  Top..RationalMap
│ │ │ │ +0009ce00: 205f 2a20 2d2d 2064 6972 6563 7420 696d   _* -- direct im
│ │ │ │ +0009ce10: 6167 6520 7669 6120 6120 7261 7469 6f6e  age via a ration
│ │ │ │ +0009ce20: 616c 206d 6170 0a2a 2a2a 2a2a 2a2a 2a2a  al map.*********
│ │ │ │  0009ce30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0009ce40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0009ce50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ -0009ce60: 2020 2a20 4f70 6572 6174 6f72 3a20 2a6e    * Operator: *n
│ │ │ │ -0009ce70: 6f74 6520 5f2a 3a20 284d 6163 6175 6c61  ote _*: (Macaula
│ │ │ │ -0009ce80: 7932 446f 6329 5f75 735f 7374 2c0a 2020  y2Doc)_us_st,.  
│ │ │ │ -0009ce90: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ -0009cea0: 2020 7068 695f 2a20 490a 2020 2a20 496e    phi_* I.  * In
│ │ │ │ -0009ceb0: 7075 7473 3a0a 2020 2020 2020 2a20 7068  puts:.      * ph
│ │ │ │ -0009cec0: 692c 2061 202a 6e6f 7465 2072 6174 696f  i, a *note ratio
│ │ │ │ -0009ced0: 6e61 6c20 6d61 703a 2052 6174 696f 6e61  nal map: Rationa
│ │ │ │ -0009cee0: 6c4d 6170 2c2c 2049 2061 2068 6f6d 6f67  lMap,, I a homog
│ │ │ │ -0009cef0: 656e 656f 7573 2069 6465 616c 2069 6e20  eneous ideal in 
│ │ │ │ -0009cf00: 7468 650a 2020 2020 2020 2020 636f 6f72  the.        coor
│ │ │ │ -0009cf10: 6469 6e61 7465 2072 696e 6720 6f66 2074  dinate ring of t
│ │ │ │ -0009cf20: 6865 2073 6f75 7263 6520 6f66 2070 6869  he source of phi
│ │ │ │ -0009cf30: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -0009cf40: 2020 2020 2a20 616e 202a 6e6f 7465 2069      * an *note i
│ │ │ │ -0009cf50: 6465 616c 3a20 284d 6163 6175 6c61 7932  deal: (Macaulay2
│ │ │ │ -0009cf60: 446f 6329 4964 6561 6c2c 2c20 7468 6520  Doc)Ideal,, the 
│ │ │ │ -0009cf70: 6964 6561 6c20 6f66 2074 6865 2063 6c6f  ideal of the clo
│ │ │ │ -0009cf80: 7375 7265 206f 6620 7468 650a 2020 2020  sure of the.    
│ │ │ │ -0009cf90: 2020 2020 6469 7265 6374 2069 6d61 6765      direct image
│ │ │ │ -0009cfa0: 206f 6620 5628 4929 2076 6961 2070 6869   of V(I) via phi
│ │ │ │ -0009cfb0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ -0009cfc0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6e20 6d6f  =========..In mo
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│ │ │ │ -0009cfe0: 2065 7175 6976 616c 656e 7420 746f 2070   equivalent to p
│ │ │ │ -0009cff0: 6869 2049 2c20 7768 6963 6820 6973 2066  hi I, which is f
│ │ │ │ -0009d000: 6173 7465 7220 6275 7420 6d61 7920 6e6f  aster but may no
│ │ │ │ -0009d010: 7420 7461 6b65 0a69 6e74 6f20 6163 636f  t take.into acco
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│ │ │ │ -0009d030: 656e 7461 7469 6f6e 7320 6f66 2074 6865  entations of the
│ │ │ │ -0009d040: 206d 6170 2e0a 0a53 6565 2061 6c73 6f0a   map...See also.
│ │ │ │ -0009d050: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ -0009d060: 6f74 6520 5261 7469 6f6e 616c 4d61 7020  ote RationalMap 
│ │ │ │ -0009d070: 5e2a 2a20 4964 6561 6c3a 2052 6174 696f  ^** Ideal: Ratio
│ │ │ │ -0009d080: 6e61 6c4d 6170 205e 5f73 745f 7374 2049  nalMap ^_st_st I
│ │ │ │ -0009d090: 6465 616c 2c20 2d2d 2069 6e76 6572 7365  deal, -- inverse
│ │ │ │ -0009d0a0: 2069 6d61 6765 0a20 2020 2076 6961 2061   image.    via a
│ │ │ │ -0009d0b0: 2072 6174 696f 6e61 6c20 6d61 700a 2020   rational map.  
│ │ │ │ -0009d0c0: 2a20 2a6e 6f74 6520 736f 7572 6365 2852  * *note source(R
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│ │ │ │ -0009d0e0: 7263 655f 6c70 5261 7469 6f6e 616c 4d61  rce_lpRationalMa
│ │ │ │ -0009d0f0: 705f 7270 2c20 2d2d 2063 6f6f 7264 696e  p_rp, -- coordin
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│ │ │ │ -0009d110: 7468 6520 736f 7572 6365 2066 6f72 2061  the source for a
│ │ │ │ -0009d120: 2072 6174 696f 6e61 6c20 6d61 700a 0a57   rational map..W
│ │ │ │ -0009d130: 6179 7320 746f 2075 7365 2074 6869 7320  ays to use this 
│ │ │ │ -0009d140: 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d  method:.========
│ │ │ │ -0009d150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0009d160: 0a0a 2020 2a20 2a6e 6f74 6520 5261 7469  ..  * *note Rati
│ │ │ │ -0009d170: 6f6e 616c 4d61 7020 5f2a 3a20 5261 7469  onalMap _*: Rati
│ │ │ │ -0009d180: 6f6e 616c 4d61 7020 5f75 735f 7374 2c20  onalMap _us_st, 
│ │ │ │ -0009d190: 2d2d 2064 6972 6563 7420 696d 6167 6520  -- direct image 
│ │ │ │ -0009d1a0: 7669 6120 6120 7261 7469 6f6e 616c 0a20  via a rational. 
│ │ │ │ -0009d1b0: 2020 206d 6170 0a2d 2d2d 2d2d 2d2d 2d2d     map.---------
│ │ │ │ +0009ce50: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4f70  ********..  * Op
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│ │ │ │ +0009ce70: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0009ce80: 5f75 735f 7374 2c0a 2020 2a20 5573 6167  _us_st,.  * Usag
│ │ │ │ +0009ce90: 653a 200a 2020 2020 2020 2020 7068 695f  e: .        phi_
│ │ │ │ +0009cea0: 2a20 490a 2020 2a20 496e 7075 7473 3a0a  * I.  * Inputs:.
│ │ │ │ +0009ceb0: 2020 2020 2020 2a20 7068 692c 2061 202a        * phi, a *
│ │ │ │ +0009cec0: 6e6f 7465 2072 6174 696f 6e61 6c20 6d61  note rational ma
│ │ │ │ +0009ced0: 703a 2052 6174 696f 6e61 6c4d 6170 2c2c  p: RationalMap,,
│ │ │ │ +0009cee0: 2049 2061 2068 6f6d 6f67 656e 656f 7573   I a homogeneous
│ │ │ │ +0009cef0: 2069 6465 616c 2069 6e20 7468 650a 2020   ideal in the.  
│ │ │ │ +0009cf00: 2020 2020 2020 636f 6f72 6469 6e61 7465        coordinate
│ │ │ │ +0009cf10: 2072 696e 6720 6f66 2074 6865 2073 6f75   ring of the sou
│ │ │ │ +0009cf20: 7263 6520 6f66 2070 6869 0a20 202a 204f  rce of phi.  * O
│ │ │ │ +0009cf30: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ +0009cf40: 616e 202a 6e6f 7465 2069 6465 616c 3a20  an *note ideal: 
│ │ │ │ +0009cf50: 284d 6163 6175 6c61 7932 446f 6329 4964  (Macaulay2Doc)Id
│ │ │ │ +0009cf60: 6561 6c2c 2c20 7468 6520 6964 6561 6c20  eal,, the ideal 
│ │ │ │ +0009cf70: 6f66 2074 6865 2063 6c6f 7375 7265 206f  of the closure o
│ │ │ │ +0009cf80: 6620 7468 650a 2020 2020 2020 2020 6469  f the.        di
│ │ │ │ +0009cf90: 7265 6374 2069 6d61 6765 206f 6620 5628  rect image of V(
│ │ │ │ +0009cfa0: 4929 2076 6961 2070 6869 0a0a 4465 7363  I) via phi..Desc
│ │ │ │ +0009cfb0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +0009cfc0: 3d3d 3d0a 0a49 6e20 6d6f 7374 2063 6173  ===..In most cas
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│ │ │ │ +0009cff0: 7768 6963 6820 6973 2066 6173 7465 7220  which is faster 
│ │ │ │ +0009d000: 6275 7420 6d61 7920 6e6f 7420 7461 6b65  but may not take
│ │ │ │ +0009d010: 0a69 6e74 6f20 6163 636f 756e 7420 6f74  .into account ot
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│ │ │ │ +0009d030: 6f6e 7320 6f66 2074 6865 206d 6170 2e0a  ons of the map..
│ │ │ │ +0009d040: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +0009d050: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 5261  ==..  * *note Ra
│ │ │ │ +0009d060: 7469 6f6e 616c 4d61 7020 5e2a 2a20 4964  tionalMap ^** Id
│ │ │ │ +0009d070: 6561 6c3a 2052 6174 696f 6e61 6c4d 6170  eal: RationalMap
│ │ │ │ +0009d080: 205e 5f73 745f 7374 2049 6465 616c 2c20   ^_st_st Ideal, 
│ │ │ │ +0009d090: 2d2d 2069 6e76 6572 7365 2069 6d61 6765  -- inverse image
│ │ │ │ +0009d0a0: 0a20 2020 2076 6961 2061 2072 6174 696f  .    via a ratio
│ │ │ │ +0009d0b0: 6e61 6c20 6d61 700a 2020 2a20 2a6e 6f74  nal map.  * *not
│ │ │ │ +0009d0c0: 6520 736f 7572 6365 2852 6174 696f 6e61  e source(Rationa
│ │ │ │ +0009d0d0: 6c4d 6170 293a 2073 6f75 7263 655f 6c70  lMap): source_lp
│ │ │ │ +0009d0e0: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ +0009d0f0: 2d2d 2063 6f6f 7264 696e 6174 6520 7269  -- coordinate ri
│ │ │ │ +0009d100: 6e67 206f 660a 2020 2020 7468 6520 736f  ng of.    the so
│ │ │ │ +0009d110: 7572 6365 2066 6f72 2061 2072 6174 696f  urce for a ratio
│ │ │ │ +0009d120: 6e61 6c20 6d61 700a 0a57 6179 7320 746f  nal map..Ways to
│ │ │ │ +0009d130: 2075 7365 2074 6869 7320 6d65 7468 6f64   use this method
│ │ │ │ +0009d140: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +0009d150: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +0009d160: 2a6e 6f74 6520 5261 7469 6f6e 616c 4d61  *note RationalMa
│ │ │ │ +0009d170: 7020 5f2a 3a20 5261 7469 6f6e 616c 4d61  p _*: RationalMa
│ │ │ │ +0009d180: 7020 5f75 735f 7374 2c20 2d2d 2064 6972  p _us_st, -- dir
│ │ │ │ +0009d190: 6563 7420 696d 6167 6520 7669 6120 6120  ect image via a 
│ │ │ │ +0009d1a0: 7261 7469 6f6e 616c 0a20 2020 206d 6170  rational.    map
│ │ │ │ +0009d1b0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │  0009d1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009d1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009d1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009d1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009d200: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
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│ │ │ │ -0009d2a0: 6e66 6f2c 204e 6f64 653a 2052 6174 696f  nfo, Node: Ratio
│ │ │ │ -0009d2b0: 6e61 6c4d 6170 207c 2049 6465 616c 2c20  nalMap | Ideal, 
│ │ │ │ -0009d2c0: 4e65 7874 3a20 5261 7469 6f6e 616c 4d61  Next: RationalMa
│ │ │ │ -0009d2d0: 7020 7c7c 2049 6465 616c 2c20 5072 6576  p || Ideal, Prev
│ │ │ │ -0009d2e0: 3a20 5261 7469 6f6e 616c 4d61 7020 5f75  : RationalMap _u
│ │ │ │ -0009d2f0: 735f 7374 2c20 5570 3a20 546f 700a 0a52  s_st, Up: Top..R
│ │ │ │ -0009d300: 6174 696f 6e61 6c4d 6170 207c 2049 6465  ationalMap | Ide
│ │ │ │ -0009d310: 616c 202d 2d20 7265 7374 7269 6374 696f  al -- restrictio
│ │ │ │ -0009d320: 6e20 6f66 2061 2072 6174 696f 6e61 6c20  n of a rational 
│ │ │ │ -0009d330: 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  map.************
│ │ │ │ +0009d200: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
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│ │ │ │ +0009d290: 2043 7265 6d6f 6e61 2e69 6e66 6f2c 204e   Cremona.info, N
│ │ │ │ +0009d2a0: 6f64 653a 2052 6174 696f 6e61 6c4d 6170  ode: RationalMap
│ │ │ │ +0009d2b0: 207c 2049 6465 616c 2c20 4e65 7874 3a20   | Ideal, Next: 
│ │ │ │ +0009d2c0: 5261 7469 6f6e 616c 4d61 7020 7c7c 2049  RationalMap || I
│ │ │ │ +0009d2d0: 6465 616c 2c20 5072 6576 3a20 5261 7469  deal, Prev: Rati
│ │ │ │ +0009d2e0: 6f6e 616c 4d61 7020 5f75 735f 7374 2c20  onalMap _us_st, 
│ │ │ │ +0009d2f0: 5570 3a20 546f 700a 0a52 6174 696f 6e61  Up: Top..Rationa
│ │ │ │ +0009d300: 6c4d 6170 207c 2049 6465 616c 202d 2d20  lMap | Ideal -- 
│ │ │ │ +0009d310: 7265 7374 7269 6374 696f 6e20 6f66 2061  restriction of a
│ │ │ │ +0009d320: 2072 6174 696f 6e61 6c20 6d61 700a 2a2a   rational map.**
│ │ │ │ +0009d330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0009d340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0009d350: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0009d360: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4f70  ********..  * Op
│ │ │ │ -0009d370: 6572 6174 6f72 3a20 2a6e 6f74 6520 7c3a  erator: *note |:
│ │ │ │ -0009d380: 2028 4d61 6361 756c 6179 3244 6f63 297c   (Macaulay2Doc)|
│ │ │ │ -0009d390: 2c0a 2020 2a20 5573 6167 653a 200a 2020  ,.  * Usage: .  
│ │ │ │ -0009d3a0: 2020 2020 2020 5068 6920 7c20 490a 2020        Phi | I.  
│ │ │ │ -0009d3b0: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -0009d3c0: 2a20 5068 692c 2061 202a 6e6f 7465 2072  * Phi, a *note r
│ │ │ │ -0009d3d0: 6174 696f 6e61 6c20 6d61 703a 2052 6174  ational map: Rat
│ │ │ │ -0009d3e0: 696f 6e61 6c4d 6170 2c2c 2024 5c70 6869  ionalMap,, $\phi
│ │ │ │ -0009d3f0: 3a58 205c 6461 7368 7269 6768 7461 7272  :X \dashrightarr
│ │ │ │ -0009d400: 6f77 2059 240a 2020 2020 2020 2a20 492c  ow Y$.      * I,
│ │ │ │ -0009d410: 2061 6e20 2a6e 6f74 6520 6964 6561 6c3a   an *note ideal:
│ │ │ │ -0009d420: 2028 4d61 6361 756c 6179 3244 6f63 2949   (Macaulay2Doc)I
│ │ │ │ -0009d430: 6465 616c 2c2c 2061 2068 6f6d 6f67 656e  deal,, a homogen
│ │ │ │ -0009d440: 656f 7573 2069 6465 616c 206f 6620 610a  eous ideal of a.
│ │ │ │ -0009d450: 2020 2020 2020 2020 7375 6276 6172 6965          subvarie
│ │ │ │ -0009d460: 7479 2024 5a5c 7375 6273 6574 2058 240a  ty $Z\subset X$.
│ │ │ │ -0009d470: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ -0009d480: 2020 202a 2061 202a 6e6f 7465 2072 6174     * a *note rat
│ │ │ │ -0009d490: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ -0009d4a0: 6e61 6c4d 6170 2c2c 2074 6865 2072 6573  nalMap,, the res
│ │ │ │ -0009d4b0: 7472 6963 7469 6f6e 206f 6620 245c 7068  triction of $\ph
│ │ │ │ -0009d4c0: 6924 2074 6f20 245a 242c 0a20 2020 2020  i$ to $Z$,.     
│ │ │ │ -0009d4d0: 2020 2024 5c70 6869 7c5f 7b5a 7d3a 205a     $\phi|_{Z}: Z
│ │ │ │ -0009d4e0: 205c 6461 7368 7269 6768 7461 7272 6f77   \dashrightarrow
│ │ │ │ -0009d4f0: 2059 240a 0a44 6573 6372 6970 7469 6f6e   Y$..Description
│ │ │ │ -0009d500: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d  .===========..+-
│ │ │ │ +0009d360: 2a2a 0a0a 2020 2a20 4f70 6572 6174 6f72  **..  * Operator
│ │ │ │ +0009d370: 3a20 2a6e 6f74 6520 7c3a 2028 4d61 6361  : *note |: (Maca
│ │ │ │ +0009d380: 756c 6179 3244 6f63 297c 2c0a 2020 2a20  ulay2Doc)|,.  * 
│ │ │ │ +0009d390: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +0009d3a0: 5068 6920 7c20 490a 2020 2a20 496e 7075  Phi | I.  * Inpu
│ │ │ │ +0009d3b0: 7473 3a0a 2020 2020 2020 2a20 5068 692c  ts:.      * Phi,
│ │ │ │ +0009d3c0: 2061 202a 6e6f 7465 2072 6174 696f 6e61   a *note rationa
│ │ │ │ +0009d3d0: 6c20 6d61 703a 2052 6174 696f 6e61 6c4d  l map: RationalM
│ │ │ │ +0009d3e0: 6170 2c2c 2024 5c70 6869 3a58 205c 6461  ap,, $\phi:X \da
│ │ │ │ +0009d3f0: 7368 7269 6768 7461 7272 6f77 2059 240a  shrightarrow Y$.
│ │ │ │ +0009d400: 2020 2020 2020 2a20 492c 2061 6e20 2a6e        * I, an *n
│ │ │ │ +0009d410: 6f74 6520 6964 6561 6c3a 2028 4d61 6361  ote ideal: (Maca
│ │ │ │ +0009d420: 756c 6179 3244 6f63 2949 6465 616c 2c2c  ulay2Doc)Ideal,,
│ │ │ │ +0009d430: 2061 2068 6f6d 6f67 656e 656f 7573 2069   a homogeneous i
│ │ │ │ +0009d440: 6465 616c 206f 6620 610a 2020 2020 2020  deal of a.      
│ │ │ │ +0009d450: 2020 7375 6276 6172 6965 7479 2024 5a5c    subvariety $Z\
│ │ │ │ +0009d460: 7375 6273 6574 2058 240a 2020 2a20 4f75  subset X$.  * Ou
│ │ │ │ +0009d470: 7470 7574 733a 0a20 2020 2020 202a 2061  tputs:.      * a
│ │ │ │ +0009d480: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ +0009d490: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ +0009d4a0: 2c2c 2074 6865 2072 6573 7472 6963 7469  ,, the restricti
│ │ │ │ +0009d4b0: 6f6e 206f 6620 245c 7068 6924 2074 6f20  on of $\phi$ to 
│ │ │ │ +0009d4c0: 245a 242c 0a20 2020 2020 2020 2024 5c70  $Z$,.        $\p
│ │ │ │ +0009d4d0: 6869 7c5f 7b5a 7d3a 205a 205c 6461 7368  hi|_{Z}: Z \dash
│ │ │ │ +0009d4e0: 7269 6768 7461 7272 6f77 2059 240a 0a44  rightarrow Y$..D
│ │ │ │ +0009d4f0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +0009d500: 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d  ======..+-------
│ │ │ │  0009d510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009d520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009d530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009d540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009d550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0009d560: 3120 3a20 5035 203d 205a 5a2f 3139 3031  1 : P5 = ZZ/1901
│ │ │ │ -0009d570: 3831 5b78 5f30 2e2e 785f 355d 2020 2020  81[x_0..x_5]    
│ │ │ │ +0009d550: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5035  ------+.|i1 : P5
│ │ │ │ +0009d560: 203d 205a 5a2f 3139 3031 3831 5b78 5f30   = ZZ/190181[x_0
│ │ │ │ +0009d570: 2e2e 785f 355d 2020 2020 2020 2020 2020  ..x_5]          
│ │ │ │  0009d580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009d5a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009d5a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009d5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009d5f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0009d600: 3120 3d20 5035 2020 2020 2020 2020 2020  1 = P5          
│ │ │ │ +0009d5f0: 2020 2020 2020 7c0a 7c6f 3120 3d20 5035        |.|o1 = P5
│ │ │ │ +0009d600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009d640: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009d640: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009d650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009d690: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0009d6a0: 3120 3a20 506f 6c79 6e6f 6d69 616c 5269  1 : PolynomialRi
│ │ │ │ -0009d6b0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +0009d690: 2020 2020 2020 7c0a 7c6f 3120 3a20 506f        |.|o1 : Po
│ │ │ │ +0009d6a0: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020  lynomialRing    
│ │ │ │ +0009d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009d6e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0009d6e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  0009d6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009d700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009d710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009d720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009d730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0009d740: 3220 3a20 5068 6920 3d20 7261 7469 6f6e  2 : Phi = ration
│ │ │ │ -0009d750: 616c 4d61 7020 7b78 5f34 5e32 2d78 5f33  alMap {x_4^2-x_3
│ │ │ │ -0009d760: 2a78 5f35 2c78 5f32 2a78 5f34 2d78 5f31  *x_5,x_2*x_4-x_1
│ │ │ │ -0009d770: 2a78 5f35 2c78 5f32 2a78 5f33 2d78 5f31  *x_5,x_2*x_3-x_1
│ │ │ │ -0009d780: 2a78 5f34 2c78 5f32 5e32 2d78 7c0a 7c20  *x_4,x_2^2-x|.| 
│ │ │ │ +0009d730: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5068  ------+.|i2 : Ph
│ │ │ │ +0009d740: 6920 3d20 7261 7469 6f6e 616c 4d61 7020  i = rationalMap 
│ │ │ │ +0009d750: 7b78 5f34 5e32 2d78 5f33 2a78 5f35 2c78  {x_4^2-x_3*x_5,x
│ │ │ │ +0009d760: 5f32 2a78 5f34 2d78 5f31 2a78 5f35 2c78  _2*x_4-x_1*x_5,x
│ │ │ │ +0009d770: 5f32 2a78 5f33 2d78 5f31 2a78 5f34 2c78  _2*x_3-x_1*x_4,x
│ │ │ │ +0009d780: 5f32 5e32 2d78 7c0a 7c20 2020 2020 2020  _2^2-x|.|       
│ │ │ │  0009d790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009d7d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0009d7e0: 3220 3d20 2d2d 2072 6174 696f 6e61 6c20  2 = -- rational 
│ │ │ │ -0009d7f0: 6d61 7020 2d2d 2020 2020 2020 2020 2020  map --          
│ │ │ │ +0009d7d0: 2020 2020 2020 7c0a 7c6f 3220 3d20 2d2d        |.|o2 = --
│ │ │ │ +0009d7e0: 2072 6174 696f 6e61 6c20 6d61 7020 2d2d   rational map --
│ │ │ │ +0009d7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009d820: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009d840: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │ +0009d820: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009d830: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
│ │ │ │ +0009d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009d870: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009d880: 2020 2020 736f 7572 6365 3a20 5072 6f6a      source: Proj
│ │ │ │ -0009d890: 282d 2d2d 2d2d 2d5b 7820 2c20 7820 2c20  (------[x , x , 
│ │ │ │ -0009d8a0: 7820 2c20 7820 2c20 7820 2c20 7820 5d29  x , x , x , x ])
│ │ │ │ +0009d870: 2020 2020 2020 7c0a 7c20 2020 2020 736f        |.|     so
│ │ │ │ +0009d880: 7572 6365 3a20 5072 6f6a 282d 2d2d 2d2d  urce: Proj(-----
│ │ │ │ +0009d890: 2d5b 7820 2c20 7820 2c20 7820 2c20 7820  -[x , x , x , x 
│ │ │ │ +0009d8a0: 2c20 7820 2c20 7820 5d29 2020 2020 2020  , x , x ])      
│ │ │ │  0009d8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009d8c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009d8e0: 2031 3930 3138 3120 2030 2020 2031 2020   190181  0   1  
│ │ │ │ -0009d8f0: 2032 2020 2033 2020 2034 2020 2035 2020   2   3   4   5  
│ │ │ │ +0009d8c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009d8d0: 2020 2020 2020 2020 2020 2031 3930 3138             19018
│ │ │ │ +0009d8e0: 3120 2030 2020 2031 2020 2032 2020 2033  1  0   1   2   3
│ │ │ │ +0009d8f0: 2020 2034 2020 2035 2020 2020 2020 2020     4   5        
│ │ │ │  0009d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009d910: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009d930: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │ +0009d910: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009d920: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
│ │ │ │ +0009d930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009d960: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009d970: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
│ │ │ │ -0009d980: 282d 2d2d 2d2d 2d5b 7820 2c20 7820 2c20  (------[x , x , 
│ │ │ │ -0009d990: 7820 2c20 7820 2c20 7820 2c20 7820 5d29  x , x , x , x ])
│ │ │ │ +0009d960: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ +0009d970: 7267 6574 3a20 5072 6f6a 282d 2d2d 2d2d  rget: Proj(-----
│ │ │ │ +0009d980: 2d5b 7820 2c20 7820 2c20 7820 2c20 7820  -[x , x , x , x 
│ │ │ │ +0009d990: 2c20 7820 2c20 7820 5d29 2020 2020 2020  , x , x ])      
│ │ │ │  0009d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009d9b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009d9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009d9d0: 2031 3930 3138 3120 2030 2020 2031 2020   190181  0   1  
│ │ │ │ -0009d9e0: 2032 2020 2033 2020 2034 2020 2035 2020   2   3   4   5  
│ │ │ │ +0009d9b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009d9c0: 2020 2020 2020 2020 2020 2031 3930 3138             19018
│ │ │ │ +0009d9d0: 3120 2030 2020 2031 2020 2032 2020 2033  1  0   1   2   3
│ │ │ │ +0009d9e0: 2020 2034 2020 2035 2020 2020 2020 2020     4   5        
│ │ │ │  0009d9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009da00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009da10: 2020 2020 6465 6669 6e69 6e67 2066 6f72      defining for
│ │ │ │ -0009da20: 6d73 3a20 7b20 2020 2020 2020 2020 2020  ms: {           
│ │ │ │ +0009da00: 2020 2020 2020 7c0a 7c20 2020 2020 6465        |.|     de
│ │ │ │ +0009da10: 6669 6e69 6e67 2066 6f72 6d73 3a20 7b20  fining forms: { 
│ │ │ │ +0009da20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009da30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009da40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009da50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009da50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009da60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009da70: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0009da70: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0009da80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009da90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009daa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009dab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009dac0: 2020 2020 2078 2020 2d20 7820 7820 2c20       x  - x x , 
│ │ │ │ +0009daa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009dab0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +0009dac0: 2020 2d20 7820 7820 2c20 2020 2020 2020    - x x ,       
│ │ │ │  0009dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009dae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009daf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009daf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009db00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009db10: 2020 2020 2020 3420 2020 2033 2035 2020        4    3 5  
│ │ │ │ +0009db10: 3420 2020 2033 2035 2020 2020 2020 2020  4    3 5        
│ │ │ │  0009db20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009db30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009db40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009db40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009db50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009db60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009db90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009dba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009dbb0: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ -0009dbc0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +0009db90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009dba0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +0009dbb0: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
│ │ │ │ +0009dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009dbe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009dbe0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009dbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009dc00: 2020 2020 2020 3220 3420 2020 2031 2035        2 4    1 5
│ │ │ │ +0009dc00: 3220 3420 2020 2031 2035 2020 2020 2020  2 4    1 5      
│ │ │ │  0009dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009dc30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009dc30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009dc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009dc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009dc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009dc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009dc80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009dca0: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ -0009dcb0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +0009dc80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009dc90: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +0009dca0: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
│ │ │ │ +0009dcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009dcd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009dcd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009dce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009dcf0: 2020 2020 2020 3220 3320 2020 2031 2034        2 3    1 4
│ │ │ │ +0009dcf0: 3220 3320 2020 2031 2034 2020 2020 2020  2 3    1 4      
│ │ │ │  0009dd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009dd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009dd20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009dd20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009dd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009dd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009dd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009dd70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009dd70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009dd90: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0009dd90: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0009dda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ddb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ddc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009dde0: 2020 2020 2078 2020 2d20 7820 7820 2c20       x  - x x , 
│ │ │ │ +0009ddc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009ddd0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +0009dde0: 2020 2d20 7820 7820 2c20 2020 2020 2020    - x x ,       
│ │ │ │  0009ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009de00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009de10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009de10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009de30: 2020 2020 2020 3220 2020 2030 2035 2020        2    0 5  
│ │ │ │ +0009de30: 3220 2020 2030 2035 2020 2020 2020 2020  2    0 5        
│ │ │ │  0009de40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009de50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009de60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009de60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009de90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009dea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009deb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009dec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ded0: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ -0009dee0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +0009deb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009dec0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +0009ded0: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
│ │ │ │ +0009dee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009df00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009df00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009df20: 2020 2020 2020 3120 3220 2020 2030 2034        1 2    0 4
│ │ │ │ +0009df20: 3120 3220 2020 2030 2034 2020 2020 2020  1 2    0 4      
│ │ │ │  0009df30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009df50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009df50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009df80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009df90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009dfa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009dfa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009dfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009dfc0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0009dfc0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0009dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009dff0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e010: 2020 2020 2078 2020 2d20 7820 7820 2020       x  - x x   
│ │ │ │ +0009dff0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009e000: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +0009e010: 2020 2d20 7820 7820 2020 2020 2020 2020    - x x         
│ │ │ │  0009e020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e040: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009e040: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e060: 2020 2020 2020 3120 2020 2030 2033 2020        1    0 3  
│ │ │ │ +0009e060: 3120 2020 2030 2033 2020 2020 2020 2020  1    0 3        
│ │ │ │  0009e070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e090: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e0b0: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │ +0009e090: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009e0a0: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
│ │ │ │ +0009e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e0e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009e0e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e130: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0009e140: 3220 3a20 5261 7469 6f6e 616c 4d61 7020  2 : RationalMap 
│ │ │ │ -0009e150: 2871 7561 6472 6174 6963 2072 6174 696f  (quadratic ratio
│ │ │ │ -0009e160: 6e61 6c20 6d61 7020 6672 6f6d 2050 505e  nal map from PP^
│ │ │ │ -0009e170: 3520 746f 2050 505e 3529 2020 2020 2020  5 to PP^5)      
│ │ │ │ -0009e180: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0009e130: 2020 2020 2020 7c0a 7c6f 3220 3a20 5261        |.|o2 : Ra
│ │ │ │ +0009e140: 7469 6f6e 616c 4d61 7020 2871 7561 6472  tionalMap (quadr
│ │ │ │ +0009e150: 6174 6963 2072 6174 696f 6e61 6c20 6d61  atic rational ma
│ │ │ │ +0009e160: 7020 6672 6f6d 2050 505e 3520 746f 2050  p from PP^5 to P
│ │ │ │ +0009e170: 505e 3529 2020 2020 2020 2020 2020 2020  P^5)            
│ │ │ │ +0009e180: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │  0009e190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009e1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009e1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009e1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009e1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c5f  ------------|.|_
│ │ │ │ -0009e1e0: 302a 785f 352c 785f 312a 785f 322d 785f  0*x_5,x_1*x_2-x_
│ │ │ │ -0009e1f0: 302a 785f 342c 785f 315e 322d 785f 302a  0*x_4,x_1^2-x_0*
│ │ │ │ -0009e200: 785f 337d 2020 2020 2020 2020 2020 2020  x_3}            
│ │ │ │ +0009e1d0: 2d2d 2d2d 2d2d 7c0a 7c5f 302a 785f 352c  ------|.|_0*x_5,
│ │ │ │ +0009e1e0: 785f 312a 785f 322d 785f 302a 785f 342c  x_1*x_2-x_0*x_4,
│ │ │ │ +0009e1f0: 785f 315e 322d 785f 302a 785f 337d 2020  x_1^2-x_0*x_3}  
│ │ │ │ +0009e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e220: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0009e220: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  0009e230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009e260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009e270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0009e280: 3320 3a20 4920 3d20 6964 6561 6c28 7261  3 : I = ideal(ra
│ │ │ │ -0009e290: 6e64 6f6d 2832 2c50 3529 2c72 616e 646f  ndom(2,P5),rando
│ │ │ │ -0009e2a0: 6d28 332c 5035 2929 3b20 2020 2020 2020  m(3,P5));       
│ │ │ │ +0009e270: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 4920  ------+.|i3 : I 
│ │ │ │ +0009e280: 3d20 6964 6561 6c28 7261 6e64 6f6d 2832  = ideal(random(2
│ │ │ │ +0009e290: 2c50 3529 2c72 616e 646f 6d28 332c 5035  ,P5),random(3,P5
│ │ │ │ +0009e2a0: 2929 3b20 2020 2020 2020 2020 2020 2020  ));             
│ │ │ │  0009e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e2c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009e2c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e310: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0009e320: 3320 3a20 4964 6561 6c20 6f66 2050 3520  3 : Ideal of P5 
│ │ │ │ +0009e310: 2020 2020 2020 7c0a 7c6f 3320 3a20 4964        |.|o3 : Id
│ │ │ │ +0009e320: 6561 6c20 6f66 2050 3520 2020 2020 2020  eal of P5       
│ │ │ │  0009e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e360: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0009e360: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  0009e370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009e380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009e390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009e3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0009e3c0: 3420 3a20 5068 6927 203d 2050 6869 7c49  4 : Phi' = Phi|I
│ │ │ │ +0009e3b0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 5068  ------+.|i4 : Ph
│ │ │ │ +0009e3c0: 6927 203d 2050 6869 7c49 2020 2020 2020  i' = Phi|I      
│ │ │ │  0009e3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e400: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009e400: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e450: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0009e460: 3420 3d20 2d2d 2072 6174 696f 6e61 6c20  4 = -- rational 
│ │ │ │ -0009e470: 6d61 7020 2d2d 2020 2020 2020 2020 2020  map --          
│ │ │ │ +0009e450: 2020 2020 2020 7c0a 7c6f 3420 3d20 2d2d        |.|o4 = --
│ │ │ │ +0009e460: 2072 6174 696f 6e61 6c20 6d61 7020 2d2d   rational map --
│ │ │ │ +0009e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e4a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009e4a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009e4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e4d0: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │ +0009e4c0: 2020 2020 2020 2020 2020 205a 5a20 2020             ZZ   
│ │ │ │ +0009e4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e4f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009e500: 2020 2020 736f 7572 6365 3a20 7375 6276      source: subv
│ │ │ │ -0009e510: 6172 6965 7479 206f 6620 5072 6f6a 282d  ariety of Proj(-
│ │ │ │ -0009e520: 2d2d 2d2d 2d5b 7820 2c20 7820 2c20 7820  -----[x , x , x 
│ │ │ │ -0009e530: 2c20 7820 2c20 7820 2c20 7820 5d29 2064  , x , x , x ]) d
│ │ │ │ -0009e540: 6566 696e 6564 2062 7920 2020 7c0a 7c20  efined by   |.| 
│ │ │ │ +0009e4f0: 2020 2020 2020 7c0a 7c20 2020 2020 736f        |.|     so
│ │ │ │ +0009e500: 7572 6365 3a20 7375 6276 6172 6965 7479  urce: subvariety
│ │ │ │ +0009e510: 206f 6620 5072 6f6a 282d 2d2d 2d2d 2d5b   of Proj(------[
│ │ │ │ +0009e520: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ +0009e530: 7820 2c20 7820 5d29 2064 6566 696e 6564  x , x ]) defined
│ │ │ │ +0009e540: 2062 7920 2020 7c0a 7c20 2020 2020 2020   by   |.|       
│ │ │ │  0009e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e560: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -0009e570: 3930 3138 3120 2030 2020 2031 2020 2032  90181  0   1   2
│ │ │ │ -0009e580: 2020 2033 2020 2034 2020 2035 2020 2020     3   4   5    
│ │ │ │ -0009e590: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009e5a0: 2020 2020 2020 2020 2020 2020 7b20 2020              {   
│ │ │ │ +0009e560: 2020 2020 2020 2020 2031 3930 3138 3120           190181 
│ │ │ │ +0009e570: 2030 2020 2031 2020 2032 2020 2033 2020   0   1   2   3  
│ │ │ │ +0009e580: 2034 2020 2035 2020 2020 2020 2020 2020   4   5          
│ │ │ │ +0009e590: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009e5a0: 2020 2020 2020 7b20 2020 2020 2020 2020        {         
│ │ │ │  0009e5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e5e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009e5f0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0009e600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e610: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0009e5e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009e5f0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0009e600: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0009e610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e630: 2020 2020 2020 3220 2020 2020 7c0a 7c20        2     |.| 
│ │ │ │ -0009e640: 2020 2020 2020 2020 2020 2020 2078 2020               x  
│ │ │ │ -0009e650: 2b20 3136 3536 3678 2078 2020 2d20 3437  + 16566x x  - 47
│ │ │ │ -0009e660: 3436 3878 2020 2d20 3730 3135 3878 2078  468x  - 70158x x
│ │ │ │ -0009e670: 2020 2b20 3930 3632 3678 2078 2020 2d20    + 90626x x  - 
│ │ │ │ -0009e680: 3234 3231 3278 2020 2d20 2020 7c0a 7c20  24212x  -   |.| 
│ │ │ │ -0009e690: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -0009e6a0: 2020 2020 2020 2020 3020 3120 2020 2020          0 1     
│ │ │ │ -0009e6b0: 2020 2020 3120 2020 2020 2020 2020 3020      1         0 
│ │ │ │ -0009e6c0: 3220 2020 2020 2020 2020 3120 3220 2020  2         1 2   
│ │ │ │ -0009e6d0: 2020 2020 2020 3220 2020 2020 7c0a 7c20        2     |.| 
│ │ │ │ +0009e630: 3220 2020 2020 7c0a 7c20 2020 2020 2020  2     |.|       
│ │ │ │ +0009e640: 2020 2020 2020 2078 2020 2b20 3136 3536         x  + 1656
│ │ │ │ +0009e650: 3678 2078 2020 2d20 3437 3436 3878 2020  6x x  - 47468x  
│ │ │ │ +0009e660: 2d20 3730 3135 3878 2078 2020 2b20 3930  - 70158x x  + 90
│ │ │ │ +0009e670: 3632 3678 2078 2020 2d20 3234 3231 3278  626x x  - 24212x
│ │ │ │ +0009e680: 2020 2d20 2020 7c0a 7c20 2020 2020 2020    -   |.|       
│ │ │ │ +0009e690: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ +0009e6a0: 2020 3020 3120 2020 2020 2020 2020 3120    0 1         1 
│ │ │ │ +0009e6b0: 2020 2020 2020 2020 3020 3220 2020 2020          0 2     
│ │ │ │ +0009e6c0: 2020 2020 3120 3220 2020 2020 2020 2020      1 2         
│ │ │ │ +0009e6d0: 3220 2020 2020 7c0a 7c20 2020 2020 2020  2     |.|       
│ │ │ │  0009e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e720: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e740: 3220 2020 2020 2020 2020 3320 2020 2020  2         3     
│ │ │ │ -0009e750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e760: 2020 3220 2020 2020 2020 2020 2020 2032    2            2
│ │ │ │ -0009e770: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009e780: 2020 2020 2020 2020 2020 2020 2078 2078               x x
│ │ │ │ -0009e790: 2020 2d20 3531 3631 3078 2020 2d20 3830    - 51610x  - 80
│ │ │ │ -0009e7a0: 3531 3778 2078 2078 2020 2b20 3231 3132  517x x x  + 2112
│ │ │ │ -0009e7b0: 3378 2078 2020 2d20 3334 3932 7820 7820  3x x  - 3492x x 
│ │ │ │ -0009e7c0: 202b 2038 3534 3234 7820 2020 7c0a 7c20   + 85424x   |.| 
│ │ │ │ -0009e7d0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -0009e7e0: 3120 2020 2020 2020 2020 3120 2020 2020  1         1     
│ │ │ │ -0009e7f0: 2020 2020 3020 3120 3220 2020 2020 2020      0 1 2       
│ │ │ │ -0009e800: 2020 3120 3220 2020 2020 2020 2030 2032    1 2        0 2
│ │ │ │ -0009e810: 2020 2020 2020 2020 2031 2020 7c0a 7c20           1  |.| 
│ │ │ │ -0009e820: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ +0009e720: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009e730: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +0009e740: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ +0009e750: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +0009e760: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0009e770: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009e780: 2020 2020 2020 2078 2078 2020 2d20 3531         x x  - 51
│ │ │ │ +0009e790: 3631 3078 2020 2d20 3830 3531 3778 2078  610x  - 80517x x
│ │ │ │ +0009e7a0: 2078 2020 2b20 3231 3132 3378 2078 2020   x  + 21123x x  
│ │ │ │ +0009e7b0: 2d20 3334 3932 7820 7820 202b 2038 3534  - 3492x x  + 854
│ │ │ │ +0009e7c0: 3234 7820 2020 7c0a 7c20 2020 2020 2020  24x   |.|       
│ │ │ │ +0009e7d0: 2020 2020 2020 2020 3020 3120 2020 2020          0 1     
│ │ │ │ +0009e7e0: 2020 2020 3120 2020 2020 2020 2020 3020      1         0 
│ │ │ │ +0009e7f0: 3120 3220 2020 2020 2020 2020 3120 3220  1 2         1 2 
│ │ │ │ +0009e800: 2020 2020 2020 2030 2032 2020 2020 2020         0 2      
│ │ │ │ +0009e810: 2020 2031 2020 7c0a 7c20 2020 2020 2020     1  |.|       
│ │ │ │ +0009e820: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │  0009e830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e860: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e880: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │ +0009e860: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009e870: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
│ │ │ │ +0009e880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e8b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009e8c0: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
│ │ │ │ -0009e8d0: 282d 2d2d 2d2d 2d5b 7820 2c20 7820 2c20  (------[x , x , 
│ │ │ │ -0009e8e0: 7820 2c20 7820 2c20 7820 2c20 7820 5d29  x , x , x , x ])
│ │ │ │ +0009e8b0: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ +0009e8c0: 7267 6574 3a20 5072 6f6a 282d 2d2d 2d2d  rget: Proj(-----
│ │ │ │ +0009e8d0: 2d5b 7820 2c20 7820 2c20 7820 2c20 7820  -[x , x , x , x 
│ │ │ │ +0009e8e0: 2c20 7820 2c20 7820 5d29 2020 2020 2020  , x , x ])      
│ │ │ │  0009e8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e900: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e920: 2031 3930 3138 3120 2030 2020 2031 2020   190181  0   1  
│ │ │ │ -0009e930: 2032 2020 2033 2020 2034 2020 2035 2020   2   3   4   5  
│ │ │ │ +0009e900: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009e910: 2020 2020 2020 2020 2020 2031 3930 3138             19018
│ │ │ │ +0009e920: 3120 2030 2020 2031 2020 2032 2020 2033  1  0   1   2   3
│ │ │ │ +0009e930: 2020 2034 2020 2035 2020 2020 2020 2020     4   5        
│ │ │ │  0009e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e950: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009e960: 2020 2020 6465 6669 6e69 6e67 2066 6f72      defining for
│ │ │ │ -0009e970: 6d73 3a20 7b20 2020 2020 2020 2020 2020  ms: {           
│ │ │ │ +0009e950: 2020 2020 2020 7c0a 7c20 2020 2020 6465        |.|     de
│ │ │ │ +0009e960: 6669 6e69 6e67 2066 6f72 6d73 3a20 7b20  fining forms: { 
│ │ │ │ +0009e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e9a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009e9a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e9c0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0009e9c0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0009e9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009e9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009e9f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009ea00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ea10: 2020 2020 2078 2020 2d20 7820 7820 2c20       x  - x x , 
│ │ │ │ +0009e9f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009ea00: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +0009ea10: 2020 2d20 7820 7820 2c20 2020 2020 2020    - x x ,       
│ │ │ │  0009ea20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ea30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ea40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009ea40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009ea50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ea60: 2020 2020 2020 3420 2020 2033 2035 2020        4    3 5  
│ │ │ │ +0009ea60: 3420 2020 2033 2035 2020 2020 2020 2020  4    3 5        
│ │ │ │  0009ea70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ea80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ea90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009ea90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009eaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009eae0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009eaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009eb00: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ -0009eb10: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +0009eae0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009eaf0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +0009eb00: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
│ │ │ │ +0009eb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009eb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009eb30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009eb30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009eb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009eb50: 2020 2020 2020 3220 3420 2020 2031 2035        2 4    1 5
│ │ │ │ +0009eb50: 3220 3420 2020 2031 2035 2020 2020 2020  2 4    1 5      
│ │ │ │  0009eb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009eb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009eb80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009eb80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009eb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009eba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ebc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ebd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009ebe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ebf0: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ -0009ec00: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +0009ebd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009ebe0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +0009ebf0: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
│ │ │ │ +0009ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ec10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ec20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009ec20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009ec30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ec40: 2020 2020 2020 3220 3320 2020 2031 2034        2 3    1 4
│ │ │ │ +0009ec40: 3220 3320 2020 2031 2034 2020 2020 2020  2 3    1 4      
│ │ │ │  0009ec50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ec60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ec70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009ec70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009ec80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ec90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009eca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ecb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ecc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009ecc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009ecd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ece0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0009ece0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0009ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ed00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ed10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009ed20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ed30: 2020 2020 2078 2020 2d20 7820 7820 2c20       x  - x x , 
│ │ │ │ +0009ed10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009ed20: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +0009ed30: 2020 2d20 7820 7820 2c20 2020 2020 2020    - x x ,       
│ │ │ │  0009ed40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ed50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ed60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009ed60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ed80: 2020 2020 2020 3220 2020 2030 2035 2020        2    0 5  
│ │ │ │ +0009ed80: 3220 2020 2030 2035 2020 2020 2020 2020  2    0 5        
│ │ │ │  0009ed90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009eda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009edb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009edb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009edc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009edd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ede0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009edf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ee00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009ee10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ee20: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ -0009ee30: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +0009ee00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009ee10: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +0009ee20: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
│ │ │ │ +0009ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ee50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009ee50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009ee60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ee70: 2020 2020 2020 3120 3220 2020 2030 2034        1 2    0 4
│ │ │ │ +0009ee70: 3120 3220 2020 2030 2034 2020 2020 2020  1 2    0 4      
│ │ │ │  0009ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ee90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009eea0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009eea0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009eed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009eee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009eef0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009eef0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009ef00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ef10: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0009ef10: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0009ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ef30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ef40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ef60: 2020 2020 2078 2020 2d20 7820 7820 2020       x  - x x   
│ │ │ │ +0009ef40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009ef50: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ +0009ef60: 2020 2d20 7820 7820 2020 2020 2020 2020    - x x         
│ │ │ │  0009ef70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ef80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ef90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009ef90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009efb0: 2020 2020 2020 3120 2020 2030 2033 2020        1    0 3  
│ │ │ │ +0009efb0: 3120 2020 2030 2033 2020 2020 2020 2020  1    0 3        
│ │ │ │  0009efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009efe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f000: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │ +0009efe0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0009eff0: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
│ │ │ │ +0009f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f030: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009f030: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f080: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0009f090: 3420 3a20 5261 7469 6f6e 616c 4d61 7020  4 : RationalMap 
│ │ │ │ -0009f0a0: 2871 7561 6472 6174 6963 2072 6174 696f  (quadratic ratio
│ │ │ │ -0009f0b0: 6e61 6c20 6d61 7020 6672 6f6d 2074 6872  nal map from thr
│ │ │ │ -0009f0c0: 6565 666f 6c64 2069 6e20 5050 5e35 2074  eefold in PP^5 t
│ │ │ │ -0009f0d0: 6f20 5050 5e35 2920 2020 2020 7c0a 7c2d  o PP^5)     |.|-
│ │ │ │ +0009f080: 2020 2020 2020 7c0a 7c6f 3420 3a20 5261        |.|o4 : Ra
│ │ │ │ +0009f090: 7469 6f6e 616c 4d61 7020 2871 7561 6472  tionalMap (quadr
│ │ │ │ +0009f0a0: 6174 6963 2072 6174 696f 6e61 6c20 6d61  atic rational ma
│ │ │ │ +0009f0b0: 7020 6672 6f6d 2074 6872 6565 666f 6c64  p from threefold
│ │ │ │ +0009f0c0: 2069 6e20 5050 5e35 2074 6f20 5050 5e35   in PP^5 to PP^5
│ │ │ │ +0009f0d0: 2920 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d  )     |.|-------
│ │ │ │  0009f0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009f0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009f100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009f110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009f120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0009f120: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │  0009f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f150: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0009f150: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  0009f160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f170: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ -0009f180: 3831 3438 7820 7820 202d 2033 3137 3739  8148x x  - 31779
│ │ │ │ -0009f190: 7820 7820 202b 2031 3936 3035 7820 7820  x x  + 19605x x 
│ │ │ │ -0009f1a0: 202d 2037 3639 7820 202d 2037 3738 3634   - 769x  - 77864
│ │ │ │ -0009f1b0: 7820 7820 202b 2031 3036 3630 7820 7820  x x  + 10660x x 
│ │ │ │ -0009f1c0: 202b 2020 2020 2020 2020 2020 7c0a 7c20   +          |.| 
│ │ │ │ -0009f1d0: 2020 2020 2030 2033 2020 2020 2020 2020       0 3        
│ │ │ │ -0009f1e0: 2031 2033 2020 2020 2020 2020 2032 2033   1 3         2 3
│ │ │ │ -0009f1f0: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ -0009f200: 2030 2034 2020 2020 2020 2020 2031 2034   0 4         1 4
│ │ │ │ -0009f210: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009f170: 2020 2020 2020 7c0a 7c33 3831 3438 7820        |.|38148x 
│ │ │ │ +0009f180: 7820 202d 2033 3137 3739 7820 7820 202b  x  - 31779x x  +
│ │ │ │ +0009f190: 2031 3936 3035 7820 7820 202d 2037 3639   19605x x  - 769
│ │ │ │ +0009f1a0: 7820 202d 2037 3738 3634 7820 7820 202b  x  - 77864x x  +
│ │ │ │ +0009f1b0: 2031 3036 3630 7820 7820 202b 2020 2020   10660x x  +    
│ │ │ │ +0009f1c0: 2020 2020 2020 7c0a 7c20 2020 2020 2030        |.|      0
│ │ │ │ +0009f1d0: 2033 2020 2020 2020 2020 2031 2033 2020   3         1 3  
│ │ │ │ +0009f1e0: 2020 2020 2020 2032 2033 2020 2020 2020         2 3      
│ │ │ │ +0009f1f0: 2033 2020 2020 2020 2020 2030 2034 2020   3         0 4  
│ │ │ │ +0009f200: 2020 2020 2020 2031 2034 2020 2020 2020         1 4      
│ │ │ │ +0009f210: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f260: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009f270: 3220 2020 2020 2020 2033 2020 2020 2020  2        3      
│ │ │ │ -0009f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f290: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0009f260: 2020 2020 2020 7c0a 7c20 3220 2020 2020        |.| 2     
│ │ │ │ +0009f270: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +0009f280: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0009f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f2b0: 2020 2020 2020 2020 2020 2020 7c0a 7c78              |.|x
│ │ │ │ -0009f2c0: 2020 2b20 3336 3035 7820 202b 2031 3231    + 3605x  + 121
│ │ │ │ -0009f2d0: 3234 7820 7820 7820 202d 2033 3035 3438  24x x x  - 30548
│ │ │ │ -0009f2e0: 7820 7820 202d 2035 3737 3433 7820 7820  x x  - 57743x x 
│ │ │ │ -0009f2f0: 7820 202d 2031 3235 3930 7820 7820 7820  x  - 12590x x x 
│ │ │ │ -0009f300: 202d 2020 2020 2020 2020 2020 7c0a 7c20   -          |.| 
│ │ │ │ -0009f310: 3220 2020 2020 2020 2032 2020 2020 2020  2        2      
│ │ │ │ -0009f320: 2020 2030 2031 2033 2020 2020 2020 2020     0 1 3        
│ │ │ │ -0009f330: 2031 2033 2020 2020 2020 2020 2030 2032   1 3         0 2
│ │ │ │ -0009f340: 2033 2020 2020 2020 2020 2031 2032 2033   3         1 2 3
│ │ │ │ -0009f350: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0009f2b0: 2020 2020 2020 7c0a 7c78 2020 2b20 3336        |.|x  + 36
│ │ │ │ +0009f2c0: 3035 7820 202b 2031 3231 3234 7820 7820  05x  + 12124x x 
│ │ │ │ +0009f2d0: 7820 202d 2033 3035 3438 7820 7820 202d  x  - 30548x x  -
│ │ │ │ +0009f2e0: 2035 3737 3433 7820 7820 7820 202d 2031   57743x x x  - 1
│ │ │ │ +0009f2f0: 3235 3930 7820 7820 7820 202d 2020 2020  2590x x x  -    
│ │ │ │ +0009f300: 2020 2020 2020 7c0a 7c20 3220 2020 2020        |.| 2     
│ │ │ │ +0009f310: 2020 2032 2020 2020 2020 2020 2030 2031     2         0 1
│ │ │ │ +0009f320: 2033 2020 2020 2020 2020 2031 2033 2020   3         1 3  
│ │ │ │ +0009f330: 2020 2020 2020 2030 2032 2033 2020 2020         0 2 3    
│ │ │ │ +0009f340: 2020 2020 2031 2032 2033 2020 2020 2020       1 2 3      
│ │ │ │ +0009f350: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │  0009f360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009f370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009f380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009f390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009f3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0009f3a0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │  0009f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f3c0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +0009f3c0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  0009f3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f3f0: 2020 2020 2020 2020 2020 2020 7c0a 7c37              |.|7
│ │ │ │ -0009f400: 3431 3078 2078 2020 2d20 3636 3536 3578  410x x  - 66565x
│ │ │ │ -0009f410: 2078 2020 2d20 3439 3139 3278 2020 2d20   x  - 49192x  - 
│ │ │ │ -0009f420: 3731 3332 3178 2078 2020 2b20 3138 3433  71321x x  + 1843
│ │ │ │ -0009f430: 3378 2078 2020 2d20 3436 3436 3078 2078  3x x  - 46460x x
│ │ │ │ -0009f440: 2020 2b20 2020 2020 2020 2020 7c0a 7c20    +         |.| 
│ │ │ │ -0009f450: 2020 2020 3220 3420 2020 2020 2020 2020      2 4         
│ │ │ │ -0009f460: 3320 3420 2020 2020 2020 2020 3420 2020  3 4         4   
│ │ │ │ -0009f470: 2020 2020 2020 3020 3520 2020 2020 2020        0 5       
│ │ │ │ -0009f480: 2020 3120 3520 2020 2020 2020 2020 3220    1 5         2 
│ │ │ │ -0009f490: 3520 2020 2020 2020 2020 2020 7c0a 7c20  5           |.| 
│ │ │ │ +0009f3f0: 2020 2020 2020 7c0a 7c37 3431 3078 2078        |.|7410x x
│ │ │ │ +0009f400: 2020 2d20 3636 3536 3578 2078 2020 2d20    - 66565x x  - 
│ │ │ │ +0009f410: 3439 3139 3278 2020 2d20 3731 3332 3178  49192x  - 71321x
│ │ │ │ +0009f420: 2078 2020 2b20 3138 3433 3378 2078 2020   x  + 18433x x  
│ │ │ │ +0009f430: 2d20 3436 3436 3078 2078 2020 2b20 2020  - 46460x x  +   
│ │ │ │ +0009f440: 2020 2020 2020 7c0a 7c20 2020 2020 3220        |.|     2 
│ │ │ │ +0009f450: 3420 2020 2020 2020 2020 3320 3420 2020  4         3 4   
│ │ │ │ +0009f460: 2020 2020 2020 3420 2020 2020 2020 2020        4         
│ │ │ │ +0009f470: 3020 3520 2020 2020 2020 2020 3120 3520  0 5         1 5 
│ │ │ │ +0009f480: 2020 2020 2020 2020 3220 3520 2020 2020          2 5     
│ │ │ │ +0009f490: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f4e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009f4f0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0009f500: 2020 2032 2020 2020 2020 2020 2020 2032     2           2
│ │ │ │ -0009f510: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0009f520: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ -0009f530: 2020 2020 2020 2020 2020 2020 7c0a 7c31              |.|1
│ │ │ │ -0009f540: 3933 3537 7820 7820 202d 2032 3535 3035  9357x x  - 25505
│ │ │ │ -0009f550: 7820 7820 202b 2031 3135 3038 7820 7820  x x  + 11508x x 
│ │ │ │ -0009f560: 202b 2033 3530 3236 7820 7820 202d 2037   + 35026x x  - 7
│ │ │ │ -0009f570: 3930 3838 7820 202b 2032 3636 3135 7820  9088x  + 26615x 
│ │ │ │ -0009f580: 7820 7820 2020 2020 2020 2020 7c0a 7c20  x x         |.| 
│ │ │ │ -0009f590: 2020 2020 2032 2033 2020 2020 2020 2020       2 3        
│ │ │ │ -0009f5a0: 2030 2033 2020 2020 2020 2020 2031 2033   0 3         1 3
│ │ │ │ -0009f5b0: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
│ │ │ │ -0009f5c0: 2020 2020 2033 2020 2020 2020 2020 2030       3         0
│ │ │ │ -0009f5d0: 2031 2034 2020 2020 2020 2020 7c0a 7c2d   1 4        |.|-
│ │ │ │ +0009f4e0: 2020 2020 2020 7c0a 7c20 2020 2020 2032        |.|      2
│ │ │ │ +0009f4f0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +0009f500: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0009f510: 2020 2020 2032 2020 2020 2020 2020 2033       2         3
│ │ │ │ +0009f520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009f530: 2020 2020 2020 7c0a 7c31 3933 3537 7820        |.|19357x 
│ │ │ │ +0009f540: 7820 202d 2032 3535 3035 7820 7820 202b  x  - 25505x x  +
│ │ │ │ +0009f550: 2031 3135 3038 7820 7820 202b 2033 3530   11508x x  + 350
│ │ │ │ +0009f560: 3236 7820 7820 202d 2037 3930 3838 7820  26x x  - 79088x 
│ │ │ │ +0009f570: 202b 2032 3636 3135 7820 7820 7820 2020   + 26615x x x   
│ │ │ │ +0009f580: 2020 2020 2020 7c0a 7c20 2020 2020 2032        |.|      2
│ │ │ │ +0009f590: 2033 2020 2020 2020 2020 2030 2033 2020   3         0 3  
│ │ │ │ +0009f5a0: 2020 2020 2020 2031 2033 2020 2020 2020         1 3      
│ │ │ │ +0009f5b0: 2020 2032 2033 2020 2020 2020 2020 2033     2 3         3
│ │ │ │ +0009f5c0: 2020 2020 2020 2020 2030 2031 2034 2020           0 1 4  
│ │ │ │ +0009f5d0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │  0009f5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009f5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009f600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009f610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009f620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0009f620: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │  0009f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f640: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +0009f640: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  0009f650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f670: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ -0009f680: 3639 3878 2078 2020 2d20 3736 3839 3078  698x x  - 76890x
│ │ │ │ -0009f690: 2078 2020 2d20 3535 3537 3478 202c 2020   x  - 55574x ,  
│ │ │ │ +0009f670: 2020 2020 2020 7c0a 7c34 3639 3878 2078        |.|4698x x
│ │ │ │ +0009f680: 2020 2d20 3736 3839 3078 2078 2020 2d20    - 76890x x  - 
│ │ │ │ +0009f690: 3535 3537 3478 202c 2020 2020 2020 2020  55574x ,        
│ │ │ │  0009f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f6c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009f6d0: 2020 2020 3320 3520 2020 2020 2020 2020      3 5         
│ │ │ │ -0009f6e0: 3420 3520 2020 2020 2020 2020 3520 2020  4 5         5   
│ │ │ │ +0009f6c0: 2020 2020 2020 7c0a 7c20 2020 2020 3320        |.|     3 
│ │ │ │ +0009f6d0: 3520 2020 2020 2020 2020 3420 3520 2020  5         4 5   
│ │ │ │ +0009f6e0: 2020 2020 2020 3520 2020 2020 2020 2020        5         
│ │ │ │  0009f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f710: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0009f710: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0009f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f760: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009f770: 2020 2020 2020 2020 3220 2020 2020 2020          2       
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│ │ │ │ +0009f770: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0009f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f790: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0009f790: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0009f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009f7b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │ -0009f7d0: 3435 3778 2078 2078 2020 2b20 3837 3978  457x x x  + 879x
│ │ │ │ -0009f7e0: 2078 2078 2020 2b20 3738 3536 3078 2078   x x  + 78560x x
│ │ │ │ -0009f7f0: 2020 2b20 3833 3939 3778 2078 2078 2020    + 83997x x x  
│ │ │ │ -0009f800: 2d20 2020 2020 2020 2020 2020 7c0a 7c20  -           |.| 
│ │ │ │ -0009f810: 2020 2020 2020 2020 3120 3420 2020 2020          1 4     
│ │ │ │ -0009f820: 2020 2020 3020 3220 3420 2020 2020 2020      0 2 4       
│ │ │ │ -0009f830: 3120 3220 3420 2020 2020 2020 2020 3220  1 2 4         2 
│ │ │ │ -0009f840: 3420 2020 2020 2020 2020 3020 3320 3420  4         0 3 4 
│ │ │ │ -0009f850: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0009f7b0: 2020 2020 2020 7c0a 7c20 2d20 3130 3033        |.| - 1003
│ │ │ │ +0009f7c0: 3678 2078 2020 2b20 3339 3435 3778 2078  6x x  + 39457x x
│ │ │ │ +0009f7d0: 2078 2020 2b20 3837 3978 2078 2078 2020   x  + 879x x x  
│ │ │ │ +0009f7e0: 2b20 3738 3536 3078 2078 2020 2b20 3833  + 78560x x  + 83
│ │ │ │ +0009f7f0: 3939 3778 2078 2078 2020 2d20 2020 2020  997x x x  -     
│ │ │ │ +0009f800: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
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│ │ │ │ +0009f820: 3220 3420 2020 2020 2020 3120 3220 3420  2 4       1 2 4 
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│ │ │ │ +0009f850: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
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│ │ │ │  0009f880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0009f8d0: 2032 2020 2020 2020 2020 2020 2020 2032   2             2
│ │ │ │ -0009f8e0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0009f8f0: 2020 2020 2020 2032 2020 2020 7c0a 7c34         2    |.|4
│ │ │ │ -0009f900: 3235 3532 7820 7820 7820 202b 2036 3937  2552x x x  + 697
│ │ │ │ -0009f910: 3532 7820 7820 7820 202d 2037 3837 3133  52x x x  - 78713
│ │ │ │ -0009f920: 7820 7820 202b 2035 3236 3938 7820 7820  x x  + 52698x x 
│ │ │ │ -0009f930: 202d 2034 3635 3937 7820 7820 202d 2034   - 46597x x  - 4
│ │ │ │ -0009f940: 3236 3636 7820 7820 202b 2020 7c0a 7c20  2666x x  +  |.| 
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│ │ │ │ -0009f960: 2020 2032 2033 2034 2020 2020 2020 2020     2 3 4        
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│ │ │ │ -0009f980: 2020 2020 2020 2020 2031 2034 2020 2020           1 4    
│ │ │ │ -0009f990: 2020 2020 2032 2034 2020 2020 7c0a 7c2d       2 4    |.|-
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│ │ │ │ +0009f940: 7820 202b 2020 7c0a 7c20 2020 2020 2031  x  +  |.|      1
│ │ │ │ +0009f950: 2033 2034 2020 2020 2020 2020 2032 2033   3 4         2 3
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│ │ │ │ +0009f990: 2034 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   4    |.|-------
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│ │ │ │ -0009fa10: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0009f9e0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +0009f9f0: 2032 2020 2020 2020 2020 2033 2020 2020   2         3    
│ │ │ │ +0009fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0009fa10: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  0009fa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009fa30: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ -0009fa40: 3431 3430 7820 7820 202d 2032 3932 3036  4140x x  - 29206
│ │ │ │ -0009fa50: 7820 202d 2038 3834 3035 7820 7820 7820  x  - 88405x x x 
│ │ │ │ -0009fa60: 202d 2031 3537 3730 7820 7820 202b 2031   - 15770x x  + 1
│ │ │ │ -0009fa70: 3830 3539 7820 7820 7820 202d 2031 3433  8059x x x  - 143
│ │ │ │ -0009fa80: 3632 7820 7820 7820 202b 2020 7c0a 7c20  62x x x  +  |.| 
│ │ │ │ -0009fa90: 2020 2020 2033 2034 2020 2020 2020 2020       3 4        
│ │ │ │ -0009faa0: 2034 2020 2020 2020 2020 2030 2031 2035   4         0 1 5
│ │ │ │ -0009fab0: 2020 2020 2020 2020 2031 2035 2020 2020           1 5    
│ │ │ │ -0009fac0: 2020 2020 2030 2032 2035 2020 2020 2020       0 2 5      
│ │ │ │ -0009fad0: 2020 2031 2032 2035 2020 2020 7c0a 7c2d     1 2 5    |.|-
│ │ │ │ +0009fa30: 2020 2020 2020 7c0a 7c33 3431 3430 7820        |.|34140x 
│ │ │ │ +0009fa40: 7820 202d 2032 3932 3036 7820 202d 2038  x  - 29206x  - 8
│ │ │ │ +0009fa50: 3834 3035 7820 7820 7820 202d 2031 3537  8405x x x  - 157
│ │ │ │ +0009fa60: 3730 7820 7820 202b 2031 3830 3539 7820  70x x  + 18059x 
│ │ │ │ +0009fa70: 7820 7820 202d 2031 3433 3632 7820 7820  x x  - 14362x x 
│ │ │ │ +0009fa80: 7820 202b 2020 7c0a 7c20 2020 2020 2033  x  +  |.|      3
│ │ │ │ +0009fa90: 2034 2020 2020 2020 2020 2034 2020 2020   4         4    
│ │ │ │ +0009faa0: 2020 2020 2030 2031 2035 2020 2020 2020       0 1 5      
│ │ │ │ +0009fab0: 2020 2031 2035 2020 2020 2020 2020 2030     1 5         0
│ │ │ │ +0009fac0: 2032 2035 2020 2020 2020 2020 2031 2032   2 5         1 2
│ │ │ │ +0009fad0: 2035 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   5    |.|-------
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│ │ │ │ +0009fb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0009fb70: 2020 2020 2020 2020 2020 2020 7c0a 7c38              |.|8
│ │ │ │ -0009fb80: 3739 3335 7820 7820 202b 2036 3733 3731  7935x x  + 67371
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│ │ │ │ -0009fba0: 7820 7820 202d 2038 3134 3831 7820 7820  x x  - 81481x x 
│ │ │ │ -0009fbb0: 7820 202d 2036 3132 3836 7820 7820 202b  x  - 61286x x  +
│ │ │ │ -0009fbc0: 2036 3338 3738 7820 7820 7820 7c0a 7c20   63878x x x |.| 
│ │ │ │ -0009fbd0: 2020 2020 2032 2035 2020 2020 2020 2020       2 5        
│ │ │ │ -0009fbe0: 2030 2033 2035 2020 2020 2020 2020 2031   0 3 5         1
│ │ │ │ -0009fbf0: 2033 2035 2020 2020 2020 2020 2032 2033   3 5         2 3
│ │ │ │ -0009fc00: 2035 2020 2020 2020 2020 2033 2035 2020   5         3 5  
│ │ │ │ -0009fc10: 2020 2020 2020 2030 2034 2035 7c0a 7c2d         0 4 5|.|-
│ │ │ │ +0009fb60: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0009fb70: 2020 2020 2020 7c0a 7c38 3739 3335 7820        |.|87935x 
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│ │ │ │ +0009fb90: 202d 2036 3035 3834 7820 7820 7820 202d   - 60584x x x  -
│ │ │ │ +0009fba0: 2038 3134 3831 7820 7820 7820 202d 2036   81481x x x  - 6
│ │ │ │ +0009fbb0: 3132 3836 7820 7820 202b 2036 3338 3738  1286x x  + 63878
│ │ │ │ +0009fbc0: 7820 7820 7820 7c0a 7c20 2020 2020 2032  x x x |.|      2
│ │ │ │ +0009fbd0: 2035 2020 2020 2020 2020 2030 2033 2035   5         0 3 5
│ │ │ │ +0009fbe0: 2020 2020 2020 2020 2031 2033 2035 2020           1 3 5  
│ │ │ │ +0009fbf0: 2020 2020 2020 2032 2033 2035 2020 2020         2 3 5    
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│ │ │ │ +0009fc10: 2030 2034 2035 7c0a 7c2d 2d2d 2d2d 2d2d   0 4 5|.|-------
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│ │ │ │ -0009fcb0: 2020 2020 2020 2020 2020 2032 7c0a 7c2d             2|.|-
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│ │ │ │ -0009fd00: 202b 2031 3031 3637 7820 7820 7c0a 7c20   + 10167x x |.| 
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│ │ │ │ -0009fd50: 2020 2020 2020 2020 2031 2035 7c0a 7c2d           1 5|.|-
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│ │ │ │ +0009fcf0: 2034 3538 3539 7820 7820 202b 2031 3031   45859x x  + 101
│ │ │ │ +0009fd00: 3637 7820 7820 7c0a 7c20 2020 2020 2020  67x x |.|       
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│ │ │ │ +0009fd20: 2034 2035 2020 2020 2020 2020 2033 2034   4 5         3 4
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│ │ │ │ -0009fdf0: 2020 2020 2020 2020 2020 2020 7c0a 7c2b              |.|+
│ │ │ │ -0009fe00: 2038 3939 3378 2078 2020 2d20 3737 3232   8993x x  - 7722
│ │ │ │ -0009fe10: 3278 2078 2020 2d20 3431 3039 7820 7820  2x x  - 4109x x 
│ │ │ │ -0009fe20: 202d 2031 3630 3238 7820 2020 2020 2020   - 16028x       
│ │ │ │ +0009fdf0: 2020 2020 2020 7c0a 7c2b 2038 3939 3378        |.|+ 8993x
│ │ │ │ +0009fe00: 2078 2020 2d20 3737 3232 3278 2078 2020   x  - 77222x x  
│ │ │ │ +0009fe10: 2d20 3431 3039 7820 7820 202d 2031 3630  - 4109x x  - 160
│ │ │ │ +0009fe20: 3238 7820 2020 2020 2020 2020 2020 2020  28x             
│ │ │ │  0009fe30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009fe40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009fe50: 2020 2020 2020 3220 3520 2020 2020 2020        2 5       
│ │ │ │ -0009fe60: 2020 3320 3520 2020 2020 2020 2034 2035    3 5        4 5
│ │ │ │ -0009fe70: 2020 2020 2020 2020 2035 2020 2020 2020           5      
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│ │ │ │ +0009fe90: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
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│ │ │ │  0009fec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0009fed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0009fee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0009fef0: 3520 3a20 6465 7363 7269 6265 2050 6869  5 : describe Phi
│ │ │ │ +0009fee0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 6465  ------+.|i5 : de
│ │ │ │ +0009fef0: 7363 7269 6265 2050 6869 2020 2020 2020  scribe Phi      
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│ │ │ │  0009ff10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009ff20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0009ff50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0009ff70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ff80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0009ff90: 3520 3d20 7261 7469 6f6e 616c 206d 6170  5 = rational map
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│ │ │ │ -0009ffb0: 7320 6f66 2064 6567 7265 6520 3220 2020  s of degree 2   
│ │ │ │ +0009ff80: 2020 2020 2020 7c0a 7c6f 3520 3d20 7261        |.|o5 = ra
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│ │ │ │ +0009ffb0: 6567 7265 6520 3220 2020 2020 2020 2020  egree 2         
│ │ │ │  0009ffc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0009ffd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0009ffe0: 2020 2020 736f 7572 6365 2076 6172 6965      source varie
│ │ │ │ -0009fff0: 7479 3a20 5050 5e35 2020 2020 2020 2020  ty: PP^5        
│ │ │ │ +0009ffd0: 2020 2020 2020 7c0a 7c20 2020 2020 736f        |.|     so
│ │ │ │ +0009ffe0: 7572 6365 2076 6172 6965 7479 3a20 5050  urce variety: PP
│ │ │ │ +0009fff0: 5e35 2020 2020 2020 2020 2020 2020 2020  ^5              
│ │ │ │  000a0000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a0010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a0020: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000a0030: 2020 2020 7461 7267 6574 2076 6172 6965      target varie
│ │ │ │ -000a0040: 7479 3a20 5050 5e35 2020 2020 2020 2020  ty: PP^5        
│ │ │ │ +000a0020: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ +000a0030: 7267 6574 2076 6172 6965 7479 3a20 5050  rget variety: PP
│ │ │ │ +000a0040: 5e35 2020 2020 2020 2020 2020 2020 2020  ^5              
│ │ │ │  000a0050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a0060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a0070: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000a0080: 2020 2020 636f 6566 6669 6369 656e 7420      coefficient 
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│ │ │ │  000a00f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  000a0140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a0150: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000a0160: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
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│ │ │ │  000a01a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000a0250: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │  000a0280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a0290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a02a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000a02b0: 2020 2020 636f 6566 6669 6369 656e 7420      coefficient 
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│ │ │ │ +000a02a0: 2020 2020 2020 7c0a 7c20 2020 2020 636f        |.|     co
│ │ │ │ +000a02b0: 6566 6669 6369 656e 7420 7269 6e67 3a20  efficient ring: 
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│ │ │ │  000a02d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a02e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a02f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
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│ │ │ │ -000a0380: 3a20 5261 7469 6f6e 616c 4d61 7020 7c7c  : RationalMap ||
│ │ │ │ -000a0390: 2049 6465 616c 2c20 2d2d 2072 6573 7472   Ideal, -- restr
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│ │ │ │ +000a03e0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  =========..  * *
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│ │ │ │ +000a0400: 207c 2049 6465 616c 3a20 5261 7469 6f6e   | Ideal: Ration
│ │ │ │ +000a0410: 616c 4d61 7020 7c20 4964 6561 6c2c 202d  alMap | Ideal, -
│ │ │ │ +000a0420: 2d20 7265 7374 7269 6374 696f 6e20 6f66  - restriction of
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│ │ │ │ +000a0450: 6c4d 6170 207c 2052 696e 6722 0a20 202a  lMap | Ring".  *
│ │ │ │ +000a0460: 2022 5261 7469 6f6e 616c 4d61 7020 7c20   "RationalMap | 
│ │ │ │ +000a0470: 5269 6e67 456c 656d 656e 7422 0a2d 2d2d  RingElement".---
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│ │ │ │  000a04a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a04b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000a0590: 743a 2072 6174 696f 6e61 6c4d 6170 5f6c  t: rationalMap_l
│ │ │ │ -000a05a0: 7049 6465 616c 5f63 6d5a 5a5f 636d 5a5a  pIdeal_cmZZ_cmZZ
│ │ │ │ -000a05b0: 5f72 702c 2050 7265 763a 2052 6174 696f  _rp, Prev: Ratio
│ │ │ │ -000a05c0: 6e61 6c4d 6170 207c 2049 6465 616c 2c20  nalMap | Ideal, 
│ │ │ │ -000a05d0: 5570 3a20 546f 700a 0a52 6174 696f 6e61  Up: Top..Rationa
│ │ │ │ -000a05e0: 6c4d 6170 207c 7c20 4964 6561 6c20 2d2d  lMap || Ideal --
│ │ │ │ -000a05f0: 2072 6573 7472 6963 7469 6f6e 206f 6620   restriction of 
│ │ │ │ -000a0600: 6120 7261 7469 6f6e 616c 206d 6170 0a2a  a rational map.*
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│ │ │ │ +000a0560: 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 653a  mona.info, Node:
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│ │ │ │ +000a0580: 4964 6561 6c2c 204e 6578 743a 2072 6174  Ideal, Next: rat
│ │ │ │ +000a0590: 696f 6e61 6c4d 6170 5f6c 7049 6465 616c  ionalMap_lpIdeal
│ │ │ │ +000a05a0: 5f63 6d5a 5a5f 636d 5a5a 5f72 702c 2050  _cmZZ_cmZZ_rp, P
│ │ │ │ +000a05b0: 7265 763a 2052 6174 696f 6e61 6c4d 6170  rev: RationalMap
│ │ │ │ +000a05c0: 207c 2049 6465 616c 2c20 5570 3a20 546f   | Ideal, Up: To
│ │ │ │ +000a05d0: 700a 0a52 6174 696f 6e61 6c4d 6170 207c  p..RationalMap |
│ │ │ │ +000a05e0: 7c20 4964 6561 6c20 2d2d 2072 6573 7472  | Ideal -- restr
│ │ │ │ +000a05f0: 6963 7469 6f6e 206f 6620 6120 7261 7469  iction of a rati
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│ │ │ │  000a0610: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000a0620: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000a0630: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000a0640: 2a2a 2a2a 0a0a 2020 2a20 4f70 6572 6174  ****..  * Operat
│ │ │ │ -000a0650: 6f72 3a20 2a6e 6f74 6520 7c7c 3a20 284d  or: *note ||: (M
│ │ │ │ -000a0660: 6163 6175 6c61 7932 446f 6329 7c7c 2c0a  acaulay2Doc)||,.
│ │ │ │ -000a0670: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ -000a0680: 2020 2020 5068 6920 7c7c 204a 0a20 202a      Phi || J.  *
│ │ │ │ -000a0690: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -000a06a0: 2050 6869 2c20 6120 2a6e 6f74 6520 7261   Phi, a *note ra
│ │ │ │ -000a06b0: 7469 6f6e 616c 206d 6170 3a20 5261 7469  tional map: Rati
│ │ │ │ -000a06c0: 6f6e 616c 4d61 702c 2c20 245c 7068 693a  onalMap,, $\phi:
│ │ │ │ -000a06d0: 5820 5c64 6173 6872 6967 6874 6172 726f  X \dashrightarro
│ │ │ │ -000a06e0: 7720 5924 0a20 2020 2020 202a 204a 2c20  w Y$.      * J, 
│ │ │ │ -000a06f0: 616e 202a 6e6f 7465 2069 6465 616c 3a20  an *note ideal: 
│ │ │ │ -000a0700: 284d 6163 6175 6c61 7932 446f 6329 4964  (Macaulay2Doc)Id
│ │ │ │ -000a0710: 6561 6c2c 2c20 6120 686f 6d6f 6765 6e65  eal,, a homogene
│ │ │ │ -000a0720: 6f75 7320 6964 6561 6c20 6f66 2061 0a20  ous ideal of a. 
│ │ │ │ -000a0730: 2020 2020 2020 2073 7562 7661 7269 6574         subvariet
│ │ │ │ -000a0740: 7920 245a 5c73 7562 7365 7420 5924 0a20  y $Z\subset Y$. 
│ │ │ │ -000a0750: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -000a0760: 2020 2a20 6120 2a6e 6f74 6520 7261 7469    * a *note rati
│ │ │ │ -000a0770: 6f6e 616c 206d 6170 3a20 5261 7469 6f6e  onal map: Ration
│ │ │ │ -000a0780: 616c 4d61 702c 2c20 7468 6520 7265 7374  alMap,, the rest
│ │ │ │ -000a0790: 7269 6374 696f 6e20 6f66 2024 5c70 6869  riction of $\phi
│ │ │ │ -000a07a0: 2420 746f 0a20 2020 2020 2020 2024 7b5c  $ to.        ${\
│ │ │ │ -000a07b0: 7068 697d 5e7b 282d 3129 7d20 5a24 2c20  phi}^{(-1)} Z$, 
│ │ │ │ -000a07c0: 247b 7b5c 7068 697d 7c7d 5f7b 7b5c 7068  ${{\phi}|}_{{\ph
│ │ │ │ -000a07d0: 697d 5e7b 282d 3129 7d20 5a7d 3a20 7b5c  i}^{(-1)} Z}: {\
│ │ │ │ -000a07e0: 7068 697d 5e7b 282d 3129 7d20 5a0a 2020  phi}^{(-1)} Z.  
│ │ │ │ -000a07f0: 2020 2020 2020 5c64 6173 6872 6967 6874        \dashright
│ │ │ │ -000a0800: 6172 726f 7720 5a24 0a0a 4465 7363 7269  arrow Z$..Descri
│ │ │ │ -000a0810: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -000a0820: 3d0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  =..+------------
│ │ │ │ +000a0630: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +000a0640: 2020 2a20 4f70 6572 6174 6f72 3a20 2a6e    * Operator: *n
│ │ │ │ +000a0650: 6f74 6520 7c7c 3a20 284d 6163 6175 6c61  ote ||: (Macaula
│ │ │ │ +000a0660: 7932 446f 6329 7c7c 2c0a 2020 2a20 5573  y2Doc)||,.  * Us
│ │ │ │ +000a0670: 6167 653a 200a 2020 2020 2020 2020 5068  age: .        Ph
│ │ │ │ +000a0680: 6920 7c7c 204a 0a20 202a 2049 6e70 7574  i || J.  * Input
│ │ │ │ +000a0690: 733a 0a20 2020 2020 202a 2050 6869 2c20  s:.      * Phi, 
│ │ │ │ +000a06a0: 6120 2a6e 6f74 6520 7261 7469 6f6e 616c  a *note rational
│ │ │ │ +000a06b0: 206d 6170 3a20 5261 7469 6f6e 616c 4d61   map: RationalMa
│ │ │ │ +000a06c0: 702c 2c20 245c 7068 693a 5820 5c64 6173  p,, $\phi:X \das
│ │ │ │ +000a06d0: 6872 6967 6874 6172 726f 7720 5924 0a20  hrightarrow Y$. 
│ │ │ │ +000a06e0: 2020 2020 202a 204a 2c20 616e 202a 6e6f       * J, an *no
│ │ │ │ +000a06f0: 7465 2069 6465 616c 3a20 284d 6163 6175  te ideal: (Macau
│ │ │ │ +000a0700: 6c61 7932 446f 6329 4964 6561 6c2c 2c20  lay2Doc)Ideal,, 
│ │ │ │ +000a0710: 6120 686f 6d6f 6765 6e65 6f75 7320 6964  a homogeneous id
│ │ │ │ +000a0720: 6561 6c20 6f66 2061 0a20 2020 2020 2020  eal of a.       
│ │ │ │ +000a0730: 2073 7562 7661 7269 6574 7920 245a 5c73   subvariety $Z\s
│ │ │ │ +000a0740: 7562 7365 7420 5924 0a20 202a 204f 7574  ubset Y$.  * Out
│ │ │ │ +000a0750: 7075 7473 3a0a 2020 2020 2020 2a20 6120  puts:.      * a 
│ │ │ │ +000a0760: 2a6e 6f74 6520 7261 7469 6f6e 616c 206d  *note rational m
│ │ │ │ +000a0770: 6170 3a20 5261 7469 6f6e 616c 4d61 702c  ap: RationalMap,
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│ │ │ │ +000a07a0: 2020 2020 2020 2024 7b5c 7068 697d 5e7b         ${\phi}^{
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│ │ │ │ +000a0820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000a0870: 2d2b 0a7c 6931 203a 2050 3520 3d20 5a5a  -+.|i1 : P5 = ZZ
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│ │ │ │  000a08a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000a08d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000a09d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000a0a50: 2d2b 0a7c 6932 203a 2050 6869 203d 2072  -+.|i2 : Phi = r
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│ │ │ │ +000a0a50: 203a 2050 6869 203d 2072 6174 696f 6e61   : Phi = rationa
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│ │ │ │ +000a11e0: 2020 2020 7820 7820 202d 2078 2078 202c      x x  - x x ,
│ │ │ │ +000a11f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1220: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a1230: 2020 2020 2020 2020 2020 2031 2032 2020             1 2  
│ │ │ │ -000a1240: 2020 3020 3420 2020 2020 2020 2020 2020    0 4           
│ │ │ │ +000a1210: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a1220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a1230: 2020 2020 2031 2032 2020 2020 3020 3420       1 2    0 4 
│ │ │ │ +000a1240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1270: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a1260: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a1270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a12a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a12b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a12c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a12d0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +000a12b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a12c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a12d0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  000a12e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a12f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1310: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a1320: 2020 2020 2020 2020 2020 7820 202d 2078            x  - x
│ │ │ │ -000a1330: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │ +000a1300: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a1310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a1320: 2020 2020 7820 202d 2078 2078 2020 2020      x  - x x    
│ │ │ │ +000a1330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1360: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a1370: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -000a1380: 3020 3320 2020 2020 2020 2020 2020 2020  0 3             
│ │ │ │ +000a1350: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a1360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a1370: 2020 2020 2031 2020 2020 3020 3320 2020       1    0 3   
│ │ │ │ +000a1380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a13a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a13b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a13c0: 2020 2020 2020 2020 207d 2020 2020 2020           }      
│ │ │ │ +000a13a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a13b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a13c0: 2020 207d 2020 2020 2020 2020 2020 2020     }            
│ │ │ │  000a13d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a13e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a13f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1400: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a13f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a1400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1450: 207c 0a7c 6f32 203a 2052 6174 696f 6e61   |.|o2 : Rationa
│ │ │ │ -000a1460: 6c4d 6170 2028 7175 6164 7261 7469 6320  lMap (quadratic 
│ │ │ │ -000a1470: 7261 7469 6f6e 616c 206d 6170 2066 726f  rational map fro
│ │ │ │ -000a1480: 6d20 5050 5e35 2074 6f20 5050 5e35 2920  m PP^5 to PP^5) 
│ │ │ │ -000a1490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a14a0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ +000a1440: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +000a1450: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ +000a1460: 7175 6164 7261 7469 6320 7261 7469 6f6e  quadratic ration
│ │ │ │ +000a1470: 616c 206d 6170 2066 726f 6d20 5050 5e35  al map from PP^5
│ │ │ │ +000a1480: 2074 6f20 5050 5e35 2920 2020 2020 2020   to PP^5)       
│ │ │ │ +000a1490: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ +000a14a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a14b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a14c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a14d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a14e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a14f0: 2d7c 0a7c 5f30 2a78 5f35 2c78 5f31 2a78  -|.|_0*x_5,x_1*x
│ │ │ │ -000a1500: 5f32 2d78 5f30 2a78 5f34 2c78 5f31 5e32  _2-x_0*x_4,x_1^2
│ │ │ │ -000a1510: 2d78 5f30 2a78 5f33 7d20 2020 2020 2020  -x_0*x_3}       
│ │ │ │ +000a14e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 5f30  -----------|.|_0
│ │ │ │ +000a14f0: 2a78 5f35 2c78 5f31 2a78 5f32 2d78 5f30  *x_5,x_1*x_2-x_0
│ │ │ │ +000a1500: 2a78 5f34 2c78 5f31 5e32 2d78 5f30 2a78  *x_4,x_1^2-x_0*x
│ │ │ │ +000a1510: 5f33 7d20 2020 2020 2020 2020 2020 2020  _3}             
│ │ │ │  000a1520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1540: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000a1530: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000a1540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a1550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a1560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a1570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a1580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a1590: 2d2b 0a7c 6933 203a 204a 203d 2069 6465  -+.|i3 : J = ide
│ │ │ │ -000a15a0: 616c 2072 616e 646f 6d28 312c 5035 293b  al random(1,P5);
│ │ │ │ +000a1580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ +000a1590: 203a 204a 203d 2069 6465 616c 2072 616e   : J = ideal ran
│ │ │ │ +000a15a0: 646f 6d28 312c 5035 293b 2020 2020 2020  dom(1,P5);      
│ │ │ │  000a15b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a15c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a15d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a15e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a15d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a15e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a15f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1630: 207c 0a7c 6f33 203a 2049 6465 616c 206f   |.|o3 : Ideal o
│ │ │ │ -000a1640: 6620 5035 2020 2020 2020 2020 2020 2020  f P5            
│ │ │ │ +000a1620: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +000a1630: 203a 2049 6465 616c 206f 6620 5035 2020   : Ideal of P5  
│ │ │ │ +000a1640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1680: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000a1670: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000a1680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a1690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a16a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a16b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a16c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a16d0: 2d2b 0a7c 6934 203a 2050 6869 2720 3d20  -+.|i4 : Phi' = 
│ │ │ │ -000a16e0: 5068 697c 7c4a 2020 2020 2020 2020 2020  Phi||J          
│ │ │ │ +000a16c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +000a16d0: 203a 2050 6869 2720 3d20 5068 697c 7c4a   : Phi' = Phi||J
│ │ │ │ +000a16e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a16f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1720: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a1710: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a1720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1770: 207c 0a7c 6f34 203d 202d 2d20 7261 7469   |.|o4 = -- rati
│ │ │ │ -000a1780: 6f6e 616c 206d 6170 202d 2d20 2020 2020  onal map --     
│ │ │ │ +000a1760: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +000a1770: 203d 202d 2d20 7261 7469 6f6e 616c 206d   = -- rational m
│ │ │ │ +000a1780: 6170 202d 2d20 2020 2020 2020 2020 2020  ap --           
│ │ │ │  000a1790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a17a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a17b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a17c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a17b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a17c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a17d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a17e0: 2020 2020 2020 5a5a 2020 2020 2020 2020        ZZ        
│ │ │ │ +000a17e0: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │  000a17f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1810: 207c 0a7c 2020 2020 2073 6f75 7263 653a   |.|     source:
│ │ │ │ -000a1820: 2073 7562 7661 7269 6574 7920 6f66 2050   subvariety of P
│ │ │ │ -000a1830: 726f 6a28 2d2d 2d2d 2d2d 5b78 202c 2078  roj(------[x , x
│ │ │ │ +000a1800: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a1810: 2020 2073 6f75 7263 653a 2073 7562 7661     source: subva
│ │ │ │ +000a1820: 7269 6574 7920 6f66 2050 726f 6a28 2d2d  riety of Proj(--
│ │ │ │ +000a1830: 2d2d 2d2d 5b78 202c 2078 2020 2020 2020  ----[x , x      
│ │ │ │  000a1840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1860: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a1870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1880: 2020 2020 3139 3031 3831 2020 3020 2020      190181  0   
│ │ │ │ -000a1890: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -000a18a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a18b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a18c0: 207b 2020 2020 2020 2020 2020 2020 2020   {              
│ │ │ │ +000a1850: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a1860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a1870: 2020 2020 2020 2020 2020 2020 2020 3139                19
│ │ │ │ +000a1880: 3031 3831 2020 3020 2020 3120 2020 2020  0181  0   1     
│ │ │ │ +000a1890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a18a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a18b0: 2020 2020 2020 2020 2020 207b 2020 2020             {    
│ │ │ │ +000a18c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a18d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a18e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a18f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1900: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a1910: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -000a1920: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000a18f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a1900: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +000a1910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a1920: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  000a1930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1950: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a1960: 2020 7820 202b 2039 3730 3278 2078 2020    x  + 9702x x  
│ │ │ │ -000a1970: 2d20 3934 3239 3478 2020 2d20 7820 7820  - 94294x  - x x 
│ │ │ │ +000a1940: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a1950: 2020 2020 2020 2020 2020 2020 7820 202b              x  +
│ │ │ │ +000a1960: 2039 3730 3278 2078 2020 2d20 3934 3239   9702x x  - 9429
│ │ │ │ +000a1970: 3478 2020 2d20 7820 7820 2020 2020 2020  4x  - x x       
│ │ │ │  000a1980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a19a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a19b0: 2020 2031 2020 2020 2020 2020 3120 3220     1        1 2 
│ │ │ │ -000a19c0: 2020 2020 2020 2020 3220 2020 2030 2033          2    0 3
│ │ │ │ +000a1990: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a19a0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +000a19b0: 2020 2020 2020 3120 3220 2020 2020 2020        1 2       
│ │ │ │ +000a19c0: 2020 3220 2020 2030 2033 2020 2020 2020    2    0 3      
│ │ │ │  000a19d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a19e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a19f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a1a00: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │ +000a19e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a19f0: 2020 2020 2020 2020 2020 207d 2020 2020             }    
│ │ │ │ +000a1a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1a40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a1a30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a1a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1a60: 2020 2020 2020 5a5a 2020 2020 2020 2020        ZZ        
│ │ │ │ +000a1a60: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │  000a1a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1a90: 207c 0a7c 2020 2020 2074 6172 6765 743a   |.|     target:
│ │ │ │ -000a1aa0: 2073 7562 7661 7269 6574 7920 6f66 2050   subvariety of P
│ │ │ │ -000a1ab0: 726f 6a28 2d2d 2d2d 2d2d 5b78 202c 2078  roj(------[x , x
│ │ │ │ +000a1a80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a1a90: 2020 2074 6172 6765 743a 2073 7562 7661     target: subva
│ │ │ │ +000a1aa0: 7269 6574 7920 6f66 2050 726f 6a28 2d2d  riety of Proj(--
│ │ │ │ +000a1ab0: 2d2d 2d2d 5b78 202c 2078 2020 2020 2020  ----[x , x      
│ │ │ │  000a1ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1ae0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a1af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1b00: 2020 2020 3139 3031 3831 2020 3020 2020      190181  0   
│ │ │ │ -000a1b10: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -000a1b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1b30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a1b40: 207b 2020 2020 2020 2020 2020 2020 2020   {              
│ │ │ │ +000a1ad0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a1ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a1af0: 2020 2020 2020 2020 2020 2020 2020 3139                19
│ │ │ │ +000a1b00: 3031 3831 2020 3020 2020 3120 2020 2020  0181  0   1     
│ │ │ │ +000a1b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a1b20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a1b30: 2020 2020 2020 2020 2020 207b 2020 2020             {    
│ │ │ │ +000a1b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a1b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1b80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a1b90: 2020 7820 202b 2031 3635 3636 7820 202d    x  + 16566x  -
│ │ │ │ -000a1ba0: 2037 3031 3538 7820 202d 2033 3831 3438   70158x  - 38148
│ │ │ │ -000a1bb0: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ -000a1bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1bd0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a1be0: 2020 2030 2020 2020 2020 2020 2031 2020     0         1  
│ │ │ │ -000a1bf0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +000a1b70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a1b80: 2020 2020 2020 2020 2020 2020 7820 202b              x  +
│ │ │ │ +000a1b90: 2031 3635 3636 7820 202d 2037 3031 3538   16566x  - 70158
│ │ │ │ +000a1ba0: 7820 202d 2033 3831 3438 7820 2020 2020  x  - 38148x     
│ │ │ │ +000a1bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a1bc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a1bd0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +000a1be0: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ +000a1bf0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  000a1c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a1c20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a1c30: 207d 2020 2020 2020 2020 2020 2020 2020   }              
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│ │ │ │  000a3460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3480: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a3470: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a34a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a34b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a34c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a34d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a34c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a34d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a34e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a34f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3520: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a3510: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3570: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a3560: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a35a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a35b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a35c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a35b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a35c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a35d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a35e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a35f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3610: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a3600: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3660: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a3650: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a36a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a36b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a36a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a36b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a36c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a36d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a36e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a36f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3700: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a36f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3750: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a3740: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a37a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a3790: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a37a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a37b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a37c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a37d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a37e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a37f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a37e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a37f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3840: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a3830: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3890: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a3880: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a38a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a38b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a38c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a38d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a38e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a38d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a38e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a38f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3930: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a3920: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3980: 207c 0a7c 3933 3539 3378 2078 2020 2d20   |.|93593x x  - 
│ │ │ │ -000a3990: 3533 3235 3178 2078 2020 2020 2020 2020  53251x x        
│ │ │ │ +000a3970: 2020 2020 2020 2020 2020 207c 0a7c 3933             |.|93
│ │ │ │ +000a3980: 3539 3378 2078 2020 2d20 3533 3235 3178  593x x  - 53251x
│ │ │ │ +000a3990: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │  000a39a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a39b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a39c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a39d0: 207c 0a7c 2020 2020 2020 3120 3520 2020   |.|      1 5   
│ │ │ │ -000a39e0: 2020 2020 2020 3320 3520 2020 2020 2020        3 5       
│ │ │ │ +000a39c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a39d0: 2020 2020 3120 3520 2020 2020 2020 2020      1 5         
│ │ │ │ +000a39e0: 3320 3520 2020 2020 2020 2020 2020 2020  3 5             
│ │ │ │  000a39f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3a20: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000a3a10: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000a3a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a3a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a3a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a3a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a3a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a3a70: 2d2b 0a7c 6935 203a 2064 6573 6372 6962  -+.|i5 : describ
│ │ │ │ -000a3a80: 6520 5068 6920 2020 2020 2020 2020 2020  e Phi           
│ │ │ │ +000a3a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ +000a3a70: 203a 2064 6573 6372 6962 6520 5068 6920   : describe Phi 
│ │ │ │ +000a3a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3ac0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a3ab0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3b10: 207c 0a7c 6f35 203d 2072 6174 696f 6e61   |.|o5 = rationa
│ │ │ │ -000a3b20: 6c20 6d61 7020 6465 6669 6e65 6420 6279  l map defined by
│ │ │ │ -000a3b30: 2066 6f72 6d73 206f 6620 6465 6772 6565   forms of degree
│ │ │ │ -000a3b40: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -000a3b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3b60: 207c 0a7c 2020 2020 2073 6f75 7263 6520   |.|     source 
│ │ │ │ -000a3b70: 7661 7269 6574 793a 2050 505e 3520 2020  variety: PP^5   
│ │ │ │ +000a3b00: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +000a3b10: 203d 2072 6174 696f 6e61 6c20 6d61 7020   = rational map 
│ │ │ │ +000a3b20: 6465 6669 6e65 6420 6279 2066 6f72 6d73  defined by forms
│ │ │ │ +000a3b30: 206f 6620 6465 6772 6565 2032 2020 2020   of degree 2    
│ │ │ │ +000a3b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a3b50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3b60: 2020 2073 6f75 7263 6520 7661 7269 6574     source variet
│ │ │ │ +000a3b70: 793a 2050 505e 3520 2020 2020 2020 2020  y: PP^5         
│ │ │ │  000a3b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3bb0: 207c 0a7c 2020 2020 2074 6172 6765 7420   |.|     target 
│ │ │ │ -000a3bc0: 7661 7269 6574 793a 2050 505e 3520 2020  variety: PP^5   
│ │ │ │ +000a3ba0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3bb0: 2020 2074 6172 6765 7420 7661 7269 6574     target variet
│ │ │ │ +000a3bc0: 793a 2050 505e 3520 2020 2020 2020 2020  y: PP^5         
│ │ │ │  000a3bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3c00: 207c 0a7c 2020 2020 2063 6f65 6666 6963   |.|     coeffic
│ │ │ │ -000a3c10: 6965 6e74 2072 696e 673a 205a 5a2f 3139  ient ring: ZZ/19
│ │ │ │ -000a3c20: 3031 3831 2020 2020 2020 2020 2020 2020  0181            
│ │ │ │ +000a3bf0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3c00: 2020 2063 6f65 6666 6963 6965 6e74 2072     coefficient r
│ │ │ │ +000a3c10: 696e 673a 205a 5a2f 3139 3031 3831 2020  ing: ZZ/190181  
│ │ │ │ +000a3c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3c50: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000a3c40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000a3c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a3c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a3c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a3c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a3c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a3ca0: 2d2b 0a7c 6936 203a 2064 6573 6372 6962  -+.|i6 : describ
│ │ │ │ -000a3cb0: 6520 5068 6927 2020 2020 2020 2020 2020  e Phi'          
│ │ │ │ +000a3c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ +000a3ca0: 203a 2064 6573 6372 6962 6520 5068 6927   : describe Phi'
│ │ │ │ +000a3cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3cf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a3ce0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3d40: 207c 0a7c 6f36 203d 2072 6174 696f 6e61   |.|o6 = rationa
│ │ │ │ -000a3d50: 6c20 6d61 7020 6465 6669 6e65 6420 6279  l map defined by
│ │ │ │ -000a3d60: 2066 6f72 6d73 206f 6620 6465 6772 6565   forms of degree
│ │ │ │ -000a3d70: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -000a3d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3d90: 207c 0a7c 2020 2020 2073 6f75 7263 6520   |.|     source 
│ │ │ │ -000a3da0: 7661 7269 6574 793a 2073 6d6f 6f74 6820  variety: smooth 
│ │ │ │ -000a3db0: 7175 6164 7269 6320 6879 7065 7273 7572  quadric hypersur
│ │ │ │ -000a3dc0: 6661 6365 2069 6e20 5050 5e35 2020 2020  face in PP^5    
│ │ │ │ -000a3dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3de0: 207c 0a7c 2020 2020 2074 6172 6765 7420   |.|     target 
│ │ │ │ -000a3df0: 7661 7269 6574 793a 2068 7970 6572 706c  variety: hyperpl
│ │ │ │ -000a3e00: 616e 6520 696e 2050 505e 3520 2020 2020  ane in PP^5     
│ │ │ │ +000a3d30: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ +000a3d40: 203d 2072 6174 696f 6e61 6c20 6d61 7020   = rational map 
│ │ │ │ +000a3d50: 6465 6669 6e65 6420 6279 2066 6f72 6d73  defined by forms
│ │ │ │ +000a3d60: 206f 6620 6465 6772 6565 2032 2020 2020   of degree 2    
│ │ │ │ +000a3d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a3d80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3d90: 2020 2073 6f75 7263 6520 7661 7269 6574     source variet
│ │ │ │ +000a3da0: 793a 2073 6d6f 6f74 6820 7175 6164 7269  y: smooth quadri
│ │ │ │ +000a3db0: 6320 6879 7065 7273 7572 6661 6365 2069  c hypersurface i
│ │ │ │ +000a3dc0: 6e20 5050 5e35 2020 2020 2020 2020 2020  n PP^5          
│ │ │ │ +000a3dd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3de0: 2020 2074 6172 6765 7420 7661 7269 6574     target variet
│ │ │ │ +000a3df0: 793a 2068 7970 6572 706c 616e 6520 696e  y: hyperplane in
│ │ │ │ +000a3e00: 2050 505e 3520 2020 2020 2020 2020 2020   PP^5           
│ │ │ │  000a3e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3e30: 207c 0a7c 2020 2020 2063 6f65 6666 6963   |.|     coeffic
│ │ │ │ -000a3e40: 6965 6e74 2072 696e 673a 205a 5a2f 3139  ient ring: ZZ/19
│ │ │ │ -000a3e50: 3031 3831 2020 2020 2020 2020 2020 2020  0181            
│ │ │ │ +000a3e20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a3e30: 2020 2063 6f65 6666 6963 6965 6e74 2072     coefficient r
│ │ │ │ +000a3e40: 696e 673a 205a 5a2f 3139 3031 3831 2020  ing: ZZ/190181  
│ │ │ │ +000a3e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a3e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a3e80: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000a3e70: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000a3e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a3e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a3ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a3eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a3ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a3ed0: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d  -+..See also.===
│ │ │ │ -000a3ee0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -000a3ef0: 2052 6174 696f 6e61 6c4d 6170 207c 2049   RationalMap | I
│ │ │ │ -000a3f00: 6465 616c 3a20 5261 7469 6f6e 616c 4d61  deal: RationalMa
│ │ │ │ -000a3f10: 7020 7c20 4964 6561 6c2c 202d 2d20 7265  p | Ideal, -- re
│ │ │ │ -000a3f20: 7374 7269 6374 696f 6e20 6f66 2061 0a20  striction of a. 
│ │ │ │ -000a3f30: 2020 2072 6174 696f 6e61 6c20 6d61 700a     rational map.
│ │ │ │ -000a3f40: 0a57 6179 7320 746f 2075 7365 2074 6869  .Ways to use thi
│ │ │ │ -000a3f50: 7320 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d  s method:.======
│ │ │ │ -000a3f60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000a3f70: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 5261  ==..  * *note Ra
│ │ │ │ -000a3f80: 7469 6f6e 616c 4d61 7020 7c7c 2049 6465  tionalMap || Ide
│ │ │ │ -000a3f90: 616c 3a20 5261 7469 6f6e 616c 4d61 7020  al: RationalMap 
│ │ │ │ -000a3fa0: 7c7c 2049 6465 616c 2c20 2d2d 2072 6573  || Ideal, -- res
│ │ │ │ -000a3fb0: 7472 6963 7469 6f6e 206f 6620 610a 2020  triction of a.  
│ │ │ │ -000a3fc0: 2020 7261 7469 6f6e 616c 206d 6170 0a20    rational map. 
│ │ │ │ -000a3fd0: 202a 2022 5261 7469 6f6e 616c 4d61 7020   * "RationalMap 
│ │ │ │ -000a3fe0: 7c7c 2052 696e 6722 0a20 202a 2022 5261  || Ring".  * "Ra
│ │ │ │ -000a3ff0: 7469 6f6e 616c 4d61 7020 7c7c 2052 696e  tionalMap || Rin
│ │ │ │ -000a4000: 6745 6c65 6d65 6e74 220a 2d2d 2d2d 2d2d  gElement".------
│ │ │ │ +000a3ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365  -----------+..Se
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│ │ │ │ +000a3f00: 5261 7469 6f6e 616c 4d61 7020 7c20 4964  RationalMap | Id
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│ │ │ │ +000a3f30: 696f 6e61 6c20 6d61 700a 0a57 6179 7320  ional map..Ways 
│ │ │ │ +000a3f40: 746f 2075 7365 2074 6869 7320 6d65 7468  to use this meth
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│ │ │ │ +000a3f60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +000a3f70: 2a20 2a6e 6f74 6520 5261 7469 6f6e 616c  * *note Rational
│ │ │ │ +000a3f80: 4d61 7020 7c7c 2049 6465 616c 3a20 5261  Map || Ideal: Ra
│ │ │ │ +000a3f90: 7469 6f6e 616c 4d61 7020 7c7c 2049 6465  tionalMap || Ide
│ │ │ │ +000a3fa0: 616c 2c20 2d2d 2072 6573 7472 6963 7469  al, -- restricti
│ │ │ │ +000a3fb0: 6f6e 206f 6620 610a 2020 2020 7261 7469  on of a.    rati
│ │ │ │ +000a3fc0: 6f6e 616c 206d 6170 0a20 202a 2022 5261  onal map.  * "Ra
│ │ │ │ +000a3fd0: 7469 6f6e 616c 4d61 7020 7c7c 2052 696e  tionalMap || Rin
│ │ │ │ +000a3fe0: 6722 0a20 202a 2022 5261 7469 6f6e 616c  g".  * "Rational
│ │ │ │ +000a3ff0: 4d61 7020 7c7c 2052 696e 6745 6c65 6d65  Map || RingEleme
│ │ │ │ +000a4000: 6e74 220a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  nt".------------
│ │ │ │  000a4010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a4020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a4030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a4040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000a4100: 7469 6f6e 616c 4d61 705f 6c70 4964 6561  tionalMap_lpIdea
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│ │ │ │ -000a4140: 6e67 5f63 6d4c 6973 745f 7270 2c20 5072  ng_cmList_rp, Pr
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│ │ │ │ -000a41a0: 6d61 7020 6672 6f6d 2061 6e20 6964 6561  map from an idea
│ │ │ │ -000a41b0: 6c0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  l.**************
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│ │ │ │ +000a40b0: 322f 7061 636b 6167 6573 2f43 7265 6d6f  2/packages/Cremo
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│ │ │ │ +000a4110: 5f63 6d5a 5a5f 7270 2c20 4e65 7874 3a20  _cmZZ_rp, Next: 
│ │ │ │ +000a4120: 7261 7469 6f6e 616c 4d61 705f 6c70 506f  rationalMap_lpPo
│ │ │ │ +000a4130: 6c79 6e6f 6d69 616c 5269 6e67 5f63 6d4c  lynomialRing_cmL
│ │ │ │ +000a4140: 6973 745f 7270 2c20 5072 6576 3a20 5261  ist_rp, Prev: Ra
│ │ │ │ +000a4150: 7469 6f6e 616c 4d61 7020 7c7c 2049 6465  tionalMap || Ide
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│ │ │ │ +000a4170: 696f 6e61 6c4d 6170 2849 6465 616c 2c5a  ionalMap(Ideal,Z
│ │ │ │ +000a4180: 5a2c 5a5a 2920 2d2d 206d 616b 6573 2061  Z,ZZ) -- makes a
│ │ │ │ +000a4190: 2072 6174 696f 6e61 6c20 6d61 7020 6672   rational map fr
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│ │ │ │ +000a41b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000a41c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000a41d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000a41e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000a41f0: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20  ..  * Function: 
│ │ │ │ -000a4200: 2a6e 6f74 6520 7261 7469 6f6e 616c 4d61  *note rationalMa
│ │ │ │ -000a4210: 703a 2072 6174 696f 6e61 6c4d 6170 2c0a  p: rationalMap,.
│ │ │ │ -000a4220: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ -000a4230: 2020 2020 7261 7469 6f6e 616c 4d61 7028      rationalMap(
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│ │ │ │ -000a43b0: 696e 2061 0a20 2020 2020 2020 2070 726f  in a.        pro
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│ │ │ │ +000a42a0: 3244 6f63 2949 6465 616c 2c2c 2061 2068  2Doc)Ideal,, a h
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│ │ │ │ +000a42d0: 636f 6f72 6469 6e61 7465 2072 696e 6720  coordinate ring 
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│ │ │ │ +000a4370: 6163 6175 6c61 7932 446f 6329 4c69 7374  acaulay2Doc)List
│ │ │ │ +000a4380: 2c20 6f66 2069 6e74 6567 6572 7320 696e  , of integers in
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│ │ │ │ +000a43a0: 2065 6d62 6564 6465 6420 696e 2061 0a20   embedded in a. 
│ │ │ │ +000a43b0: 2020 2020 2020 2070 726f 6475 6374 206f         product o
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│ │ │ │  000a47f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a4800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a4810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  000a4850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a4860: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000a4860: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │  000a4880: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000a48b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a48c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a48d0: 2020 2020 5a5a 2020 2020 2020 2020 2020      ZZ          
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│ │ │ │ +000a48c0: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │ +000a48d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000a4900: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000a4950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a4960: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a4970: 2020 3333 3333 3120 2030 2020 2036 2020    33331  0   6  
│ │ │ │ +000a4950: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a4960: 2020 2020 2020 2020 2020 2020 3333 3333              3333
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│ │ │ │  000a4980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a4990: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000a49b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ +000a49a0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ +000a49b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a49c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a49d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a49e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a49f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000a4a30: 785f 312a 785f 342c 785f 325e 322d 785f  x_1*x_4,x_2^2-x_
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│ │ │ │ -000a4a50: 307c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  0|.|------------
│ │ │ │ +000a49f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 6964  -----------|.|id
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│ │ │ │ +000a4a30: 342c 785f 325e 322d 785f 302a 785f 352c  4,x_2^2-x_0*x_5,
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│ │ │ │ +000a4a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a4a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a4a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a4a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a4a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000a4ab0: 2c78 5f36 293b 2020 2020 2020 2020 2020  ,x_6);          
│ │ │ │  000a4ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a4ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a4ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a4af0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000a4ae0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000a4af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a4b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a4b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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(
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│ │ │ │ +000a4b40: 203a 2074 696d 6520 7068 6920 3d20 7261   : time phi = ra
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│ │ │ │ +000a4b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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
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│ │ │ │ -000a4bb0: 3039 3533 3231 3573 2028 7468 7265 6164  0953215s (thread
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│ │ │ │ -000a4bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a4be0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a4b80: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +000a4b90: 2d20 7573 6564 2030 2e31 3333 3034 3873  - used 0.133048s
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│ │ │ │ +000a4bb0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +000a4bc0: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +000a4bd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a4be0: 2020 2020 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
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│ │ │ │ +000a4c30: 203d 202d 2d20 7261 7469 6f6e 616c 206d   = -- rational m
│ │ │ │ +000a4c40: 6170 202d 2d20 2020 2020 2020 2020 2020  ap --           
│ │ │ │  000a4c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a4c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a4c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a4c80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a4c90: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │ +000a4c70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a4c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a4c90: 2020 5a5a 2020 2020 2020 2020 2020 2020    ZZ            
│ │ │ │  000a4ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a4cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a4cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a4cd0: 207c 0a7c 2020 2020 2073 6f75 7263 653a   |.|     source:
│ │ │ │ -000a4ce0: 2050 726f 6a28 2d2d 2d2d 2d5b 7820 2c20   Proj(-----[x , 
│ │ │ │ -000a4cf0: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -000a4d00: 7820 2c20 7820 5d29 2020 2020 2020 2020  x , x ])        
│ │ │ │ -000a4d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a4d20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a4d30: 2020 2020 2020 3333 3333 3120 2030 2020        33331  0  
│ │ │ │ -000a4d40: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -000a4d50: 2035 2020 2036 2020 2020 2020 2020 2020   5   6          
│ │ │ │ -000a4d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a4d70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a4d80: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │ +000a4cc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a4cd0: 2020 2073 6f75 7263 653a 2050 726f 6a28     source: Proj(
│ │ │ │ +000a4ce0: 2d2d 2d2d 2d5b 7820 2c20 7820 2c20 7820  -----[x , x , x 
│ │ │ │ +000a4cf0: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ +000a4d00: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ +000a4d10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a4d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a4d30: 3333 3333 3120 2030 2020 2031 2020 2032  33331  0   1   2
│ │ │ │ +000a4d40: 2020 2033 2020 2034 2020 2035 2020 2036     3   4   5   6
│ │ │ │ +000a4d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a4d60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a4d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a4d80: 2020 5a5a 2020 2020 2020 2020 2020 2020    ZZ            
│ │ │ │  000a4d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a4da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a4db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a4dc0: 207c 0a7c 2020 2020 2074 6172 6765 743a   |.|     target:
│ │ │ │ -000a4dd0: 2050 726f 6a28 2d2d 2d2d 2d5b 7920 2c20   Proj(-----[y , 
│ │ │ │ -000a4de0: 7920 2c20 7920 2c20 7920 2c20 7920 2c20  y , y , y , y , 
│ │ │ │ -000a4df0: 7920 2c20 7920 2c20 7920 2c20 7920 2c20  y , y , y , y , 
│ │ │ │ -000a4e00: 7920 2c20 7920 202c 2079 2020 2c20 2020  y , y  , y  ,   
│ │ │ │ -000a4e10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a4e20: 2020 2020 2020 3333 3333 3120 2030 2020        33331  0  
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│ │ │ │ +000a5a00: 2020 2020 2032 2033 2036 2020 2020 3120       2 3 6    1 
│ │ │ │ +000a5a10: 3420 3620 2020 2020 2020 2020 2020 2020  4 6             
│ │ │ │  000a5a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5a40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a5a30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5a90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a5aa0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +000a5a80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a5aa0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  000a5ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5ae0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a5af0: 2020 2020 2020 2020 2020 7820 7820 202d            x x  -
│ │ │ │ -000a5b00: 2078 2078 2078 202c 2020 2020 2020 2020   x x x ,        
│ │ │ │ +000a5ad0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a5af0: 2020 2020 7820 7820 202d 2078 2078 2078      x x  - x x x
│ │ │ │ +000a5b00: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  000a5b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5b30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a5b40: 2020 2020 2020 2020 2020 2032 2036 2020             2 6  
│ │ │ │ -000a5b50: 2020 3020 3520 3620 2020 2020 2020 2020    0 5 6         
│ │ │ │ +000a5b20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a5b40: 2020 2020 2032 2036 2020 2020 3020 3520       2 6    0 5 
│ │ │ │ +000a5b50: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │  000a5b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5b80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a5b70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5bd0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a5be0: 2020 2020 2020 2020 2020 7820 7820 7820            x x x 
│ │ │ │ -000a5bf0: 202d 2078 2078 2078 202c 2020 2020 2020   - x x x ,      
│ │ │ │ +000a5bc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a5be0: 2020 2020 7820 7820 7820 202d 2078 2078      x x x  - x x
│ │ │ │ +000a5bf0: 2078 202c 2020 2020 2020 2020 2020 2020   x ,            
│ │ │ │  000a5c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5c20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a5c30: 2020 2020 2020 2020 2020 2031 2032 2036             1 2 6
│ │ │ │ -000a5c40: 2020 2020 3020 3420 3620 2020 2020 2020      0 4 6       
│ │ │ │ +000a5c10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a5c30: 2020 2020 2031 2032 2036 2020 2020 3020       1 2 6    0 
│ │ │ │ +000a5c40: 3420 3620 2020 2020 2020 2020 2020 2020  4 6             
│ │ │ │  000a5c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5c70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a5c60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5cc0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a5cd0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +000a5cb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a5cd0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  000a5ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5d10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a5d20: 2020 2020 2020 2020 2020 7820 7820 202d            x x  -
│ │ │ │ -000a5d30: 2078 2078 2078 202c 2020 2020 2020 2020   x x x ,        
│ │ │ │ +000a5d00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a5d20: 2020 2020 7820 7820 202d 2078 2078 2078      x x  - x x x
│ │ │ │ +000a5d30: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  000a5d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5d60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a5d70: 2020 2020 2020 2020 2020 2031 2036 2020             1 6  
│ │ │ │ -000a5d80: 2020 3020 3320 3620 2020 2020 2020 2020    0 3 6         
│ │ │ │ +000a5d50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a5d70: 2020 2020 2031 2036 2020 2020 3020 3320       1 6    0 3 
│ │ │ │ +000a5d80: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │  000a5d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5db0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a5da0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5e00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a5e10: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -000a5e20: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -000a5e30: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -000a5e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5e50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a5e60: 2020 2020 2020 2020 2020 7820 7820 202d            x x  -
│ │ │ │ -000a5e70: 2032 7820 7820 7820 202b 2078 2078 2020   2x x x  + x x  
│ │ │ │ -000a5e80: 2b20 7820 7820 202d 2078 2078 2078 2020  + x x  - x x x  
│ │ │ │ -000a5e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5ea0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a5eb0: 2020 2020 2020 2020 2020 2032 2033 2020             2 3  
│ │ │ │ -000a5ec0: 2020 2031 2032 2034 2020 2020 3020 3420     1 2 4    0 4 
│ │ │ │ -000a5ed0: 2020 2031 2035 2020 2020 3020 3320 3520     1 5    0 3 5 
│ │ │ │ -000a5ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5ef0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000a5f00: 2020 2020 2020 2020 207d 2020 2020 2020           }      
│ │ │ │ +000a5df0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a5e10: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +000a5e20: 2020 2020 2020 2020 3220 2020 2032 2020          2    2  
│ │ │ │ +000a5e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a5e40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a5e60: 2020 2020 7820 7820 202d 2032 7820 7820      x x  - 2x x 
│ │ │ │ +000a5e70: 7820 202b 2078 2078 2020 2b20 7820 7820  x  + x x  + x x 
│ │ │ │ +000a5e80: 202d 2078 2078 2078 2020 2020 2020 2020   - x x x        
│ │ │ │ +000a5e90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a5eb0: 2020 2020 2032 2033 2020 2020 2031 2032       2 3     1 2
│ │ │ │ +000a5ec0: 2034 2020 2020 3020 3420 2020 2031 2035   4    0 4    1 5
│ │ │ │ +000a5ed0: 2020 2020 3020 3320 3520 2020 2020 2020      0 3 5       
│ │ │ │ +000a5ee0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a5f00: 2020 207d 2020 2020 2020 2020 2020 2020     }            
│ │ │ │  000a5f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5f40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a5f30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a5f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a5f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5f90: 207c 0a7c 6f33 203a 2052 6174 696f 6e61   |.|o3 : Rationa
│ │ │ │ -000a5fa0: 6c4d 6170 2028 6375 6269 6320 7261 7469  lMap (cubic rati
│ │ │ │ -000a5fb0: 6f6e 616c 206d 6170 2066 726f 6d20 5050  onal map from PP
│ │ │ │ -000a5fc0: 5e36 2074 6f20 5050 5e31 3329 2020 2020  ^6 to PP^13)    
│ │ │ │ -000a5fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a5fe0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ +000a5f80: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +000a5f90: 203a 2052 6174 696f 6e61 6c4d 6170 2028   : RationalMap (
│ │ │ │ +000a5fa0: 6375 6269 6320 7261 7469 6f6e 616c 206d  cubic rational m
│ │ │ │ +000a5fb0: 6170 2066 726f 6d20 5050 5e36 2074 6f20  ap from PP^6 to 
│ │ │ │ +000a5fc0: 5050 5e31 3329 2020 2020 2020 2020 2020  PP^13)          
│ │ │ │ +000a5fd0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ +000a5fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a5ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a6000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a6010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a6020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a6030: 2d7c 0a7c 7920 202c 2079 2020 5d29 2020  -|.|y  , y  ])  
│ │ │ │ +000a6020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7920  -----------|.|y 
│ │ │ │ +000a6030: 202c 2079 2020 5d29 2020 2020 2020 2020   , y  ])        
│ │ │ │  000a6040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a6050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a6060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6080: 207c 0a7c 2031 3220 2020 3133 2020 2020   |.| 12   13    
│ │ │ │ +000a6070: 2020 2020 2020 2020 2020 207c 0a7c 2031             |.| 1
│ │ │ │ +000a6080: 3220 2020 3133 2020 2020 2020 2020 2020  2   13          
│ │ │ │  000a6090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a60a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a60b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a60c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a60d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000a60c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000a60d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a60e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a60f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a6100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a6110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a6120: 2d2b 0a7c 6934 203a 2064 6573 6372 6962  -+.|i4 : describ
│ │ │ │ -000a6130: 6520 7068 6921 2020 2020 2020 2020 2020  e phi!          
│ │ │ │ +000a6110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +000a6120: 203a 2064 6573 6372 6962 6520 7068 6921   : describe phi!
│ │ │ │ +000a6130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a6140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a6150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6170: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a6160: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a6170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a6180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a6190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a61a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a61b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a61c0: 207c 0a7c 6f34 203d 2072 6174 696f 6e61   |.|o4 = rationa
│ │ │ │ -000a61d0: 6c20 6d61 7020 6465 6669 6e65 6420 6279  l map defined by
│ │ │ │ -000a61e0: 2066 6f72 6d73 206f 6620 6465 6772 6565   forms of degree
│ │ │ │ -000a61f0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -000a6200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6210: 207c 0a7c 2020 2020 2073 6f75 7263 6520   |.|     source 
│ │ │ │ -000a6220: 7661 7269 6574 793a 2050 505e 3620 2020  variety: PP^6   
│ │ │ │ +000a61b0: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +000a61c0: 203d 2072 6174 696f 6e61 6c20 6d61 7020   = rational map 
│ │ │ │ +000a61d0: 6465 6669 6e65 6420 6279 2066 6f72 6d73  defined by forms
│ │ │ │ +000a61e0: 206f 6620 6465 6772 6565 2033 2020 2020   of degree 3    
│ │ │ │ +000a61f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a6200: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a6210: 2020 2073 6f75 7263 6520 7661 7269 6574     source variet
│ │ │ │ +000a6220: 793a 2050 505e 3620 2020 2020 2020 2020  y: PP^6         
│ │ │ │  000a6230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a6240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6260: 207c 0a7c 2020 2020 2074 6172 6765 7420   |.|     target 
│ │ │ │ -000a6270: 7661 7269 6574 793a 2050 505e 3133 2020  variety: PP^13  
│ │ │ │ +000a6250: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a6260: 2020 2074 6172 6765 7420 7661 7269 6574     target variet
│ │ │ │ +000a6270: 793a 2050 505e 3133 2020 2020 2020 2020  y: PP^13        
│ │ │ │  000a6280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a6290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a62a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a62b0: 207c 0a7c 2020 2020 2069 6d61 6765 3a20   |.|     image: 
│ │ │ │ -000a62c0: 362d 6469 6d65 6e73 696f 6e61 6c20 7661  6-dimensional va
│ │ │ │ -000a62d0: 7269 6574 7920 6f66 2064 6567 7265 6520  riety of degree 
│ │ │ │ -000a62e0: 3136 2069 6e20 5050 5e31 3320 6375 7420  16 in PP^13 cut 
│ │ │ │ -000a62f0: 6f75 7420 6279 2032 3120 2020 2020 2020  out by 21       
│ │ │ │ -000a6300: 207c 0a7c 2020 2020 2064 6f6d 696e 616e   |.|     dominan
│ │ │ │ -000a6310: 6365 3a20 6661 6c73 6520 2020 2020 2020  ce: false       
│ │ │ │ +000a62a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a62b0: 2020 2069 6d61 6765 3a20 362d 6469 6d65     image: 6-dime
│ │ │ │ +000a62c0: 6e73 696f 6e61 6c20 7661 7269 6574 7920  nsional variety 
│ │ │ │ +000a62d0: 6f66 2064 6567 7265 6520 3136 2069 6e20  of degree 16 in 
│ │ │ │ +000a62e0: 5050 5e31 3320 6375 7420 6f75 7420 6279  PP^13 cut out by
│ │ │ │ +000a62f0: 2032 3120 2020 2020 2020 207c 0a7c 2020   21        |.|  
│ │ │ │ +000a6300: 2020 2064 6f6d 696e 616e 6365 3a20 6661     dominance: fa
│ │ │ │ +000a6310: 6c73 6520 2020 2020 2020 2020 2020 2020  lse             
│ │ │ │  000a6320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a6330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6350: 207c 0a7c 2020 2020 2062 6972 6174 696f   |.|     biratio
│ │ │ │ -000a6360: 6e61 6c69 7479 3a20 6661 6c73 6520 2020  nality: false   
│ │ │ │ +000a6340: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a6350: 2020 2062 6972 6174 696f 6e61 6c69 7479     birationality
│ │ │ │ +000a6360: 3a20 6661 6c73 6520 2020 2020 2020 2020  : false         
│ │ │ │  000a6370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a6380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a63a0: 207c 0a7c 2020 2020 2064 6567 7265 6520   |.|     degree 
│ │ │ │ -000a63b0: 6f66 206d 6170 3a20 3120 2020 2020 2020  of map: 1       
│ │ │ │ +000a6390: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a63a0: 2020 2064 6567 7265 6520 6f66 206d 6170     degree of map
│ │ │ │ +000a63b0: 3a20 3120 2020 2020 2020 2020 2020 2020  : 1             
│ │ │ │  000a63c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a63d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a63e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a63f0: 207c 0a7c 2020 2020 2070 726f 6a65 6374   |.|     project
│ │ │ │ -000a6400: 6976 6520 6465 6772 6565 733a 207b 312c  ive degrees: {1,
│ │ │ │ -000a6410: 2033 2c20 362c 2031 322c 2031 362c 2031   3, 6, 12, 16, 1
│ │ │ │ -000a6420: 362c 2031 367d 2020 2020 2020 2020 2020  6, 16}          
│ │ │ │ -000a6430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6440: 207c 0a7c 2020 2020 206e 756d 6265 7220   |.|     number 
│ │ │ │ -000a6450: 6f66 206d 696e 696d 616c 2072 6570 7265  of minimal repre
│ │ │ │ -000a6460: 7365 6e74 6174 6976 6573 3a20 3120 2020  sentatives: 1   
│ │ │ │ +000a63e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a63f0: 2020 2070 726f 6a65 6374 6976 6520 6465     projective de
│ │ │ │ +000a6400: 6772 6565 733a 207b 312c 2033 2c20 362c  grees: {1, 3, 6,
│ │ │ │ +000a6410: 2031 322c 2031 362c 2031 362c 2031 367d   12, 16, 16, 16}
│ │ │ │ +000a6420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000a6430: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a6440: 2020 206e 756d 6265 7220 6f66 206d 696e     number of min
│ │ │ │ +000a6450: 696d 616c 2072 6570 7265 7365 6e74 6174  imal representat
│ │ │ │ +000a6460: 6976 6573 3a20 3120 2020 2020 2020 2020  ives: 1         
│ │ │ │  000a6470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6490: 207c 0a7c 2020 2020 2064 696d 656e 7369   |.|     dimensi
│ │ │ │ -000a64a0: 6f6e 2062 6173 6520 6c6f 6375 733a 2034  on base locus: 4
│ │ │ │ +000a6480: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a6490: 2020 2064 696d 656e 7369 6f6e 2062 6173     dimension bas
│ │ │ │ +000a64a0: 6520 6c6f 6375 733a 2034 2020 2020 2020  e locus: 4      
│ │ │ │  000a64b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a64c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a64d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a64e0: 207c 0a7c 2020 2020 2064 6567 7265 6520   |.|     degree 
│ │ │ │ -000a64f0: 6261 7365 206c 6f63 7573 3a20 3320 2020  base locus: 3   
│ │ │ │ +000a64d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a64e0: 2020 2064 6567 7265 6520 6261 7365 206c     degree base l
│ │ │ │ +000a64f0: 6f63 7573 3a20 3320 2020 2020 2020 2020  ocus: 3         
│ │ │ │  000a6500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a6510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6530: 207c 0a7c 2020 2020 2063 6f65 6666 6963   |.|     coeffic
│ │ │ │ -000a6540: 6965 6e74 2072 696e 673a 205a 5a2f 3333  ient ring: ZZ/33
│ │ │ │ -000a6550: 3333 3120 2020 2020 2020 2020 2020 2020  331             
│ │ │ │ +000a6520: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000a6530: 2020 2063 6f65 6666 6963 6965 6e74 2072     coefficient r
│ │ │ │ +000a6540: 696e 673a 205a 5a2f 3333 3333 3120 2020  ing: ZZ/33331   
│ │ │ │ +000a6550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a6560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6580: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ +000a6570: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ +000a6580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a6590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a65a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a65b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a65c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a65d0: 2d7c 0a7c 6879 7065 7273 7572 6661 6365  -|.|hypersurface
│ │ │ │ -000a65e0: 7320 6f66 2064 6567 7265 6520 3220 2020  s of degree 2   
│ │ │ │ +000a65c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 6879  -----------|.|hy
│ │ │ │ +000a65d0: 7065 7273 7572 6661 6365 7320 6f66 2064  persurfaces of d
│ │ │ │ +000a65e0: 6567 7265 6520 3220 2020 2020 2020 2020  egree 2         
│ │ │ │  000a65f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a6600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6620: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000a6610: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000a6620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a6630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a6640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a6650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a6660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a6670: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d  -+..See also.===
│ │ │ │ -000a6680: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -000a6690: 2074 6f4d 6170 3a20 746f 4d61 702c 202d   toMap: toMap, -
│ │ │ │ -000a66a0: 2d20 7261 7469 6f6e 616c 206d 6170 2064  - rational map d
│ │ │ │ -000a66b0: 6566 696e 6564 2062 7920 6120 6c69 6e65  efined by a line
│ │ │ │ -000a66c0: 6172 2073 7973 7465 6d0a 2020 2a20 2a6e  ar system.  * *n
│ │ │ │ -000a66d0: 6f74 6520 7261 7469 6f6e 616c 4d61 703a  ote rationalMap:
│ │ │ │ -000a66e0: 2072 6174 696f 6e61 6c4d 6170 2c20 2d2d   rationalMap, --
│ │ │ │ -000a66f0: 206d 616b 6573 2061 2072 6174 696f 6e61   makes a rationa
│ │ │ │ -000a6700: 6c20 6d61 700a 2020 2a20 2a6e 6f74 6520  l map.  * *note 
│ │ │ │ -000a6710: 7261 7469 6f6e 616c 4d61 7028 5461 6c6c  rationalMap(Tall
│ │ │ │ -000a6720: 7929 3a20 7261 7469 6f6e 616c 4d61 705f  y): rationalMap_
│ │ │ │ -000a6730: 6c70 5269 6e67 5f63 6d54 616c 6c79 5f72  lpRing_cmTally_r
│ │ │ │ -000a6740: 702c 202d 2d20 7261 7469 6f6e 616c 206d  p, -- rational m
│ │ │ │ -000a6750: 6170 0a20 2020 2064 6566 696e 6564 2062  ap.    defined b
│ │ │ │ -000a6760: 7920 616e 2065 6666 6563 7469 7665 2064  y an effective d
│ │ │ │ -000a6770: 6976 6973 6f72 0a0a 5761 7973 2074 6f20  ivisor..Ways to 
│ │ │ │ -000a6780: 7573 6520 7468 6973 206d 6574 686f 643a  use this method:
│ │ │ │ -000a6790: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -000a67a0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ -000a67b0: 7261 7469 6f6e 616c 4d61 7028 4964 6561  rationalMap(Idea
│ │ │ │ -000a67c0: 6c29 220a 2020 2a20 2272 6174 696f 6e61  l)".  * "rationa
│ │ │ │ -000a67d0: 6c4d 6170 2849 6465 616c 2c4c 6973 7429  lMap(Ideal,List)
│ │ │ │ -000a67e0: 220a 2020 2a20 2272 6174 696f 6e61 6c4d  ".  * "rationalM
│ │ │ │ -000a67f0: 6170 2849 6465 616c 2c5a 5a29 220a 2020  ap(Ideal,ZZ)".  
│ │ │ │ -000a6800: 2a20 2a6e 6f74 6520 7261 7469 6f6e 616c  * *note rational
│ │ │ │ -000a6810: 4d61 7028 4964 6561 6c2c 5a5a 2c5a 5a29  Map(Ideal,ZZ,ZZ)
│ │ │ │ -000a6820: 3a20 7261 7469 6f6e 616c 4d61 705f 6c70  : rationalMap_lp
│ │ │ │ -000a6830: 4964 6561 6c5f 636d 5a5a 5f63 6d5a 5a5f  Ideal_cmZZ_cmZZ_
│ │ │ │ -000a6840: 7270 2c20 2d2d 206d 616b 6573 0a20 2020  rp, -- makes.   
│ │ │ │ -000a6850: 2061 2072 6174 696f 6e61 6c20 6d61 7020   a rational map 
│ │ │ │ -000a6860: 6672 6f6d 2061 6e20 6964 6561 6c0a 2d2d  from an ideal.--
│ │ │ │ +000a6660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365  -----------+..Se
│ │ │ │ +000a6670: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ +000a6680: 0a20 202a 202a 6e6f 7465 2074 6f4d 6170  .  * *note toMap
│ │ │ │ +000a6690: 3a20 746f 4d61 702c 202d 2d20 7261 7469  : toMap, -- rati
│ │ │ │ +000a66a0: 6f6e 616c 206d 6170 2064 6566 696e 6564  onal map defined
│ │ │ │ +000a66b0: 2062 7920 6120 6c69 6e65 6172 2073 7973   by a linear sys
│ │ │ │ +000a66c0: 7465 6d0a 2020 2a20 2a6e 6f74 6520 7261  tem.  * *note ra
│ │ │ │ +000a66d0: 7469 6f6e 616c 4d61 703a 2072 6174 696f  tionalMap: ratio
│ │ │ │ +000a66e0: 6e61 6c4d 6170 2c20 2d2d 206d 616b 6573  nalMap, -- makes
│ │ │ │ +000a66f0: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ +000a6700: 2020 2a20 2a6e 6f74 6520 7261 7469 6f6e    * *note ration
│ │ │ │ +000a6710: 616c 4d61 7028 5461 6c6c 7929 3a20 7261  alMap(Tally): ra
│ │ │ │ +000a6720: 7469 6f6e 616c 4d61 705f 6c70 5269 6e67  tionalMap_lpRing
│ │ │ │ +000a6730: 5f63 6d54 616c 6c79 5f72 702c 202d 2d20  _cmTally_rp, -- 
│ │ │ │ +000a6740: 7261 7469 6f6e 616c 206d 6170 0a20 2020  rational map.   
│ │ │ │ +000a6750: 2064 6566 696e 6564 2062 7920 616e 2065   defined by an e
│ │ │ │ +000a6760: 6666 6563 7469 7665 2064 6976 6973 6f72  ffective divisor
│ │ │ │ +000a6770: 0a0a 5761 7973 2074 6f20 7573 6520 7468  ..Ways to use th
│ │ │ │ +000a6780: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d  is method:.=====
│ │ │ │ +000a6790: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000a67a0: 3d3d 3d0a 0a20 202a 2022 7261 7469 6f6e  ===..  * "ration
│ │ │ │ +000a67b0: 616c 4d61 7028 4964 6561 6c29 220a 2020  alMap(Ideal)".  
│ │ │ │ +000a67c0: 2a20 2272 6174 696f 6e61 6c4d 6170 2849  * "rationalMap(I
│ │ │ │ +000a67d0: 6465 616c 2c4c 6973 7429 220a 2020 2a20  deal,List)".  * 
│ │ │ │ +000a67e0: 2272 6174 696f 6e61 6c4d 6170 2849 6465  "rationalMap(Ide
│ │ │ │ +000a67f0: 616c 2c5a 5a29 220a 2020 2a20 2a6e 6f74  al,ZZ)".  * *not
│ │ │ │ +000a6800: 6520 7261 7469 6f6e 616c 4d61 7028 4964  e rationalMap(Id
│ │ │ │ +000a6810: 6561 6c2c 5a5a 2c5a 5a29 3a20 7261 7469  eal,ZZ,ZZ): rati
│ │ │ │ +000a6820: 6f6e 616c 4d61 705f 6c70 4964 6561 6c5f  onalMap_lpIdeal_
│ │ │ │ +000a6830: 636d 5a5a 5f63 6d5a 5a5f 7270 2c20 2d2d  cmZZ_cmZZ_rp, --
│ │ │ │ +000a6840: 206d 616b 6573 0a20 2020 2061 2072 6174   makes.    a rat
│ │ │ │ +000a6850: 696f 6e61 6c20 6d61 7020 6672 6f6d 2061  ional map from a
│ │ │ │ +000a6860: 6e20 6964 6561 6c0a 2d2d 2d2d 2d2d 2d2d  n ideal.--------
│ │ │ │  000a6870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a6880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a6890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a68a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a68b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ -000a68c0: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ -000a68d0: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ -000a68e0: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ -000a68f0: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ -000a6900: 6179 322d 312e 3235 2e31 312b 6473 2f4d  ay2-1.25.11+ds/M
│ │ │ │ -000a6910: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ -000a6920: 6167 6573 2f43 7265 6d6f 6e61 2f0a 646f  ages/Cremona/.do
│ │ │ │ -000a6930: 6375 6d65 6e74 6174 696f 6e2e 6d32 3a38  cumentation.m2:8
│ │ │ │ -000a6940: 3230 3a30 2e0a 1f0a 4669 6c65 3a20 4372  20:0....File: Cr
│ │ │ │ -000a6950: 656d 6f6e 612e 696e 666f 2c20 4e6f 6465  emona.info, Node
│ │ │ │ -000a6960: 3a20 7261 7469 6f6e 616c 4d61 705f 6c70  : rationalMap_lp
│ │ │ │ -000a6970: 506f 6c79 6e6f 6d69 616c 5269 6e67 5f63  PolynomialRing_c
│ │ │ │ -000a6980: 6d4c 6973 745f 7270 2c20 4e65 7874 3a20  mList_rp, Next: 
│ │ │ │ -000a6990: 7261 7469 6f6e 616c 4d61 705f 6c70 5269  rationalMap_lpRi
│ │ │ │ -000a69a0: 6e67 5f63 6d54 616c 6c79 5f72 702c 2050  ng_cmTally_rp, P
│ │ │ │ -000a69b0: 7265 763a 2072 6174 696f 6e61 6c4d 6170  rev: rationalMap
│ │ │ │ -000a69c0: 5f6c 7049 6465 616c 5f63 6d5a 5a5f 636d  _lpIdeal_cmZZ_cm
│ │ │ │ -000a69d0: 5a5a 5f72 702c 2055 703a 2054 6f70 0a0a  ZZ_rp, Up: Top..
│ │ │ │ -000a69e0: 7261 7469 6f6e 616c 4d61 7028 506f 6c79  rationalMap(Poly
│ │ │ │ -000a69f0: 6e6f 6d69 616c 5269 6e67 2c4c 6973 7429  nomialRing,List)
│ │ │ │ -000a6a00: 202d 2d20 7261 7469 6f6e 616c 206d 6170   -- rational map
│ │ │ │ -000a6a10: 2064 6566 696e 6564 2062 7920 7468 6520   defined by the 
│ │ │ │ -000a6a20: 6c69 6e65 6172 2073 7973 7465 6d20 6f66  linear system of
│ │ │ │ -000a6a30: 2068 7970 6572 7375 7266 6163 6573 2070   hypersurfaces p
│ │ │ │ -000a6a40: 6173 7369 6e67 2074 6872 6f75 6768 2072  assing through r
│ │ │ │ -000a6a50: 616e 646f 6d20 706f 696e 7473 2077 6974  andom points wit
│ │ │ │ -000a6a60: 6820 6d75 6c74 6970 6c69 6369 7479 0a2a  h multiplicity.*
│ │ │ │ +000a68b0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ +000a68c0: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ +000a68d0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ +000a68e0: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ +000a68f0: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ +000a6900: 3235 2e31 312b 6473 2f4d 322f 4d61 6361  25.11+ds/M2/Maca
│ │ │ │ +000a6910: 756c 6179 322f 7061 636b 6167 6573 2f43  ulay2/packages/C
│ │ │ │ +000a6920: 7265 6d6f 6e61 2f0a 646f 6375 6d65 6e74  remona/.document
│ │ │ │ +000a6930: 6174 696f 6e2e 6d32 3a38 3230 3a30 2e0a  ation.m2:820:0..
│ │ │ │ +000a6940: 1f0a 4669 6c65 3a20 4372 656d 6f6e 612e  ..File: Cremona.
│ │ │ │ +000a6950: 696e 666f 2c20 4e6f 6465 3a20 7261 7469  info, Node: rati
│ │ │ │ +000a6960: 6f6e 616c 4d61 705f 6c70 506f 6c79 6e6f  onalMap_lpPolyno
│ │ │ │ +000a6970: 6d69 616c 5269 6e67 5f63 6d4c 6973 745f  mialRing_cmList_
│ │ │ │ +000a6980: 7270 2c20 4e65 7874 3a20 7261 7469 6f6e  rp, Next: ration
│ │ │ │ +000a6990: 616c 4d61 705f 6c70 5269 6e67 5f63 6d54  alMap_lpRing_cmT
│ │ │ │ +000a69a0: 616c 6c79 5f72 702c 2050 7265 763a 2072  ally_rp, Prev: r
│ │ │ │ +000a69b0: 6174 696f 6e61 6c4d 6170 5f6c 7049 6465  ationalMap_lpIde
│ │ │ │ +000a69c0: 616c 5f63 6d5a 5a5f 636d 5a5a 5f72 702c  al_cmZZ_cmZZ_rp,
│ │ │ │ +000a69d0: 2055 703a 2054 6f70 0a0a 7261 7469 6f6e   Up: Top..ration
│ │ │ │ +000a69e0: 616c 4d61 7028 506f 6c79 6e6f 6d69 616c  alMap(Polynomial
│ │ │ │ +000a69f0: 5269 6e67 2c4c 6973 7429 202d 2d20 7261  Ring,List) -- ra
│ │ │ │ +000a6a00: 7469 6f6e 616c 206d 6170 2064 6566 696e  tional map defin
│ │ │ │ +000a6a10: 6564 2062 7920 7468 6520 6c69 6e65 6172  ed by the linear
│ │ │ │ +000a6a20: 2073 7973 7465 6d20 6f66 2068 7970 6572   system of hyper
│ │ │ │ +000a6a30: 7375 7266 6163 6573 2070 6173 7369 6e67  surfaces passing
│ │ │ │ +000a6a40: 2074 6872 6f75 6768 2072 616e 646f 6d20   through random 
│ │ │ │ +000a6a50: 706f 696e 7473 2077 6974 6820 6d75 6c74  points with mult
│ │ │ │ +000a6a60: 6970 6c69 6369 7479 0a2a 2a2a 2a2a 2a2a  iplicity.*******
│ │ │ │  000a6a70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000a6a80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000a6a90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000a6aa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000a6ab0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000a6ac0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000a6ad0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000a6ae0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000a6af0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20  *************.. 
│ │ │ │ -000a6b00: 202a 2046 756e 6374 696f 6e3a 202a 6e6f   * Function: *no
│ │ │ │ -000a6b10: 7465 2072 6174 696f 6e61 6c4d 6170 3a20  te rationalMap: 
│ │ │ │ -000a6b20: 7261 7469 6f6e 616c 4d61 702c 0a20 202a  rationalMap,.  *
│ │ │ │ -000a6b30: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -000a6b40: 2072 6174 696f 6e61 6c4d 6170 2852 2c7b   rationalMap(R,{
│ │ │ │ -000a6b50: 612c 692c 6a2c 6b2c 2e2e 2e7d 290a 2020  a,i,j,k,...}).  
│ │ │ │ -000a6b60: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -000a6b70: 2a20 522c 2061 202a 6e6f 7465 2070 6f6c  * R, a *note pol
│ │ │ │ -000a6b80: 796e 6f6d 6961 6c20 7269 6e67 3a20 284d  ynomial ring: (M
│ │ │ │ -000a6b90: 6163 6175 6c61 7932 446f 6329 506f 6c79  acaulay2Doc)Poly
│ │ │ │ -000a6ba0: 6e6f 6d69 616c 5269 6e67 2c0a 2020 2020  nomialRing,.    
│ │ │ │ -000a6bb0: 2020 2a20 6120 2a6e 6f74 6520 6c69 7374    * a *note list
│ │ │ │ -000a6bc0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -000a6bd0: 4c69 7374 2c2c 2061 206c 6973 7420 245c  List,, a list $\
│ │ │ │ -000a6be0: 7b61 2c69 2c6a 2c6b 2c5c 6c64 6f74 735c  {a,i,j,k,\ldots\
│ │ │ │ -000a6bf0: 7d24 206f 660a 2020 2020 2020 2020 6e6f  }$ of.        no
│ │ │ │ -000a6c00: 6e6e 6567 6174 6976 6520 696e 7465 6765  nnegative intege
│ │ │ │ -000a6c10: 7273 0a20 202a 202a 6e6f 7465 204f 7074  rs.  * *note Opt
│ │ │ │ -000a6c20: 696f 6e61 6c20 696e 7075 7473 3a20 284d  ional inputs: (M
│ │ │ │ -000a6c30: 6163 6175 6c61 7932 446f 6329 7573 696e  acaulay2Doc)usin
│ │ │ │ -000a6c40: 6720 6675 6e63 7469 6f6e 7320 7769 7468  g functions with
│ │ │ │ -000a6c50: 206f 7074 696f 6e61 6c20 696e 7075 7473   optional inputs
│ │ │ │ -000a6c60: 2c3a 0a20 2020 2020 202a 202a 6e6f 7465  ,:.      * *note
│ │ │ │ -000a6c70: 2044 6f6d 696e 616e 743a 2044 6f6d 696e   Dominant: Domin
│ │ │ │ -000a6c80: 616e 742c 203d 3e20 2e2e 2e2c 2064 6566  ant, => ..., def
│ │ │ │ -000a6c90: 6175 6c74 2076 616c 7565 206e 756c 6c2c  ault value null,
│ │ │ │ -000a6ca0: 200a 2020 2a20 4f75 7470 7574 733a 0a20   .  * Outputs:. 
│ │ │ │ -000a6cb0: 2020 2020 202a 2061 202a 6e6f 7465 2072       * a *note r
│ │ │ │ -000a6cc0: 6174 696f 6e61 6c20 6d61 703a 2052 6174  ational map: Rat
│ │ │ │ -000a6cd0: 696f 6e61 6c4d 6170 2c2c 2074 6865 2072  ionalMap,, the r
│ │ │ │ -000a6ce0: 6174 696f 6e61 6c20 6d61 7020 6465 6669  ational map defi
│ │ │ │ -000a6cf0: 6e65 6420 6279 2074 6865 0a20 2020 2020  ned by the.     
│ │ │ │ -000a6d00: 2020 206c 696e 6561 7220 7379 7374 656d     linear system
│ │ │ │ -000a6d10: 206f 6620 6879 7065 7273 7572 6661 6365   of hypersurface
│ │ │ │ -000a6d20: 7320 6f66 2064 6567 7265 6520 2461 2420  s of degree $a$ 
│ │ │ │ -000a6d30: 696e 2024 5072 6f6a 2852 2924 2068 6176  in $Proj(R)$ hav
│ │ │ │ -000a6d40: 696e 6720 2469 240a 2020 2020 2020 2020  ing $i$.        
│ │ │ │ -000a6d50: 7261 6e64 6f6d 2062 6173 6520 706f 696e  random base poin
│ │ │ │ -000a6d60: 7473 206f 6620 6d75 6c74 6970 6c69 6369  ts of multiplici
│ │ │ │ -000a6d70: 7479 2031 2c20 246a 2420 7261 6e64 6f6d  ty 1, $j$ random
│ │ │ │ -000a6d80: 2062 6173 6520 706f 696e 7473 206f 660a   base points of.
│ │ │ │ -000a6d90: 2020 2020 2020 2020 6d75 6c74 6970 6c69          multipli
│ │ │ │ -000a6da0: 6369 7479 2032 2c20 246b 2420 7261 6e64  city 2, $k$ rand
│ │ │ │ -000a6db0: 6f6d 2062 6173 6520 706f 696e 7473 206f  om base points o
│ │ │ │ -000a6dc0: 6620 6d75 6c74 6970 6c69 6369 7479 2033  f multiplicity 3
│ │ │ │ -000a6dd0: 2c20 616e 6420 736f 206f 6e0a 2020 2020  , and so on.    
│ │ │ │ -000a6de0: 2020 2020 756e 7469 6c20 7468 6520 6c61      until the la
│ │ │ │ -000a6df0: 7374 2069 6e74 6567 6572 2069 6e20 7468  st integer in th
│ │ │ │ -000a6e00: 6520 6769 7665 6e20 6c69 7374 2e0a 0a44  e given list...D
│ │ │ │ -000a6e10: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -000a6e20: 3d3d 3d3d 3d3d 0a0a 496e 2074 6865 2065  ======..In the e
│ │ │ │ -000a6e30: 7861 6d70 6c65 2062 656c 6f77 2c20 7765  xample below, we
│ │ │ │ -000a6e40: 2074 616b 6520 7468 6520 7261 7469 6f6e   take the ration
│ │ │ │ -000a6e50: 616c 206d 6170 2064 6566 696e 6564 2062  al map defined b
│ │ │ │ -000a6e60: 7920 7468 6520 6c69 6e65 6172 2073 7973  y the linear sys
│ │ │ │ -000a6e70: 7465 6d20 6f66 0a73 6570 7469 6320 706c  tem of.septic pl
│ │ │ │ -000a6e80: 616e 6520 6375 7276 6573 2077 6974 6820  ane curves with 
│ │ │ │ -000a6e90: 3320 7261 6e64 6f6d 2073 696d 706c 6520  3 random simple 
│ │ │ │ -000a6ea0: 6261 7365 2070 6f69 6e74 7320 616e 6420  base points and 
│ │ │ │ -000a6eb0: 3920 7261 6e64 6f6d 2064 6f75 626c 650a  9 random double.
│ │ │ │ -000a6ec0: 706f 696e 7473 2e0a 0a2b 2d2d 2d2d 2d2d  points...+------
│ │ │ │ +000a6af0: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 756e  *******..  * Fun
│ │ │ │ +000a6b00: 6374 696f 6e3a 202a 6e6f 7465 2072 6174  ction: *note rat
│ │ │ │ +000a6b10: 696f 6e61 6c4d 6170 3a20 7261 7469 6f6e  ionalMap: ration
│ │ │ │ +000a6b20: 616c 4d61 702c 0a20 202a 2055 7361 6765  alMap,.  * Usage
│ │ │ │ +000a6b30: 3a20 0a20 2020 2020 2020 2072 6174 696f  : .        ratio
│ │ │ │ +000a6b40: 6e61 6c4d 6170 2852 2c7b 612c 692c 6a2c  nalMap(R,{a,i,j,
│ │ │ │ +000a6b50: 6b2c 2e2e 2e7d 290a 2020 2a20 496e 7075  k,...}).  * Inpu
│ │ │ │ +000a6b60: 7473 3a0a 2020 2020 2020 2a20 522c 2061  ts:.      * R, a
│ │ │ │ +000a6b70: 202a 6e6f 7465 2070 6f6c 796e 6f6d 6961   *note polynomia
│ │ │ │ +000a6b80: 6c20 7269 6e67 3a20 284d 6163 6175 6c61  l ring: (Macaula
│ │ │ │ +000a6b90: 7932 446f 6329 506f 6c79 6e6f 6d69 616c  y2Doc)Polynomial
│ │ │ │ +000a6ba0: 5269 6e67 2c0a 2020 2020 2020 2a20 6120  Ring,.      * a 
│ │ │ │ +000a6bb0: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163  *note list: (Mac
│ │ │ │ +000a6bc0: 6175 6c61 7932 446f 6329 4c69 7374 2c2c  aulay2Doc)List,,
│ │ │ │ +000a6bd0: 2061 206c 6973 7420 245c 7b61 2c69 2c6a   a list $\{a,i,j
│ │ │ │ +000a6be0: 2c6b 2c5c 6c64 6f74 735c 7d24 206f 660a  ,k,\ldots\}$ of.
│ │ │ │ +000a6bf0: 2020 2020 2020 2020 6e6f 6e6e 6567 6174          nonnegat
│ │ │ │ +000a6c00: 6976 6520 696e 7465 6765 7273 0a20 202a  ive integers.  *
│ │ │ │ +000a6c10: 202a 6e6f 7465 204f 7074 696f 6e61 6c20   *note Optional 
│ │ │ │ +000a6c20: 696e 7075 7473 3a20 284d 6163 6175 6c61  inputs: (Macaula
│ │ │ │ +000a6c30: 7932 446f 6329 7573 696e 6720 6675 6e63  y2Doc)using func
│ │ │ │ +000a6c40: 7469 6f6e 7320 7769 7468 206f 7074 696f  tions with optio
│ │ │ │ +000a6c50: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020  nal inputs,:.   
│ │ │ │ +000a6c60: 2020 202a 202a 6e6f 7465 2044 6f6d 696e     * *note Domin
│ │ │ │ +000a6c70: 616e 743a 2044 6f6d 696e 616e 742c 203d  ant: Dominant, =
│ │ │ │ +000a6c80: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ +000a6c90: 616c 7565 206e 756c 6c2c 200a 2020 2a20  alue null, .  * 
│ │ │ │ +000a6ca0: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +000a6cb0: 2061 202a 6e6f 7465 2072 6174 696f 6e61   a *note rationa
│ │ │ │ +000a6cc0: 6c20 6d61 703a 2052 6174 696f 6e61 6c4d  l map: RationalM
│ │ │ │ +000a6cd0: 6170 2c2c 2074 6865 2072 6174 696f 6e61  ap,, the rationa
│ │ │ │ +000a6ce0: 6c20 6d61 7020 6465 6669 6e65 6420 6279  l map defined by
│ │ │ │ +000a6cf0: 2074 6865 0a20 2020 2020 2020 206c 696e   the.        lin
│ │ │ │ +000a6d00: 6561 7220 7379 7374 656d 206f 6620 6879  ear system of hy
│ │ │ │ +000a6d10: 7065 7273 7572 6661 6365 7320 6f66 2064  persurfaces of d
│ │ │ │ +000a6d20: 6567 7265 6520 2461 2420 696e 2024 5072  egree $a$ in $Pr
│ │ │ │ +000a6d30: 6f6a 2852 2924 2068 6176 696e 6720 2469  oj(R)$ having $i
│ │ │ │ +000a6d40: 240a 2020 2020 2020 2020 7261 6e64 6f6d  $.        random
│ │ │ │ +000a6d50: 2062 6173 6520 706f 696e 7473 206f 6620   base points of 
│ │ │ │ +000a6d60: 6d75 6c74 6970 6c69 6369 7479 2031 2c20  multiplicity 1, 
│ │ │ │ +000a6d70: 246a 2420 7261 6e64 6f6d 2062 6173 6520  $j$ random base 
│ │ │ │ +000a6d80: 706f 696e 7473 206f 660a 2020 2020 2020  points of.      
│ │ │ │ +000a6d90: 2020 6d75 6c74 6970 6c69 6369 7479 2032    multiplicity 2
│ │ │ │ +000a6da0: 2c20 246b 2420 7261 6e64 6f6d 2062 6173  , $k$ random bas
│ │ │ │ +000a6db0: 6520 706f 696e 7473 206f 6620 6d75 6c74  e points of mult
│ │ │ │ +000a6dc0: 6970 6c69 6369 7479 2033 2c20 616e 6420  iplicity 3, and 
│ │ │ │ +000a6dd0: 736f 206f 6e0a 2020 2020 2020 2020 756e  so on.        un
│ │ │ │ +000a6de0: 7469 6c20 7468 6520 6c61 7374 2069 6e74  til the last int
│ │ │ │ +000a6df0: 6567 6572 2069 6e20 7468 6520 6769 7665  eger in the give
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│ │ │ │ +000a6e10: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ +000a6e20: 0a0a 496e 2074 6865 2065 7861 6d70 6c65  ..In the example
│ │ │ │ +000a6e30: 2062 656c 6f77 2c20 7765 2074 616b 6520   below, we take 
│ │ │ │ +000a6e40: 7468 6520 7261 7469 6f6e 616c 206d 6170  the rational map
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│ │ │ │ +000a6e70: 0a73 6570 7469 6320 706c 616e 6520 6375  .septic plane cu
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│ │ │ │ +000a6e90: 6f6d 2073 696d 706c 6520 6261 7365 2070  om simple base p
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│ │ │ │ +000a6eb0: 6f6d 2064 6f75 626c 650a 706f 696e 7473  om double.points
│ │ │ │ +000a6ec0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │  000a6ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a6ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a6ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a6f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000a6f30: 5b76 6172 7328 302e 2e32 295d 3b20 2020  [vars(0..2)];   
│ │ │ │ +000a6f10: 2d2b 0a7c 6931 203a 2072 696e 6750 3220  -+.|i1 : ringP2 
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│ │ │ │ +000a6f30: 302e 2e32 295d 3b20 2020 2020 2020 2020  0..2)];         
│ │ │ │  000a6f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a6f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a6f60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000a6f60: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  000a6f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a6f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a6f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a6fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000a6fb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2070  -------+.|i2 : p
│ │ │ │ -000a6fc0: 6869 203d 2072 6174 696f 6e61 6c4d 6170  hi = rationalMap
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│ │ │ │ +000a6fb0: 2d2b 0a7c 6932 203a 2070 6869 203d 2072  -+.|i2 : phi = r
│ │ │ │ +000a6fc0: 6174 696f 6e61 6c4d 6170 2872 696e 6750  ationalMap(ringP
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│ │ │ │  000a6ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000a7000: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000a7010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7040: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000a7070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a70a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000a70b0: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │ +000a70a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a70b0: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │  000a70c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a70d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a70e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a70f0: 2020 2020 2020 207c 0a7c 2020 2020 2073         |.|     s
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│ │ │ │ +000a70f0: 207c 0a7c 2020 2020 2073 6f75 7263 653a   |.|     source:
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│ │ │ │  000a7120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a7140: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000a7150: 2020 2020 2020 2020 2020 2020 3635 3532              6552
│ │ │ │ -000a7160: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +000a7140: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a7150: 2020 2020 2020 3635 3532 3120 2020 2020        65521     
│ │ │ │ +000a7160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a7190: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000a71a0: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │ +000a7190: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a71a0: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │  000a71b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a71c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a71d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a71e0: 2020 2020 2020 207c 0a7c 2020 2020 2074         |.|     t
│ │ │ │ -000a71f0: 6172 6765 743a 2050 726f 6a28 2d2d 2d2d  arget: Proj(----
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│ │ │ │ -000a7210: 2c20 7420 2c20 7420 5d29 2020 2020 2020  , t , t ])      
│ │ │ │ +000a71e0: 207c 0a7c 2020 2020 2074 6172 6765 743a   |.|     target:
│ │ │ │ +000a71f0: 2050 726f 6a28 2d2d 2d2d 2d5b 7420 2c20   Proj(-----[t , 
│ │ │ │ +000a7200: 7420 2c20 7420 2c20 7420 2c20 7420 2c20  t , t , t , t , 
│ │ │ │ +000a7210: 7420 5d29 2020 2020 2020 2020 2020 2020  t ])            
│ │ │ │  000a7220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a7230: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000a7240: 2020 2020 2020 2020 2020 2020 3635 3532              6552
│ │ │ │ -000a7250: 3120 2030 2020 2031 2020 2032 2020 2033  1  0   1   2   3
│ │ │ │ -000a7260: 2020 2034 2020 2035 2020 2020 2020 2020     4   5        
│ │ │ │ +000a7230: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a7240: 2020 2020 2020 3635 3532 3120 2030 2020        65521  0  
│ │ │ │ +000a7250: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ +000a7260: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │  000a7270: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000a7290: 6566 696e 696e 6720 666f 726d 733a 207b  efining forms: {
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│ │ │ │ -000a72e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000a7300: 2020 2020 2020 2020 2037 2020 2020 2020           7      
│ │ │ │ -000a7310: 2020 2036 2020 2020 2020 2020 2020 3520     6          5 
│ │ │ │ -000a7320: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
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│ │ │ │ +000a72d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a72e0: 2020 2020 2020 2020 2020 2032 2035 2020             2 5  
│ │ │ │ +000a72f0: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ +000a7300: 2020 2037 2020 2020 2020 2020 2036 2020     7         6  
│ │ │ │ +000a7310: 2020 2020 2020 2020 3520 2020 2020 2020          5       
│ │ │ │ +000a7320: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a7330: 2020 2020 2020 2020 2020 6120 6220 202d            a b  -
│ │ │ │ +000a7340: 2031 3736 3137 612a 6220 202b 2031 3239   17617a*b  + 129
│ │ │ │ +000a7350: 3030 6220 202b 2031 3933 3037 6120 6320  00b  + 19307a c 
│ │ │ │ +000a7360: 2d20 3138 3232 3961 2062 2a20 2020 2020  - 18229a b*     
│ │ │ │ +000a7370: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000a7380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a73a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a73b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a73c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000a73d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a73e0: 2033 2034 2020 2020 2020 2020 2020 3620   3 4          6 
│ │ │ │ -000a73f0: 2020 2020 2020 2020 3720 2020 2020 2020          7       
│ │ │ │ -000a7400: 2020 3620 2020 2020 2020 2020 2035 2020    6          5  
│ │ │ │ -000a7410: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000a7420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a7430: 6120 6220 202d 2036 3431 3761 2a62 2020  a b  - 6417a*b  
│ │ │ │ -000a7440: 2d20 3331 3339 3662 2020 2d20 3238 3838  - 31396b  - 2888
│ │ │ │ -000a7450: 3661 2063 202d 2032 3134 3430 6120 622a  6a c - 21440a b*
│ │ │ │ -000a7460: 6320 2020 2020 207c 0a7c 2020 2020 2020  c      |.|      
│ │ │ │ +000a73c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a73d0: 2020 2020 2020 2020 2020 2033 2034 2020             3 4  
│ │ │ │ +000a73e0: 2020 2020 2020 2020 3620 2020 2020 2020          6       
│ │ │ │ +000a73f0: 2020 3720 2020 2020 2020 2020 3620 2020    7         6   
│ │ │ │ +000a7400: 2020 2020 2020 2035 2020 2020 2020 2020         5        
│ │ │ │ +000a7410: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a7420: 2020 2020 2020 2020 2020 6120 6220 202d            a b  -
│ │ │ │ +000a7430: 2036 3431 3761 2a62 2020 2d20 3331 3339   6417a*b  - 3139
│ │ │ │ +000a7440: 3662 2020 2d20 3238 3838 3661 2063 202d  6b  - 28886a c -
│ │ │ │ +000a7450: 2032 3134 3430 6120 622a 6320 2020 2020   21440a b*c     
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│ │ │ │  000a7470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a74a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a74b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000a74c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000a74e0: 2020 2020 2020 2020 2037 2020 2020 2020           7      
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│ │ │ │ -000a7550: 6320 2020 2020 207c 0a7c 2020 2020 2020  c      |.|      
│ │ │ │ +000a74b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a74c0: 2020 2020 2020 2020 2020 2034 2033 2020             4 3  
│ │ │ │ +000a74d0: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ +000a74e0: 2020 2037 2020 2020 2020 2020 3620 2020     7        6   
│ │ │ │ +000a74f0: 2020 2020 2020 2035 2020 2020 2020 2020         5        
│ │ │ │ +000a7500: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a7510: 2020 2020 2020 2020 2020 6120 6220 202b            a b  +
│ │ │ │ +000a7520: 2032 3038 3636 612a 6220 202b 2033 3131   20866a*b  + 311
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│ │ │ │  000a7560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a75a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000a75b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a75c0: 2035 2032 2020 2020 2020 2020 2020 3620   5 2          6 
│ │ │ │ -000a75d0: 2020 2020 2020 2037 2020 2020 2020 2020         7        
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│ │ │ │ -000a7630: 6120 6320 2d20 3133 3137 6120 622a 6320  a c - 1317a b*c 
│ │ │ │ -000a7640: 2d20 2020 2020 207c 0a7c 2020 2020 2020  -      |.|      
│ │ │ │ +000a75a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000a75b0: 2020 2020 2020 2020 2020 2035 2032 2020             5 2  
│ │ │ │ +000a75c0: 2020 2020 2020 2020 3620 2020 2020 2020          6       
│ │ │ │ +000a75d0: 2037 2020 2020 2020 2020 2036 2020 2020   7         6    
│ │ │ │ +000a75e0: 2020 2020 2035 2020 2020 2020 2020 2020       5          
│ │ │ │ +000a75f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │  000a7660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7680: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000a76a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a76b0: 2036 2020 2020 2020 2020 2020 2020 3620   6            6 
│ │ │ │ -000a76c0: 2020 2020 2020 2037 2020 2020 2020 2020         7        
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│ │ │ │ -000a7710: 2b20 3239 3732 6220 202b 2035 3435 3761  + 2972b  + 5457a
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│ │ │ │ -000a7730: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
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│ │ │ │ +000a76a0: 2020 2020 2020 2020 2020 2036 2020 2020             6    
│ │ │ │ +000a76b0: 2020 2020 2020 2020 3620 2020 2020 2020          6       
│ │ │ │ +000a76c0: 2037 2020 2020 2020 2020 3620 2020 2020   7        6     
│ │ │ │ +000a76d0: 2020 2020 3520 2020 2020 2020 2020 2020      5           
│ │ │ │ +000a76e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
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│ │ │ │ +000a7710: 6220 202b 2035 3435 3761 2063 202b 2032  b  + 5457a c + 2
│ │ │ │ +000a7720: 3536 3261 2062 2a63 202d 2020 2020 2020  562a b*c -      
│ │ │ │ +000a7730: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000a7740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a7780: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000a7790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a77a0: 2037 2020 2020 2020 2020 2020 2036 2020   7           6  
│ │ │ │ -000a77b0: 2020 2020 2020 3720 2020 2020 2020 2020        7         
│ │ │ │ -000a77c0: 3620 2020 2020 2020 2020 2035 2020 2020  6          5    
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│ │ │ │ -000a77e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000a7830: 2020 2020 2020 2020 2020 2020 2020 207d                 }
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│ │ │ │ +000a7830: 2020 2020 2020 2020 207d 2020 2020 2020           }      
│ │ │ │  000a7840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a7870: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000a7870: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000a7880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a7890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a78a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a78b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a78c0: 2020 2020 2020 207c 0a7c 6f32 203a 2052         |.|o2 : R
│ │ │ │ -000a78d0: 6174 696f 6e61 6c4d 6170 2028 7261 7469  ationalMap (rati
│ │ │ │ -000a78e0: 6f6e 616c 206d 6170 2066 726f 6d20 5050  onal map from PP
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│ │ │ │ +000a78c0: 207c 0a7c 6f32 203a 2052 6174 696f 6e61   |.|o2 : Rationa
│ │ │ │ +000a78d0: 6c4d 6170 2028 7261 7469 6f6e 616c 206d  lMap (rational m
│ │ │ │ +000a78e0: 6170 2066 726f 6d20 5050 5e32 2074 6f20  ap from PP^2 to 
│ │ │ │ +000a78f0: 5050 5e35 2920 2020 2020 2020 2020 2020  PP^5)           
│ │ │ │  000a7900: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000a7930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a7940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a7950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000a7970: 2020 2020 3420 3220 2020 2020 2020 2020      4 2         
│ │ │ │ -000a7980: 3320 3320 2020 2020 2020 2020 2032 2034  3 3          2 4
│ │ │ │ -000a7990: 2020 2020 2020 2020 2020 2035 2020 2020             5    
│ │ │ │ -000a79a0: 2020 2020 2020 3620 2020 2020 2020 2020        6         
│ │ │ │ -000a79b0: 2035 2020 2020 207c 0a7c 6320 2d20 3330   5     |.|c - 30
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│ │ │ │  000a7c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000a7e60: 207c 0a7c 2033 3533 3261 2062 2063 202b   |.| 3532a b c +
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│ │ │ │ -000a7f10: 2020 2020 3420 2020 3220 2020 2020 2020      4   2       
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│ │ │ │ -000a7f40: 2032 2020 2020 2020 2020 2035 2032 2020   2         5 2  
│ │ │ │ -000a7f50: 2020 2020 2020 207c 0a7c 2020 2b20 3139         |.|  + 19
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│ │ │ │ -000a84e0: 2033 2020 2020 2020 2020 3320 3420 2020   3        3 4   
│ │ │ │ -000a84f0: 2020 2020 2020 327c 0a7c 2063 2020 2b20        2|.| c  + 
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│ │ │ │ +000a85a0: 2020 3320 2020 2020 2020 2020 3220 3220    3         2 2 
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│ │ │ │ +000a85c0: 2020 2020 2020 2034 2033 2020 2020 2020         4 3      
│ │ │ │ +000a85d0: 2020 2033 2034 2020 2020 2020 2020 2032     3 4         2
│ │ │ │ +000a85e0: 207c 0a7c 2020 2d20 3136 3732 3361 2062   |.|  - 16723a b
│ │ │ │ +000a85f0: 2a63 2020 2b20 3234 3536 3461 2062 2063  *c  + 24564a b c
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│ │ │ │ +000a8610: 2d20 3833 3339 6220 6320 202d 2031 3836  - 8339b c  - 186
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│ │ │ │ +000a8630: 627c 0a7c 2020 2020 2020 2020 2020 2020  b|.|            
│ │ │ │  000a8640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a8650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a8660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a8670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a8680: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000a8690: 2020 3320 2020 3320 2020 2020 2020 2020    3   3         
│ │ │ │ -000a86a0: 3220 3220 3320 2020 2020 2020 2020 2020  2 2 3           
│ │ │ │ -000a86b0: 3320 3320 2020 2020 2020 2034 2033 2020  3 3        4 3  
│ │ │ │ -000a86c0: 2020 2020 2033 2034 2020 2020 2020 2020       3 4        
│ │ │ │ -000a86d0: 2032 2020 2034 207c 0a7c 2d20 3138 3432   2   4 |.|- 1842
│ │ │ │ -000a86e0: 3161 2062 2a63 2020 2d20 3130 3630 3061  1a b*c  - 10600a
│ │ │ │ -000a86f0: 2062 2063 2020 2d20 3239 3636 3461 2a62   b c  - 29664a*b
│ │ │ │ -000a8700: 2063 2020 2d20 3634 3335 6220 6320 202d   c  - 6435b c  -
│ │ │ │ -000a8710: 2036 3330 6120 6320 202b 2032 3035 3036   630a c  + 20506
│ │ │ │ -000a8720: 6120 622a 6320 207c 0a7c 2020 2020 2020  a b*c  |.|      
│ │ │ │ +000a8680: 207c 0a7c 2020 2020 2020 2020 3320 2020   |.|        3   
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│ │ │ │ +000a86a0: 2020 2020 2020 2020 2020 3320 3320 2020            3 3   
│ │ │ │ +000a86b0: 2020 2020 2034 2033 2020 2020 2020 2033       4 3       3
│ │ │ │ +000a86c0: 2034 2020 2020 2020 2020 2032 2020 2034   4         2   4
│ │ │ │ +000a86d0: 207c 0a7c 2d20 3138 3432 3161 2062 2a63   |.|- 18421a b*c
│ │ │ │ +000a86e0: 2020 2d20 3130 3630 3061 2062 2063 2020    - 10600a b c  
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│ │ │ │ +000a8700: 3634 3335 6220 6320 202d 2036 3330 6120  6435b c  - 630a 
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│ │ │ │  000a8730: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000a8750: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000a8770: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
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│ │ │ │ -000a8790: 3220 3220 3320 2020 2020 2020 2020 2020  2 2 3           
│ │ │ │ -000a87a0: 3320 3320 2020 2020 2020 2020 3420 3320  3 3         4 3 
│ │ │ │ -000a87b0: 2020 2020 2020 2020 3320 3420 2020 2020          3 4     
│ │ │ │ -000a87c0: 2020 2020 3220 207c 0a7c 2d20 3133 3434      2  |.|- 1344
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│ │ │ │ -000a8800: 2d20 3133 3835 3761 2063 2020 2b20 3136  - 13857a c  + 16
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│ │ │ │ +000a8790: 2020 2020 2020 2020 2020 3320 3320 2020            3 3   
│ │ │ │ +000a87a0: 2020 2020 2020 3420 3320 2020 2020 2020        4 3       
│ │ │ │ +000a87b0: 2020 3320 3420 2020 2020 2020 2020 3220    3 4         2 
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│ │ │ │  000a8820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a8830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a8840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a8850: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000a8880: 3220 3320 2020 2020 2020 2020 2020 3320  2 3           3 
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│ │ │ │ -000a88a0: 2020 2020 2020 3320 3420 2020 2020 2020        3 4       
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│ │ │ │ -000a88f0: 3134 3031 3461 2063 2020 2d20 3331 3934  14014a c  - 3194
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│ │ │ │ -000a8980: 3320 3320 2020 2020 2020 2020 3420 3320  3 3         4 3 
│ │ │ │ -000a8990: 2020 2020 2020 2020 3320 3420 2020 2020          3 4     
│ │ │ │ -000a89a0: 2020 2020 3220 207c 0a7c 2d20 3236 3538      2  |.|- 2658
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│ │ │ │ -000a89f0: 3935 3961 2062 2a7c 0a7c 2d2d 2d2d 2d2d  959a b*|.|------
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│ │ │ │ -000a8ad0: 2020 2d20 3236 3633 3362 2063 2020 2d20    - 26633b c  - 
│ │ │ │ -000a8ae0: 3138 3034 3361 207c 0a7c 2020 2020 2020  18043a |.|      
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│ │ │ │ -000a8c50: 2020 2020 2020 2020 2020 2035 2020 2020             5    
│ │ │ │ -000a8c60: 2020 2020 2032 2035 2020 2020 2020 2020       2 5        
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│ │ │ │  000a8cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000a8d40: 3520 2020 2020 2020 2020 2020 2035 2020  5            5  
│ │ │ │ -000a8d50: 2020 2020 2020 3220 3520 2020 2020 2020        2 5       
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│ │ │ │ +000a8e40: 3520 2020 2020 2020 2020 2020 3620 2020  5           6   
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│ │ │ │ +000a9770: 6365 3a20 6661 6c73 6520 2020 2020 2020  ce: false       
│ │ │ │  000a9780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ 

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